Functional-IntegralRepresentation ofQuantumFieldTheoryusers.physik.fu-berlin.de › ~kleinert › b6...

Perfection is achieved, not when there is nothing more to add,but when there is nothing left to take away.

Antoine de Saint-Exupery (1900–1944)

14

Functional-Integral Representation

of Quantum Field Theory

In Chapter 7 we have quantized various fields with the help of canonical commutationrules between field variables and their canonical conjugate field momenta. Fromthese and the temporal behavior of the fields determined by the field equationsof motion we have derived the Green functions of the theory. These contain allexperimentally measurable informations on the quantum field theory. They can allbe derived from functional derivatives of certain generating functionals. For a realscalar field this was typically an expectation value

Z[j] ≡ 〈0|T [j]|0〉 (14.1)

where T [j] was the time-ordered product in (7.827)

T [j] ≡ T ei∫

d4x j(x)φ(x). (14.2)

Here the Green functions can all be obtained from functional derivatives of Z[j] ofthe type (7.841).

For complex scalar fields, the corresponding generating functional is given bythe expectation value (7.849). Now the Green functions can all be obtained fromfunctional derivatives of the type (7.850).

In theoretical physics, Fourier transformations have always played an importantrole in yielding complementary insights into mathematical structures. Due to theconjugate appearance of fields φ(x) and sources j(x) in expressions like (14.2), this isalso true for generating functionals and constitutes a basis for the functional-integralformalism of quantum field theory.

14.1 Functional Fourier Transformations

An important observation is now that instead of calculating these generating func-tionals as done in Chapter 7 from a formalism of field theory, in which φ(x) is a field

926

14.1 Functional Fourier Transformations 927

operator, they can also be derived from a functional Fourier transform of anotherfunctional Z[φ] that depends on a classical field φ(x):

Z[j] ≡∫

DφZ[φ] ei∫

dDx j(x)φ(x). (14.3)

The symbol∫ Dφ(x) in this expression is called a functional integral.

The mathematics of functional integration is an own discipline that is presentedin many textbooks [1, 2]. Functional integrals were first introduced in ordinary quan-tum mechanics by R.P. Feynman [3], who used them to express physical amplitudeswithout employing operators. The uncertainty relation that can be expressed by anequal-time commutation relation [x(t), p(t)] = ih between x(t) and the conjugatevariable p(t) of quantum mechanics is the consequence of quantum fluctuations ofthe classical variables x(t) and p(t). In quantum mechanics, the functional integralis merely a path integral of a fluctuation variable x(t). In field theory, there is afluctuating path for each space point x. Instead of a time-dependent variable x(t)one deals with more general dynamical variables φx(t) = φ(x, t) = φ(x), one foreach spacepoint x. The path integrals over all x(t) go over into functional integralsover all fluctuating fields φ(x).

Functional integrals may be defined most simply in a discretized approximation.Spacetime is grated into a fine spacetime lattice. For every spacetime coordinate xµ

we introduce a discrete lattice point close to it

xµ → xµn ≡ nǫ, n = 0, ± 1, ± 2, ± 3, (14.4)

where ǫ is a very small lattice spacing. Then we may approximate integrals by sums:∫

dDx j(x)φ(x) ≈ ǫD∑

n

j(xn)φ(xn) ≡ ǫD∑

n

jnφn (14.5)

where n is to be read as a D-dimensional index (n0, n1, . . . , nD), one for eachcomponent of the spacetime vector xµ. Now we define

∫ Dφ(x) as the infinite productof integrals over φn at each point xn:

∫

Dφ(x) =∏

n

∫

dφn√

2πi/ǫD. (14.6)

Operations with functional integrals are very similar to those with ordinary inte-grals. For example, the Fourier transform of (14.3) can be inverted, by analogy withordinary integrals, to obtain:

Z[φ] ≡∫

Dj(x)Z[j]e−i∫

dDx j(x)φ(x). (14.7)

There are functional analogs of the Dirac δ-function:∫

Dj(x) e−i∫

dDx j(x)φ(x) = δ[φ], (14.8)∫

Dφ(x) ei∫

dDx j(x)φ(x) = δ[j], (14.9)

928 14 Functional-Integral Representation of Quantum Field Theory

called δ-functionals . In the lattice approximation corresponding to (14.6), they aredefined as infinite products of ordinary δ-functions

δ[φ] =∏

n

√

2πi/ǫDδ(φn), δ[j] =∏

n

√

2πi/ǫDδ(jn). (14.10)

They have the obvious property

∫

Dφ δ[φ] = 1,∫

Dj δ[j] = 1. (14.11)

A commonly used notation for the measure (14.6) of functional integrals employscontinuously infinite product of integrals which must be imagined as the continuumlimit of the lattice product (14.6). In this notation one writes

∫

Dφ(x) =∏

x

∫

dφ(x)√2πi

,∫

Dj(x) =∏

x

∫

dj(x)√2πi

, (14.12)

and the associated δ-functionals as

δ[φ] =∏

x

√2πi δ(φ(x)), δ[j] =

∏

x

√2πi δ(j(x)). (14.13)

14.2 Gaussian Functional Integral

Only very few functional integrals can be solved explicitly. The simplest nontrivialexample is the Gaussian integral1

∫

Dj(x)e− i2

∫

dDxdDx′ j(x)M(x,x′)j(x′). (14.14)

In the discretized form, this can be written as

∏

n

∫

djn√

2πi/ǫD

e−i2ǫ2D∑

n,mjnMnmjm. (14.15)

We may assumeM to be a real symmetric functional matrix, since its antisymmetricpart would not contribute to (14.14). Such a matrix may be diagonalized by arotation

jn → j′n = Rnmjm, (14.16)

which leaves the measure of integration invariant

∂ (j1, . . . , jn)

∂ (j′1, . . . , j′n)

= detR−1 = 1. (14.17)

1Mathematically speaking, integrals with an imaginary quadratic exponent are more accuratelycalled Fresnel integrals, but field theorists do not make this distinction.

14.2 Gaussian Functional Integral 929

In the diagonal form, the multiple integral (14.15) factorizes into a product of Gaus-sian integrals, which are easily calculated:

∏

n

∫ dj′n√

2πi/ǫDe−

i2ǫ2D∑

nj′nMnj′n =

∏

n

1√−ǫDMn

= det −1/2(−ǫDM). (14.18)

On the right-hand side we have used the fact that the product of diagonal valuesMn is equal to the determinant of M . The final result

∏

n

∫

djn√

2πi/ǫD

e− i

2ǫ2D∑

n,mjnMnmjm = det −1/2(−ǫDM) (14.19)

is invariant under rotations, so that it holds also without diagonalizing the matrix.This formula can be taken to the continuum limit of infinitely fine gratings ǫ→ 0.

Recall the well-known matrix formula

detA = elog detA = etr logA, (14.20)

and expand tr logA into power series as follows

tr logA = tr log [1 + (A− 1)] = −tr∞∑

k=1

(−)k

k(A− 1)k . (14.21)

The advantage of this expansion is that when approximating the functional matrixA by the discrete matrix ǫDM , the traces of powers of ǫDM remain well-definedobjects in the continuum limit ǫ→ 0:

tr(

ǫDM)

= ǫD∑

n

Mnn →∫

dDxM(x, x) ≡ TrM,

tr(

ǫDM)2

= ǫ2D∑

n,m

MnmMmm →∫

dDx dDx′M(x, x′)M(x′, x) ≡ TrM2, (14.22)

... .

We therefore rewrite the right-hand side of (14.19) as exp[−(1/2)tr log(−ǫDM)], andexpand

tr log(−ǫDM) = tr log[

1 +(

−ǫDM − 1)]

−−−→ǫ→0

−∞∑

k=1

(−)k

k

∫

dDx1 · · · dDxk[

−M(x1, x2)− δ(D)(x1 − x2)]

× · · ·

×[

−M(x2, x3)− δ(D)(x2 − x3)] [

−M(xk, x1)− δ(D)(xk − x1)]

. (14.23)

The expansion on the right-hand side defines the trace of the logarithm of thefunctional matrix −M(x, x′), and will be denoted by Tr log(−M). This, in turn,


serves to define the functional determinant of −M(x, x′) by generalizing formula(14.20) to functional matrices:

Det (−M) = elog det (−M) = eTr log(−M). (14.24)

Thus we obtain for the functional integral (14.14) the result:∫

Dj(x)e− i2

∫

dDx dDx′ j(x)M(x,x′)j(x′) = Det −1/2(−M) = e−12Tr log(−M). (14.25)

This formula can be generalized to complex integration variables by separatingthe currents into real and imaginary parts, j(x) = [j1(x)+ ij2(x)]/

√2. Each integral

gives the same functional determinant so that∫

Dj∗(x)Dj(x)e−i∫

dDxdDx′ j∗(x)M(x,x′)j(x′) = e−Tr logM . (14.26)

Here M(x, x′) is an arbitrary Hermitian matrix, and the measure of integrationfor complex variables j(x) is defined as the product of the measures for real andimaginary parts: Dj∗(x)Dj(x) ≡ Dj1(x)Dj2(x).

14.3 Functional Formulation for Free Quantum Fields

Having calculated the Gaussian functional integrals (14.25) and (14.26) we are ableto perform the functional integrations over the generating functional (14.3) to deriveits Fourier transform Z[φ]. First we shall do so only for the free-field generatingfunctional (7.843), which we shall equip with a subscript 0 to emphasize the freesituation:

Z0[j] = e−12

∫

d4y1d4y2 j(y1)G0(y1,y2)j(y2). (14.27)

By writing M(x, x′) as

M(x, x′) = −iG0(x, x′), (14.28)

the Gaussian functional integral (14.14) becomes∫

Dj(x)e− 12

∫

dDx dDx′ j(x)G0(x,x′)j(x′) = Det −1/2(−iG0). (14.29)

This result can immediately be extended to calculate the functional Fourier tran-form of the generating functional Z0[j] defined in (14.3). Thus we want to form

Z0[φ] =∫

Dj(x)e− 12

∫

dDx dDx′ j(x)G0(x,x′)j(x′)+i∫

j(x)φ(x). (14.30)

The extra term linear in j(x) does not change the harmonic nature of the exponent.The integral can be reduced to the Gaussian form (14.29) by a simple quadraticcompletion process. For this manipulation it is useful to omit the spacetime indices,and use an obvious functional vector notation to rewrite (14.30) as

Z0[φ] =∫

Dje− i2jT 1

iG0j+ijTφ. (14.31)

14.3 Functional Formulation for Free Quantum Fields 931

The exponent may be completed quadratically as

− i

2

(

jT + iG−10 φ

)T 1

iG0

(

j + iG−10 φ

)

+i

2φT iG−1

0 φ. (14.32)

We now replace the variable j + iG−10 φ by j′ which, in each of the infinite integrals,

amounts only to a trivial shift of the center of integration. Thus

Z0[φ] =(∫

Dj′e− i2j′ 1iG0j′

)

ei2φT iG−1

0 φ. (14.33)

We can now apply formula (14.26) and find

Z0[φ] = Det (M)1/2ei2

∫

dDxdDx′ φ(x)M−1(x,x′)φ(x′), (14.34)

Inserting for M(x, x′) the functional matrix (14.28) we see that the exponent con-tains the free-particle Green function:

M−1(x, x′) = −iG0(x, x′) = (−∂2 −m2)δ(D)(x− x′). (14.35)

The determinant is a constant prefactor which does not depend on the field φ. Weshall abbreviate it as a normalization factor

N = Det −1/2(−iG0) = Det−1/2(−∂2 −m2). (14.36)

With the help of Eq. (14.20), this can be rewritten as

N = exp

− 1

2Tr log(−∂2 −m2)

. (14.37)

Inserting into (14.34) the differential operator (14.35), and performing a partialintegration in spacetime, we can rewrite it as

Z0[φ] = N ei∫

dDx [ 12(∂φ)2−m2

2φ2] = N ei

∫

dDxL0(φ,∂φ). (14.38)

Thus the Fourier-transformed generating functional is, up to the normalization fac-tor N , just the exponential of the classical free-field action under consideration.

By Fourier-transforming Z[φ] according to formula (14.3), we recover the initialgenerating functional Z0[j]. This procedure yields the famous functional integralrepresentation for the free-particle generating functional

Z0[j] = N∫

Dφ(x)ei∫

dDx [L0(φ,∂φ)+j(x)φ(x)]. (14.39)

In this representation, the field is no longer an operator but a real variable thatcontains all quantum information via its field fluctuations. The functional integralover the φ-field is defined in (14.6). By analogy with (14.25), we have for the realfield fluctuations the Gaussian formula

∫

Dφ(x)e i2∫

dDx dDx′ φ(x)M−1(x,x′)φ(x′) = (DetM)12 , (14.40)


valid for real symmetric functional matrices M(x, x′).For complex fields φ = (φ1 + iφ2)/

√2, there is a similar functional integral

formula∫

Dφ∗(x)Dφ(x)ei∫

dDxdDx′ φ∗(x)M−1(x,x′)φ(x′) = DetM, (14.41)

in which M(x, x′) is a Hermitian functional matrix. This follows directly from(14.26).

Let us use the formula (14.40) to cross-check the proper normalization of (14.39).At zero source, Z0[j] has to be equal to unity [compare (14.27)]. This is ensured if

N−1 = Det 1/2(−iG0) =∫

Dφ(x)ei∫

dDxL0(φ,∂φ). (14.42)

Indeed, after a spacetime integration by parts, the right-hand side may be writtenas∫

Dφ ei∫

dDxL0(φ,∂φ)=∫

Dφ ei∫

dDx 12[(∂φ)2−m2φ2]=

∫

Dφ ei∫

dDxφ(x)M−1(x,x′)φ(x′). (14.43)

Applying now formula (14.39), and using (14.36), we verify that Z0[0] = 1.Using the expression (14.42) for the normalization factor, we can rewrite

Eq. (14.38) for the Fourier transform Z0[φ] as

Z0[φ] =ei∫

dDxL0(φ,∂φ)

∫ Dφ(x)ei∫

dDxL0(φ,∂φ). (14.44)

This ratio is obviously normalized:

∫

Dφ(x)Z0[φ] = 1. (14.45)

Note that the expression (14.44) has precisely the form of the quantum mechanicalversion (1.491) of the thermodynamical Gibbs distribution (1.489):

wn ≡ Z−1QM(tb − ta)e

−iEn(tb−ta)/h. (14.46)

There exists a useful formula for harmonically fluctuating fields which are en-countered in many physical contexts that can be derived immediately from this.Consider the correlation function of two exponentials of a free field φ(x). Insertinginto Eq. (14.59) the special current

j12(x) = a∫

dDx[aφ(x− x1)− bφ(x− x2)], (14.47)

the partition function (14.27) reads

Z0[j12] = Det −1/2(−iG0)∫

Dφ(x) eiaφ(x1)e−ibφ(x2)ei∫

dDx [L0(φ,∂φ)]. (14.48)

14.4 Interactions 933

As such it coincides with the harmonic expectation value

〈eiaφ(x1)e−ibφ(x2)〉. (14.49)

Inserting (14.43) and performing the functional integral in (14.48) yields

〈eiaφ(x1)e−ibφ(x2)〉 = e−12 [a

2G(x1,x1)−2abG(x1,x2)+b2G(x2,x2)]. (14.50)

In the brackets of the exponent we recognize the expectation values of pairs of fields

a2〈φ(x1)φ(x1)〉 − 2ab〈φ(x1)φ(x2)〉+ b2〈φ(x2)φ(x2)〉, (14.51)

such that we may also write (14.50) as

〈eiaφ(x1)e−ibφ(x2)〉 = 〈eiaφ(x1)−ibφ(x2)〉 = e−12〈[aφ(x1)−bφ(x2)]2〉. (14.52)

Of course, this result may also be derived by using field operators andWick’s theoremalong the lines of Subsection 7.17.1.

14.4 Interactions

Let us now include interactions. We have seen in Eq. (10.24) that the generat-ing functional in the interaction picture [more precisely the functional ZD[j] ofEq. (10.22)] may simply be written as

Z[j] = ei∫

dDxLint(−iδ/δj(x))Z0[j]. (14.53)

This can immediately be Fourier-transformed to

Z[φ] =∫

Dj(x)e−i∫

dDx j(x)φ(x)[

ei∫

dDxLint(φ)(−iδ/δj)Z0[j]]

. (14.54)

Removing the second exponential by a partial functional integration, we obtain

Z[φ] = ei∫

dDxLint(φ)∫

Dj(x)e−i∫

dDx j(x)φ(x)Z0[j]

= ei∫

dDxLint(φ,∂φ)N ei∫

dDx 12 [(∂φ)2−m2φ2] (14.55)

=ei∫

dDxL(φ,∂φ)

∫ Dφ(x)ei∫

dDxL0(φ,∂φ).

The functional Fourier transform of this renders a generalization of (14.39) thatincludes interactions. In this way we have derived functional integral representationof the interacting theory:

Z[j] = N∫

Dφ(x)ei∫

dDx [L(φ,∂φ)+j(x)φ(x)]

=

∫ Dφ(x)ei∫

dDx [L(φ,∂φ)+j(x)φ(x)]

∫ Dφ(x)ei∫

dDxL0(φ,∂φ)(φ). (14.56)


This representation may be compared with the perturbation theoretic formula ofoperator quantum field theory

Z[j] = 〈0|Tei∫

dDx [Lint(φ)+j(x)φ(x)]|0〉, (14.57)

where the symbol φ(x) denotes free-field operators, and the vacuum expectationvalue of products of these fields follow Wick’s theorem. In the functional integralrepresentations (14.54)–(14.56), on the other hand, φ(x) is a classical c-number field.All quantum properties of Z[j] arise from the infinitely many integrals over φ(x),one at each spacetime point x, rather than from field operators.

Note that in formula (14.56), the full action appears in the exponent, whereasin (14.57), only the interacting part appears.

In contrast to Z0[φ] of Eqs. (14.44) and (14.45), the amplitude for the interactingtheory is no longer properly normalized. In fact, we know from the perturbativeevaluation of (14.57) that it represents the sum of all vacuum diagrams displayed in(10.81). Since the denominator does not normalize A[φ] anyhow, it is convenient todrop it and work with the numerator only, using the unnormalized

Z[j] =∫

Dφ(x)ei∫

dDx [L(φ,∂φ)+j(x)φ(x)] (14.58)

as the generating functional.

The normalization in (14.58) has an important advantage over the previous onein (14.56). In the euclidean formulation of the theory to be discussed in Section 14.5,it makes Z[0] equal to the thermodynamic partition function of the system.

For free fields, Z[0] is equal to the partition function of a set of harmonic oscil-lators of frequencies ω(k) for all momenta k. This statement can be proved only inthe lattice version of the theory. In the continuum limit the statement is nontrivial,since the determinants on the right-hand sides are infinite. However, we shall seein Section 14.7, Eqs. (14.123)–(14.133), that correct finite partition functions areobtained if the infinities are removed by the method of dimensional regularization,that was used in Section 11.5 to remove divergences from Feynman integrals.

Even though the operator formula (14.57) and the functional integral formula(14.58) are completely equivalent, there are important advantages of the latter. Insome theories it may be difficult to find a canonically quantized set of free fieldson which to construct an interaction representation for Z[j] following Eq. (14.57).The photon field is an important example where it was quite hard to interpretthe Hilbert space. In particular, we remind the reader of the problem that in theGupta-Bleuler quantization scheme, the vacuum energy contains the quanta of twounphysical polarization states of the photon. Within the functional approach, thisproblem can easily be avoided as will be explained in Chapter 17.

A second and very important advantage is the possibility of deriving the Feynmanrules, without any knowledge of the Hilbert space, directly from the representation(14.58). For this we simply take the non-quadratic piece of the action, which defines

14.4 Interactions 935

the vertices of the perturbation expansion, outside the functional integral as in(14.53), i.e., we rewrite (14.58) as:

Z[j] = ei∫

dDxLint(−iδ/δj(x))∫

Dφ(x)ei∫

dDx [L0(φ,∂φ)+j(x)φ(x)]. (14.59)

Using (14.34), this reads more explicitly

Z[j] = ei∫

dDxLint(−iδ/δj(x))∫

Dφ(x)e[ i2∫

dDx dDx′ φ(x)iG−10 (x,x′)φ(x′)+j(x)φ(x)]. (14.60)

A shift in the field variables to

φ′(x) = φ(x) +∫

dy G0(x, x′)j(x′), (14.61)

and a quadratic completion lead to

Z[j] = ei∫

dDxLint(−iδ/δj(x))e−12

∫

dDxdDx′ j(x)G0(x,x′)j(x′)

×∫

Dφ′(x)ei2

∫

dDxdDx′ φ′(x)iG−10 (x,x′)φ′(x′). (14.62)

The functional integral over the shifted field φ′(x) can now be performed with thehelp of formula (14.25), inserting there M(x, x′) = −iG0(x, x

′). The result is, re-calling (14.36),

∫

Dφ′(x)ei2

∫

dDx dDx′ φ′(x)iG−10 (x,x′)φ′(x′) = Det−1/2(−iG0), (14.63)

such that we find

Z[j] = Det −1/2(−iG0) ei∫

dDxLint(−iδ/δj(x))e−12

∫

dDxdDx′ j(x)G0(x,x′)j(x′). (14.64)

Expanding the prefactor in (14.60) in a power series yields all terms of theperturbation expansion (10.29). They correspond to the Wick contractions in Sec-tion 10.3.1, with the associated Feynman diagrams. The free-field propagators arethe functional inverse of the operators between the fields in the quadratic part ofthe Lagrangian. If this is written as

i

2

∫

dDx dDx′ φ(x)D(x, x′)φ(x′), (14.65)

then

G0(x, x′) = iD−1(x, x′). (14.66)

This formal advantage of obtaining perturbation expansions from the functional in-tegral representations has far-reaching consequences. We have seen in Chapter 11that the evaluation of the perturbation series proceeds most conveniently by Wick-rotating all energy integrations to make them run along the imaginary axes in thecomplex energy plane. In this way one avoids the singularities in the propagators


that would be encountered at the physical particle energies. In Section 10.7, on theother hand, these singularities were shown to be responsible for the fact that particlesleave a scattering region and form asymptotic states. Hence, Wick-rotated pertur-bation expansions describe a theory which does not possess any particles states. Infact, they cannot be described by field operators creating particle states in a Hilbertspace. Such states can be obtained from a simple modification of the above func-tional integral representation of the generating functional Z[j]. We simply performthe x-space version of the Wick-rotation that was illustrated before on page 495 inFig. 7.2.

14.5 Euclidean Quantum Field Theory

In Eq. (7.136), we replaced the coordinates xµ (µ = 0, 1, 2, 3) in D = 4 spacetimedimensions by the euclidean coordinates xµE = (x1, . . . , x3, x4 = −ix0). The sameoperation may be done to the time x0 in any dimensions. Under this replacement,the action

A ≡∫

dDxL(φ, ∂Eφ) ≡∫

dDx

[

1

2(∂φ)2 − m2

2φ2

]

+ Lint(φ)

(14.67)

goes over into i times the euclidean action

AE =∫

dDxE

[

1

2(∂Eφ)

2 +m2

2φ2

]

+ Lint(φ)

. (14.68)

The euclidean versions of the Gaussian integral formulas (14.40) and (14.89) are∫

Dφ(x)e− 12

∫

dDxdDx′ φ(x)M(x,x′)φ(x′) = (DetM)∓12 , (14.69)

and∫

Dφ∗(x)Dφ(x)e−∫

dDx dDx′ φ∗(x)M(x,x′)φ(x′) = (DetM)∓1 , (14.70)

for complex fields φ = (φ1 + iφ2)/√2, where Dφ∗(x)Dφ(x) ≡ Dφ1(x)Dφ2(x).

The amplitude (14.55) becomes

wE [φ] =e−∫

dDxE LE(φ,∂φ)

∫ Dφ(x)e−∫

dDxE LE 0(φ,∂φ). (14.71)

The normalized version of this,

w[φ] =e−∫

dDxE LE(φ,∂φ)

∫ Dφ(x)e−∫

dDxE LE(φ,∂φ), (14.72)

represents the functional version of the proper quantum statistical Gibbs distributioncorresponding to (14.46) [recall (1.489)]:

wn =e−En/kBT

∑

n e−En/kBT

. (14.73)

14.6 Functional Integral Representation for Fermions 937

The functional integral representation for the unnormalized generating functionalof all Wick-rotated Green functions corresponding to (14.58) is then

ZE[j] =∫

Dφ(x)e−∫

dDxE [LE(φ,∂Eφ)−j(x)φ(x)]. (14.74)

The euclidean action corresponds to an energy of a field configuration. Theintegrand plays the role of a Boltzmann factor and gives the relative probability forthis configuration to occur in a thermodynamic ensemble.

We now understand the advantage of working with the unnormalized functionalintegral: At zero external source, ZE[j] corresponds precisely to the thermodynamicpartition function of the system. This will be seen explicitly in the examples inSection 14.7.

14.6 Functional Integral Representation for Fermions

If we want to use the functional technique to also describe the statistical proper-ties of fermions, some modifications are necessary. Then the fields must be takento be anticommuting c-numbers. In mathematics, such objects form a so-calledGrassmann algebra G. If ξ, ξ′ are real elements of G, then

θθ′ = −θ′θ. (14.75)

A trivial consequence of this condition is that the square of each Grassmann elementvanishes, i.e., θ2 = 0. If θ = θ1 + iθ2 is a complex element of G, then θ2 = −θ∗θ =−2iθ1θ2 is nonzero, but (θ∗θ)2 = (θθ)2 = 0.

All properties of operator quantum field theory for fermions can be derived fromfunctional integrals if we find an appropriate extension of the integral formulasin the previous sections to Grassmann variables. Integrals are linear functionals.For Grassmann variables, these are completely determined from the following basicintegration rules, which for real θ are

∫

dθ√2π

≡ 0,∫

dθ√2πθ ≡ 1,

∫

dθ√2πθn ≡ 0, n > 1. (14.76)

For complex variable θ = (θ1 + iθ2)/√2, these lead to

∫

dθ√2π

≡ 0,∫

dθ∗√2πθ ≡ 1,

∫

dθ√2πθ∗ ≡ 1,

∫

dθ√2π

dθ∗√2π

(θ∗θ)n ≡ −δn1 , (14.77)

with the definition dθdθ∗ ≡ −idθ1dθ2.Note that these integration rules make the linear operation of integration in

(14.76) coincide with the linear operation of differentiation. A function F (θ) of areal Grassmann variable θ, is determined by only two parameters: the zeroth- andthe first-order Taylor coefficients. Indeed, due to the property θ2 = 0, the Taylor


series has only two terms F (θ) = F0 + F ′θ, where F0 = F (0) and F ′ ≡ dF (θ)/dθ.But according to (14.76), also the integral gives F ′:

∫

dθ√2πF (θ) = F ′. (14.78)

The coincidence of integration and differentiation has the important consequencethat any changes in integration variables will not transform with the Jacobian, butrather with the inverse Jacobian:

∫

dθ√2π

= a∫

d(aθ)√2π

, (14.79)

We shall use this transformation property below in Eq. (14.85).As far as perturbation theory is concerned, it is sufficient to define only Gaussian

functional integrals such as (14.40) and (14.89). In the discretized form, we mayderive the formula

∏

n

[

∫ ∞

−∞

dθn√2πiǫD

]

exp

(

i

2ǫ2D

∑

m,n

θmMmnθn

)

= det 1/2(ǫDM). (14.80)

The right-hand side is the inverse of the bosonic result (14.40). In addition, thereis an important difference: only the antisymmetric part of the functional matrixcontributes.

If the matrix Mmn is Hermitian, complex Grassmann variables are necessary toproduce a nonzero Gaussian integral. For complex variables we have

∏

n

[

∫

dθndθ∗n√

2πiǫD√2πiǫD

]

exp

(

iǫ2D∑

m,n

θ∗mMmnθn

)

= det (ǫDM), (14.81)

the right-hand side being again the inverse of the corresponding bosonic result(14.89):

We first prove the latter formula. After bringing the matrix Mmn to a diagonalform via a unitary transformation, we obtain the product of integrals

∏

n

[

∫

dθndθ∗n√

2πiǫD√2πiǫD

]

exp

(

iǫ2D∑

n

θ∗nMnθn

)

. (14.82)

Expanding the exponentials into a power series leaves only the first two terms, since(θ∗nθn)

2 = 0, so that the integral reduces to

∏

m

∫ dθndθ∗n√

2πiǫD√2πiǫD

(1 + iǫ2Dθ∗nMnθn). (14.83)

Each of these integrals is performed via the formulas (14.77), and we obtain theproduct of eigenvalues Mn, which is the determinant:

∏

m

Mn = detM. (14.84)

14.6 Functional Integral Representation for Fermions 939

For real fermion fields, we observe that an arbitrary real antisymmetric matrixMmn can always be brought to a canonical form C, that is zero except for 2 × 2matrices c = iσ2 along the diagonal, by a real orthogonal transformation T . ThusM = T TCT . The matrix C has a unit determinant so that det T = det 1/2(M).Let θ′m ≡ Tmnθn, then the measure of integration in (14.80) changes according to(14.79) as follows:

∏

n

dθn = det T∏

n

dθ′n. (14.85)

Applying now the formulas (14.76), the Grassmannian functional integral (14.80)can be evaluated as follows:

∏

n

[

∫

dθn√2πiǫD

]

exp

iǫ2D∑

k,l

θmMmnθn

=det T∏

n

[

∫

dθ′n√2πiǫD

]

exp

(

∑

m,n

θ′mCmnθ′n

)

= det 1/2(iǫDM). (14.86)

The integrals over θn in one dimension decompose into a product of two-dimensionalGrassmannian integrals involving the antisymmetric unit matrix c = iσ2. They havethe generic form

∏

n

[

∫ dθ′2n√2πiǫD

]

dθ′2n+1√2πiǫD

(

1 + iǫ2Dθ′2nθ′2n+1

)

= ǫ2D. (14.87)

There is one such factor for every second lattice site, which changes det T = det 1/2Minto det 1/2(ǫDM), thus proving (14.80). See [4]. .

In the continuum limit, the result of this discussion can be summarized in anextension of the Gaussian functional integral formulas (14.40) and (14.26) to

∫

Dφ(x)e i2∫

dDxdDx′ φ(x)M(x,x′)φ(x′) = Det ∓ 12M, (14.88)

where M(x, x′) is real symmetric or antisymmetric for bosons or fermions, respec-tively, and

∫

Dφ∗(x)Dφ(x)ei∫

dDx dDx′ φ∗(x)M(x,x′)φ(x′) = Det∓1M, (14.89)

where the matrix is Hermitian.The functional integral formulation of fermions follows now closely that of bosons.

For N relativistic real fermion fields χa, we can obtain an amplitude Z[χ] from theFourier transform

Z[χ] =∫

Dj(x)Z[j]e−i∫

dDx ja(x)χa(x), (14.90)

and find

Z[χ] =ei∫

dDxL(χ,∂χ)[

∫ Dχei∫

dDxL0(χ,∂χ)] . (14.91)


The functional

Z[j] = N∏

a

[∫

Dχa]

ei∫

dDxL(χ,∂χ), (14.92)

with a normalization factor

N−1 =∏

a

[∫

Dχa]

ei∫

dDxL0(χ,∂χ), (14.93)

provides us with a functional integral representation of the generating functional ofall fermionic Green functions.

An obvious extension of this holds for complex fermion fields. In the case of aDirac field, for example, where the sources j are commonly denoted by η, we obtain

Z[η, η] = N∫

DψDψ∗ei∫

dDx [ψ(i/∂ −m)ψ+Lint(ψ)+ηψ+ψη] (14.94)

with

N−1 =∫

DψDψ∗eiψ(i/∂−m)ψ = Det iG−10 (x, x′) = Det (i/∂ −m). (14.95)

As in the boson case, we shall from now on work with the unnormalized functionalwithout the factor N ,

Z[η, η] =∫

Dψei∫

dDx [ψ(i/∂−m)ψ+Lint(ψ)+ηψ+ψη], (14.96)

which again has the advantage that the euclidean version of Z[0, 0] becomes directlythe thermodynamic partition function of the system.

The functional representations of the generating functionals can of course becontinued to a euclidean form, as in Section 14.5, thereby replacing operator quan-tum physics by statistical physics. The corresponding Gaussian formulas for bosonand fermion fields are the obvious generalization of Eqs. (14.88) and (14.89):

∫

Dφ(x)e− 12

∫

dDxdDx′ φ(x)M(x,x′)φ(x′) = Det∓ 12M, (14.97)

and∫

Dφ(x)Dφ∗(x)e−∫

dDx dDx′ φ∗(x)M(x,x′)φ(x′) = Det ∓1M. (14.98)

Here we have defined the measure of the euclidean functional integration in the sameway as before in Eqs. (14.88) and (14.89), except without the factors i under thesquare roots. The euclidean version of the generating functional (14.94) can be usedto obtain all Wick-rotated Green functions from functional derivatives Z[η, η].

14.7 Relation Between Z[j] and the Partition Function 941

As a side result of the above development we can state the following functionalintegral formulas known under the name Hubbard-Stratonovich transformations:

∫

Dϕ(x, t)e i2∫

d3xdtd3x′dt′ [ϕ(x,t)A(x,t;x′,t′)ϕ(x′,t′)+2j(x,t)ϕ(x,t)δ3(x−x′,t)δ(t−t′)]

= ei(± i

2Trlog

1i

A)− i2

∫

d3xdtd3x′dt′ j(x,t)A−1(x,t;x′,t′)j(x′,t′), (14.99)

∫

Dψ∗(x, t)Dψ(x, t)ei∫

d3xdtd3x′dt′ ψ∗(x,t)A(x,t;x′,t′)ψ(x,t′)+[η∗(x,t)ψ(x)δ3(x−x′)δ(t−t′)+c.c.]

= ei(±iTrlogA)−i∫

d3xdtd3x′dtη∗(x,t)A−1(x,t;x′,t′)η(x′,t′). (14.100)

These formulas will be needed repeatedly in the remainder of this text. They arethe basis for the reformulation of many interacting quantum field theories in termsof collective quantum fields .

14.7 Relation Between Z[j] and the Partition Function

The introduction of the unnormalized functional integral representation (14.58) forZ[j] was motivated by the fact that, in the euclidean version (14.74), Z[0] is equalto the thermodynamic partition function of the system, except for a trivial overallfactor. Let us verify this for a free field theory in D = 1 dimension. Then ZE [0] ofEq. (14.74) becomes

Zω =∫

Dx exp

−∫ β

0dτ

[

1

2x2(τ) +

ω2

2x2(τ)

]

. (14.101)

For D = 1, the fields φ(τ) may be interpreted as paths x(τ) in imaginary timeτ = −it, and we have changed the notation accordingly. In the exponent, werecognize the euclidean version

AE =∫ β

0dτ

[

1

2x2(τ) +

ω2

2x2(τ)

]

(14.102)

of the action of the harmonic oscillator:

A =∫ tb

tadt

[

1

2x2(t)− ω2

2x2(t)

]

, (14.103)

for tb − ta = −ihβ = −ih/kBT . Thus Zω in expression (14.114) is a quantum-statistical path integral for a harmonic oscillator. The measure of path integration isdefined as a product of integrals on a lattice of points τn = nǫ with n = 0, . . . , N +1on the τ -axis:

∫

Dx(τ) =N∏

n=0

∫

dxn√2πǫ

, (14.104)


where xn ≡ x(τn) and N + 1 = hβ/ǫ. For a finite τ -interval hβ, the paths have tosatisfy periodic boundary conditions

x(hβ) = x(0), (14.105)

as a reflection of the quantum-mechanical trace. On the τ -lattice, this impliesxN+1 = x0, and the action becomes

ANE =

1

2ǫN+1∑

n=1

[

(xn − xn−1)2

ǫ2+ ω2x2n

]

. (14.106)

This can be rewritten as

ANE =

1

2ǫ

N+1∑

n=1

xn(−ǫ2∇∇+ ǫ2ω2)xn, (14.107)

where ∇∇x denotes the lattice version of x. It may be represented as an (N + 1)×(N + 1)-matrix

−ǫ2∇∇ =

2 −1 0 . . . 0 0 −1−1 2 −1 . . . 0 0 0...

...0 0 0 . . . −1 2 −1−1 0 0 . . . 0 −1 2

. (14.108)

The Gaussian functional integral can now be evaluated using formula (14.40), andwe obtain

Zω = detN+1[−ǫ2∇∇+ ǫω2]−1/2. (14.109)

The determinant is calculated recursively,2 and yields

Zω =1

2 sinh(hωβ/2), (14.110)

where ω is the auxiliary frequency

ω ≡ 2

ǫarsinh

ωǫ

2. (14.111)

In the continuum limit ǫ→ 0, the frequency ω goes against ω, and Zω becomes

Zω =1

2 sinh(hωβ/2). (14.112)

This can be expanded as

Zω = e−hω/2kBT + e−3hω/2kBT + e−5hω/2kBT + . . . , (14.113)

2See the textbook [1] and Section 2.12 of the textbook [2].


which is the quantum statistical partition function of the harmonic oscillator, as wewanted to prove. The ground state has a nonzero energy, as observed in the operatordiscussion in Chapter 7.32.

For the quantum-mechanical version of the functional integral (14.114)

Zω =∫

Dx exp

i∫ tb

tadt

[

1

2x2(t)− ω2

2x2(t)

]

, (14.114)

we obtain, with the measure of functional integration analog to (14.104)

∫

Dx(t) =N∏

n=0

∫

dxn√2πiǫ

, (14.115)

and the use of the Gaussian integral formula (14.40), the result

Zω = detN+1[−ǫ2∇∇− ǫω2]−1/2. (14.116)

Thus we only have to replace ω → iω in (14.111)–(14.112). In the continuum limit,we therefore obtain

Zω =1

2 sin(hωβ/2). (14.117)

We end this section by mentioning that the path-integral representation of thepartition function (14.114) with the integration measure (14.104) can be obtainedfrom a euclidean phase space path integral3

Zω =∫ ∫

Dx(τ)Dp(τ)2πh

exp

∫ β

0dτ

[

ipx− 1

2p2 − ω2

2x2]

(14.118)

by going to a τ -lattice and integrating out the momentum variables. The momentumintegrals give the factors

√2πǫ in the denominators of the measure (14.104). In the

quantum-mechanical version of (14.118)

Zω =∫ ∫

Dx(t)Dp(t)2πh

exp

i∫ tb

tadt

[

px− 1

2p2 − ω2

2x2]

, (14.119)

the p-integrals produce the denominators containing the factors√i in the measure

(14.104).The measure of functional integration on the τ -lattice

N∏

n=0

[∫ ∞

−∞dxn

]N+1∏

n=1

[

∫ ∞

−∞

dpn2πh

]

(14.120)

is the obvious generalization of the classical statistical weight in phase space

∫ ∞

−∞dx∫ ∞

−∞

dp

2πh(14.121)

3For a detailed discussion of the measure see Chapters 2 and 7 in the textbook [2].


to fluctuating paths with many variables xn = x(τn).For completeness, we write down the free energy Fω = −β logZω associated with

the partition function (14.112):

Fω =1

βlog[2 sinh(hωβ/2)] =

hω

2+

1

βlog(1− e−βhω). (14.122)

At zero temperatures, only the ground-state oscillations contribute.The detailed time-sliced calculation of the partition functions has to be compared

with the formal evaluation of the partition function (14.114) according to formula(14.69), which would yield

Zω =∫

Dx(τ) exp

−∫ β

0dτ

[

1

2x2(τ) +

ω2

2x2(τ)

]

= Det−1/2(−∂2τ + ω2). (14.123)

In this continuum formulation, the right-hand side is at first meaningless. It differsfrom the results obtained by proper time-sliced calculations (14.109)–(14.112) by atemperature-dependent infinite overall factor. In order to see what this factor is, wenote that for a periodic boundary condition (14.105), the eigenvectors of the matrix−∂2τ + ω2 are e−iωmτ with eigenvalues ω2

m + ω2. Since ωm grows with m like m2,the functional determinant Det −1/2(−∂2τ + ω2) is strongly divergent. Indeed, thebosonic lattice result (14.112) can only be obtained in the limit ǫ→ 0 after dividingout of the the lattice result (14.109) an equally divergent ω-independent functionaldeterminant and calculating the ratio

detN+1[−ǫ2∇∇− ǫω2]−1/2

hβdet′N+1[−ǫ2∇∇− ǫω2]−1/2−−−→ǫ→0

Det −1/2(−∂2τ + ω2)

hβDet ′−1/2(−∂2τ ). (14.124)

The prime in the denominator indicates that the zero-frequency ω0 must be omittedto obtain a finite result. The factor 1/hβ is the regularized contribution of the zerofrequency. This follows from a simple integral consideration. An integral

∫ b/2

−b/2

dx√2πe−ω

2x2/2 (14.125)

gives 1/ω only for finite ω(≫ 1/b). In the limit ω → 0, it gives b/√2π. In the path

integral, the zero mode is associated with the fluctuations of the average of the pathx(τ). To indicate this origin of the factor hβ in the ratio (14.124), we may write(14.124) as

Det−1/2(−∂2τ + ω2)

Det−1/2R (−∂2τ )

, (14.126)

where the subscript R indicates the above regularization.Explicitly, the ratio (14.124) is calculated from the product of eigenvalues:

Det−1/2(−∂2τ + ω2)

hβDet ′−1/2(−∂2τ )=

1

hβω

∏

m>0

ω2m

ω2m + ω2

. (14.127)


The product can be found in the tables4:

∏

m6=0

ω2m

ω2m + ω2

=hβω/2

sinh(hβω/2), (14.128)

so that (14.127) is equal to 1/2 sinh(hβω/2), and the properly renormalized partitionfunction (14.123) yields the lattice result (14.112). In a lattice calculation, the

determinant Det−1/2R (−∂2τ ) for β → ∞ is equal to unity.

It is curious to see that a formal evaluation of the functional determinant via theanalytic regularization procedure of Sections 7.12 and 11.5 is indifferent to this de-nominator. It produces precisely the same result from the formal expression (14.123)as from the proper lattice calculation. Recalling formula (11.134) and setting D = 1,we find

Det−1/2(−∂2τ + ω2) = exp

−1

2Tr log(−∂2τ + ω2)

= exp

−1

2βh

∫

dω′

2πlog(ω′2 + ω2)

= e−βhω2 . (14.129)

The exponent gives precisely the free energy at zero temperatures in Eq. (14.122).We now admit finite temperatures. For this purpose, we have to replace the

integral over ω′ by a sum over Matsubara frequencies (2.415), and evaluate

1

hβ

∑

m

log(ω2m + ω2) =

∫

dω′

2πlog(ω′2 + ω2) +

(

1

hβ

∑

m

−∫

dωm2π

)

log(ω2m + ω2).

(14.130)

In contrast to the first term whose evaluation required the analytic regularization,the second term is finite. It can be rewritten as

−(

1

hβ

∑

m

−∫

dωm2π

)

∫ ∞

ω2dω′2 1

ω2m + ω′2 . (14.131)

Recalling the summation formula (2.424), this becomes

−∫ ∞

ω2

dω′2

2ω′

[

coth(hβω′/2)tanh(hβω′/2)

− 1

]

= ±∫ ∞

ω2

dω′2

2ω′2

eβhω′ ∓ 1

= kBT 2 log(1∓ e−βhω). (14.132)

For later applications, we have treated also the case of fermionic Matsubara frequen-cies (the lower case). Together with the zero-temperature result (14.129), we findat any temperature

Zω = e−βFω = Det−1/2(−∂2τ + ω2) = exp

−1

2Tr log(−∂2τ + ω2)

= exp

−[

βhω

2+ log(1− e−βhω)

]

, (14.133)

in agreement with (14.122).

4I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.431.2: sinhx =∏∞

m=1(1 + x2/m2π2).


14.8 Bosons and Fermions in a Single State

The discussion in the last section cannot be taken over to fermionic variables x(τ).For fermions the action (14.114) vanishes identically, as a consequence of the sym-metry of the functional matrix D(τ, τ ′) = (−∂2τ + ω2)δ(τ − τ ′). A fermionic versionof the above path integral can only be introduced within the canonical formulation(14.119) of the harmonic path integral. With the help of a canonical transformation

a† =√

1/2hω(ωx− ip), a =√

1/2hω(ωx+ ip), (14.134)

this may be transformed into the action

AωQM =∫ tb

tadτ (a∗i∂ta− ωa∗a). (14.135)

A canonical quantization with commutation and anticommutation rules

[a(t), a†(t)]∓ = 1,

[a†(t), a†(t)]∓ = 0, (14.136)

[a(t), a(t)]∓ = 0

produces a second-quantized Hilbert space of the type discussed in Chapter 2. Sincea†(t) and a(t) carry no space variables. They describe Bose and Fermi particles ata single point.

The path-integral representation of the quantum-mechanical partition functionof this system is

ZωQM ≡∫ Da(t)Da∗(t)

2πeiAωQM. (14.137)

The measure of integration is∫ ∫ Da∗Da

2π≡∫ ∞

−∞

∫ ∞

−∞

D a1D a22π

, (14.138)

where a = (a1 + ia2)/√2 and a† = (a1 − ia2)/

√2 are directly obtained from the

canonical measure of path integration in (14.118).At a euclidean time τ = −it, the action of the free nonrelativistic field becomes

Aω =∫ hβ

0dτ (a∗∂τa+ ωa∗a), (14.139)

and the thermodynamic partition function has the path-integral representation

Zω ≡∫ Da(τ)Da∗(τ)

2πe−Aω . (14.140)

In this formulation, there is no problem of treating both bosons or fermions atthe same time. We simply have to assume the fields a(τ), a∗(τ) to be periodic orantiperiodic, respectively, in the imaginary-time interval hβ:

a(hβ) = ±a(0), (14.141)

14.9 Free Energy of Free Scalar Fields 947

or in the sliced formaN+1 = ±a0. (14.142)

Using formula (14.98), the partition function can be written as

Zω =∫ Da∗Da

2πexp

[

−∫ hβ

0dτ (a∗∂τa + ωa∗a)

]

= Det ∓1/2(∂τ + ω). (14.143)

Contact with the previous oscillator calculation is established by observing thatin the determinant, the operator ∂τ + ω can be replaced by the conjugate operator−∂τ+ω, since all eigenvalues come in complex-conjugate pairs, except for them = 0 -value, which is real. Hence the determinant of ∂τ +ω can be substituted everywhereby

det (∂τ + ω) = det (−∂τ + ω) =√

det (−∂2τ + ω2). (14.144)

In the boson case, we thus reobtain the result (14.123). In both cases, we maytherefore write

Zω = Det∓1/2(−∂2τ + ω2). (14.145)

The right-hand side was evaluated for bosons in Eqs. (14.129)–(14.133). In anticipa-tion of the present result, we have calculated the Matsubara sums up to Eq. (14.132)for bosons and fermions, with the results

Zω = e−βFω = Det∓1/2(−∂2τ + ω2) = exp

∓1

2Tr log(−∂2τ + ω2)

= exp

∓[

βhω

2+ log(1∓ e−βhω)

]

. (14.146)

This can be written as

Zω =

[sinh(hβω/2)]−1

cosh(hβω/2)

for

bosons,fermions.

. (14.147)

For bosons, the physical interpretation of this expression was given afterEq. (14.110). The analog sum of Boltzmann factors for fermions is

Zω = ehω/2kBT + e−hω/2kBT . (14.148)

This shows that the fermionic system at a point has two states, one with no particleand one with a single particle, where the no-particle state has a negative vacuumenergy, as observed in the operator discussion in Chapter 2.

14.9 Free Energy of Free Scalar Fields

The results of the last section are easily applied to fluctuating scalar fields. Considerthe free-field partition function

Z0 =∫

Dφ(x)e−∫

dDx 12 [(∂φ)2+m2φ2], (14.149)


which is of the general form (14.74) for j = 0. If the fields are decomposed into theirspatial Fourier components in a finite box of volume V ,

φ(x) =1√2V

∑

k

∫

[

eikxφk(τ) + c.c.]

, (14.150)

the partition function becomes

Z0 =∏

k

∫

Dφk(τ)e−∫ hβ

0dt 1

2|φk(τ)|2+k2|φk(τ)|2+m2|φk(τ)|2. (14.151)

For each k, the functional integral is obviously the same as in (14.114), so that

Z0 =∏

k

Zω(k), (14.152)

where ω(k) ≡√k2 +m2, and by (14.112)

Zω(k) =1

2 sinh[hω(k)β/2]. (14.153)

For a real field φ(x), the anticommuting alternative cannot be accommodatedinto the functional integral (14.149). We must first go to the field-theoretic analogof the path integral in phase space

Z0 =∫ ∫

Dφ(x)Dπ(x)2πh

× exp∫

d4x[

iπ(x)∂µφ(x)−1

2π2(x)− 1

2[(∇φ(x))2 +m2φ2(x)]

]

, (14.154)

where π(x) are the canonical field momenta (7.1). After the Fourier decomposition(14.150) and a similar one for π(x), we perform again a canonical transformationcorresponding to (14.134),

a†k(τ)=1√2hω

[ω(k)φk(τ)−iπk(τ)], ak(τ)=1√2hω

[ω(k)φk(τ)+iπk(τ)], (14.155)

and arrive at the analog of (14.143):

Zω =∏

k

∫ Da∗kDak2π

exp

[

−∫ hβ

0dτ (a∗k∂τak + ωa∗kak)

]

=∏

k

Det∓1/2(∂τ + ω(k)). (14.156)

This can be evaluated as in (14.147) to yield

Zω =∏

k

2 sinh[hβω(k)/2]−1

2 cosh[hβω(k)/2]

for

bosons,fermions.

. (14.157)

14.10 Interacting Nonrelativistic Fields 949

The associated free energies are

F0 = −β logZ0 = ±∑

k

ω(k)

2+ log

[

1∓ e−βhω(k)]

. (14.158)

For a complex field, the canonical transformation is superfluous. The field in thepartition function

Z0 =∫

Dφ(x)Dφ∗(x)e−∫

dDx 12(∂φ∗∂φ+m2φ∗φ) (14.159)

can directly be assumed to be of the bosonic or of the fermionic type. A directapplication of the Gaussian formula (14.98) leads to

Zω =∏

k

Det [−∂2τ + ω2(k)]∓1

=∏

k

2 sinh[hβω(k)/2]−2

2 cosh[hβω(k)/2]2

for

bosons,fermions.

, (14.160)

with free energies twice as large as (14.158).

14.10 Interacting Nonrelativistic Fields

The quantization of nonrelativistic particles was amply discussed in Chapter 2 andapplied to many-body Bose and Fermi systems in Chapter 3. Her we shall demon-strate that a completely equivalent formulation of the second-quantized nonrela-tivistic field theory is possible with the help of functional integrals.

Consider a many-fermion system described by an action

A ≡ A0 +Aint =∫

d3xdtψ∗(x, t) [i∂t − ǫ(−i∇)]ψ(x, t) (14.161)

−1

2

∫

d3xdtd3x′dt′ψ∗(x′, t′)ψ∗(x, t)V (x, t;x′t′)ψ(x, t)ψ(x′, t′)

with a translationally invariant two-body potential

V (x, t;x′, t′) = V (x− x′, t− t′). (14.162)

In the systems to be treated in this text we shall be concerned with a potential thatis, in addition, instantaneous in time

V (x, t;x′, t′) = δ(t− t′)V (x− x′). (14.163)

This property will greatly simplify the discussion.The fundamental field ψ(x) may describe bosons or fermions. The complete

information on the physical properties of the system resides in the Green functions.


In the operator Heisenberg picture, these are given by the expectation values of thetime-ordered products of the field operators

G (x1, t1, . . . ,xn, tn;xn′ , tn′, . . . ,x1′, t1′) (14.164)

= 〈0|T(

ψH(x1, t1) · · · ψH(xn, tn)ψ†H(xn′ , tn′) · · · ψ†

H(x1′ , t1′))

|0〉.

The time-ordering operator T changes the position of the operators behind it insuch a way that earlier times stand to the right of later times. To achieve the finalordering, a number of field transmutations are necessary. If F denotes the numberof transmutations of Fermi fields, the final product receives a sign factor (−1)F .

It is convenient to view all Green functions (14.164) as derivatives of the gener-ating functional

Z[η∗, η] = 〈0|T exp

i∫

d3xdt[

ψ†H(x, t)η(x, t) + η∗(x, t)ψH(x, t)

]

|0〉, (14.165)

namely

G (x1, t1, . . . ,xn, tn;xn′, tn′, . . . ,x1′ , t1′) (14.166)

= (−i)n+n′ δn+n′

Z[η∗, η]

δη∗(x1, t1) · · · δη∗(xn, tn)δη(xn′, tn′) · · · δη(x1′, tn′)

∣

∣

∣

∣

∣

η=η∗≡0

.

Physically, the generating functional (14.165) describes the probability amplitudefor the vacuum to remain a vacuum in the presence of external sources η∗(x, t) andη∗(x, t).

The calculation of these Green functionals is usually performed in the interactionpicture which can be summarized by the operator expression for Z:

Z[η∗, η] = N〈0|T exp

iAint[ψ†, ψ] + i

∫

d3xdt[

ψ†(x, t)η(x, t) + h.c.]

|0〉. (14.167)

In the interaction picture, the fields ψ(x, t) possess free-field propagators and thenormalization constant N is determined by the condition [which is trivially true for(14.165)]:

Z[0, 0] = 1. (14.168)

The standard perturbation theory is obtained by expanding expiAint in (14.167)in a power series and bringing the resulting expression to normal order via Wick’sexpansion technique. The perturbation expansion of (14.167) often serves conve-niently to define an interacting theory. Every term can be pictured graphically andhas a physical interpretation as a virtual process.

Unfortunately, the perturbation series up to a certain order in the coupling con-stant is unable to describe several important physical phenomena. Examples are theformation of bound states in the vacuum, or the existence and properties of collectiveexcitations in many-body systems. Those require the summation of infinite subsets


of diagrams to all orders. In many situations it is well-known which subsets have tobe taken in order to account approximately for specific effects. What is not so clearis how such lowest approximations can be improved in a systematic manner. Thepoint is that, as soon as a selective summation is performed, the original couplingconstant has lost its meaning as an organizer of the expansion and there is need fora new systematics of diagrams. This will be presented in the sequel.

As soon as bound states or collective excitations are formed, it is very suggestiveto use them as new quantum fields rather than the original fundamental particles ψ.The goal would then be to rewrite the expression (14.167) for Z[η∗, η] in terms ofnew fields whose unperturbed propagator has the free energy spectrum of the boundstates or collective excitations and whose Aint describes their mutual interactions.In the operator form (14.167), however, such changes of fields are hard to conceive.

14.10.1 Functional Formulation

In the functional integral approach, the generating functional (14.165) is given by[5]:

Z[η∗, η] = N∫

Dψ∗(x, t)Dψ(x, t)

× exp

iA[ψ∗, ψ] + i∫

d3xdt [ψ∗(x, t)η(x, t) + c.c.]

. (14.169)

It is worth emphasizing that the field ψ(x, t) in the path-integral formulation is acomplex number and not an operator. All quantum effects are accounted for byfluctuations; the path integral includes not only the classical field configurations butalso all classically forbidden ones, i.e., all those which do not run through the valleyof extremal action in the exponent.

By analogy with the development in Section 14.4 we take the interactions outsidethe integral and write the functional integral (14.169) as

Z[η∗, η] = exp

iAint

[

1

i

δ

δη,1

i

δ

δη∗

]

Z0[η∗, η], (14.170)

where Z0 is the generating functional of the free-field correlation functions, whosefunctional integral looks like (14.169), but with the action being only the free-particleexpression

A0[ψ∗, ψ] =

∫

dxdt ψ∗(x, t) [i∂t − ǫ(−i∇)]ψ(x, t), (14.171)

rather than the full A[ψ∗, ψ]=A0[ψ∗, ψ]+Aint[ψ

∗, ψ] of Eq. (14.161) in the exponent.The functional integral is of the Gaussian type (14.89) with a matrix

A(x, t;x′, t′) = [i∂t − ǫ(−i∇)] δ(3)(x− x′)δ(t− t′). (14.172)

This matrix is the inverse of the free propagator

A(x, t;x′, t′) = iG−10 (x, t;x′, t′) (14.173)


where

G0(x, t;x′, t′) =

∫ dE

2π

∫ d3p

(2π)4e−i[E(t−t′)−p(x−x′)] i

E − ǫ(p) + iη.

(14.174)

Inserting this into (14.100), we see that

Z0[η∗, η] = N exp

[

i(

±iTr log iG−10

)

−∫

d3xdtd3x′dt′η∗(x, t)G0(x′, t′)η(x′, t′)

]

.

(14.175)We now fix N in accordance with the normalization (14.168) to

N = exp [i (±iTr log iG0)] (14.176)

and arrive at

Z0[η∗, η] = exp

[

−∫

d3xdtd3x′dt′η∗(x, t)G0(x, t;x′, t′)η(x′, t′)

]

. (14.177)

This coincides exactly with what would have been obtained from the operator ex-pression (14.167) for Z0[η

∗, η] (i.e., without Aint).Indeed, according to Wick’s theorem [2, 5, 6], any time ordered product can

be expanded as a sum of normal products with all possible contractions taken viaFeynman propagators. The formula for an arbitrary functional of free fields ψ, ψ∗ is

TF [ψ∗, ψ] = exp∫

d3xdtd3x′dt′δ

δψ(x, t)G0(x, t;x

′, t′)δ

δψ∗(x, t′):F [ψ∗, ψ] : . (14.178)

Applying this to

〈0|TF [ψ∗, ψ]|0〉 = 〈0|T exp[

i∫

dxdt(ψ∗η + η∗ψ)]

|0〉 (14.179)

one finds:

Z0[η∗, η] = exp

[

−∫

dxdtdx′dt′η∗(x, t)G0(x, t;x′, t′)η(x′, t′)

]

× 〈0| : exp[

i∫

dxdt(ψ∗η + η∗ψ)]

: |0〉. (14.180)

The second factor is equal to unity, thus proving the equality of this operatoriallydefined Z0[η

∗, η] with the path-integral expression (14.177). Because of (14.170),this equality holds also for the interacting geerating functional Z[η∗, η].

14.10.2 Grand-Canonical Ensembles at Zero Temperature

All these results are easily generalized from vacuum expectation values to thermo-dynamic averages at fixed temperatures T and chemical potential µ. The change


at T = 0 is trivial: The single particle energies in the action (14.161) have to bereplaced by

ξ(−i∇) = ǫ(−i∇)− µ (14.181)

and new boundary conditions have to be imposed upon all Green functions via anappropriate iǫ prescription in G0(x, t;x

′, t′) of (14.174) [see [2, 7]]:

T=0G0(x, t;x′, t′) =

∫

dEd3p

(2π)4e−iE(t−t′)+ip(x−x′) i

E − ξ(p) + iη sgn ξ(p). (14.182)

Note that, as a consequence of the chemical potential, fermions with ξ < 0 insidethe Fermi sea propagate backwards in time. Bosons, on the other hand, have ingeneral ξ > 0 and, hence, always propagate forward in time.

In order to simplify the notation we shall often use four-vectors p = (p0,p) andwrite the measure of integration in (14.182) as

∫

dEd3p

(2π)4=∫

d4p

(2π)4. (14.183)

Note that in a solid, the momentum integration is really restricted to a Brillouinzone. If the solid has a finite volume V , the integral over spacial momenta becomesa sum over momentum vectors,

∫

d3p

(2π)3=

1

V

∑

p

, (14.184)

and the Green function (14.182) reads

T=0G0(x, t;x′, t′) ≡

∫

dE

2π

1

V

∑

p

e−ip(x−x′) i

p0 − ξ(p) + iη sgn ξ(p). (14.185)

The resulting power series expansions for the generating functional at zero-temperature T=0Z[η∗, η] and nonzero coupling can be written down as before afterperforming a Wick rotation in the complex energy plane in all energy integrals oc-curring in the expansions of formulas (14.180) and (14.167) in powers of the sourcesη(x, t) and η∗(x, t). For this, one sets E = p0 ≡ iω and replaces

∫ ∞

−∞

dE

2π→ i

∫ ∞

−∞

dω

2π. (14.186)

Then the Green function (14.182) becomes

T=0G0(x, t;x′, t′) = −

∫

dω

2π

d3p

(2π)3eω(t−t

′)+ip(x−x′) 1

iω − ξ(p). (14.187)

Note that with formulas (14.170) and (14.177) the generating functional T=0Z[η∗, η]is the grand-canonical partition function in the presence of sources [7].


Finally, we have to introduce arbitrary temperatures T . According to the stan-dard rules of quantum field theory (for an elementary introduction see Chapter 2 inRef. [2]), we must continue all times to imaginary values t = iτ , restrict the imag-inary time interval to the inverse temperature5 β ≡ 1/T , and impose periodic orantiperiodic boundary conditions upon the fields ψ(x,−iτ) of bosons and fermions,respectively [2, 7]:

ψ(x,−iτ) = ±ψ(x,−i(τ + 1/T )). (14.188)

When there is no danger of confusion, we shall drop the factor −i in front of theimaginary time in field arguments, for brevity. The same thing will be done withGreen functions.

By virtue of (14.177), these boundary conditions wind up in all free Green func-tions, i.e., they have the property

TG0 (x, τ + 1/T ;x′, τ ′) ≡ ±T

G0(x,−iτ ;x′,−iτ ′). (14.189)

This property is enforced automatically by replacing the energy integrations∫∞−∞ dω/2π in (14.187) by a summation over the discrete Matsubara frequencies [byanalogy with the momentum sum (14.184), the temporal “volume” being β = 1/T ]

∫ ∞

−∞

dω

2π→ T

∑

ωn

, (14.190)

which are even or odd multiples of πT

ωn =

2n2n+ 1

πT for

bosonsfermions

. (14.191)

The prefactor T of the sum over the discrete Matsubara frequencies accounts for thedensity of these frequencies yielding the correct T → 0-limit.

Thus we obtain the following expression for the imaginary-time Green functionof a free nonrelativistic field at finite temperature (the so-called free thermal Green

function)

TG0(x, τ,x

′, τ ′) =− T∑

ωn

∫

d3p

(2π)3e−iωn(τ−τ

′)+ip(x−x′) 1

iωn − ξ(p). (14.192)

Incorporating the Wick rotation in the sum notation we may write

T∑

p0

= −iT∑

ωn

= −iT∑

p4

, (14.193)

where p4 = −ip0 = ω. If temperature and volume are both finite, the Green functionis written as

TG0(x, τ,x

′, τ ′) =− T

V

∑

p0

∑

p

e−iωn(τ−τ′)+ip(x−x′) 1

iωn − ξ(p). (14.194)

5We use natural units in this chapter, so that kB = 1, h = 1.


At equal space points and equal imaginary times, the sum can easily be evaluated.One must, however, specify the order in which τ → τ ′. Let η denote an infinitesimalpositive number and consider the case τ ′ = τ + η, where the Green function is

TG0(x, τ,x, τ + η) =− T

∑

ωn

∫ d3p

(2π)3eiωnη

1

iωn − ξ(p). (14.195)

Then the sum is found after converting it into a contour integral

T∑

ωn

eiωnη1

iωn − ξ(p)=

T

2πi

∫

Cdz

eηz

ez/T ∓ 1

1

z − ξ. (14.196)

The upper sign holds for bosons, the lower for fermions. The contour of integrationC encircles the imaginary z-axis in a positive sense, thereby enclosing all integer orhalf-integer valued poles of the integrand at the Matsubara frequencies z = iωm (seeFig. 2.8). The factor eηz ensures that the contour in the left half-plane does notcontribute.

By deforming the contour C into C ′ and contracting C ′ to zero we pick up thepole at z = ξ and find

T∑

ωn

eiωnη1

iωn − ξ(p)= ∓ 1

eξ(p)/T ∓ 1= ∓ 1

eξ(p)/T ∓ 1= ∓n(ξ(p)). (14.197)

The phase eηz ensures that the contour in the left half-plane does not contribute.The function on the right is known as the Bose or Fermi distribution function.

By subtracting from (14.197) the sum with ξ replaced by −ξ, we obtain theimportant sum formula

T∑

ωn

1

ω2n + ξ2(p)

=1

2ξ(p)coth±1 ξ(p)

T. (14.198)

In the opposite limit with τ ′ = τ − η, the phase factor in the sum would bee−iωmη leading to a contour integral

−kBT∑

ωm

eiωmη1

iωm − ξ(p)= ±kBT

2πi

∫

Cdz

e−ηz

e−z/kBT ∓ 1

1

z − ξ, (14.199)

and we would find 1± nξ(p).In the operator language, these limits correspond to the expectation values of

free non-relativistic field operators

TG0 (x, τ ;x, τ + η) = 〈0|T

(

ψH(x, τ)ψ†H(x, τ + η)

)

|0〉 = ±〈0|ψ†H(x, τ)ψH(x, τ)|0〉

TG0 (x, τ ;x, τ − η) = 〈0|T

(

ψH(x, τ)ψ†H(x, τ − η)

)

|0〉 = 〈0|ψH(x, τ)ψ†H(x, τ)|0〉

= 1± 〈0|ψ†H(x, τ)ψH(x, τ ∓ η)|0〉 .

The function n(ξ(p)) is the thermal expectation value of the number operator


N = ψ†H(x, τ)ψH(x, τ). (14.200)

In the case of T 6= 0 ensembles, it is also useful to employ a four-vector notation.The four-vector

pE ≡ (p4,p) = (ω,p) (14.201)

is called the euclidean four-momentum. Correspondingly, we define the euclidean

spacetime coordinate

xE ≡ (−τ,x). (14.202)

The exponential in (14.192) can be written as

pExE = −ωτ + px. (14.203)

Collecting integral and sum in a single four-dimensional summation symbol, we shallwrite (14.192) as

TG0(xE − x′) ≡ −T

V

∑

pE

exp [−ipE(xE − x′E)]1

ip4 − ξ(p). (14.204)

It is quite straightforward to derive the general T 6= 0 Green function from apath-integral formulation analogous to (14.169). For this we consider classical fieldsψ(x, τ) with the periodicity or anti-periodicity

ψ(x, τ) = ±ψ (x, τ + 1/T ) . (14.205)

They can be Fourier-decomposed as

ψ(x, τ) =T

V

∑

ωn

∑

p

e−iωnτ+ipxa(ωn,p) ≡T

V

∑

pE

e−ipExEa(pE) (14.206)

with a sum over even or odd Matsubara frequencies ωn. If a free action is nowdefined as

A0[ψ∗, ψ] = −i

∫ 1/2T

−1/2Tdτ∫

d3xψ∗(x, τ) [−∂τ − ξ (−i∇)]ψ(x, τ), (14.207)

formula (14.100) renders [1, 8]

TZ0[η

∗, η] = e∓Tr logA+

∫ ∫ 1/2T

−1/2Tdτdτ ′

∫

d3xd3x′η∗(x,τ)A−1(x,τ,x′,τ ′)η(x′,τ ′), (14.208)

with the functional matrix

A(x, τ ;x′, τ ′) = [∂τ + ξ (−i∇)] δ(3)(x− x′)δ(τ − τ ′). (14.209)


Its inverse A−1 is equal to the propagator (14.192), the Matsubara frequencies arisingdue to the finite-τ interval of euclidean space together with the periodic boundarycondition (14.205).

Again, interactions are taken care of by multiplying TZ0[η∗η] with the factor

(14.170). In terms of the fields ψ(x, τ), the exponent has the form:

Aint =1

2

∫ ∫ 1/2T

−1/2Tdτdτ ′

×∫

d3xd3x′ψ∗(x, τ)ψ∗(x′, τ ′)ψ(x′, τ ′)ψ(x, τ)V (x,−iτ ;x′,−iτ ′). (14.210)

In the case of a potential (14.163) that is instantaneous in time t, the potential ofthe euclidean formulation becomes instantaneous in τ :

V (x,−iτ ;x′,−iτ ′) = V (x− x′) iδ(τ − τ ′). (14.211)

In this case Aint can be written in terms of the interaction Hamiltonian as

Aint = i∫ 1/2T

−1/2TdτHint(τ). (14.212)

Thus the grand-canonical partition function in the presence of external sources maybe calculated from the path integral [8]:

TZ[η∗, η] =

∫

Dψ∗(x, τ)Dψ(x, τ)eiTA+

∫ 1/2T

−1/2Tdτ∫

d3x[ψ∗(x,τ)η(x,τ)+c.c.], (14.213)

where the grand-canonical action is

iTA[ψ∗, ψ] = −

∫ 1/2T

−1/2Tdτ∫

d3xψ∗(x, τ) [∂τ + ξ(−i∇)]ψ(x, τ) (14.214)

+i

2

∫ 1/2T

−1/2Tdτdτ ′

∫

d3xd3x′ψ∗(x, τ)ψ∗(x′, τ ′)ψ(x, τ ′)ψ(x, τ)V (x,−iτ ;x,−iτ ′).

The Green functions of the fully interacting theory are obtained from the func-tional derivatives

G (x1, τ1, . . . ,xn, τn;xn′ , τn′, . . . ,x1′, τ1′) (14.215)

= (−i)n+n′ δn+n′

Z[η∗, η]

δη∗(x1, τ1) · · · δη∗(xn, τn)δη(xn′, τn′) · · · δη(x1′, τn′)

∣

∣

∣

∣

∣

η=η∗≡0

.

Explicitly, the right-hand side represent the functional integrals

N∫

Dψ∗(x, t)Dψ(x, t)ψ(x1, τ1)· · ·ψ(xn, τn)ψ∗(xn′ , τn′)· · ·ψ∗(x1′ , τ1′)eiTA[ψ∗,ψ].(14.216)

In the sequel, we shall always assume the normalization factor to be chosen in sucha way that Z[0, 0] is normalized to unity. Then the functional integrals (14.216) areobviously the correlation function of the fields:

〈ψ(x1, τ1) · · · ψ(xn, τn)ψ∗(xn′ , τn′) · · · ψ∗(x1′ , τ1′)〉. (14.217)


In contrast to Section 1.2, the bra and ket symbols denote now a thermal averageof the classical fields.

If the generating functional of the interacting theory is evaluated in a pertur-bation expansion using formula (14.170), the periodic boundary conditions for thefree Green functions (14.189) will go over to the fully interacting Green functions(14.215).

The functional integral expression (14.213) for the generating functional havea great advantage in comparison to the equivalent operator formulation based on(14.180) and (14.170). They share with ordinary integrals the extreme flexibilitywith respect to changes in the field variables.

Summarizing we have seen that the functional (14.213) defines the most generaltype of theory involving two-body forces. It contains all information on a physicalsystem in the vacuum as well as in thermodynamic ensembles. The vacuum theoryis obtained by setting T = 0 and µ = 0, and by continuing the result back fromT to physical times. Conversely, the functional (14.169) in the vacuum can begeneralized to ensembles in a straight-forward manner by first continuing the timest to imaginary values −iτ via a Wick rotation in all energy integrals and then goingto periodic functions in τ .

There is a complete correspondence between the real-time generating functional(14.169) and the thermodynamic imaginary-time expression (14.213). For this rea-son it will be sufficient to exhibit all techniques only in one version, for which weshall choose (14.169). Note, however, that due to the singular nature of the propa-gators (14.174) in real energy-momentum, the thermodynamic formulation specifiesthe way how to avoid singularities.

14.11 Interacting Relativistic Fields

Let us see how this formalism works for relativistic boson and fermion systems.Consider a Lagrangian of Klein-Gordon and Dirac particles consisting of a sum

L(

ψ, ψ, ϕ)

= L0 + Lint. (14.218)

As in the case of nonrelativistic fields, all time ordered Green’s functions can beobtained from the derivatives with respect to the external sources of the generatingfunctional

Z [η, η, j] = const × 〈0|Tei∫

dx(Lint+ηψ+ψη+jϕ)|0〉. (14.219)

The fields in the exponent follow free equations of motion and |0〉 is the free-fieldvacuum. The constant is conventionally chosen to make Z [0, 0, 0] = 1, i. e.

const =[

〈0|Tei∫

dxLint(ψ,ψ,ϕ)|0〉]−1

. (14.220)

This normalization may always be enforced at the very end of any calculation suchthat Z [η, η, j] is only interesting as far as its functional dependence is concerned.Any constant prefactor is irrelevant.

14.11 Interacting Relativistic Fields 959

It is then straight-forward to show that Z [η, η, j] can alternatively be computedvia the Feynman path-integral formula

Z [η, η, j] = const ×∫

DψDψDϕei∫

dx[L0(ψ,ψ,ϕ)+Lint+ηψ+ψη+jϕ]. (14.221)

Here the fields are no more operators but classical functions (with the mental reser-vation that classical Fermi fields are anticommuting objects). Notice that contraryto the operator formula (14.219) the full action appears in the exponent.

For simplicity, we demonstrate the equivalence only for one real scalar field ϕ(x).The extension to other fields is immediate [9, 10, 51]. Note that it is sufficient togive the proof for free fields, where

Z0 [j] = 〈0|Tei∫

dxj(x)ϕ(x)|0〉= const ×

∫

Dϕei∫

dx[ 12ϕ(x)(−x−µ2)ϕ(x)+j(x)ϕ(x)]. (14.222)

For if it holds there, a simple multiplication on both sides of (14.222) by the differ-ential operator

ei∫

dxLint( 1i

δδj(x)) (14.223)

would extend it to the interacting functionals (14.219) or (14.221). But (14.222)follows directly from Wick’s theorem, according to which any time-ordered productof a free field can be expanded into a sum of normal products with all possible timeordered contractions. This statement can be summarized in operator form valid forany functional F [ϕ] of a free field ϕ(x):

TF [ϕ] = e12

∫

dxdy δδϕ(x)

D(x−y) δδϕ(y) : F [ϕ] : , (14.224)

where D(x− y) is the free-field propagator

D(x− y) =i

−x − µ2 + iǫδ(x− y) =

∫ d4q

(2π)4e−iq(x−y)

i

q2 − µ2 + iǫ. (14.225)

Applying this to (14.224) gives

Z0 = e12

∫

dxdy δδϕ(x)

D(x−y) δδϕ(y) 〈0| : ei

∫

dxj(x)ϕ(x) : |0〉= e−

12

∫

dxdyj(x)D(x−y)j(y)〈0| : ei∫

dxj(x)ϕ(x) : |0〉= e−

12

∫

dxdyj(x)D(x−y)j(y). (14.226)

The last part of the equation follows from the vanishing of all normal products ofϕ(x) between vacuum states.

Exactly the same result is obtained by performing the functional integral in(14.222) and by using the functional integral formula (14.99). The matrix A is equalto A(x, y) = (−x − µ2) δ(x− y), and its inverse yields the propagator D(x− y):

A−1(x, y) =1

−x − µ2 + iǫδ(x− y) = −iD(x− y), (14.227)


yielding again (14.226).The generating functional of a free Dirac field theory reads

Z0 [η, η] = 〈0|Tei∫

(ηψ+ˆψη)dx|0〉= const ×

∫

DψDψei∫

dx[L0(ψ,ψ)+ηψ+ψη]. (14.228)

where L0(x) is the free-field Lagrangian

L0(x) = ψ(x) (iγµ∂µ −M)ψ(x) = ψ(x)A(x.y)ψ(x), (14.229)

By analogy with the bosonic expression (14.226) we obtain for Dirac particles

Z0[η, η] = e12

∫

dxdy δδψ(x)

G0(x−y) δδψ(y) 〈0| : ei

∫

dx(ηψ+ˆψ)η : |0〉= e−

12

∫

dxdyη(x)G0(x−y)η(y)〈0| : ei∫

dx(ηψ+ˆψ)η : |0〉= e−

12

∫

dxdyη(x)G0(x−y)η(y), (14.230)

where G0(x− y) is the free fermion propagator, related to the functional inverse ofthe matrix A(x, y) by

A−1(x, y) =1

iγµ∂µ −M + iǫδ(x− y) = −iG0(x− y). (14.231)

Note that it is Wick’s expansion which supplies the free part of the Lagrangianwhen going from the operator form (14.224) to the functional version (14.221).

14.12 Plasma Oscillation

The functional formulation of second-quantized many-body systems allows us totreat efficiently various collective phenomena. As a first example we shall considera many-electron system that interacts only via long-range Coulomb forces. TheCoulomb forces give rise to collective modes called plasmons.

The other extremely important example caused by attractive short-range inter-actions will be treated in the next chapter.

14.12.1 Plasmon Fields

Let us give a first application of the functional method by transforming the grandpartition function (14.213) to plasmon coordinates.

For this, we make use of the Hubbard-Stratonovich transformation (14.99) andobserve that a two-body interaction (14.161) in the generating functional (14.213)can always be produced (following Maxwell’s original ideas in electromagnetism) byan auxiliary field ϕ(x) as follows:

exp[

− i

2

∫

dxdx′ψ∗(x)ψ∗(x′)ψ(x)ψ(x′)V (x, x′)]

(14.232)

= const×∫

Dϕ

i

2

∫

dxdx′[

ϕ(x)V −1(x, x′)ϕ(x′)−2ϕ(x)ψ∗(x)ψ(x)δ(x− x′)]

.

14.12 Plasma Oscillations 961

To abbreviate the notation, we have used a four-vector notation in which

x ≡ (x, t), dx ≡ d3xdt, δ(x) ≡ δ3(x)δ(t).

The symbol V −1(x, x′) denotes the functional inverse of the matrix V (x, x′), i.e., thesolution of the equation

∫

dx′V −1(x, x′)V (x′, x′′) = δ(x− x′′). (14.233)

The constant prefactor in (14.232) is [det V ]−1/2. Absorbing this into the al-ways omitted normalization factor N of the functional integral, the grand-canonicalpartition function Ω = Z becomes

Z[η∗, η] =∫

Dψ∗DψDϕ exp[

iA+ i∫

dx (η∗(x)ψ(x) + ψ∗(x)η(x))]

, (14.234)

where the new action is

A[ψ∗, ψ, ϕ] =∫

dxdx′

ψ∗(x) [i∂t − ξ(−i∇)− ϕ(x)] δ(x− x′)ψ(x′) (14.235)

+1

2ϕ(x)V −1(x, x′)ϕ(x′)

.

Note that the effect of using formula (14.99) in the generating functional amountsto the addition of the complete square in ϕ in the exponent:

1

2

∫

dxdx′[

ϕ(x)−∫

dyV (x, y)ψ∗(y)ψ(y)]

V −1(x, x′)

×[

ϕ(x′)−∫

dy′V (x′, y′)ψ∗(y′)ψ(y′)]

, (14.236)

together with the additional integration over Dϕ. This procedure of going from(14.161) to (14.235) is probably simpler mnemonically than via the formula (14.99).The fact that the functional Z remains unchanged by this addition is obvious, sincethe integral Dϕ produces only the irrelevant constant [det V ]−1/2.

The physical significance of the new field ϕ(x) is easy to understand: ϕ(x) isdirectly related to the particle density. At the classical level this is seen immediatelyby extremizing the action (14.235) with respect to variations δϕ(x):

δA∂ϕ(x)

= ϕ(x)−∫

dyV (x, y)ψ∗(y)ψ(y) = 0. (14.237)

Quantum mechanically, there will be fluctuations around the field configurationϕ(x) determined by Eq. (14.237), causing a difference between the Green functionsinvolving the fields ϕ(x) versus those involving the associated composite field oper-ators

∫

dyV (x, y)ψ∗(y)ψ(y). Due to the Gaussian nature of the Dϕ integration, the


difference between the two is quite simple. for example, one can easily see that thepropagators of the two fields differ merely by the direct interaction:

〈T (ϕ(x)ϕ(x′))〉 (14.238)

= V (x− x′) +⟨

T[∫

dyV (x, y)ψ(y)] [∫

dy′V (x′, y′)ψ∗(y′)ψ(y′)]⟩

.

For the proof, the reader is referred to Appendix 14A. Note that for a potentialV which is dominantly caused by a single fundamental-particle exchange, the fieldϕ(x) coincides with the field of this particle: If, for example, V (x, y) represents theCoulomb interaction

V (x, x′) =e2

|x− x′|δ(t− t′), (14.239)

then Eq. (14.237) amounts to

ϕ(x, t) = −4πe2

∇2 ψ

∗(x, t)ψ(x, t), (14.240)

revealing the auxiliary field as the electric potential.If the particles ψ(x) have spin indices, the potential will, in this example, be

thought of as spin conserving at every vertex, and Eq. (14.237) must be read as spincontracted: ϕ(x) ≡ ∫

d4yV (x, y)ψ∗α(y)ψα(y). This restriction is initially appliedonly for convenience, and can easily be dropped later. Nothing in our proceduredepends on this particular property of V and ϕ. In fact, V could arise from theexchange of many different fundamental particles and their multiparticle configura-tions (for example π, ππ, σ, ϕ, etc. in nuclei) so that the spin dependence is the rulerather than the exception.

The important point is now that the entire theory can be rewritten as a fieldtheory of only the auxiliary field ϕ(x). For this we integrate out ψ∗ and ψ inEq. (14.234), and make use of formula (14.100) to obtain

Z[η∗, η] ≡ Ω[η∗, η] = NeiA, (14.241)

where the new action is

A[ϕ] = ±Tr log(

iG−1ϕ

)

+1

2

∫

dxdx′η∗(x)Gϕ(x, x′)η(x′), (14.242)

with Gϕ being the Green function of the fundamental particles in an external clas-sical field ϕ(x):

[i∂t − χ(−i∇)− ϕ(x)]Gϕ(x, x′) = iδ(x− x′). (14.243)

The field ϕ(x) is called a plasmon field. The new plasmon action can easily beinterpreted graphically. For this, one expands Gϕ in powers of ϕ:

Gϕ(x, x′) = G0(x− x′)− i

∫

dx1G0(x− x1)ϕ(x1 − x′) + . . . (14.244)


iE−ζ(p) ϕ ϕ ϕ

Figure 14.1 The last pure-current term of collective action (14.242). The original funda-

mental particle (straight line) can enter and leave the diagrams only via external currents,

emitting an arbitrary number of plasmons (wiggly lines) on its way.

Hence the couplings to the external currents η∗, η in (14.242) amount to radiatingone, two, etc. ϕ fields from every external line of fundamental particles (see Fig.14.1).

A functional expansion of the Tr log expression in powers of ϕ gives

±iTr log(iG−1ϕ ) = ±iTr log(iG−1

0 )± iTr log(1 + iG0ϕ)

= ±iTr log(iG−10 )∓ iTr

∞∑

n=1

(−iG0ϕ)n 1

n. (14.245)

The first term leads to an irrelevant multiplicative factor in (14.241). The nth termcorresponds to a loop of the original fundamental particle emitting nϕ lines (see Fig.14.2).

Figure 14.2 Non-polynomial self-interaction terms of plasmons arising from the Tr log

in (14.242). The nth term presents a single-loop diagram emitting n plasmons.

Let us now use the action (14.242) to construct a quantum field theory of plas-mons. For this we may include the quadratic term

±iTr(G0ϕ)21

2(14.246)

into the free part of ϕ in (14.242) and treat the remainder perturbatively. The freepropagator of the plasmon becomes

0|Tϕ(x)ϕ(x′)|0 ≡ (2s+ 1)G0(x′, x). (14.247)

This corresponds to an inclusion into the V propagator of all ring graphs (see Fig.14.3). It is worth pointing out that the propagator in momentum space Gpl(k)contains actually two important physical informations. From the derivation at fixedtemperature it appears in the transformed action (14.242) as a function of discrete


Figure 14.3 Free plasmon propagator containing an infinite sequence of single-loop cor-

rections (“bubblewise summation”).

euclidean frequencies νn = 2πnT only. In this way it serves to set up a time-independent fixed-T description of the system. The calculation (2.16), however,renders it as a function in the whole complex energy plane. It is this function whichdetermines by analytic continuation the time-dependent collective phenomena forreal times6.

With the propagator (2.16) and the interactions given by (2.13), the original the-ory of fundamental fields ψ∗, ψ has been transformed into a theory of ϕ fields whosebare propagator accounts for the original potential which has absorbed ringwise aninfinite sequence of fundamental loops.

This transformation is exact. Nothing in our procedure depends on the statisticsof the fundamental particles nor on the shape of the potential. Such propertiesare important when it comes to solving the theory perturbatively. Only underappropriate physical circumstances will the field ϕ represent important collectiveexcitations with weak residual interactions. Then the new formulation is of greatuse in understanding the dynamics of the system. As an illustration consider a dilutefermion gas of very low temperature. Then the function ξ(−i∇) is ǫ(−i∇)−µ withǫ(−i∇) = −∇

2/2m.

14.12.2 Physical Consequences

Let the potential be translationally invariant and instantaneous:

V (x, x′) = δ(t− t′)V (x− x′). (14.248)

Then the plasmon propagator (2.16) reads in momentum space

Gpl(ν,k) = V (k)1

1− V (k)π(ν,k)(14.249)

where the single electron loop symbolizes the analytic expression7

π(ν,k) = 2T

V

∑

p

1

iω − p2/2m+ µ

1

i(ω + ν)− (p+ k)2/2m+ µ. (14.250)

6See the discussion in Chapter 9 of the last of Ref. [7] and G. Baym and N.D. Mermin, J. Math.Phys. 2, 232 (1961).

7The factor 2 stems from the trace over the electron spin.


The frequencies ω and ν are odd and even multiples of πT . In order to calculatethe sum we introduce a convergence factor eiωη, and rewrite (14.250) as

π(ν,k) = 2∫

d3p

(2π)31

ξ(p+ k)− ξ(p)− iν(14.251)

× T∑

ωn

eiωnη[

1

i(ωn + ν)− ξ(p+ k)− 1

iωn − ξ(p)

]

.

Using the summation formula (14.197), this becomes

π(ν,k) = 2∫

d3p

(2π)3n(p+ k)− n(p)

ǫ(p+ k)− ǫ(p)− iν, (14.252)

or, after some rearrangement,

π(ν,k) = −2∫ d3p

(2π)3n(p)

[

1

ǫ(p+ k)− ǫ(p)− iν+

1

ǫ(p− k)− ǫ(p) + iν

]

.(14.253)

Let us study this function for real physical frequencies ω = iν where we rewrite itas

π(ω,k)=−2∫ d3p

(2π)3n(p)

[

1

ǫ(p+k)−ǫ(p)− ω+

1

ǫ(p−k)−ǫ(p) + ω

]

, (14.254)

which can be brought to the form

π(ω,k)=2k2

mω2

∫

d3p

(2π)3n(p)

1

(ω − p · k/m+ iη)2 − (k2/2M)2. (14.255)

For |ω| > pFk/m+ k2/2m, the integrand is real and we can expand

π(ω,k)= 2k2

mω2

∫

d3p

(2π)3n(p)

1 +2p · kmω

+ 3

(

p · kmω

)2

+

(

p · kmω

)3

+80(p · k)4+m2ω2k4

16m2ω4+ . . .

. (14.256)

Zero Temperature

For zero temperature, the chemical potential coincides with the radius of the Fermisphere µ = pF , and all levels below the Fermi momentum are occupied, to that theFermi distribuion function is n(p) = Θ(p − pF ). Then the integral in (14.256) canbe performed trivially using the integral

2∫

d3p

(2π)3nT=0(p) =

N

V= n =

p3F3π2

, (14.257)


and we obtain

π(ω,k)=k2

ω2

n

m

1 +3

5

(

pFk

mω

)2

+1

5

(

pFk

mω

)4

+1

16

k4

m2ω2+ . . .

. (14.258)

Inserting this into (14.249) we find, for long wavelengths, the Green function

Gpl(ν,k) ≈ V (k)

[

1− V (k)

ω2

n

m+ . . .

]−1

. (14.259)

Thus the original propagator is modified by a factor

ǫ(ω,k) = 1− 4πe2

ω2

n

m+ . . . . (14.260)

The dielectric constant vanishes at the frequency

ω = ωpl =

√

4πe2

m, (14.261)

which is the famous plasma frequency of the electron gas. At this frequency, theplasma propagator (14.249) has a pole on the real-ω axis, implying the existence ofan undamped excitation of the system.

For an electron gas we insert the Coulomb interaction (14.240), and obtain

Gpl(ν,k) ≈ 4πe2

k2

[

1− 4πe2

mω2n + . . .

]−1

. (14.262)

Thus the original Coulomb propagator is modified by a factor

ǫ(ω,k) = 1− 4πe2

mω2n+ . . . , (14.263)

which is simply the dielectric constant.The zero temperature limit can also be calculated exactly starting from the

expression (14.256), written in the form

π(ω,k) = −2∫ d3p

(2π)3Θ(p− pF )

[

1

p · k+ k2/2m− ω+ (ω → −ω)

]

. (14.264)

Performing the integral yields

π(ω,k) = −mpF2π2

1− 1

2kpF

p2F −(

k

2+mω

k

)2

+ p2F

logk2 + 2mω − 2kpFk2 + 2mω + 2kpF

+ (ω → −ω). (14.265)

The lowest terms of a Taylor expansion in powers of k agree with (14.258).


Short-Range Potential

Let us also find the real poles of Gpl(ν,k) for a short-rang potential where thesingularity at k = 0 is absent. Then a rotationally invariant [V (k)]−1 has the long-wavelength expansion

[V (k)]−1 = [V (0)]−1 + ak2 + . . . , (14.266)

as long as [V (0)]−1 is finite and positive, i.e., for a well behaved overall repulsivepotential satisfying V (0) =

∫

d3xV (x) > 0. Then the Green function (14.249)becomes

Gpl(ω,k) = ω2

ω2 [V (0)]−1 + aω2k2 − k2n

m

1 +3

5

(

pFk

mω

)2

+ . . .

−1

. (14.267)

This has a pole at ω = ±c0k where

c0 = V (0)n

m. (14.268)

This is the spectrum of zero sound with the velocity c0.In the neighbourhood of the positive-energy pole, the propagator has the form

Gpl(k0, k) ≈ V (0)× |k|ω − c0|k|

. (14.269)

Nonzero Temperature

In order to discuss the case of nonzero temperature it is convenient to add andsubtract a term −n(p+ k)n(k) in the numerator of (14.252), and rewrite it as

π(ν,k) = 2∫ d3p

(2π)3n(p+ k) [1− n(p)]− n(p) [1− n(p+ k)]

ǫ(p+ k)− ǫ(p)− iν, (14.270)

which can be rearranged to

π(ν,k) = −4∫ d3p

(2π)3n(p) [1− n(p+ k)]

ǫ(p+ k)− ǫ(p)

[ǫ(p+ k)− ǫ(p]2 + ν2. (14.271)

In the high-temperature limit the Fermi distribution becomes Boltzmannian,n(p) ≈ e−β(p

2/2−µ), and we evaluate again most easily expression (14.256) as follows:

π(ω,k) = −2∫ ∞

0dσ

∫

d3p

(2π)3e−β(p

2/2−µ)−σ[ǫ(p+k)−ǫ(p)−ω] + (ω → −ω) . (14.272)

The right-hand side is equal to

−∫ ∞

0dσ∫ 1

−1

d cos θ

2

∫

d3p

(2π)3e−β(p

2/2−µ)−σ[(pk cos θ/m+k2/2m)−ω] + (ω → −ω). (14.273)

Performing the angular integral yields

−m2

k2

∫ ∞

0

dσ

2π2

∫

d3p

(2π)3e−β(p

2/2−µ)+σω 1

p2(pk cosh pk − sinh pk) + (ω → −ω). (14.274)


14.13 Pair Fields

The introduction of a scalar field ϕ(x) was historically the first way, invented byMaxwell, to convert the Coulomb interaction in a theory (14.232) into a local fieldtheory. The resulting plasmon action depends only on the local field ϕ. There existsan alternative way of converting the interaction between four fermions in (14.232)into a new field theory. That is based on introducing a bilocal scalar field which hasbeen very successful to understand the properties of electrons in of superconductors.It is a collective field complementary to the plasmon field. Generically it will becalled a pair field . It describes the dominant low-energy collective excitations insystems such as type II superconductors, superfluid 3He, excitonic insulators, etc.The pair field is originally a bilocal field and will be denoted by ∆(x t;x′t′), withtwo space arguments and two time arguments. It is introduced into the generatingfunctional by rewriting the exponential of the interaction term in (14.232) in thepartition function (14.169) as a functional integral

exp[

− i

2

∫

dxdx′ψ∗(x)ψ∗(x′)ψ(x′)ψ(x)V (x, x′)]

= const×∫

D∆(x, x′)D∆∗(x, x′)

× ei2

∫

dxdx′

|∆(x,x′)|2 1V (x,x′)

−∆∗(x,x′)ψ(x)ψ(x′)−ψ∗(x)ψ∗(x′)∆(x,x′)

. (14.275)

In contrast to the similar-looking plasmon expression (14.232), the inverse 1/V (x, x′)in (14.275) is understood as a numeric division for each x, y, not as a functionalinverse. Hence the grand-canonical potential becomes

Z[η, η∗] =∫

Dψ∗DψD∆∗D∆ eiA[ψ∗,ψ,∆∗,∆]+i∫

dx(ψ∗(x)η(x)+c.c.), (14.276)

with the action

A[ψ∗, ψ,∆∗,∆] =∫

dxdx′ ψ∗(x) [i∂t − ξ(−i∇)] δ(x− x′)ψ(x′) (14.277)

−1

2∆∗(x, x′)ψ(x)ψ(x′)− 1

2ψ∗(x)ψ∗(x′)∆(x, x′) +

1

2|∆(x, x′)|2 1

V (x, x′)

,

where ξp ≡ εp − µ is the grand-canonical single particle energy (2.256). This newaction arises from the original one in (14.169) by adding to it the complete square

i

2

∫

dxdx′ |∆(x, x)− V (x′, x)ψ(x′)ψ(x)|2 1

V (x, x′), (14.278)

which removes the fourth-order interaction term and gives, upon functional integra-tion over

∫ D∆∗D∆, merely an irrelevant constant factor to the generating func-tional.

At the classical level, the field ∆(x, x′) is nothing but a convenient abbreviationfor the composite field V (x, x′)ψ(x)ψ(x′). This follows from the equation of motionobtained by extremizing the new action with respect to δ∆∗(x, x′). This yields

δAδ∆∗(x, x′)

=1

2V (x, x′)[∆(x, x′)− V (x, x′)ψ(x)ψ(x′)] ≡ 0. (14.279)

14.13 Pair Fields 969

Quantum mechanically, there are Gaussian fluctuations around this solution whichare discussed in Appendix 14B.

Taking care of the spin components of the Fermi field, we can rewrite the firstline in the expression (14.278), which are quadratic in the fundamental fields ψ(x),in a matrix form as

1

2f ∗(x)A(x, x′)f(x′)

=1

2f †(x)

(

[i∂t − ξ(−i∇)] δ(x− x′) −∆(x, x′)−∆∗(x, x′) ∓ [i∂t + ξ(i∇)] δ(x− x′)

)

f(x′),(14.280)

where f(x) denotes the fundamental field doublet

f(x) =

(

ψ(x)ψ∗(x)

)

(14.281)

and f † ≡ f ∗T , as usual. Here the field f ∗(x) is not independent of f(x). Indeed,there is an identity

f †Af = fT(

0 11 0

)

Af. (14.282)

Therefore, the real-field formula (14.99) must be used to evaluate the functionalintegral for the generating functional

Z[η∗, η] =∫

D∆∗D∆ eiA[∆∗,∆]− 12

∫

dx∫

dx′j†(x)G∆(x,x′)j(x′), (14.283)

where j(x) collects the external source η(x) and its complex conjugate, j(x) ≡(

η(x)η∗(x)

)

. Then the collective action (14.278) reads

A[∆∗,∆] = ± i

2Tr log

[

iG−1∆ (x, x′)

]

+1

2

∫

dxdx′|∆(x, x′)|2 1

V (x, x′). (14.284)

The 2 × 2 matrix G∆ denotes the propagator iA−1 which satisfies the functionalequation

∫

dx′′(

[i∂t − ξ(−i∇)] δ(x−x′′) −∆(x, x′′)−∆∗(x, x′′) ∓ [i∂t + ξ(i∇)] δ(x−x′′)

)

G∆(x′′, x′)= iδ(x−x′).

(14.285)

Writing G∆ as a matrix

(

G G∆

G†∆ G

)

, the mean-field equations associated with this

action are precisely the equations used by Gorkov to study the behavior of type IIsuperconductors [12]. With Z[η∗, η] being the full partition function of the system,the fluctuations of the collective field ∆(x, x′) can now be incorporated, at least inprinciple, thereby yielding corrections to these equations.


Let us set the sources in the generating functional Z[η∗, η] equal to zero andinvestigate the behavior of the collective quantum field ∆. In particular, we want todevelop Feynman rules for a perturbative treatment of the fluctuations of ∆(x, x′).As a first step we expand the Green function G∆ in powers of ∆ as

G∆ = G0− iG0

(

0 ∆∆∗ 0

)

G0−G0

(

0 ∆∆∗ 0

)

G0

(

0 ∆∆∗ 0

)

G0+ . . . (14.286)

with

G0(x, x′) =

i

i∂t − ξ(−i∇)δ(x− x′) 0

0 ∓ i

i∂t + ξ(i∇)δ(x− x′)

. (14.287)

We shall see later that this expansion is applicable only close to the critical tem-perature Tc. Inserting this expansion into (14.283), the source term can be inter-preted graphically by the absorption and emission of lines ∆(k) and ∆∗(k), respec-tively, from virtual zig-zag configurations of the underlying particles ψ(k), ψ∗(k) (seeFig. 14.4)

iν−ν′+ω−ξ(q−q′+p)

iν′−ω−ξ(q′−p)

iω−ξ(p)∆∗(ν′,q′)

∆(ν,q)

Figure 14.4 Fundamental particles (fat lines) entering any diagram only via the external

currents in the last term of (14.283), absorbing n pairs from the right (the past) and

emitting the same number from the left (the future).

(14.286)+ + . . .

The functional submatrices in G0 have the Fourier representation

G0(x, x′) =

T

V

∑

p

i

p0 − ξpe−i(p

0t−px), (14.289)

G0(x, x′) = ±T

V

∑

p

i

−p0 − ξ−p

e−i(p0t−px), (14.290)

where we have used the notation ξp for the Fourier components ξ(p) of ξ(−i∇).The first matrix coincides with the operator Green function

G0(x− x′) = 〈0|Tψ(x)ψ†(x′)|0〉. (14.291)

The second one corresponds to

G0(x− x′) = 〈0|Tψ†(x)ψ(x′)|0〉 = ±〈0|T(

ψ(x′)ψ†(x))

|0〉= ±G0(x

′ − x) ≡ ±[G0(x, x′)]T , (14.292)


where T denotes the transposition in the functional sense (i.e., x and x′ are in-terchanged). After a Wick rotation of the energy integration contour, the Fouriercomponents of the Green functions at fixed energy read

G0(x− x′, ω) = −∑

p

1

iω − ξpeip(x−x′) (14.293)

G0(x− x′, ω) = ∓∑

p

1

−iω − ξ−p

eip(x−x′) = ∓G0(x′ − x,−ω). (14.294)

The Tr log term in Eq. (14.284) can be interpreted graphically just as easily byexpanding as in (4.15):

± i

2Tr log

(

iG−1∆

)

= ± i

2Tr log

(

iG−10

)

∓ i

2Tr

[

−iG0

(

0 ∆∆∗ 0

)

∆∗]n

1

n.(14.295)

The first term only changes the irrelevant normalization N of Z. To the remainingsum only even powers can contribute so that we can rewrite

A[∆∗,∆] = ∓i∞∑

n=1

(−)n

2nTr

[(

i

i∂t − ξ(−i∇)δ

)

∆

(

∓ii∂t + ξ(i∇)

δ

)

∆∗]n

+1

2

∫

dxdx′|∆(x, x′)|2 1

V (x, x′)

=∞∑

n=1

An[∆∗,∆] +

1

2

∫

dxdx′|∆(x, x′)|2 1

V (x, x′). (14.296)

This form of the action allows an immediate quantization of the collective field ∆.The graphical rules are slightly more involved technically than in the plasmon casesince the pair field is bilocal. Consider at first the free collective fields which can beobtained from the quadratic part of the action:

A2[∆∗,∆] = − i

2Tr

[(

i

i∂t − ξ(−i∇)δ

)

∆

(

i

i∂t + ξ(i∇)δ

)

∆∗]

. (14.297)

Variation with respect to ∆ displays the equations of motion

∆(x, x′) = iV (x, x′)

[(

i

i∂t − ξ(−i∇)δ

)

∆

(

i

i∂t + ξ(i∇)δ

)]

. (14.298)

This equation coincides exactly with the Bethe-Salpeter equation [13] in the lad-der approximation. Originally, this was set up for two-body bound-state vertexfunctions, usually denoted in momentum space by

Γ(p, p′) =∫

dxdx′ exp[i(px+ p′x′)]∆(x, x′). (14.299)

Thus the free excitations of the field ∆(x, x′) consist of bound pairs of the originalfundamental particles. The field ∆(x, x′) will consequently be called pair field. If


we introduce total and relative momenta q and P = (p − p′)/2, then (14.298) canbe written as8

Γ(P |q) = −i∫

d4P ′

(2π)4V (P − P ′)

i

q0/2 + P ′0 − ξq/2+P′ + iη sgn ξ

× Γ(P ′|q) i

q0/2− P ′0 − ξq/2−P′ + iη sgn ξ

. (14.300)

Graphically this formula can be represented as follows: The Bethe-Salpeter wave

Figure 14.5 Free pair field following the Bethe-Salpeter equation as pictured in this

diagram.

function is related to the vertex Γ(P |q) by

Φ(P |q) = Ni

q0/2 + P0 − ξq/2+P + iη sgn ξ

× i

q0/2 + P0 − ξq/2+P + iη sgn ξΓ(P |q). (14.301)

It satisfies

G0 (q/2 + P )G0 (q/2− P )Φ(P |q) = −i∫ dP ′

(2π)4V (P, P ′)Φ(P ′|q), (14.302)

thus coinciding, up to a normalization, with the Fourier transform of the two-bodystate wave functions

ψ(x, t;x′, t′) = 〈0|T (ψ(x, t)ψ(x′, t′)) |B(q)〉. (14.303)

If the potential is instantaneous, then (14.298) shows ∆(x, x′) to be factorizableaccording to

∆(x, x′) = δ(t− t′)∆(x,x′; t) (14.304)

so that Γ(P |q) becomes independent of P0.

8Here q abbreviates the four-vector qµ = (q0,q) with q0 = E.


Consider now the system at T = 0 in the vacuum. Then µ = 0 and ξp = εp > 0.One can perform the P0 integral in (14.300) with the result

Γ(P|q) =∫

d3P ′

(2π)4V (P−P′)

1

q0 − εq/2+P′ − εq/2−P′ + iηΓ(P′|q). (14.305)

Now the equal-time Bethe-Salpeter wave function

ψ(x,x′; t) ≡ N∫

d3Pdq0d3q

(2π)7exp

[

−i(

q0t− q · x + x′

2−P · (x− x′)

)]

× 1

q0 − εq/2+P − εq/2−P + iηΓ(P|q) (14.306)

satisfies

[

i∂t − ǫ(−i∇) − ǫ(−i∇′)

]

ψ(x,x′; t) = V (x− x)ψ(x,x′; t), (14.307)

which is simply the Schrodinger equation of the two-body system. Thus, in theinstantaneous case, the free collective excitations in ∆(x, x′) are the bound statesderived from the Schrodinger equation.

In a thermal ensemble, the energies in (14.300) have to be summed over theMatsubara frequencies only. First, we write the Schrodinger equation as

Γ(P|q) = −∫

d3P′

(2π)3V (P−P′)l(P′|q)Γ(P′|q) (14.308)

with

l(P|q) = −i∑

P0

G0 (q/2 + P ) G0 (P − q/2) (14.309)

= −i∑

P0

i

q0/2+P0−ξq/2+P+iη sgn ξ

i

q0/2− P0−ξq/2−P+iη sgn ξ.

After a Wick rotation and setting q0 ≡ iν, the replacement of the energy integrationby a Matsubara sum leads to

l(P|q) = −T∑

ωn

1

i (ωn + ν/2)− ξq/2+P

1

i (ωn − ν/2) + ξq/2−P

= T∑

ωn

1

iν − ξq/2+P − ξq/2−P

×[

1

i(ωn + ν/2)− ξq/2+P

− 1

i(ωn − ν/2) + ξq/2−P

]

= −±[

nq/2+P + nq/2−P

]

iν − ξq/2+P − ξq/2−P

. (14.310)


Here we have used the frequency sum [see (14.197)]

T∑

ωn

1

iωn − ξp= ∓ 1

eξp/T ∓ 1≡ ∓np. (14.311)

with np being the occupation number of the state of energy ξp. Using the identitynp → ∓1− np, the expression in brackets can be rewritten as −N(P,q) where

N(P|q) ≡ 1±(

nq/2+P + nq/2−P

)

=1

2

(

tanh ∓1 ξq/2+P

2T+ tanh ∓1 ξq/2−P

2T

)

, (14.312)

so that

l(P|q) = − N(P|q)iν − ξq/2+P − ξ(q/2−P

. (14.313)

Defining again a Schrodinger type wave function for T 6= 0 as in (14.306), thebound-state problem can be brought to the form (14.305) but with a momentumdependent potential V (P−P′)×N(P′|q). Thus the Bethe-Salpeter equation at anytemperature reads

Γ(P|q) =∫

d3P ′

(2π)4V (P−P′)N(P′|q) 1

q0 − εq/2+P′ − εq/2−P′ + iηΓ(P′|q). (14.314)

We are now ready to construct the propagator of the pair field ∆(x, x′) for T = 0.In many cases, this is most simply done by considering Eq. (14.300) with a potentialλV (P, P ′) rather than V , and asking for all eigenvalues λn at fixed q. Let Γn(P |q)be a complete set of vertex functions for this q. Then one can write the propagatoras

∆(P |q)∆†(P ′|q′) = −i∑

n

Γn(P |q)Γ∗n(P

′|q)λ− λn(q)

∣

∣

∣

∣

∣

λ=1

(2π)4δ(4)(q − q′) (14.315)

where a hook denotes, as usual, the Wick contraction of the fields. Obviously the ver-tex functions have to be normalized in a specific way, as discussed in Appendix 14A.

An expansion of (14.315) in powers of [λ/λn(q)]n exhibits the propagator of ∆

as a ladder sum of exchanges as shown in Fig. 14.6.

(14.316)+ . . .++

Figure 14.6 Free pair propagator, amounting to a sum of all ladders of fundamental

potential exchanges. This is revealed explicitly by the expansion of (14.315) in powers of

(λ/λn(q)).

14.14 Competition of Plasmon and Pair Fields 975

For an instantaneous interaction, either side is independent of P0, P′0. Then the

propagator can be shown to coincide directly with the scattering matrix T of theSchrodinger equation (14.307) and the associated integral equation in momentumspace (14.305) [see Eq. (14A.13)].

∆∆† = iT ≡ iV + iV1

E −HV. (14.317)

Consider now the higher interactionsAn, n ≥ 3 of Eq. (14.296). They correspondto zig-zag loops shown in Fig. 14.7. These have to be calculated with every possibleΓn(P |q),Γ∗

m(P |q) entering or leaving, respectively. Due to the P dependence at

(14.318)+ . . .+

Figure 14.7 Self-interaction terms of the non-polynomial pair action (14.296) amounting

to the calculation of all single zig-zag loop diagrams absorbing and emitting n pair fields.

every vertex, the loop integrals become very involved. A slight simplification arisesfor instantaneous potentials. Then the frequency sums can be performed. Only inthe special case of a completely local action, the full P -dependence disappears andthe integrals can be calculated. See Section IV.2 in Ref. [4].

14.14 Competition of Plasmon and Pair Fields

The Hubbard-Stratonovich transformation has a well-established place in many-body theory [8, 9, 4]. After it had been successfully applied in 1957 by Bardeen,Cooper, and Schrieffer to explain the phenomenon of superconductivity by withthe so-called BCS theory [50], Nambu and Jona-Lasinio [14] discovered that thesame mechanism which explains the formation of an energy gap and Cooper pairsof electrons in a metal can be used to understand the surprising properties of quarkmasses in the physics of strongly interacting particles. This aspect of particle physicswill be explained in Chapter 26.

In many-body theory, the use of the Hubbard-Stratonovich transformation hasled to a good understanding of important collective physical phenomena such asplasma oscillations and other charge-density type of waves, for example paramagnonsin superfluid He3. It has put heuristic calculations such as the Gorkov’s derivation[12] of the Ginzburg-Landau equations [15] on a reliable theoretical ground [4]. In


addition, it is in spirit close to the density functional theory [16] via the Hohenberg-Kohn and Kohn-Sham theorems [17].

In Sections 14.12 and 14.13 we have used the Hubbard-Stratonovich transfor-mation to rewrite the many-body action in two ways. One was theory of a localplasmon field ϕ(x), the other a thery of a bilocal scalar pair field ∆(x, x′). In thetheory of collective excitation, either of the two transformations has been helpful tounderstand either the behavior of electron gases or that of superconductors. In thefirst case, one is able to deal efficiently with the oscillations of charge distributions inthe gas. In the second case one is able to see how the electrons in a superconductorbecome bound to Cooper pairs, and how the binding gives rise to a frictionless flowof the doubly charged bosonic pairs through the metal. In general, however, thereexists a competition between the two mechanisms.

The transformation was cherished by theoreticians since it allows them to re-express a four-particle interaction exactly in terms of a collective field variable whosefluctations can in principle be described by higher loop diagrams. The only bitter pillis that any approximative treatment of a many-body system can describe interestingphysics only if calculations can be restricted to a few low-order diagrams. If this isnot the case, the transformation fails.

The trouble arises in all those many-body systems in which different collectiveeffects compete with similar relevance. Historically, an important example is thefermionic superfluid He3. While BCS superconductivity is described easily via theHubbard-Stratonovich transformation which turns the four-electron interaction intoa field theory of Cooper pairs, the same approach did initially not succeed in a liq-uid of He3-atoms. Due to the strongly repulsive core of an atom, the forces in theattractive p-wave are not sufficient to bind the Cooper pairs. Only by taking intoaccount the existence of another collective field that arises in the competing para-magnon channel was it possible to explain the formation of weakly-bound Cooperpairs [18].

It is important to learn how to deal with this kind of mixture. The answeris found with the help of Variational Perturbation Theory (VPT), which has beendiscussed in Chapter 3 (see the pages 177 and 216). The key is to abandon the fluc-tuating collective quantum fields introduced by means of the Hubbard-Stratonovichtransformation. Instead one must turn to a variety of collective classical fields. Af-ter a perturbative calculation of the effective action, one obtains a functional thatdepends on these classical fields. The dependence can be optimized, usually by ex-tremizing their influence upon the effective action. In this way one is able to obtainexponentially fast converging results.

It is the purpose of this section to point out how to circumvent the fatal focussingof the Hubbard-Stratonovich transformation on a specific channel and to take intoaccount the competition between several competing channels.

14.15 Ambiguity in the Selection of Important Channels 977

14.15 Ambiguity in the Selection of Important Channels

The basic weakness of the Hubbard-Stratonovich transformation lies in the differentpossibilities of decomposing the fourth order interaction by a quadratic completionwith the help of an auxiliary field. The first is based on the introduction of ascalar plasmon field ϕ(x) and the use of the quadratic-completion formula (14.232).The other uses the introduction of a bilocal pair field ∆(x, x′) in combination withthe quadratic-completion formula (14.275). The trouble with both approaches isthat, when introducing an auxiliary field ϕ(x) or ∆(x, x′) and summing over allfluctuations of one of the fields, the effects of the other is automatically included.At first sight, this may appear as an advantage. Unfortunately, this is an illusion.In either case, even the lowest-order fluctuation effect of the other is extremely hardto calculate. That can be seen most simply in the simplest models of quantumfield theory such as the Gross-Neveu model (to be discussed in Chapter 23). Therethe propagator of the quantum field ∆(x, x′) is a very complicated object. So itis practically impossible to recover the effects OF ϕ(x) from the loop calculationswith these propagators. As a consequence, the use of a specific quantum field theorymust be abandoned whenever collective effects of different channels are important.

To be specific let us assume the fundamental interaction to be of the local form

Alocint =

g

2

∫

xψ∗αψ

∗βψβψα = g

∫

xψ∗↑ψ

∗↓ψ↓ψ↑, (14.319)

where the subscripts ↑, ↓ denote the spin directions of the fermion fields. For brevity,we have absorbed the spacetime arguments x in the spin subscripts and written thesymbol

∫

x for an integral over spacetime and a sum over the spin indices of thefermion field.

We now introduce auxiliary classical collective fields which are no longer assumedto undergo functional fluctuations, and we replace the exponential in the interactingversion of the generating functional (14.276),

Z[η, η∗] =∫

Dψ∗Dψ eiA0[ψ∗,ψ]+iAlocint+i

∫

dx(ψ∗(x)η(x)+c.c.), (14.320)

identically by [34]

eiAlocint = expi g

∫

xψ∗↑,xψ

∗↓,xψ↓,xψ↑,x = exp

− i

2

∫

xfTx Mxfx

× expiAnewint (14.321)

= exp

− i

2

∫

x

(

ψβ∆∗βαψα + ψ∗

α∆αβψ∗β + ψ∗

βρβαψα + ψ∗αραβψβ

)

×expiAnewint .

Here f(x) is here the doubled spinor field (14.281) with spin index:

f(x) =

(

ψα(x)ψ∗α(x)

)

, (14.322)


and fTx denotes the transposed fundamental field doublet fTx = (ψα, ψ∗α). The new

interaction reads

Anewint =Aloc

int +1

2

∫

xfTx Mxfx =

∫

x

[

g

2ψ∗αψ

∗βψβψα (14.323)

+1

2

(

ψβ∆∗βαψα + ψ∗

α∆αβψ∗β

)

+ ψ∗αραβψβ

]

.

We now define a further free action by the quadratic form

Anew0 ≡ A0 −

1

2

∫

xfTx Mxfx =

1

2f †xA

∆,ρx,x′fx′. (14.324)

with the functional matrix A∆,ρx,x′ being now equal to

(

[i∂t−ξ(−i∇)]δαβ+ραβ ∆αβ

∆∗αβ [i∂t+ξ(i∇)]δαβ−ραβ

)

. (14.325)

The physical properties of the theory associated with the action A0+Alocint can now be

derived as follows: first we calculate the generating functional of the new quadraticaction Anew

0 via the functional integral

Znew0 [η, η∗] =

∫

Dψ∗Dψ eiAnew0 +i

∫

dx(ψ∗(x)η(x)+c.c.). (14.326)

From its functional derivatives with respect to the sources ηα and η†α we find thenew free propagators G∆ and Gρ. To higher orders, we expand the exponentialeiA

newint in a power series and evaluate all expectation values (in/n!)〈[Anew

int ]n〉new0 with

the help of Wick’s theorem. They are expanded into sums of products of the freeparticle propagators G∆ and Gρ. The sum of all diagrams up to a certain order gN

defines an effective collective action ANeff as a function of the collective classical fields

∆αβ ,∆∗βα, ραβ ,

Obviously, if the expansion is carried to infinite order, the result must be inde-pendent of the auxiliary collective fields since they were introduced and removed in(14.322) without changing the theory. However, any calculation can only be carriedup to a finite order, and that will depend on these fields. We therefore expect thebest approximation to arise from the extremum of the effective action [6, 21, 54].

The lowest-order effective collective action is obtained from the trace of thelogarithm of the matrix (14.325):

A0∆,ρ = − i

2Tr log

[

iG−1∆,ρ

]

. (14.327)

The 2× 2 matrix G∆,ρ denotes the propagator i[A∆,ρx,x′]

−1.To first order in perturbation theory we must calculate the expectation value

〈Aint〉 of the interaction (14.324). This is done with the help of Wick contractionsin the three channels, Hartree, Fock, and Bogoliubov:

〈ψ∗↑ψ

∗↓ψ↓ψ↑〉 = 〈ψ∗

↑ψ↑〉〈ψ∗↓ψ↓〉 − 〈ψ∗

↑ψ↓〉〈ψ∗↓ψ↑〉+ 〈ψ∗

↑ψ∗↓〉〈ψ↓ψ↑〉. (14.328)

14.15 Ambiguity in the Selection of Important Channels 979

For this purpose we now introduce the expectation values

∆∗αβ ≡ g〈ψ∗

αψ∗β〉, ∆βα ≡ g〈ψβψα〉 = [∆∗

αβ ]∗, (14.329)

ραβ ≡ g〈ψ∗αψβ〉, ρ†αβ ≡ [ρβα]

∗, (14.330)

and rewrite 〈Anewint 〉 as

〈Anewint 〉 =

1

g

∫

x(∆∗

↓↑∆↓↑ − ρ↑↓ρ↓↑ + ρ↑↑ρ↓↓)−1

2g

∫

x(∆βα∆

∗βα + ∆∗

αβ∆βα + 2ραβραβ).

Due to the locality of ∆αβ , the diagonal matrix elements vanish, and ∆αβ has theform cαβ∆, where cαβ is i times the Pauli matrix σ2

αβ . In the absence of a magneticfield, the expectation values ραβ may have certain symmetries:

ρ↑↑ ≡ ρ = ρ↓↓, ρ↑↓ = ρ↓↑ ≡ 0, (14.331)

so that (14.331) simplifies to

〈Anewint 〉=

1

g

∫

x

[

(|∆|2+ρ2)− (∆∆∗ + ∆∗∆+ 2ρρ)]

. (14.332)

The total first-order collective classical action A1∆,ρ is given by the sum

A1∆,ρ=A0

∆,ρ+〈Anewint 〉. (14.333)

Now we observe that the functional derivatives of the zeroth-order action A0∆,ρ are

the free-field propagators G∆ and Gρ

δ

δ∆αβ

A0∆,ρ = [G∆]αβ ,

δ

δραβA0

∆,ρ = [Gρ]αβ . (14.334)

Then we can extremize A1∆,ρ with respect to the collective fields ∆ and ρ, and find

that to this order these fields satisfy the gap-like equations

∆x = g[G∆]x,x, ρx = g[Gρ]x,x. (14.335)

If the fields satisfy (14.335), the extremal action has the value

A1∆,ρ = A0

∆,ρ −1

g

∫

x(|∆|2 + ρ2). (14.336)

Note how the theory differs, at this level, from the collective quantum field theoryderived via the Hubbard-Stratonovich transformation. If we assume that ρ vanishesidentically, the extremum of the one-loop action A1

∆,ρ gives the same result as ofthe mean-field collective quantum field action (14.284), which reads for the presentattractive δ-function in (14.319):

A1∆,0 = A0

∆,0 −1

g

∫

x|∆|2. (14.337)


On the other hand, if we extremize the action A1∆,ρ at ∆ = 0, we find the extremum

from the expression

A10,ρ = A0

0,ρ −1

g

∫

xρ2. (14.338)

The extremum of the first-order combined collective classical action (14.336) agreeswith the good-old Hartree-Fock-Bogolioubov theory.

The essential difference between this and the new approach arises in two issues:

• First, by being able to carry the expansion to higher orders: If the collectivequantum field theory is based on the Hubbard-Stratonovich transformation,the higher-order diagrams must be calculated with the help of the propaga-tors of the collective field such as 〈∆x∆x′〉. These are extremely complicatedfunctions. For this reason, any loop diagram formed with them is practicallyimpossible to integrate. In contrast to that, the higher-order diagrams in thepresent theory need to be calulated using only ordinary particle propagatorsG∆ and Gρ of Eq. (14.334) and the interaction (14.324). Even that becomes,of course, tedious for higher orders in g. At least, there is a simple rule tofind the contributions of the quadratic terms 1

2

∫

x fTx Mxfx in (14.322), given

the diagrams without these terms. One calculates the diagrams from only thefour-particle interaction, and collects the contributions up to order gN in aneffective action AN

∆,ρ. Then one replaces AN∆,ρ by AN

∆−ǫg∆,ρ−ǫgρ and re-expands

everything in powers of g up to the order gN , forming a new series∑Ni=0 g

iAi∆,ρ.

Finally one sets ǫ equal to 1/g [23] and obtains the desired collective classicaction AN

∆,ρ as an expansion extending (14.336) to:

AN∆,ρ =

N∑

i=0

Ai∆,ρ − (1/g)

∫

x(|∆|2 + ρ2). (14.339)

Note that this action must merely be extremized. There are no more quantumfluctuations in the classical collective fields ∆, ρ. Thus, at the extremum, theaction (14.339) provides us directly with the desired grand-canonical potential.

• The second essential difference with respect to the Hubbard-Stratonovichtransformation approach is the following: It becomes possible to study a richvariety of competing collective fields without the danger of double-countingFeynman diagrams. One simply generalizes the matrix Mx subtracted fromAloc

int to define Anewint in different ways. For instance, we may subtract and add a

vector field ψ†σaψSa containing the Pauli matrices σa and study paramagnonfluctuations, thus generalizing the assumption (14.331) and allowing a sponta-neous magnetization in the ground state. Or one may do the same thing witha term ψ†σa∇iψAia + c.c. added to the previous term. In this way we derivethe Ginzburg-Landau theory of superfluid He3 as in [4, 24].

14.16 Gauge Fields and Gauge Fixing 981

An important property of the proposed procedure is that it yields good resultseven in the limit of infinitely strong coupling. It was precisely this property whichled to the successful calculation of critical exponents of all φ4-theories in the text-book [21] since critical phenomena arise in the limit in which the unrenormalizedcoupling constant goes to infinity [57]. This is in contrast to another possibility.For example that of carrying the variational approach to highers order via the so-called higher effective actions [26]. These where discussed in Chapter 13. There oneextremizes the Legendre transforms of the generating functionals of bilocal correla-tion functions, which sums up all two-particle irreducible diagrams. That does notgive physically meaningful results [27] in the strong-coupling limit, even for simplequantum-mechanical models, as we have shown in Section 13.12.

The mother of this approach is Variational Perturbation Theory (VPT). Itsorigin was a variational approach developed for quantum mechanics some yearsago by Feynman and the author [53]. in the textbook [21] to quantum field theory.It converts divergent perturbation expansions of quantum mechanical systems intoexponentially fast converging expansions for any coupling strength [54].

What we have shown in this section is that this powerful theory can easilybe transferred to many-body theory, if we identify a variety of relevant collective

classical fields, rather than a fluctuating collective quantum field suggested by theHubbard-Stratonovich Transformation. To lowerst order in the coupling constantthis starts out with the standard Hartree-Fock-Bogoliubov approximation, and al-lows to go to higher oders with arbitrarily high accuracy.

14.16 Gauge Fields and Gauge Fixing

The functional integral formalism developed in the previous sections does not imme-diately apply to electromagnetism and any other gauge fields. There are subtletieswhich we are now going to discuss. These will lead to an explanation of the mistakein the vacuum energy observed in Eq. (7.506) when quantizing the electromagneticfield via Gupta-Bleuler formalism. Consider a set of external electromagnetic cur-rents described by the current density jµ(x). Since charge is conserved, these satisfy

∂µjµ(x) = 0. (14.340)

The currents are sources of electromagnetic fields Fµν determined from the fieldequation (12.49),

∂νFνµ(x) = jµ(x), (14.341)

if we employ natural units with c = 1. The action reads

A =∫

d4x[

−1

4F 2µν(x)− jµ(x)Aµ(x)

]

(14.342)

=∫

d4x

1

2

[

E2(x)−B2(x)]

−[

ρφ(x)− 1

cj ·A(x)

]

.


The field strengths are the four-dimensional curls of the vector potential Aµ(x):

Fµν = ∂µAν − ∂νAµ, (14.343)

that satisfy, for single-valued fields Aµ, the Bianchi identity

ǫµνλκ∂νFλκ = 0. (14.344)

The decomposition (14.343) is not unique. If we add to Aµ(x) the gradient of anarbitrary function Λ(x),

Aµ(x) → AΛµ(x) = Aµ(x) + ∂µΛ(x), (14.345)

then Λ does not appear in the field strengths, assuming that it satisfies

(∂µ∂ν − ∂ν∂µ)Λ(x) = 0, (14.346)

i.e., the derivatives in front of Λ(x) commute. In the theory of partial differentialequations, this is referred to as the Schwarz integrability condition for the functionΛ(x). In general, a function Λ(x) which satisfies (14.346) in a simply-connecteddomain can be defined uniquely in this domain. Only if Λ(x) fulfills this condition,the transformation (14.345) is called a local gauge transformation.

If the domain is multiply connected, there is more than one path along whichto continue the function Λ(x) from one spatial point to another and Λ(x) becomesmulti-valued. This happens, for example, if (14.346) is nonzero on a closed line inthree-dimensional space, in which case the set of paths between two given pointsdecomposes into equivalence classes, depending on how often the closed line is en-circled. Each of these paths allows another continuation of Λ(x). By (14.346), suchfunctions are not allowed in gauge transformations (14.345).

Let us now see how we can construct the generating functional of fluctuatingfree electromagnetic fields in the presence of external currents jµ(x). As in (14.58),we would like to calculate a functional integral, now in the unnormalized versioncorresponding to (14.58):

Z0[j] =∫

DAµ(x)ei∫

d3x(−F 2µν/4−jµAµ). (14.347)

The F 2µν-terms in the exponents can be partially integrated and rewritten as

−1

4

∫

d4xF 2µν =

1

2

∫

d4xAµ(∂2gµν − ∂µ∂ν)Aν . (14.348)

Hence we can identify the functional matrix D(x, x′) in (14.34) as

Dµν(x, x′) = (∂2gµν − ∂µ∂ν)δ

(4)(x− x′). (14.349)

Recalling (14.88) it appears, at first, as though the generating functional (14.347)should simply be equal to

Z0[j] = (DetG0µν)1/2e−

12

∫

d4xd4x′jµ(x)G0 µν(x,x′)jµ(x′), (14.350)


where, by analogy with (14.35), we get

G0µν(x, x′) = iD−1

µν (x, x′). (14.351)

We have divided out a normalization factor N = (DetG0µν)−1/2 , assuming that

we are dealing with an unnormalized version of Z0[j]. Unfortunately, however, theexpression (14.350) is meaningless, since the inverse of the functional matrix (14.349)does not exist. In order to see this explicitly we diagonalize the functional part (i.e.,the x, y -part) of G0µν(x, x

′) by considering the Fourier transform

Dµν(q) = −q2gµν + qµqν . (14.352)

For every momentum q, this matrix has obviously an eigenvector with zero eigen-value, namely qµ. This prevents us from inverting the matrix Dµν(q). Correspond-ingly, when trying to form the inverse determinant of the functional matrix Dµν in(14.350), we encounter an infinite product of infinities, one for every momentum q.

The difficulty can be resolved using the fact that the action (14.342) is gaugeinvariant. For Fµν this is trivially true; for the source term jµ(x)Aµ(x) this is a con-sequence of current conservation. Indeed, if we change Aµ according to (14.345), thesource term is changed by

∫

d4x jµ(x)∂µΛ(x). With the help of a partial integration,this is equal to − ∫ d4 x ∂µjµ(x)(x)Λ(x), and this expression vanishes due to currentconservation (14.340).

Because of this invariance, not all degrees of freedom, which are integrated overin the functional integral (14.347), are associated with a Gaussian integral. Thefluctuations corresponding to pure gauge transformations leave the exponent invari-ant. Since a path integral is a product of infinitely many integrals from minus toplus infinity, the gauge invariance of the integral produces an infinite product of in-finite factors. This is precisely the origin of the infinity that occurs in the functionaldeterminant.

This infinity must be controlled by restricting the functional integrals to field fluc-tuations via some specific gauge condition. For example, the restriction is achievedby inserting into the integral gauge-fixing functionals . Several examples have beenused:

F1[A] = δ[∂µAµ(x)], (Lorenz gauge)

F2[A] = δ[∇ ·A(x)], (Coulomb gauge)F3[A] = δ[A0(x)], (Hamiltonian gauge)F3[A] = δ[A3(x)], (axial gauge)

F4[A] = exp

− i2α

∫

d4x [∂µAµ(x)]2

, (generalized Lorenz gauge)

F5[A] =∫Dζe−i

∫

d4x ζHζ/2δ[∂µAµ−ζ ]×Det−1/2H. (’t Hooft gauge)

(14.353)

The first is a δ-functional enforcing the Lorenz gauge at each spacetime point:∂µA

µ(x) = 0. The second enforces the Coulomb gauge, the third corresponds tothe axial gauge, and the fourth is a generalized form of the Lorenz gauge used


by Feynman and which also serves to derive the Feynman diagrams of the Gupta-Bleuler quantization formalism, thereby correcting the mistake in the vacuum en-ergy. The fifth, finally, is a generalization of the fourth used by ’t Hooft that arises

by rewriting the fourth as a path integral F4[A] =∫ Dζe−i

∫

d4x ζ2/2αδ[∂µAµ − ζ ] and

generalizing the constant 1/α to a functional matrix A(x, x′). If we insert any ofthese gauge-fixing functionals Fi[A] into the path integral, then gauge-transform thevector potential Aµ(x) a la (14.345), and integrate functionally over all Λ(x), theintegral receives a finite contribution from that gauge function Λ(x) which enforcesthe desired gauge. The result is a gauge-invariant functional of Aµ(x):

Φ[A] =∫

DΛF [AΛ] =∫

DΛF [A+ ∂Λ] . (14.354)

Explicitly, we find for the above cases (14.353), the normalization functionals:

Φ1[A] =∫

DΛF1[AΛ] =

∫

DΛδ[∂µAµ(x) + ∂2Λ], (14.355)

Φ2[A] =∫

DΛF2[AΛ] =

∫

DΛδ[∇ ·A(x) +∇2Λ], (14.356)

Φ3[A] =∫

DΛF3[AΛ] =

∫

Dδ[∇A0(x) + ∂0Λ], (14.357)

Φ4[A] =∫

DΛF4[AΛ] =

∫

DΛ exp

− i

2α

∫

d4x [∂µAµ + ∂2Λ]2

, (14.358)

Φ5[A] =∫

DΛF5[AΛ] =

∫

DΛ∫

Dζ exp

− i

2

∫

d4x ζHζ

δ[∂µAµ + ∂2Λ−ζ ]

×Det −1/2H ×Det (∂2). (14.359)

If we form the ratios Fi[A]/Φi[A], we obtain gauge fixing functionals which all yieldunity when integrated over all gauge transformations. If any of these are inserted intothe functional integral (14.347), they will all remove the gauge degree of freedom,and lead to a finite functional integral which is the same for each choice of Fi[A].

Let us calculate the functionals Φi[A] explicitly. For Φ1,2,3[A] we simply observea trivial identity for δ-functions

δ(ax) = a−1δ(x). (14.360)

This is proved by multiplying both sides with a smooth function f(x) and integratingover x. The functional generalization of this is

δ[OA] = Det−1O δ[A], (14.361)

where O is an arbitrary differential operator acting on the field Aµ(x). From thiswe find immediately the normalization functionals:

Φ1[A] = Det −1(∂2), (14.362)

Φ2[A] = Det −1(∇2), (14.363)

Φ3[A] = Det −1(∂0). (14.364)


The fourth functional Φ4[A] is simply a Gaussian functional integral. The addi-tive term ∂µA

µ(x) can be removed by a trivial shift of the integration variableΛ(x) → Λ′(x) = Λ(x) − ∂µA

µ(x)/∂2, under which the measure of integration re-mains invariant, DΛ = DΛ′. Using formula (14.25) we obtain

Φ4[A] = Det−1(∂2/√α). (14.365)

The functional determinants Φ−1i are called Faddeev-Popov determinants [49].

The Faddeev-Popov determinants in the four examples happen to be independentof Aµ(x), so that we shall write them as Φi without arguments. This independenceis a very useful property. Complications arising for A-dependent functionals Φ[A]will be illustrated below in an example.

We now study the consequences of inserting the gauge-fixing factors Fi[A]/Φiinto the functional integrands (14.347). For F4[A]/Φ4, the generating functionalbecomes

Z0[j] = (DetG0µν)1/2Det 1/2(∂4/α)

∫

DAµ(x)ei∫

d4x[− 14F 2µν− 1

α(∂µAµ)2−jµAµ]. (14.366)

The free-field action in the exponent can be written in the form

A =∫

d4x1

2

[

− (∂µAν)2 +

(

1− 1

α

)

(∂µAµ)2]

. (14.367)

The associated Euler-Lagrange equation is

∂2Aµ −(

1− 1

α

)

∂µ(∂A) = 0, (14.368)

which is precisely the field equation (7.381) of the covariant quantization scheme.With the additional term in the action, the matrix (14.352) becomes

Dµν(q) = −q2[

gµν −(

1− 1

α

)

qµqνq2

]

. (14.369)

This can be decomposed into projection matrices with respect to the subspacestransverse and longitudinal to the four-vector qµ,

P lµν(q) =

qµqνq2

, P lµν(q) = gµν −

qµqνq2

, (14.370)

as

Dµν(q) = −q2[

P tµν(q) +

1

αP lµν(q)

]

. (14.371)

It is easy to verify that the matrices are really projections, since they satisfy

P lµνP

lνλ = P t

µν , P lµνP

lνλ = P l

µλ, P tµνP

lνλ = 0. (14.372)


Similar projections appeared before in the three-dimensional subspace [see (4.334)and (4.336)], where the projections were indicated by capital subscripts T , L.

Due to the relations (14.372), there is no problem in inverting Dµν(q), and wefind the free photon propagator in momentum space

G0µν(q) = iD−1µν (q) = − i

q2[P tµν(q) + αP l

µν(q)]

= − i

q2

[

gµν − (1− α)qµqνq2

]

. (14.373)

For α = 0, this is the propagator Gµν derived in the Gupta-Bleuler canonical fieldquantization in Eq. (7.510), there obtained from canonical quantization rules witha certain filling of the vacuum with unphysical states.

Taking into account the Faddeev-Popov determinant Φ−14 of (14.365), we obtain

for the generating functional (14.366):

Z[j] = [Det (−iG0 µν)]1/2Det 1/2(∂4/α)e−

12

∫

d4xd4x′jµ(x)G0 µν(x,x′)jν(x′). (14.374)

Let us calculate the functional determinant DetG0µν . For this we take qµ inthe momentum representation (14.373) along the 0-direction, and see that thereare three spacelike eigenvectors of eigenvalues iq2, and one timelike eigenvector ofeigenvalue iα/q2. The total determinant is therefore

Det (−iG0 µν) =1

Det (−∂2)2Det (∂4/α). (14.375)

The two prefactors of (14.374) together yield a factor

[Det (−iG0µν)]1/2Det 1/2(∂4/α) ∝ 1

Det (−∂2) . (14.376)

Recalling the discussion of Eqs. (14.129)–(14.133), we see that the associated freeenergy is

F0 = −β logZ0 = 2∑

k

ω(k)

2+ log

[

1− e−βhω(k)]

. (14.377)

It contains precisely the energy of the two physical transverse photons. The un-physical polarizations have been eliminated by the Faddeev-Popov determinant in(14.374).

We now understand why the Gupta-Bleuler formalism failed to get the correctvacuum energy in Eq. (7.506). It has no knowledge of the Faddeev-Popov determi-nant.

Taking into account current conservation, the exponent in (14.374) reduces to

jµ(x)G0 µν(x, x′)jν(x′) = jµ(x)

i

∂2jµ(x), (14.378)


so that the generating functional becomes

Z[j] = const×Det −1(−∂2)e− 12

∫

d4xd4x′jµ(x)(i/∂2)jµ(x′), (14.379)

where const is an infinite product of identical constant factors.Let us see what happens in the other gauges (14.353). The results in the Lorenz

gauge are immediately obtained by going to the limit α → 0 in the previous calcu-lation. In this limit, the functional Φ4 coincides with Φ1, up to a trivial factor.

The Hamiltonian and the axial gauges are quite similar, so we may only discussone of them. In the Hamiltonian gauge, where the Faddeev-Popov determinant isgiven by (14.364), the generating functional (14.374) becomes

Z[j] = (DetG0µν)1/2Det (∂0)e

− 12

∫

d4xd4x′ji(x)G0 ij(x,x′)jj(x′), (14.380)

where the matrix G0 ij(q) has only spatial entries, and is equal to

G0 ij(q) = iD−1ij (q) (14.381)

withDij(q) = q2δij + qiqj (14.382)

being the spatial part of the 4×4 -matrix (14.352). With the help of 3×3 -projectionmatrices

P Tij (q) = δij − qiqj/q

2 (14.383)

PLij (q) = qiqj/q

2, (14.384)

this can be decomposed as follows:

Dij(q) = q2P Tij (q) + q0 2PL

ij (q). (14.385)

The inverse of this is

D−1ij (q) =

1

q2P Tij (q) +

1

q0 2PLij (q) =

1

q2

(

δij −qiqjq0 2

)

. (14.386)

In the exponent of (14.380) we have to evaluate iD−1ij (q) between two conserved

currents, and find

ji∗(q)G0 ij(q)jj(q) = − i

q2

[

j∗(q) · j(q)− 1

q0 2q · j∗(q)q · j(q)

]

. (14.387)

Inserting the momentum space version of the local current conservation law∂µj

µ(x) = 0:q · j(q) = q0 j0(q), (14.388)

we obtain

ji∗(q)G0 ij(q)jj(q) =

i

q2jµ∗(q)jµ(q). (14.389)


Let us calculate the functional determinant DetG0µν in this gauge. From(14.386) we see that G0µν has two eigenvectors of eigenvalue i/q2, and one eigenvec-tor of eigenvalue −i/q0 2. Hence:

DetG0µν =1

Det (−i∂2)2Det (i∂20). (14.390)

The two prefactors in (14.380) together are therefore proportional to

(DetG0µν)1/2Det (∂0) ∝

1

Det (−i∂2) , (14.391)

which is the same as (14.376). Thus the generating functional (14.380) agrees withthe previous one in (14.379).

Let us also show that the Coulomb gauge leads to the same result. We rewritethe exponent in (14.379) in momentum space as

∫

d4xjµ(x)1

−∂2 jµ(x) =∫

d4q

(2π)4

[

c2ρ(q)1

q2ρ(q)− c2j(q)

1

q2j(q)

]

. (14.392)

Here we keep explicitly the light velocity c in all formulas since we want to rederivethe interaction equivalent to Eq. (12.85) where c is not set equal to unity. Writingthe denominator as q2 = q20 − q2 and j2(q) = j2L(q) + j2T (q) with jL(q) = q · j(q)/|q|and jT (q) · jL(q) = 0, we can bring (14.392) to the form

∫ d4q

(2π)4

[

c2ρ(q)1

q20 − q2ρ(q)− jL(q)

1

q20 − q2jL(q)− jT (q)

1

q20 − q2jT (q)

]

. (14.393)

Now we use the current conservation law cq0ρ(q) = q · j(q) to rewrite (14.393) as

∫

d4q

(2π)4

[

c2ρ(q)1

q2 − q2ρ(q)− c2

q20q2ρ(q)

1

q20 − q2ρ(q)− jT (q)

1

q20 − q2jT (q)

]

. (14.394)

The first two terms can be combined to

AintL = −

∫

d4q

(2π)4c2ρ(q)

1

q2ρ(q). (14.395)

Undoing the Fourier transformation and multiplying this by e2/c2 we find as a firstterm in the interaction (12.84) the Coulomb term which is is the longitudinal partof the interaction (12.83):

AintL = −1

2

∫

d4xE2L(x) =

e2

2

∫

d4x ρ(x)1

∇2ρ(x)

= − e2

8π

∫

dt∫

d3xd3x′ ρ(x, t)1

|x− x′|ρ(x′, t). (14.396)

14.17 Nontrivial Gauge and Faddeev-Popov Ghosts 989

The third term in (14.394) involves only the transverse current

AintT = −

∫

d4q

(2π)4j∗T (q)

1

q2jT (q). (14.397)

It is the result of the transverse fields in the Lagrangian (12.54):

AintT =

1

4π

∫

d4x[E2T (x)−B2(x)] +

1

cj(x)AT (x). (14.398)

It is found by integrating out the vector potential. Using Eq. (5.56), the transverseinteraction (14.397) can be rewritten as

AintT =

∫

d4q

(2π)4

[

j∗µ(q)1

q2jµ(q) +

1

q2|j0(q)|2

]

= Ainttot −Aint

L . (14.399)

The transverse part of the electromagnetic action of a four-dimensional currentjµ(q) is the difference between the total covariant Biot-Savart interaction plus theinstantaneous Coulomb interaction.

14.17 Nontrivial Gauge and Faddeev-Popov Ghosts

The Faddeev-Popov determinants in the above examples were all independent of thefields. As such they were irrelevant for the calculation of any Green function. This,however, is not always true.

As an example, consider the following nontrivial gauge-fixing functional (see also[55])

F [A] = δ[

(∂A)2 + gA2]

. (14.400)

As in Eqs. (14.355)–(14.359), we calculate

Φ[A] =∫

DΛF [AΛ] =∫

DΛ δ[∂A + gA2 + ∂2Λ + 2gA∂Λ + g(∂Λ)2]. (14.401)

This path integral can trivially be performed by analogy with the ordinary integral

∫

dx δ(ax+ bx2) =1

a, (14.402)

and yields the result

Φ[A] = Det (∂2 + 2gAµ∂µ)−1. (14.403)

The generating functional is therefore

Z[j] =∫

DAµDet (∂2 + 2gAµ∂µ) δ[∂µAµ + gA2] exp

[

i∫

d4x(

−1

4F 2µν − jµA

µ)]

.

(14.404)


Contrary to the previous gauges, the functional determinant is no longer a trivialoverall factor, but it depends now functionally on the field Aµ. It can therefore nolonger be brought outside the functional integral.

There is a simple way of including its effect within the usual field-theoreticformalism. One introduces an auxiliary Faddeev-Popov ghost field. We may considerthe determinant as the result of a fluctuating complex fermion field c with a complex-conjugate c∗, and write

Det (∂2 + 2gAµ∂µ) =∫

Dc∗Dce−i∫

d4x(∂c∗∂c−2gAµc∗∂µc). (14.405)

Note that the Fermi fields are necessary to produce the determinant in the numer-ator; a Bose field would have put it into the denominator. A complex field is takento make the determinant appear directly rather than the square-root of it.

The ghost fields interact with the photon fields. This interaction is necessary inorder to compensate the interactions induced by the constraint δ[∂Aµ + gA2] in thefunctional integral.

It is possible to exhibit the associated cancellations order by order in perturbationtheory. For this we have to bring the integrand to a form in which all fields appearin the exponent. This can be achieved for the δ-functional by observing that thesame representation (14.404) would be true with any other choice of gauge, say

δ[∂µAµ(x) + gA2(x)− λ(x)], (14.406)

since this would lead to the same Faddeev-Popov ghost term (14.405). Thereforewe can average over all possible functions λ(x) with a Gaussian weight and replace(14.406) just as well by

∫

Dλe− i2

∫

d4xλ2(x) δ[∂µAµ(x) + gA2(x)− λ(x)]. (14.407)

Now the generating functional has the form

Z[j] =∫

DAµ∫

Dc∗Dc ei∫

d4x(L−jµAµ) (14.408)

with a Lagrangian

L = −1

4F 2µν −

1

2α

(

∂A + gA2)2 − ∂µc

∗∂µc + 2gAµc∗∂µc. (14.409)

The photon and ghost propagators are

(14.410)Aµ(x)Aν(0) = −∫

d4k

(2π)4e−ikx

i

k2

[

gµν − (1− α)kµkνk2

]

,

(14.411)c∗(x)c(0) = −∫

d4q

(2π)4e−iqx

i

q2.

14.17 Nontrivial Gauge and Faddeev-Popov Ghosts 991

Contrary to the previous gauges, there are now photon-ghost and photon-photoninteraction terms

− g

α∂µA

µA2 − g2

2α

(

A2µ

)2+ 2gAµc∗∂µc (14.412)

with the corresponding vertices

(14.413)

(14.414)

. (14.415)

It can be shown that the Faddeev-Popov ghost Lagrangian has the property ofcanceling all these unphysical contributions order by order in perturbation theory.We leave it as an exercise to show, for example, that there is no contribution ofthe ghosts to photon-photon scattering up to, say, second order in g, and that self-energy corrections to the photon propagator due to photon and ghost loops cancelexactly.

In the context of quantum electrodynamics, there is little sense in using a gaugefixing term (14.400). The present discussion is, however, a useful warm-up exerciseto gauge-fixing procedures in nonabelian gauge theories, where the Faddeev-Popovdeterminant will always be field dependent.

Of course, also the previous field-independent Faddeev-Popov determinants canbe generated from fermionic ghost fields. The determinant Φ−1

4 in (14.365), forexample, can be generated from a complex ghost field c(x) and its complex conjugatec∗ by a functional integral

Φ−14 =

∫

Dc∗Dc ei∫

d4x∂c∗∂c/√α. (14.416)

It should be pointed out that the signs of the kinetic term of these c-field La-grangians are opposite to those of a normal field. If these fields were associated withparticles, their anticommutation rules would carry the wrong sign and the stateswould have a negative norm. Such states are commonly referred to as ghosts, and


this is the reason for the name of the fields c, c∗. Note that the determinant cannotbe generated by a real fermion field via a functional integral

Φ−14

?=∫

Dc e−i∫

d4x∂2c∂2c/α. (14.417)

The reason is that the differential operator ∂4 is a symmetric functional matrix, sothat the exponent vanishes after diagonalization by an orthogonal transformation.

In the language of ghost fields, the mistake in calculating the vacuum energyin Eq. (7.506), that arose when quantizing the electromagnetic field via the Gupta-Bleuler formalism, can be phrased as follows. When fixing the gauge in the action(7.376) by adding an Lagrangian density LGF (10.86), we must also add a ghostLagrangian density

Lghost = ∂c∗∂c/√α. (14.418)

The ghost fields have to be quantized canonically, and the physical states mustsatisfy, beside the Gupta-Bleuler subsidiary condition in Eq. (7.502), the conditionof being a vacuum to the ghost fields:

c|ψ“phys”〉 = 0. (14.419)

The Faddeev-Popov formalism is extremely useful offering many other possi-bilities of fixing a gauge and performing the functional integral for the generatingfunctional over all Aµ-components.

14.18 Functional Formulation of Quantum Electrodynamics

For quantum electrodynamics, the functional integral from which we can derive alltime-ordered vacuum expectation values reads [1, 3, 5]:

Z[j, η, η] =∫

DψDψDAµDD eiA−i∫

d4x [ψ(x)η(x)+η(x)ψ(x)]−i∫

d4x jµ(x)Aµ(x), (14.420)

where A is the sum of the free-field action (12.86) and the minimal interaction(12.87), which we shall write here as

A =∫

d4x[

ψ(

i/∂ − e

c/A−m

)

ψ − 1

4F µνFµν −D∂µAµ +D2/2

]

. (14.421)

The Dirac field appears only quadratically in the action. It is therefore possible tointegrate it out using the Gaussian integral formula (14.95), and we obtain

Z[j, η, η] = Det (i/∂ −M)∫

DAµDD eiA′−∫

d4x d4x′ η(x)G0(x,x′)η(x′)ψ(x)−i∫

d4x jµ(x)Aµ(x),

(14.422)where A′ is the action

A =∫

d4x[

−1

4F µνFµν −D∂µAµ +D2/2

]

+Aeff [A]. (14.423)

14.18 Functional Formulation of Quantum Electrodynamics 993

The effects of the electron are collected in the effective action

ei∆Aeff

= exp [Tr log (i/∂ − e/A −M)− Tr log (i/∂ −M)] , (14.424)

where TrO combines the functional trace of the operator O and the matrix tracein the 4 × 4 space of Dirac matrices. We have performed a subtraction of theinfinite vacuum energy caused by the filled-negative energy states. The subtractionis compensated by the determinant in the prefactor of Eq. (14.422).

14.18.1 Decay Rate of Dirac Vacuum in Electromagnetic Fields

An important application of the functional formulation (14.422) of quantum elec-trodynamics was made by Heisenberg and Euler [32, 33]. They observed that ina constant external electric field the vacuum becomes unstable. There exists a fi-nite probability of creating an electron-positron pair. This process has to overcomea large energy barrier 2Mc2, but if the pair is separated sufficiently far, the totalenergy of the pair can be made arbitrarily low, so that the process will occur withnonzero probability. The rate can be derived from the result (6.254). For a largetotal time ∆t, the time dependence of an unstable vacuum state will have the forme−i(E0−iΓ/2)∆t, where E0 is the vacuum energy, Γ the desired decay rate, and ∆t thetotal time over which the amplitude is calculated. If Leff is the effective Lagrangiandensity causing the decay, and Aeff the associated action, we identify

Γ

V= 2 Im

Aeff

V∆t= 2 ImLeff . (14.425)

We now make use of the fact that, due to the invariance under charge conjugation,the right-hand side of (14.424) can depend only on M2. Thus we also have

exp [Tr log (i/∂ −e/A−M)]= exp [Tr log (i/∂ −e/A +M)]

= exp

1

2[Tr log (i/∂ − e/A−M) (i/∂ − e/A +M)]

, (14.426)

and may use the product relation (6.107) to calculate (14.424) from half the tracelog of the Pauli operator in Eq. (6.241) to find the effective action

i∆Aeff =1

2Tr log

[i∂ − eA(x)]2 − e

2σµνFµ

ν −M2

− 1

2Tr log

−∂2 −M2

.

(14.427)We now use the integral identity

loga

b= −

∫ ∞

0

dτ

τ

[

eiaτ − eibτ]

(14.428)

and relation (6.206) to rewrite (14.427) as

i∆Aeff = − 1

2

∫ ∞

0

dτ

τe−iτ(M

2−iǫ)tr〈x|eiτ[i∂−eA(x)]2+ e2σµνF e

µν − e−iτ∂2 |x〉. (14.429)


Recalling (6.189) and (6.206), the first, unsubtracted, term can be re-expressed as

i∆Aeff1 = −1

2

∫ ∞

0

dτ

τtr〈x|e−iτH |x〉 = −1

2

∫ ∞

0

dτ

τtr〈x, τ |x 0〉. (14.430)

Inserting (6.254), and subtracting the field-free second term in (14.429), we obtainthe contribution to the effective Lagrangian density

∆Leff =1

2(2π)2

∫ ∞

0

dτ

τ 3

(

eEτ

tanh eEτ− 1

)

e−iτ(M2−iη). (14.431)

The integral over τ is logarithmically divergent at τ = 0. We can separate thedivergent term by a further subtraction, splitting

∆Leff = ∆Leffdiv +∆Leff

R (14.432)

into a convergent integral

∆LeffR =

1

2(2π)2

∫ ∞

0

dτ

τ 3

(

eEτ

tanh eEτ− 1− e2E2τ 2

3

)

e−iτ(M2−iη), (14.433)

and a divergent one

∆Leffdiv =

e2

2E2 1

3(2π)2

∫ ∞

0

dτ

τe−iτ(M

2−iη). (14.434)

The latter is proportional to the electric part of the original Maxwell Lagrangiandensity in (4.237). It can therefore be removed by renormalization. We add (14.434)to the Maxwell Lagrangian density, and define a renormalized charge eR by theequation

1

e2R=

1

e2+

1

12π2

∫ ∞

0

dτ

τe−iτ(M

2−iη) ≡ 1

Z3e2, (14.435)

to obtain the modified electric Lagrangian density

LE =e2

2e2RE2. (14.436)

Now we redefine the electric fields by introducing renormalized fields

ER ≡ e

eRE, (14.437)

and identify these with the physical fields. In terms of these, (14.436) takes againthe usual Maxwell form

LE =1

2E2R. (14.438)


The finite effective Lagrangian density (14.433) possesses an imaginary partwhich by Eq. (14.425) determines the decay rate of the vacuum per unit volume

Γ

V= Im

1

(2π)2

∫ ∞

0

dτ

τ 3

(

eEτ

tanh eEτ− 1− e2E2τ 2

3

)

e−iτ(M2−iη). (14.439)

For comparison we mention that, for a charged boson field, the expression(14.424) is replaced by

ei∆Aeff

= exp[

−Tr log

[i∂ − eA(x)]2 −M2

+ Tr log(

−∂2 −M2)]

. (14.440)

Hence the last factor 4 cosh eEτ in (6.254) is simply replaced by −2, and the un-subtracted effective action (14.439) becomes

i∆Aeffu =

∫ ∞

0

dτ

τ〈x, τ |x 0〉 = − i

4(2π)2

∫ ∞

0

dτ

τ 3eEτ

sinh eEτe−iτ(M

2−iη), (14.441)

implying a twice subtracted effective Lagrangian density

∆LeffR = − 1

4(2π)2

∫ ∞

0

dτ

τ 3

(

eEτ

sinh eEτ− 1 +

e2E2τ 2

6

)

e−iτ(M2−iη), (14.442)

and a charge renormalization

1

e2R=

1

e2− 1

6π2

∫ ∞

0

dτ

τe−iτ(M

2−iη) ≡ 1

Z3e2. (14.443)

The decay rate per unit volume is

ΓKG

V= Im

1

4(2π)2

∫ ∞

0

dτ

τ 3

(

eEτ

sinh eEτ− 1 +

e2E2τ 2

6

)

e−iτ(M2−iη). (14.444)

The integrands in (14.439) and (14.444) are even in τ , so that the integrals forthe decay rate can be extended symmetrically to run over the entire τ -axis. Afterthis, the contour of integration can be closed in the lower half-plane and the integralcan be evaluated by Cauchy’s residue theorem. To find the pole terms we expandthe integrand for fermions in Eq. (14.439) as

eετ

tanh eετ− 1 =

∞∑

n=−∞

(eετ)2

(eετ)2 + n2π2= 2

∞∑

n=1

τ 2

τ 2 + τ 2n, τn ≡ nπ

eε. (14.445)

The relevant poles lie at τ = −iτn and yield the result

Γ

V=e2E2

4π3

∞∑

n=1

1

n2e−nπM

2/eE . (14.446)

A technical remark is necessary at this place concerning the integral (14.444).At first sight it may appear as if the second subtraction term in the integrand


e2E2τ 2/3 can be omitted [29]. First, it is unnecessary to arrive at a finite integral inthe imaginary part, and second, it seems to contribute only to the real part, sincefor all even powers α > 2, the integral

∫ ∞

0

dτ

τ 3ταe−iτ(M

2−iη) =1

(iM2)α−2Γ(α− 2) (14.447)

is real. The limit at hand α → 2, however, is an exception since for α ≈ 2, theintegral possesses an imaginary part due to the divergence at small τ

1

(iM2)α−2Γ(α− 2) ≈ −γ − logM2 − i

π

2+O(α− 1). (14.448)

The right-hand side of Eq. (14.446) is a polylogarithmic function (2.274), so thatwe may write

Γ

V=e2E2

4π3ζ2(e

−πM2/eE). (14.449)

For large fields, this has the so-called Robinson expansion [30]

ζν(e−α) = Γ(1− ν)αν−1 + ζ(ν) +

∞∑

k=1

1

k!(−α)kζ(ν − k). (14.450)

This expansion plays an important role in the discussion of Bose-Einstein conden-sation [31]. For ν → 2, the Robinson expansion becomes

ζ2(e−α) =

π2

6+ (−1 + logα) α− α2

4+α3

72+O(α5). (14.451)

Hence we find the strong-field expansion

Γ

V=e2E2

4π3

π2

6+

[

−1 + logπM2

eE

] (

πM2

eE

)

− 1

4

(

πM2

eE

)2

+1

72

(

πM2

eE

)3

+ . . .

.

(14.452)For bosons, we expand

eετ

sinh eετ−1=2

∞∑

n=1

(−1)n(eEτ)2

(eEτ)2 + n2π2=2

∞∑

n=1

(−1)nτ 2

τ 2 + τ 2n, τn ≡ nπ

eE. (14.453)

Comparison with (14.445) shows that the bosonic result for the decay rate differsfrom the fermionic (14.446) by an alternating sign, accounting for the differentstatistics. There is also a factor 1/2, since there is no spin. Thus we find the decayrate per volume

ΓKG

V=e2E2

4π3

1

2

∞∑

n=1

(−1)n−1 1

n2e−nπM

2/eE . (14.454)

The sum can again be expressed in terms of the polylogarithmic function (14.450)as follows:

ζν(z) ≡∞∑

k=1

(−1)k−1zk

kν=

∞∑

k=1

zk

kν− 2

∞∑

k=1

z2k

(2k)ν= ζν(z)− 21−νζν(z

2). (14.455)


For z = e−α ≈ 1, the Robinson expansion (14.451) yields

ζ2(e−α) =

π2

12− α log 2 +

α2

4− α3

24+

α5

960+O(α8). (14.456)

The expansion ζ2(e−α) replaces the curly bracket in (14.452), if we set α = πM2/eE.

14.18.2 Constant Electric and Magnetic Background Fields

In the presence of both E and B fields, Eq. (6.253) reads

exp(

−ie2σµνFµ

ντ)

=

(

e−ie·(B−iE)τ 00 e−ie·(B+iE)τ

)

. (14.457)

The trace of this can be found by adding the traces of the 2 × 2 block ma-trices ee·(−iB∓E)τ separately. These are equal to e−eλ1τ + e−eλ2τ and their complexconjugates, respectively, where λ1, λ2 are the eigenvalues of the matrix ·(−iB−E):

λ1,2 = ±√

(E+ iB)2 = ±√E2 −B2 + 2iEB. (14.458)

Thus we find

tr exp(

−ie2σµνFµ

ντ)

= 2(cos eλ1 + cos eλ∗1). (14.459)

The eigenvalues are, of course, Lorentz-invariant quantities. They depend only onthe two quadratic Lorentz invariants of the electromagnetic field: the scalar S andthe pseudoscalar P defined by

S ≡ −1

4FµνF

µν =1

2

(

E2 −B2)

, P ≡ −1

4FµνF

µν = EB. (14.460)

In terms of these, Eq. (14.458) reads

λ1,2 = ±√2√S + iP , (14.461)

which can be rewritten as

λ1,2 = ±√2(

S2 + P 2)1/4

(cosϕ/2 + i sinϕ/2) , (14.462)

where

tanϕ =P

S, (14.463)

so that

cosϕ =S√

S2 + P 2, sinϕ =

P√S2 + P 2

, (14.464)


implying that cosϕ/2 =√

(1 + cosϕ)/2 and sinϕ/2 =√

(1− cosϕ)/2, or

cosϕ/2

sinϕ/2

=

√√S2 + P 2 ± S√

2 (S2 + P 2)1/4. (14.465)

We shall abbreviate the result (14.462) by

λ1,2 = ± (ε+ iβ) , (14.466)

where

ε

β

≡√√

S2 + P 2 ± S =1√2

√

√

(E2 −B2)2 + 4(EB)2 ± (E2 −B2). (14.467)

In terms of ε and β, the invariants S and P in (14.460) become

S ≡ −1

4FµνF

µν =1

2

(

E2 −B2)

=1

2

(

ε2 − β2)

, P ≡ −1

4FµνF

µν = EB = εβ,

(14.468)and the trace (14.459) is simply

tr exp(

−ie2σµνFµ

ντ)

= 2 cosh (ε+ iβ) + 2 cosh (ε− iβ) = 4 cosh ε cos β. (14.469)

Some special cases will simplify the upcoming formulas:

1. If B = 0, then ε reduces to |E|, whereas for E = 0, β reduces to |B|.

2. If E 6= 0 and B 6= 0 are orthogonal to each other, then we have either β = 0for E > B, or ε = 0 for B > E. The formulas are then the same as for pureelectric or magnetic fields.

3. If E 6= 0 and B 6= 0 are parallel to each other, then ǫ = |E| 6= 0, β = |B| 6= 0.

In all these cases, the calculation of the exponential (14.457) can be done verysimply. Take the third case. Due to rotational symmetry, we can assume the fieldsto point in the z-direction, B = Bz, E = Ez, and the exponential (14.457) has thematrix form

exp(

−ie2σµνFµ

ντ)

=

(

e−ie σ3(B−iE)τ 00 e−ie σ3(B+iE)τ

)

(14.470)

=

e−ie(B−iE)τ 0 0 00 eie(B−iE)τ 0 00 0 e−ie(B+iE)τ 00 0 0 eie(B+iE)τ

.

This has the trace

tr exp(

−ie2σµνFµ

ντ)

= 2 cosh (E + iB) τ + 2 cosh (E − iB) τ = 4 coshEτ cosBτ,

(14.471)


in agreement with (14.469). In fact, given an arbitrary constant field configurationB and E, it is always possible to perform a Lorentz transformation to a coordinateframe in which the transformed fields, call them BCF and ECF, are parallel. Thisframe is called center-of-fields frame. The transformation has the form (4.285) and(4.286) with a velocity of the transformation determined by

v/c

1 + (|v|/c)2 =E×B

|E|2 + |B|2 . (14.472)

By Lorentz invariance, we see that

E2 −B2 = E2CF −B2

CF, E ·B = ECF ·BCF, (14.473)

which shows that |ECF| and |BCF| in Eq. (14.471) coincide with ε and β inEq. (14.468). This is the reason why Eq. (14.471) gives the general result for arbi-trary constant fields, if E and B are replaced by |ECF| = ε and |BCF| = β.

These considerations permit us to present a simple alternative calculation of adeterminantal prefactor that occurred much earlier in Chapter 6, in particular inEqs. (6.214) and (6.225). For a general constant field strength Fµ

ν , the basic matrixthat had to be diagonalized was eeFτ in Eq. (6.245). For an electric field pointingin the z-direction, this has the form e−iM3eEτ . In contrast to (6.253), this is a boostmatrix with rapidity ζ = eEτ in the defining 4×4 -representation [compare (4.63)].The fact that the rapidity in the Dirac representation (6.252) was twice as large, hasits origin in the value of the gyromagnetic ratio 2 of the Dirac particle in Eq. (6.119).

If the field points in any direction, the obvious generalization is e−iMEeτ . In thepresence of a magnetic field, the generators of the rotation group (4.57) enter and(6.245) can be written as eeFτ = e−ie(ME+LB)τ . This is the defining four-dimensionalrepresentation of the complex Lorentz transformation, whose chiral Dirac represen-tation was written down in (14.457), apart from the factor 2 multiplying the rapidityand the rotation vectors.

As before in the Dirac representation, much labor is saved by working in thecenter-of-fields frame where electric and magnetic fields are parallel and point inthe z-direction. Their lengths have the invariant values ε and β, respectively. Theassociated transformation eeFτ has then the simple form

eeFτ = e−i(M3ε+L3β)eτ =

cosh εeτ 0 0 − sinh εeτ0 cos βeτ − sin βeτ 00 sin βeτ cos βeτ 0

− sinh εeτ 0 cosh εeτ

. (14.474)

From this we find a matrix for sin eFτ = [e−i(M3ε+L3β)eτ − ei(M3ε+L3β)eτ ]/2:

sin eFτ =

0 0 0 − sinh εeτ0 0 − sin βeτ 00 sin βeτ 0 0

− sinh εeτ 0 1

, (14.475)


and from this

eFτ =

0 0 0 −εeτ0 0 −βeτ 00 βeτ 0 0

−εeτ 0 1

. (14.476)

This leads to the desired prefactor in the amplitude (6.225)

det −1/2

(

sinh eFτ

eFτ

)

= eεβτ 2eεβτ 2

sinh ετ sin βeτ. (14.477)

Thus we can obtain the imaginary part of the vacuum energy in a constantelectromagnetic field if we simply replace the term eEτ coth eEτ in the integrand ofthe rate formula (14.439) as follows:

eEτ

tanh eEτ→ eετ

tanh eετ

eβτ

tan eβτ. (14.478)

14.18.3 Decay Rate in a Constant Electromagnetic Field

magnetic field the effective Lagrangian density (14.433) becomes, after the replace-ment (14.478)

∆LeffR =

1

2(2π)2

∫ ∞

0

dτ

τ 3

[

eετ

tanh eετ

eβτ

tan eβτ− 1− e2(ε2 − β2)

3

]

e−iτ(M2−iη). (14.479)

The subtracted small-τ divergence lies in the integral [recall the equality ε2 − β2 =E2 −B2 from Eq. (14.468)]

∆Leffdiv =

e2

2(E2 −B2)

1

3(2π)2

∫ ∞

0

dτ

τe−iτ(M

2−iη), (14.480)

which is proportional to the full original Maxwell Lagrangian density in (4.237), andcan therefore be absorbed into it by a renormalization of the charge as in (14.435)and of the fields

BR ≡ e

eRB, ER ≡ e

eRE. (14.481)

From twice the imaginary part of (14.489) we obtain the decay rate per unit vol-ume extending Eq. (14.439) in an obvious manner. Inserting the expansion (14.445),extending the integral over the entire τ -axis and rotating the contour of integrationas we did in evaluating (14.439), we find the generalization of Eq. (14.446) to con-stant electromagnetic fields

ΓKG

V=e2E2

4π3

∞∑

n=1

1

n2

nπβ/ε

tanhnπβ/εe−nπM

2/eE . (14.482)

For bosons obeying the Klein-Gordon equation, we obtain, by analogy, the ex-tension of (14.454) to constant electromagnetic fields:

Γ

V=e2E2

4π3

1

2

∞∑

n=1

(−1)n−1

n2

nπβ/ε

sinhnπβ/εe−nπM

2/eε. (14.483)


14.18.4 Effective Action in a Purely Magnetic Field

If there is only a magnetic field, magnetic field the effective Lagrangian density(14.433) becomes, after the replacement (14.478)

∆LeffR =

1

2(2π)2

∫ ∞

0

dτ

τ 3

[

eετ

tanh eετ

eβτ

tan eβτ− 1− e2(ε2 − β2)

3

]

e−iτ(M2−iη). (14.484)


∆Leffdiv =

e2

2(E2 −B2)

1

3(2π)2

∫ ∞

0

dτ

τe−iτ(M

2−iη), (14.485)


BR ≡ e

eRB, ER ≡ e

eRE. (14.486)


ΓKG

V=e2E2

4π3

∞∑

n=1

1

n2

nπβ/ε


2/eE. (14.487)


Γ

V=e2E2

4π3

1

2

∞∑

n=1

(−1)n−1

n2

nπβ/ε


2/eε. (14.488)


If there is only a magnetic field, For a constant electromagnetic field the effectiveLagrangian density (14.433) becomes, after the replacement (14.478)

∆LeffR =

1

2(2π)2

∫ ∞

0

dτ

τ 3

[

eετ

tanh eετ

eβτ

tan eβτ− 1− e2(ε2 − β2)

3

]

e−iτ(M2−iη). (14.489)


∆Leffdiv =

e2

2(E2 −B2)

1

3(2π)2

∫ ∞

0

dτ

τe−iτ(M

2−iη), (14.490)



BR ≡ e

eRB, ER ≡ e

eRE. (14.491)


ΓKG

V=e2E2

4π3

∞∑

n=1

1

n2

nπβ/ε


2/eE . (14.492)


Γ

V=e2E2

4π3

1

2

∞∑

n=1

(−1)n−1

n2

nπβ/ε


2/eε. (14.493)


If there is only a magnetic field, the integral representation (14.489) reduces to[32, 33]

∆LeffR =

1

2(2π)2

∫ ∞

0

dτ

τ 3

(

eBτ

tan eBτ− 1 +

e2B2τ 2

3

)

e−iτ(M2−iη). (14.494)

The integral still contains a divergence at small τ . The associated divergent integralis precisely a magnetic version of Eq. (14.434). It can be removed by the same typeof subtraction as before, with E2 replaced by B2. Going to renormalized quantitiesas in (14.435) and (14.491) and rotating the contour of integration clockwise to moveit away from the poles, a trivial change of the integration variable leads to

∆LeffR = − e2B2

2(2π)2

∫ ∞

0

ds

s2

(

coth s− 1

s− s

3

)

e−sM2/eB. (14.495)

This is a typical Borel transformation of the expression in parentheses. It impliesthat its power series expansion leads to coefficients of B2n which grow like (2n)!.The expansion has therefore a vanishing radius of convergence. It is an asymptoticseries, and we shall understand later the physical origin of this.

For Klein-Gordon particles, the result becomes

∆LRKG =e2B2

(2π)2

∫ ∞

0

ds

s2

(

1

sin s− 1

s+s

6

)

e−sM2/eB. (14.496)


14.18.7 Heisenberg-Euler Lagrangian

By expanding (14.478) in powers of e, we obtain

1

τ 3eβτ

tan eβτ

eετ

tanh eετ=

1

τ 3+e2

3τ

(

ε2 − β2)

− e4τ

45

(

ε4 + 5ε2β2 + β4)

+ e6τ 3

945

(

2ε6 + 7ε4β2 − 7ε2β4 − 2β6)

+ . . . . (14.497)

Inserting this into (14.489) and performing the integral over τ leads to an expansionin powers of the fields, whose lowest terms are, with e2 = 4πα:

∆LeffR =

2α2

45M4

(E2−B2)2 + 7(EB)2

+16πα3

315M8

2(E2−B2)3 + 13(E2−B2)(EB)2

.

(14.498)In each term we can replace α by αR = α(1 + O(α)), and the fields by the re-normalized fields via (14.491). Then we obtain the same series as in (14.498) butfor the physical renormalized quantities, plus higher-order corrections in α for eachcoefficient, which we ignore in this lowest-order calculation.

Each coefficient is exact to leading order in α. To illustrate the form of thehigher-order corrections we include, without derivation, the leading correction intothe first term, which becomes (see Appendix 14C for details)

∆LeffR =

2α2

45m4e

(

1 +40α

9π

)

(E2−B2)2 + 7(

1 +1315α

252π

)

(E ·B)2

+ . . . .(14.499)

Electrons in a Constant Magnetic Field

For Dirac particles in arbitrary constant fields we expand

eετ coth eετ =∞∑

n=−∞

(eετ)2

(eετ)2 + n2π2=

∞∑

n=−∞

τ 2

τ 2 + τ 2n,ε, τn,ε ≡

nπ

eε, (14.500)

eβτ cot eβτ =∞∑

m=−∞

(eβτ)2

(eβτ)2 −m2π2=

∞∑

m=−∞

τ 2

τ 2 − τ 2m,β, τm,β ≡ mπ

eβ. (14.501)

The iη accompanying the mass term in the τ -integral (14.439) is equivalent to re-placing e−iτ(M

2−iη) by e−iτ(1−iη)M2, implying that the integral over all τ has to be

performed slightly below the real axis. Equivalently we may shift the τm,β slightlyupwards in the complex plane to τm,β+iǫ. This leads to an additional contribution tothe action corresponding to a constant electromagnetic field ∆Leff = ∆Leff

div+∆LeffR .

It contains a logarithmically divergent part

∆Leffdiv =

1

2(2π)2

∫ ∞

0

dτ

τe−iτ(1−iη)M

2

2

( ∞∑

n=1

1

τ 2n,ε−

∞∑

m=1

1

τ 2m,β

)

, (14.502)


and the finite part

∆LeffR =

1

2(2π)2

∫ ∞

0

dτ

τ 3

∞∑

n,m=−∞

τ 2

τ 2 + τ 2n,ε

τ 2

τ 2 − τ 2m,β− 1−2τ 2

( ∞∑

n=0

1

τ 2n,ε−

∞∑

m=1

1

τ 2m,β

)

× e−iτ(1−iη)M2

. (14.503)

Performing the sums

∞∑

n=1

1

τkn,ε=(

eε

π

)k

ζ(k),∞∑

m=1

1

τkm,β=

(

eβ

π

)k

ζ(k), (14.504)

we see that (14.502) coincides with (14.490), as it should. The remaining sum isfinite:

∆LeffR =

1

8π2

∞∑

n,m=−∞

′ 1

τ 2n,ε + τ 2m,β

∫ ∞

0dτ

[

τ e−τM2

τ 2 − τ 2n,ε+iη− τ e−τM

2

τ 2 + τ 2m,β−iη

]

− . . . , (14.505)

where the dots indicate the subtractions. Now we decompose a la Sochocki [recallFootnote 9 in Chapter 1]:

τ

τ 2 − τ 2n,ε + iη= i

π

2δ(τ + τn,ε)− i

π

2δ(τ − τn,ε) + τ

Pτ 2 − τ 2n,ε

, (14.506)

where P indicates the principal value under the integral. The integrals over theδ-functions contribute

∆δLeff =i

8π

∑

n>0,m6=0

1

τ 2n,ε + τ 2m,βe−τn,εM

2

= ie2ε2β2

8π3

∞∑

n>0,m≥0

1

n2β2 +m2ε2e−nπM

2/eε.

(14.507)This leads to a decay rate per volume which agrees with (14.493) if we expand inthat expression [recall (14.445) and (14.425)]:

nπβ/ε

tanhnπβ/ε= 1 +

∞∑

m6=0

(nπβ/ε)2

(nπβ/ε)2 +m2π2= 1 +

∑

m6=0

n2β2

n2β2 +m2ε2. (14.508)

It remains to do the principal-value integrals. Here we use the formula9

J(z) ≡ P∫ ∞

0dτ

τe−τ

τ 2 − z2= −1

2

[

e−zEi(z) + ezEi(−z)]

, (14.509)

where

Ei(z) ≡∫ z

−∞dt

Ptet = log(−z) +

∞∑

k=1

zk

k k!(14.510)

9I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products , Academic Press,New York, 1980, Formulas 3.354.4 and 8.211.


is the exponential integral, γ being the Euler-Mascheroni constant

γ = 0.577216 . . . . (14.511)

The function (14.509) has the small-z expansion

J(z) = −1

2

[

ez log(z) + e−z log(−z)]

− γ cosh z

−∞∑

l=1

z2l

(2l)(2l)!cosh z +

∞∑

l=1

z2l+1

(2l + 1)(2l + 1)!sinh z, (14.512)

and behaves for large z like

J(z) = − 1

z2− 6

z4− 120

z6− 5040

z8+ . . . . (14.513)

Using these formulas, we find from the principal-value integrals in (14.505) [34]:

∆PLeff = − e2

4π4

∞∑

n=1

1

n2(β2an + ε2bn), (14.514)

with

an =nπε/β

tanhnπε/β

[

Ci

(

nπm2

eβ

)

cosnπm2

eβ+ si

(

nπm2

eβ

)

sinnπm2

eβ

]

, (14.515)

bn = −1

2

nπβ/ε

tanhnπβ/ε

[

exp

(

nπm2

eε

)

Ei

(

−nπm2

eε

)

+ exp

(

−nπm2

eε

)

Ei

(

nπm2

eε

)

]

,

(14.516)

where Ci(z) ≡ 12i[Ei(iz) + Ei(−iz)] and si(z) ≡ 1

2i[Ei(iz) − Ei(−iz)] = Si(z) − π/2

are the cosine and sine integrals with the integral representations [35]:

Ci(z)≡−∫ ∞

z

dt

tcos t=γ + log z +

∫ z

0

dt

t(cos t− 1) , si(z)≡−

∫ ∞

z

dt

tsin t . (14.517)

The prefactors in (14.515) and (14.516) come once more from sums of the type(14.508).

For large arguments of J(z), the expansion (14.513) becomes

∆PLeff=1

8π2

∞∑

n,m=−∞

′ 1

τ 2n,ε + τ 2m,β

[

1

τ 2n,εM4+

6

τ 4n,εM8+

120

τ 6n,εM12

+ . . .

]

. (14.518)

Applying the summation formulas (14.504), this is seen to agree with the small-fieldexpansion (14.498).

For E = 0, the sum (14.514) contains only the n = 0 -terms. If B is large, then-sum is dominated by the first term in the expansion (14.512), and yields

∆PLeff =1

8π22

∞∑

m=1

1

τ 2m,βlog(τm,βM

2). (14.519)


The leading part of this is

∆PLeff =1

4π2

∞∑

m=1

e2B2

π2m2log

M2

eB+ . . . = − 1

24π2e2B2 log

eB

M2, (14.520)

as derived first by Weisskopf [36].Note that this logarithmic behavior can be understood as the result of a lowest

expansion in e2 of an anomalous power behavior of the effective action

∆Leff =1

2(E2 − B2)1+e

2/12π2

. (14.521)

Due to the smallness of e2, this power can be observed only at very large fieldstrengths. In self-focussing materials, however, it is visible in present-day laserbeams [37].

14.18.8 Alternative Derivation for a Constant Magnetic Field

The case of a constant magnetic field can also be treated in a different way [38].The eigenvalues of the euclidean Klein-Gordon operator of a free scalar field

OKG = −h2∂2 +M2c2 (14.522)

areλp = p2 +M2c2. (14.523)

If a constant magnetic field is present that points in the z-direction, the kineticenergy p21 + p22 in the xy-plane changes to Landau energies:

p21 + p22 → 2M × p21 + p222M

= 2M × hωL

(

n +1

2

)

, n = 0, 1, 2, . . . , (14.524)

where ωL = eB/Mc is the Landau frequency. Thus the eigenvalues become

λn,p⊥= p2

⊥ + 2he

cB(n+ 1/2) +M2c2, (14.525)

where p⊥ is the two-dimensional momentum in the x3− x4 -plane.For a Dirac electron satisfying the Pauli equation (6.110), this changes to

λn,p⊥= p2

⊥ + 2he

cB(n+ 1/2 + s3) +M2c2, s3 = ±1/2. (14.526)

In this expression, the famous g-factor which accounts for the anomalous magneticmoment is approximated by the Dirac value 2. Radiative corrections would insert afactor g/2 in front of σ3, with g = 2 + α/π + . . . .

The contribution of electrons to the effective action in Eq. (14.427) can thereforebe written as

i∆Aeff =1

2

(

∑

n,σ

∫

d2p⊥(2πh)2

log λn,p⊥−∫

d4p

(2πh)4log λp

)

. (14.527)


These are divergent expressions. In order to deal with them efficiently it is useful tointroduce the Hurwitz ζ-function employed in number theory [39, 40]:

ζH(s, z) =∞∑

n=0

1

(n + z)s, Re(s) > 1 , v 6= 0,−1,−2, . . . . (14.528)

This Hurwitz ζ-function can be analytically continued in the s-plane to define ananalytic function with a single simple pole at s = 1. In quantum field theory, onemay extend this definition to an arbitrary eigenvalue spectrum of an operator O as

ζO(s) =∑ 1

(λ/µd)s, (14.529)

where µ is some mass parameter to make the eigenvalues of mass dimension d di-mensionless. For large enough s, this is always convergent. By analytic continuationin s, one can derive a finite value for the functional determinant of O:

Det O = eTr log O, Tr log O = −ζ ′O(0). (14.530)

For the Dirac spectrum (14.526), the ζ-function becomes, in natural units,

ζD(s) =eB

2π

∞∑

n=0

∑

±

∫

d2k⊥(2π)2

[

M2 + k2⊥ + eB(2n+ 1± 1)

µ2

]−s, (14.531)

where the prefactor of eB/2π is the degeneracy of the Landau levels which ensuresthat the sum (eB/2π)

∑

n converges in the limit B → 0 against the momentumintegral

∫

dk1dk2/(2π)2 =∫

dk2/4π. Performing the integral over the momenta p⊥yields

ζD(s) =B

8π2µ2s 1

s− 1

[M2 + 2eBn]1−s + [M2 + 2eB(n+ 1)]1−s

. (14.532)

This can be expressed in terms of the Hurwitz ζ-function (14.528) as

ζD(s) =M4

4π2

(

eB

M2

)2 ∞∑

n=0

(

µ2

2eB

)s1

s− 1

2ζH

(

s− 1,M2

2eB

)

−(

M2

2eB

)1−s

. (14.533)

From this we obtain

ζ ′D(0) =(eB)2

4π2

−2ζ ′H

(

−1,M2

2eB

)

− M2

2eBlog

(

M2

2eB

)

+

1

6+

(

M2

2eB

)2

[

1 + log

(

µ2

2eB

)]

. (14.534)

Choosing µ = M and subtracting the zero-field contribution, which is simply−3M4/32π2, we find

∆Leff =(eB)2

2π2

ζ ′H

(

−1,M2

2eB

)

+ ζH

(

−1,M2

2eB

)

log

(

M2

2eB

)

− 1

12+

1

4

(

M2

2eB

)2

,

(14.535)


where we have used the property [41, 39]

ζH(−1, z) = − 1

12+z

2− z2

2. (14.536)

Contact with the previous result in (14.495) is established with the help of theintegral representation of the Hurwitz ζ-function [41]:

ζH(s, z) =1

Γ(s)

∫ ∞

0

e−z t ts−1

1− e−tdt , Re(s) > 1 , Re(z) > 0. (14.537)

The integral can be rewritten as

ζH(s, z) =z1−s

s− 1+z−s

2+sz−1−s

12+

2s−1

Γ(s)

∫ ∞

0

dt

t1−se−2z t

(

coth t− 1

t− t

3

)

,(14.538)

this expression being valid for Re(s) > −2, where the integral converges [42]. Fromthis we evaluate the s-derivative at s = −1 as follows:

ζ ′H(−1, z) =1

12− z2

4− ζH(−1, z) log z − 1

4

∫ ∞

0

dt

t2e−2z t

(

coth t− 1

t− t

3

)

. (14.539)

Inserting this into (14.535) we recover exactly the previous Eq. (14.495).Let us now derive the strong-field limit. For this we use the following relation

between the Hurwitz ζ-function and the Γ-function [41, 39]:

ζ ′H(−1, z) = ζ ′(−1)− z

2log(2π)− z

2(1− z) +

∫ z

0log Γ(x)dx . (14.540)

This identity follows from an integration of Binet’s integral representation [43] oflog Γ(z). Thus we can write

∆Leff =(eB)2

2π2

− 1

12+ ζ ′(−1)− M2

4eB+

3

4

(

M2

2eB

)2

− M2

4eBlog(2π)

+

− 1

12+M2

4eB− 1

2

(

M2

2eB

)2

log

(

M2

2eB

)

+∫ M2/2eB

0dx log Γ(x)

. (14.541)

In the strong-field limit, the range of integration in the last term vanishes, so wecan use the Taylor expansion [41, 44] of log Γ(x):

log Γ(x) = − log x− γx+∞∑

n=2

(−1)n

nζ(n) xn, (14.542)

where ζ(n) is the usual Riemann ζ-function. This leads to :

∆Leff =(eB)2

2π2

− 1

12+ ζ ′(−1)− M2

4eB+

3

4

(

M2

2eB

)2

− M2

4eBlog(2π)

+

− 1

12+M2

4eB− 1

2

(

M2

2eB

)2

log

(

M2

2eB

)

− γ

2

(

M2

2eB

)2

+M2

2eB

[

1− log

(

M2

2eB

)]

+∞∑

n=2

(−1)nζ(n)

n(n + 1)

(

M2

2eB

)n+1

. (14.543)


The leading behavior in the strong-field limit is

∆Leff =(eB)2

24π2log

2eB

M2+ . . . , (14.544)

in agreement with Weisskopf’s result (14.520).

Charged Scalar Field in a Constant Magnetic Field

For a charged scalar field obeying the Klein-Gordon equation there is no spin sumand the ζ-function (14.531) becomes

ζKG(s) =eB

2π

∞∑

n=0

∫

d2k⊥(2π)2

[

M2 + k2⊥ + eB(2n+ 1)

µ2

]−s

=(eB)2

4π2

[(

µ2

2eB

)s1

(s− 1)ζH

(

s− 1,1

2+M2

2eB

)]

. (14.545)

Setting again µ = m, and subtracting the zero field contribution 3m4/64π2, weobtain

∆LeffKG = −(eB)2

4π2

ζ ′H

(

−1,1

2+M2

2eB

)

+3

4

(

M2

2eB

)2

+

[

1 + log

(

M2

2eB

)]

ζH

(

−1,1

2+M2

2eB

)

. (14.546)

In the Bose case, the equivalence with the previous result is shown with the help ofthe integral representations of the Hurwitz ζ-function [compare (14.538)]

ζH(s, 1/2 + z) =1

Γ(s)

∫ ∞

0e−z t ts−1 e−t/2

1− e−tdt , Re(s) > 1,Re(z) > −1

2

=z1−s

s− 1− sz−1−s

24+

2s−1

Γ(s)

∫ ∞

0

dt

t1−se−2z t

(

1

sinh t− 1

t+t

6

)

.(14.547)

The second expression is valid for Re(s) > −2, where the subtracted integralconverges. We can therefore use it to find the derivative at s = −1 required inEq. (14.546):

ζ ′H(−1, 1/2 + z) = − 1

24(1 + log z)− z2

4+z2

2log z

−1

4

∫ ∞

0

dt

t2e−2z t

(

1

sinh t− 1

t+t

6

)

. (14.548)

Inserting this into (14.546), we recover exactly the previous result (14.496).In order to find a strong-field expansion of ∆Leff

KG we use the following relationbetween the Hurwitz ζ-function and the log of the Γ-function [41, 43]:

ζ ′H(−1, 1/2 + z)= −1

2ζ ′(−1)− z

2log 2π − log 2

24+z2

2+∫ z

0dz log Γ(x+ 1

2), (14.549)


where we have used the formula [42, 45, 46]

∫ 12

0log Γ(x) dx =

5

24log 2 +

1

4log π +

1

8− 3

2ζ ′(−1) . (14.550)

Then we expand

log Γ(

x+ 1/2)

=1

2log π − (γ + 2 log 2) x+

∞∑

n=2

(−1)n−1(1− 2n)

nζ(n) xn, (14.551)

and obtain the strong-field expansion

∆LeffKG = −(eB)2

4π2

5

4

(

M2

2eB

)2

+

1

24− 1

2

(

M2

2eB

)2

[

1 + log

(

M2

2eB

)]

−1

2ζ ′(−1)− log 2

24− M2

4eBlog 2− 1

2

(

M2

2eB

)2

(γ + 2 log 2)

+∞∑

n=2

(−1)n−1(1− 2n)ζ(n)

n(n + 1)

(

M2

2eB

)n+1

, (14.552)

with the leading behavior

∆LeffKG =

(eB)2

96π2log

2eB

M2+ . . . . (14.553)

Appendix 14A Propagator of the Bilocal Pair Field

Consider the Bethe-Salpeter equation (14.300) with a potential λV instead of V

Γ = −iλV G0G0Γ. (14A.1)

Take this as an eigenvalue problem in λ at fixed energy-momentum q = (q0,q)= (E,q) of the boundstates. Let Γn(P |q) be all solutions, with eigenvalues λn(q). Then the convenient normalization ofΓn is:

−i∫

d4P

(2π)4Γ†n (P |q)G0

(q

2+ P

)

G0

(q

2− P

)

Γn′(P |q) = δnn′ . (14A.2)

If all solutions are known, there is a corresponding completeness relation (the sum may comprisean integral over a continuous part of the spectrum)

−i∑

n

G0

( q

2+ P

)

G0

( q

2− P

)

Γn(P |q)Γ†n(P

′|q) = (2π)4δ(4)(P − P ′). (14A.3)

This completeness relation makes the object given in (14.315) the correct propagator of ∆. Inorder to see this, write the free ∆-action A2[∆

†∆] as

A2 =1

2∆†

(

1

λV+ iG0 ×G0

)

∆ (14A.4)

Appendix 14A Propagator of the Bilocal Pair Field 1011

where we have used λV instead of V . The propagator of ∆ would have to satisfy(

1

λV+ iG0 ×G0

)

∆∆† = i. (14A.5)

Indeed, by performing a short calculation, we can verify that this equation is fulfilled by thespectral expansion (14.315). We merely have to use the fact that Γn and λn are eigenfunctionsand eigenvalues of Eq. (14A.1), and find that that

(

1

λV+ iG0 ×G0

)

×

−iλ∑

n

ΓnΓ†n

λ− λn(q)

= −iλ∑

n

1λV ΓnΓ

†n + iG0 ×G0ΓnΓ

†n

λ− λn(q)

= iλ∑

n

−λn(q)λ + 1

λ− λn(q)(−iG0 ×G0ΓnΓ

†n)

= i

(

−i∑

i

G0 ×G0ΓnΓ†n

)

= i. (14A.6)

Note that the expansion of the spectral representation of the propagator in powers of λ

∆∆† = −iλ∑

n

ΓnΓ†n

λ− λn(q)= i∑

k

(

∑

n

(

λ

λn(q)

)k

ΓnΓ†n

)

(14A.7)

corresponds to the graphical sum over one, two, three, etc. exchanges of the potential λV . Forn = 1 this is immediately obvious since (14A.1) implies that

i∑

n

λ

λn(q)ΓnΓ

†n =

∑ λ

λn(q)λn(q)V G0 ×G0ΓnΓ

†n = iλV. (14A.8)

For n = 2 one can rewrite, using the orthogonality relation,

i∑

n

(

λ

λn(q)

)2

ΓnΓ†n =

∑

nn′

λ

λn(q)ΓnΓ

†nG0 ×G0Γn′Γ†

n′

λ

λn′(q)= λV G0 ×G0λV . (14A.9)

This displays the exchange of two λV terms with particles propagating in between. The sameprocedure applies at any order in λ. Thus the propagator has the expansion

∆∆† = iλV − iλV G0 ×G0iλV + . . . . (14A.10)

If the potential is instantaneous, the intermediate∫

dP0/2π can be performed replacing

G0 ×G0 → i1

E − E0(P|q) (14A.11)

where

E0(P|q) = ξ(q

2+P

)

+ ξ(q

2−P

)

is the free particle energy which may be considered as the eigenvalue of an operator H0. In thiscase the expansion (14A.10) reads

∆∆† = i

(

λV + λV1

E −H0λV + . . .

)

= iλVE −H0

E −H0 − λV. (14A.12)


We see it related to the resolvent of the complete Hamiltonian as

∆∆† = iλV (RλV + 1) (14A.13)

where

R ≡ 1

E −H0 − λV=∑

n

ψnψ†n

E − En(14A.14)

with ψn being the Schrodinger amplitudes in standard normalization. We can now easily determinethe normalization factor N in the connection between Γn and the Schrodinger amplitude ψn.Eq. (29A.3) gives in the instantaneous case

∫

d3P

(2π)3Γ†n(P|q) 1

E −H0Γn′(P|q) = δnn′ . (14A.15)

Inserting ψ from (14.306) renders

1

N2

∫

d3P

(2π)ψ†n(P|q)(E −H0)ψn′(P|q) = δnn′ . (14A.16)

Using finally the Schrodinger equation

(E −H0)ψ = λV ψ, (14A.17)

we find

1

N2

∫

d3P

(2π)3ψ†n (P|q) λV ψn′ (P|q) = δnn′ . (14A.18)

For the wave functions ψn (P|q) in standard normalization, the integral is equal to the energydifferential

λdE

dλ.

For a typical calculation of a resolvent, the reader is referred to Schwinger’s treatment [47] of theCoulomb problem. His result may directly be used for a propagator of electron hole pairs boundto excitons.

Appendix 14B Fluctuations around the Composite Field

Here we show that the quantum mechanical fluctuations around the classical equations of motion(14.237) are quite simple to calculate. The exponent of (14.232) is extremized by the field

ϕ(x) =

∫

dyV (x, y)ψ†(y)ψ(y). (14B.1)

Similarly, the extremum of the exponent of (14.278) yields

∆(x, y) = V (x− y)ψ(x)ψ(y). (14B.2)

For this let us compare the Green functions of ϕ(x) or ∆(x, y) with those of the composite operatorson the right-hand side of Eqs. (14B.1) or (14B.2). The Green functions of ϕ(x) or ∆(x, y) aregenerated by adding to the final actions (14.242) or (14.284) external currents

∫

dxϕ(x)I(x) or

Appendix 14B Fluctuations around the Composite Field 1013

1/2∫

dxdy(∆(y, x)I†(x, y) + h.c.), respectively, and by forming functional derivatives δ/δI. TheGreen functions of the composite operators, on the other hand, are obtained by adding

∫

dx

(∫

dyV (x, y)ψ†(y)ψ(y)

)

K(x) (14B.3)

1

2

∫

dx

∫

dyV (x− y)ψ(x)ψ(y)K†(x, y) + h.c. (14B.4)

to the original actions (14.235) or (14.278), respectively, and by forming functional derivativesδ/δK. It is obvious that the sources K(x) nad K(x, y) can be included in the final actions (14.242)and (14.284) by simply replacing

ϕ(x) → ϕ′(x) = ϕ(x) −∫

dx′K(x′)V (x′, x), (14B.5)

or

∆(x, y) → ∆′(x, y) = ∆(x, y)−K(x, y). (14B.6)

If one now shifts the functional integrations to these new translated variables and drops the irrel-evant superscript “prime”, the actions can be rewritten as

A[ϕ] = ±iTr log(iG−1ϕ )+

1

2

∫

dxdx′ϕ(x)V −1(x, x′)ϕ(x′)+i

∫

dxdx′η†(x)Gϕ(x, x′)η(x)

+

∫

dxϕ(x) [I(x) +K(x)] +1

2

∫

dxdx′K(x)V (x, x′)K(x′), (14B.7)

or

A[∆] = ± i

2Tr log

(

iG−1∆

)

+1

2

∫

dxdx′|∆(x, x′)|2 1

V (x, x′)

+i

2

∫

dxdx′j†(x)G∆(x, x′)1

V (x, x′)

+1

2

∫

dxdx′

∆(y, x)[

I†(x, y) +K†(x, y)]

+ h.c.

+1

2

∫

dxdx′|K(x, x′)|2V (x, x′). (14B.8)

In this form the actions display clearly the fact that derivatives with respect to the sources Kor I coincide exactly, except for all possible insertions of the direct interaction V . For example,the propagators of the plasmon field ϕ(x) and of the composite operator

∫

dyV (x, y)ψ†(y)ψ(y) arerelated by

ϕ(x)ϕ(x′) = − δ(2)Z

δI(x)δI(x′)= V −1(x, x′)− δ(2)Z

δK(x)δK(x′)(14B.9)

= V −1(x, x′) + 〈0|(∫

dyV (x, y)ψ†(y)ψ(ϕ))(

∫

dy′V (x′y′)ψ†(y′)ψ†(y′)ψ(y′))|0〉,

in agreement with (14.238). Similarly, one finds for the pair fields:

∆(x, x′)∆†(y, y′) = δ(x− y)δ(x′ − y′)iV (x− x′)

+ 〈0|(V (x′, x)ψ(x′)ψ(x))(V (y′, y)ψ†(y)ψ†(y′))|0〉. (14B.10)

Note that the latter relation is manifestly displayed in the representation (14A.10) of the propagator∆. Since

∆∆† = iV G(4)V, (14B.11)


one has from (14B.10)

〈0|V (ψψ)(ψ†ψ†V )|0〉 = V G(4)V, (14B.12)

which is correct remembering that G(4) is the full four-point Green function. In the equal-timesituation relevant for an instantaneous potential, G(4) is replaced by the resolvent R.

Appendix 14C Two-Loop Heisenberg-EulerEffective Action

The next correction to the Heisenberg-Euler Lagrangian density (14.489) is [38, 48]

∆(2)Leff = − ie2

128π4

∫ ∞

0

dτ

∫ ∞

0

dτ ′e4β2ε2

sin eβτ sin eβτ ′ sinh eετ sinh eετ ′e−i(M2−iη)(τ+τ ′)

×

4M2 [S(τ)S(τ ′) + P (τ)P (τ ′)] I0 − i I

, (14C.1)

where

S(τ) ≡ cos eβτ cosh eετ , P (τ) ≡ sin eβτ sinh eετ ,

and

I0 ≡ 1

b− alog

b

a, I ≡ (q − p)

(b− a)2log

b

a− aq − bp

ba(b− a),

a ≡ eβ (cot eβτ + cot eβτ ′) , b ≡ eε (coth eετ + coth eετ ′) ,

p ≡ 2e2β2 cosh eε(τ − τ ′)

sin eβτ sin(eβτ ′), q ≡ 2e2ε2 cos eβ(τ − τ ′)

sinh eετ sinh eετ ′. (14C.2)

This expression contains divergences which require renormalization. First, there is a subtractionof an infinity to make ∆(2)Leff vanish for zero fields. Then there are both charge and wavefunction renormalizations, just as for the one-loop effective Lagrangian, which involves identifyinga divergent term in ∆(2)Leff of the form of the zero-loop Maxwell Lagrangian. This is done simplyby expanding the integrand to quadratic order in the fields β and ε. This divergence can beabsorbed by redefining the electric charge and the fields as

eR = e Z1/23 , BR = BZ

−1/23 , ER = EZ

−1/23 (14C.3)

where Z3 is some divergent normalization constant, which was given by (14.435) in the previous

result (14.491). The invariants β and ε are renormalized accordingly: βR = βZ−1/23 , εR = εZ

−1/23 .

Then we re-express ∆(2)Leff in terms of the renormalized charges and fields. Finally, we haveto renormalize the mass:

m2R = m2

0 + δM2 ,

∆(1)LeffR (m2

R) = L(1)R (m2

0) + δM2 ∂L(1)R (m2

0)

∂m20

. (14C.4)

The second term in (14C.4) is of the order α2, since δM2 and ∆(1)LeffR are both of order α. For

details of removing the divergencies, see the original papers in Refs. [48]. The final answer for therenormalized two loop effective Lagrangian is

∆(2)LeffR = − ie2

64π4

∫ ∞

0

dτ

∫ τ

0

dτ ′[

K(τ, τ ′)− K0(τ)

τ ′

]

− ie2

64π4

∫ ∞

0

dτK0(τ)

[

log(iM2τ) + γ − 5

6

]

(14C.5)

Notes and References 1015

where γ ≈ 0.577... is Euler’s constant, and the functions K(τ, τ ′) and K0(τ) are

K(τ, τ ′) = e−iM2(τ+τ ′)

(eβ)2(eε)2

P (τ)P (τ ′)

[

4M2(S(τ)S(τ ′) + P (τ)P (τ ′))I0 − iI]

− 1

ττ ′(τ+τ ′)

[

4M2 − 2i

τ + τ ′+e2(β2 − ε2)

3

(

2M2(ττ ′−2τ2−2τ ′2)− 5iττ ′

τ+τ ′

)]

K0(τ) = e−iM2τ

(

4M2 + i∂

∂τ

)

1

τ2

[

eβτ

tan eβτ

eετ

tanh eετ− 1 +

e2(β2 − ε2)τ2

3

]

. (14C.6)

The fields in this expression can be replaced by the renormalized fields, and everything is finite.The lowest contribution is of the fourth power in the fields and reads

∆(2)Leff =e6

64π4m4

[

16

81(β2 − ε2)2 +

263

162(β ε)2

]

+ . . . , (14C.7)

which has been added to the one-loop result in Eq. (14.499). In the limit of strong magnetic fieldsit yields

∆(2)Leff =e4β2

128π4

[

log

(

eβ

πM2

)

+ constant

]

+ . . . . (14C.8)

Notes and References

[1] R.P. Feynman and A.R. Hibbs, Path Integrals and Quantum Mechanics, McGraw-Hill, NewYork (1968);

[2] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, WorldScientific Publishing Co., Singapore 1995, pp. 1–890.

[3] R.P. Feynman, Rev. Mod. Phys. 20, 367 (1948);

[4] H. Kleinert, Collective Quantum Fields, Fortschr. Physik 26, 565 (1978) (http://klnrt.de/55). See also (http://klnrt.de/b7/psfiles/sc.pdf).

[5] J. Rzewuski, Quantum Field Theory II, Hefner, New York (1968).

[6] S. Coleman, Erice Lectures 1974, in Laws of Hadronic Matter, ed. by A. Zichichi, p. 172.

[7] See for example:A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory inStatistical Physics, Dover, New York (1975);L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics, Benjamin, New York (1962);A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, NewYork (1971).

[8] R.L. Stratonovich, Sov. Phys. Dokl. 2, 416 (1958);J. Hubbard, Phys. Rev. Letters 3, 77 (1959);B. Muhlschlegel, J. Math. Phys. 3, 522 (1962);J. Langer, Phys. Rev. 134, A 553 (1964);T.M. Rice, Phys. Rev. 140 A 1889 (1965); J. Math. Phys. 8, 1581 (1967);A.V. Svidzinskij, Teor. Mat. Fiz. 9, 273 (1971);D. Sherrington, J. Phys. C4 401 (1971).

[9] The first authors to employ such identities wereP.T. Mathews, A. Salam, Nuovo Cimento 12, 563 (1954), 2, 120 (1955).

[10] H.E. Stanley, Phase Transitions and Critical Phenomena, Clarendon Press, Oxford, 1971.


[11] For the introduction of collective bilocal fields in particle physics and applications seeH. Kleinert, On the Hadronization of Quark Theories , Erice Lectures 1976 on ParticlePhysics, publ. in Understanding the Fundamental Constituents of Matter , Plenum Press1078, (ed. by A. Zichichi). See alsoH. Kleinert, Phys. Letters B 62, 429 (1976), B 59, 163 (1975).

[12] The mean-field equations associated with the pair fields of the electrons in a metal areprecisely the equations used by Gorkov to study the behavior of type II superconductors.See, for example, p. 444 in the third of Refs. [52].

[13] H.A. Bethe and E.E. Salpeter in Encyclopedia of Physics (Handbuch der Physik), Springer,Berlin, 1957, p. 405.

[14] Y. Nambu and G. Jona Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961).

[15] V.L. Ginzburg and L.D. Landau, Eksp. Teor. Fiz. 20, 1064 (1950).

[16] R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford, Ox-ford, 1989; K.U. Gross and R.M. Dreizler, Density Functional Theory, NATO ScienceSeries B, 1995.

[17] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).

[18] A. Leggett, Rev. Mod. Phys. 47, 331 (1975).

[19] Note that the hermitian adjoint ∆∗↑↓ comprises transposition in the spin indices, i.e., ∆∗

↑↓ =

[∆↓↑]∗.

[20] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Finan-cial Markets , World Scientific, Singapore 2004 (http://klnrt.de/b5).

[21] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4-Theories, World Scientific,2001 (klnrt.de/b8).

[22] H. Kleinert, Converting Divergent Weak-Coupling into Exponentially Fast ConvergentStrong-Coupling Expansions, Lecture presented at the Summer School on ”Approximationand extrapolation of convergent and divergent sequences and series” in Luminy bei Marseillein 2009 (arXiv:1006.2910).

[23] The alert reader will recognize her the so-called square-root trick of Chapter 5 in the textbookRef. [6].

[24] See the www page (http://klnrt.de/b7/psfiles/hel.pdf).

[25] H. Kleinert, Phys. Rev. D 57, 2264 (1998); Phys.Rev. D 60, 085001 (1999). (See alsoklnrt.de/critical).

[26] C. De Dominicis, J. Math. Phys. 3, 938 (1962); C. De Dominicis and P.C. Martin, J.Math. Phys. 5, 16, 31 (1964); J.M. Cornwall, R. Jackiw, and E.T. Tomboulis, Phys. Rev. D10, 2428 (1974); H. Kleinert, Fortschr. Phys. 30, 187 (1982) (klnrt.de/82); Lett. NuovoCimento 31, 521 (1981) (klnrt.de/77).

[27] H. Kleinert, Annals of Physics 266, 135 (1998) (klnrt.de/255).

[28] R.P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (klnrt.de/159).

[29] Such an omission was done in Eq. (4.117) of the textbook [55].

[30] J.E. Robinson, Phys. Rev. 83, 678 (1951).

[31] See Chapter 7 in the textbook [6].

Notes and References 1017

[32] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). English translation available athttp://klnrt.de/files/heisenbergeuler.pdf.

[33] J. Schwinger, Phys. Rev. 84, 664 (1936); 93, 615; 94, 1362 (1954).

[34] U.D. Jentschura, H. Gies, S.R. Valluri, D.R. Lamm, E.J. Weniger, Canadian J. Phys. 80,267 (2002) (hep-th/0107135).

[35] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York,1972. See Section 5.2.

[36] V. Weisskopf, The electrodynamics of the vacuum based on the quantum theory of the elec-tron, Kong. Dans. Vid. Selsk. Math.-Fys. Medd. XIV No. 6 (1936); English translation in:Early Quantum Electrodynamics: A Source Book, A.I. Miller, Cambridge University Press,1994.

[37] G. Mourou, T. Tajima, and S.V. Bulanov, Reviews of Modern Physics, 78, 309, (2006); andreferences therein.

[38] G.V. Dunne, Heisenberg-Euler Effective Actions: 75 years on, Int.J. Mod. Phys. A 27 (2012)1260004.

[39] E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed., Cambridge, 1950, pp.268–269.

[40] http://mathworld.wolfram.com/HurwitzZetaFunction.html.

[41] A. Erdelyi (ed.), Higher Transcendental Functions, Vol. I, Kreiger, Florida, 1981.

[42] V. Adamchik, Symbolic and Numeric Computation of the Barnes Function, Conference onapplications of Computer Algebra, Albuquerque, June 2001; Contributions to the Theory ofthe Barnes Function, (math.CA/0308086).

[43] E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed., Cambridge, 1950.

[44] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York,1972.

[45] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press,New York, 1972; Formula 6.441.1.

[46] E.W. Barnes, The theory of the G-function, Quart. J. Pure Appl. Math XXXI (1900) 264.

[47] J. Schwinger, J. Math. Phys. 5, 1606 (1964).

[48] V.I. Ritus, Lagrangian of an intense electromagnetic field and quantum electrodynamics atshort distances , Zh. Eksp. Teor. Fiz 69, 1517 (1975) [Sov. Phys. JETP 42, 774 (1975)];Connection between strong-field quantum electrodynamics with short-distance quantum elec-trodynamics , Zh. Eksp. Teor. Fiz 73, 807 (1977) [Sov. Phys. JETP 46, 423 (1977)]; TheLagrangian Function of an Intense Electromagnetic Field , in Proc. Lebedev Phys. Inst. Vol.168, Issues in Intense-field Quantum Electrodynamics, V.L. Ginzburg, ed., (NovaSciencePub., NY 1987); Effective Lagrange function of intense electromagnetic field in QED, (hep-th/9812124).

[49] L.D. Faddeev and V.N. Popov Phys. Lett. B 25 29 (1967); See alsoM. Ornigotti and A. Aiello, (arXiv:1407.7256).

[50] J. Bardeen, L. N. Cooper, and J.R. Schrieffer: Phys. Rev. 108, 1175 (1957).See also the little textbook from the russian school:N.N. Bogoliubov, E. A. Tolkachev, and D.V. Shirkov A New Method in the Theory o Su-perconductivity, Consultnts Bureau, New York, 1959.


[51] For the introduction of collective bilocal fields in particle physics and applications seeH. Kleinert, On the Hadronization of Quark Theories , Erice Lectures 1976 on ParticlePhysics, publ. in Understanding the Fundamental Constituents of Matter , Plenum Press1078, (ed. by A. Zichichi). See alsoH. Kleinert, Phys. Letters B 62, 429 (1976), B 59, 163 (1975).

[52] See for example:A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory inStatistical Physics, Dover, New York (1975);L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics, Benjamin, New York (1962);A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Paricle Systems, McGraw-Hill, NewYork (1971).

[53] R.P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (klnrt.de/159).

[54] H. Kleinert, Converting Divergent Weak-Coupling into Exponentially Fast ConvergentStrong-Coupling Expansions, Lecture presented at the Summer School on ”Approximationand extrapolation of convergent and divergent sequences and series” in Luminy bei Marseillein 2009 (arXiv:1006.2910).

[55] C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill (1985). See Section 12.2.

[56] For more details see (http://klnrt.de/b8/crit.htm).

[57] H. Kleinert, Phys. Rev. D 57, 2264 (1998); Phys.Rev. D 60, 085001 (1999). (See alsoklnrt.de/critical).

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