Hadamard renormalisation for charged scalar fields · 2021. 1. 11. · Visakan Balakumar (She eld)...

Hadamard renormalisation for charged scalar fields

Visakan Balakumar

Supervised by Elizabeth Winstanley

School of Mathematics and StatisticsUniversity of Sheffield

CRAG seminar on 3rd June 2020

Visakan Balakumar (Sheffield) Hadamard renormalisation for charged scalar CRAG 1 / 28

Introduction

The stress-energy tensor of a QFT gives important information aboutthe particle content, or flux of energy.

We would like to explicitly calculate 〈Tµν〉 to use as a source term inEinstein’s semiclassical field equations.

Gµν := Rµν −1

2gµνR = 〈Tµν〉 . (1)

〈Tµν〉 is formed from products of field operators evaluated at thesame spacetime point x , causing it to be formally divergent.

Need to renormalise 〈Tµν〉 - we do this by Hadamard renormalisationdeveloped in conjunction with Wald’s axioms.

Uses point-splitting approach that involves taking one field operatorto a small, but finite, distance away.


Geometrical preliminaries

Hadamard parametrix depends upon Synge’s world function σ(x , x ′).

Defined as half the square of the geodesic distance between x and x ′:

2σ = σ ;µσ ;µ , (2)

where σ ;µ = ∇µσ.

Also require van Vleck-Morette determinant ∆(x , x ′), which gives therate at which geodesics converge or diverge from each other.

In D dimensions, it is related to the world function by:

∇µ∇µσ = D − 2∆−12 ∆

12;µσ

;µ, (3)

with boundary condition limx→x ′ ∆(x , x ′) = 1.


Hadamard expansion of Feynmann propagator

Choosing an appropriate vacuum state |ψ〉, 〈Tµν〉 is defined to be:

〈ψ|Tµν(x) |ψ〉 = limx ′→x

Tµν(x , x ′)[− i GF (x , x ′)

], (4)

where GF is the Feynmann Green’s function.

In D = 4, short-distance singularity structure of GF given by:

GF(x , x ′) =i

8π2

[U(x , x ′)

[σ + iε]+ V (x , x ′) ln[σ + iε] + W (x , x ′)

], (5)

where U(x , x ′), V (x , x ′) and W (x , x ′) are biscalar functions, regular asx ′ → x , that admit power series expansions in σ.

Explicitly we have U = U0 and V = Σ∞n=0Vnσn.


Hadamard renormalisation

U(x , x ′) and V (x , x ′) are uniquely-determined geometric quantities.

They contain the singular behaviour, so we can write GF as:

GF(x , x ′) = GFsing(x , x ′) + GF

reg(x , x ′) . (6)

Then 〈Tµν〉ren is given by:

〈ψ|Tµν(x) |ψ〉ren = limx ′→x

Tµν[− i(GF(x , x ′)− GF

sing(x , x ′))]. (7)

Means we need to find U0(x , x ′) and Vn(x , x ′) explicitly.

Decanini and Folacci have given the general Hadamard procedure fora neutral scalar field in 2008.

We would like to extend this for a scalar field with arbitrary charge q.


Governing equations for Hadamard coefficients

We employ a covariant Taylor expansion method. With a fixed, classicalbackground gauge field Aµ, the gauge covariant derivative is defined as:

Dµ = ∇µ − iqAµ . (8)

Then, the inhomogenous Klein-Gordon equation is given by:

(DµDµ −m2 − ξR)GF(x , x ′) = − 1√

−g(x)δ4(x − x ′) , (9)

where g(x) = det[ gµν(x) ].

Substituting Hadamard form of GF into (9) gives us expression interms of σ.

Equating powers of σ gives us equations U and V must satisfy.


Finding U(x , x ′)

We require U0 to fourth order to obtain the RSET.

Collecting terms proportional to σ−2 gives equation for U0:

U0 ;µ σ;µ −

((∆−

12 )∆

12 ;µσ

;µ)U0 − iqAµU0 σ

;µ = 0 . (10)

In the uncharged case, the exact solution to (10) is U0 = ∆12 .

To find U0 in the charged case, we expand as a covariant Taylorexpansion:

U0 = U00 + U01µσ;µ + U02(µν)σ

;µσ;ν + U03(µνρ)σ;µσ;νσ;ρ

+ U04(µνρτ)σ;µσ;νσ;ρσ;τ + . . .

(11)


Coefficients of U0(x , x ′) I

Considering powers of σ in the equation for U0, we find the followingexpressions for the coefficients:

U00 = 1; (12)

U01µ = iqAµ; (13)

U02(µν) =1

12Rµν −

1

2iq∇(µAν) −

1

2q2A(µAν); (14)

U03(µνρ) = − 1

24R(µν;ρ) +

1

12iq A(µRνρ) +

1

6iq∇(µ∇νAρ)

+1

2q2A(µ∇νAρ) −

1

6iq3A(µAνAρ). (15)

The expression for U04(µνρτ) is too long to fit on the slide!


Coefficients of U0(x , x ′) II

U04µνρτ =1

80∇(µ∇νRρτ) +

1

360Rθ

(µ|φ|νRφρ|θ|τ) +

1

288R(µνRρτ)

− 1

24iqA(µ∇νRρτ) −

1

24iq(∇(µAν

)Rρτ) −

1

24q2A(µAνRρτ)

− 1

24iq∇(µ∇ν∇ρAτ) −

1

6q2A(µ∇ν∇ρAτ)

− 1

8q2(∇(µAν

) (∇ρAτ)

)+

1

4iq3A(µAν∇ρAτ)

+1

24q4AµAνAρAτ . (16)


Coefficients of U0(x , x ′) III

We can simplify the expressions by writing them in terms of the gaugecovariant derivative Dµ:

(DµU0)σ ;µ −((∆−

12 )∆

12 ;µσ

;µ)U0 = 0 . (17)

U00 = 1; (18)

U01µ = iqAµ; (19)

U02(µν) =1

12Rµν −

iq

2D(µAν); (20)

U03(µνρ) = − 1

24R(µν;ρ) +

iq

6D(µDνAρ) +

iq

12A(µRνρ); (21)

U04(µνρτ) =1

80R(µν;ρτ) +

1

360Rθ

(µ|φ|νRφρ|θ|τ) +

1

288R(µνRρτ)

− iq

24D(µDνDρAτ) −

iq

24D(µ[AνRρτ)]. (22)


Finding V (x , x ′)

Collecting terms proportional to lnσ gives recursion relation for Vn:

2(n + 1)2Vn+1 + 2(n + 1) (DµVn+1)σ ;µ − 2(n + 1)Vn+1∆−12 ∆

12

;µσ;µ

+ (DµDµ −m2 − ξR)Vn = 0 .(23)

We require V0 to second order to obtain the RSET:

V0 = V00 + V01µσ;µ + V02(µν)σ

;µσ;ν + . . . (24)

We only require lowest order term of V1 to find 〈Tµν〉ren:

V1 = V10 + . . . (25)


Coefficients of V0

Then for the coefficients of V0 we obtain:

V00 =1

2

[m2 +

(ξ − 1

6

)R

]; (26)

V01µ = −1

4

(ξ − 1

6

)R;µ +

iq

2

[m2 +

(ξ − 1

6

)R

]Aµ −

iq

12∇αFαµ;(27)

V02(µν) =1

24

[m2 +

(ξ − 1

6

)R

]Rµν +

1

12

(ξ − 3

20

)R;µν −

1

240�Rµν

+1

180Rα

µRαν −1

360RαβRαµβν −

1

360Rαβγ

µRαβγν

− iq

4

[m2 +

(ξ − 1

6

)R

]D(µAν) −

iq

4

(ξ − 1

6

)A(µR;ν)

− q2

24Fα

µFνα −q2

12A(µ∇αFν)α −

iq

24∇(µ∇αFν)α. (28)


Coefficient of V1

For the lowest order term in V1 we obtain:

V10 =1

8

[m2 +

(ξ − 1

6

)R

]2

− 1

24

(ξ − 1

5

)�R − 1

720RαβRαβ

+1

720RαβγδRαβγδ +

q2

48FαβFαβ. (29)


Real symmetric biscalars in the uncharged case

Given a real, symmetric biscalar K (x , x ′) = K (x ′, x), with a covariantTaylor expansion given by:

K (x , x ′) = k0(x) + k1µ(x)σ ;µ + k2(µν)(x)σ ;µσ ;ν + k3(µνρ)σ;µσ ;νσ ;ρ + . . .

(30)we can express odd coefficients in terms of even ones. At lowest orders:

k1µ = −1

2k0;µ , (31)

k3(µνρ) = −1

2k2(µν;ρ) +

1

24k;(µνρ) . (32)

In the case of U0, we have U00 = 1 while U01µ = 0. Also:

U03(µν) = − 1

24R(µν;ρ) = −1

2∇(ρ

(1

12Rµν)

)= −1

2U02(µν;ρ) +

1

24U0;(µνρ) .

(33)


Complex sequisymmetric biscalars in the charged case

In the charged case, U0 and Vn are complex biscalars sequisymmetricin the exchange of x and x ′, which satisfy:

K (x , x ′) = K ∗(x ′, x) . (34)

The expressions relating even and odd coefficients become:

<[k1µ

]= −1

2k0;µ , (35)

<[k3(µνρ)

]= −1

2R[k2(µν;ρ)

]+

1

24k;(µνρ) . (36)

Considering the charged corrections to U0, we can verify that:

<[U03(µνρ)

]=

q2

2A(µ∇νAρ) = −1

2∇(ρ

(− q2

2AµAν)

)= −1

2<[U02(µν;ρ)

].

(37)


Renormalised expectation values

We subtract the divergent parts from GF(x , x ′) to give a regularisedGreen’s function:

−i GFreg(x , x ′) = −i

[GF(x , x ′)− GF

sing(x , x ′)]

= αW (x , x ′) . (38)

W (x , x ′) depends on the quantum state under consideration. We canexpand it as:

W (x , x ′) = w0(x) + w1µσ;µ + w2µνσ

;µσ ;ν + w3µνρσ;µσ ;νσ ;ρ + . . . (39)

The vacuum polarisation is then given as:

〈ΦΦ†〉ren = limx ′→x

R[−i GF

reg(x , x ′)]

= αw0(x) . (40)


Identities concerning W (x , x ′) I

W (x , x ′) satisfies the equation:

0 = [DµDµ − (m2 + ξR)]W + 2[σ ;µDµ + 3]V1 +O(σ) . (41)

We can generate equations that W must satisfy in terms of V byinserting their expansions. At zeroth order in σ we have:

0 = DµDµw0 + 2Dµw1µ + 2gµνw2µν −

(m2 + ξR

)w0

+ 2 (p + 1)V10 . (42)

Taking real and imaginary parts of (42), we obtain:

0 = 2gµν<[w2µν ] + 2qAµ=(w1µ)− [m2 + ξR + q2AµAµ]w0 (43)

+ 2 (p + 1)V10 ,

0 = ∇µ= (w1µ)− qAµ∇µw0 − q (∇µAµ)w0 . (44)


Identities concerning W (x , x ′) II

Evaluating (41) to order σ12 , we obtain:

0 = DµDµw1α + 4Dµw2αµ + 6gµνw3αµν +

1

3Rµ

αw1µ

−(m2 + ξR

)w1α. (45)

We only require the real part of (45):

0 = 2∇µ< (w2αµ) + qAµ∇µ= (w1α) + 2q(∇(αA

µ)=(w1µ)

)− 1

4∇α�w0 −

1

2Rµ

αw0;µ −[

1

2ξR;α + q2Aµ∇αAµ

]w0

+ ∇αV10 . (46)

We can now generate expressions for 〈Jµ〉ren and 〈Tµν〉ren in terms ofthe coefficients of W (x , x ′).


Renormalised expectation value of the current

The classical current Jµ for a charged complex scalar field Φ is:

Jµ =iq

8π[Φ∗DµΦ− Φ (DµΦ)∗] = − q

4π= [Φ∗DµΦ] . (47)

Therefore the renormalised expectation value is given by:

〈Jµ〉ren = − q

4πlimx ′→x

={Dµ[−iGR(x , x ′)

]}=αq

4π{qAµw0 −= [w1µ]} .

(48)

We require (48) to be conserved, that is:

0 = ∇µ〈Jµ〉ren =q

4π{q (∇µAµ)w0 + qAµ∇µw0 −∇µ= [w1µ]} , (49)

which holds from (43).


Renormalised stress-energy tensor I

The RSET is given by:

〈Tµν〉 = limx ′→x

<{Tµν(x , x ′)

[−iGR(x , x ′)

]}, (50)

where Tµν is a second order differential operator given by:

Tµν = (1− 2ξ) gνν′DµD

∗ν′ +

(2ξ − 1

2

)gµνg

ρτ ′DρD∗τ ′ − 2ξDµDν

+ 2ξgµνDρDρ + ξ

(Rµν −

1

2gµνR

)− 1

2m2gµν . (51)

It is unique only up to the addition of a local conserved tensor:

〈Tµν〉ren = α limx ′→x

<[Tµν(x , x ′)W (x , x ′)

]+ Θµν , (52)

where Θµν will be constrained by considering the divergence of 〈Tµν〉ren.


Renormalised stress-energy tensor II

Evaluating (52), we obtain:

〈Tµν〉ren = α

{−2< (w2µν)− 2qA(µ=

(w1ν)

)−(ξ − 1

2

)w0;µν

+(ξRµν + q2AµAν

)w0 + gµν [gρτ< (w2ρτ ) + qAρ= (w1ρ)

+

(ξ − 1

4

)�w0 −

1

2

(m2 + ξR + q2AρA

ρ)w0

]}+ Θµν . (53)

This expression is manifestly symmetric in µ and ν and reduces to theuncharged case when Aµ = 0.


Conservation of the RSET I

Taking the divergence of (53) and using (43) to (45), we get:

∇µ〈Tµν〉ren = −2α∇νV10 + 4πFµν〈Jµ〉ren +∇µΘµν . (54)

We can define:

Θµν = 2αgµνV10 + Θµν . (55)

where Θµν is a local conserved tensor, giving the expected renormalisationambiguity in the RSET.

However, we now have:

∇µ〈Tµν〉ren = 4πFµν〈Jµ〉ren (56)

leading to the nonconservation of the RSET of the charged scalar field.


Conservation of the RSET II

There are two matter fields in our system - the quantum chargedscalar field and a classical background electromagnetic field.

Only the total stress-energy tensor will be conserved.

The stress-energy tensor due to the electromagnetic field is:

TFµν = FµρFν

ρ − 1

4gµνFρτF

ρτ . (57)

Taking the divergence gives:

∇µTFµν = Fνρ∇µF

µρ = 4πFνρ〈Jρ〉ren, (58)

where we have used Maxwell’s equation 0 = ∇[µFρτ ] (and the secondequality follows from the semiclassical Maxwell equation).

Since the electromagnetic field Fµν is antisymmetric, the totalstress-energy tensor TF

µν + 〈Tµν〉ren is conserved, as required.


Renormalisation ambiguities I

There is an additional ambiguity due to the choice of renormalisationlength scale ` in the Hadamard parametrix.

This leads to an ambiguity in the Hadamard coefficient W (x , x ′)corresponding to the freedom to make the replacement:

W (x , x ′)→W (x , x ′) + V (x , x ′) ln `2. (59)

Ambiguities in 〈ΦΦ†〉ren, 〈Jµ〉ren and 〈Tµν〉ren therefore arise.

For the scalar condensate, we find:

〈ΦΦ†〉ren → αw0 + V00 ln `2. (60)

From (26), this depends on R, m and ε but not the electromagneticpotential.


Renormalisation ambiguities II

For the current, we find:

〈Jµ〉ren →αq

4π

{qAµw0 −= [w1µ] +

q

12(∇ρFρµ) ln `2

}. (61)

The current acts as a source for the semiclassical Maxwell equationsthrough ∇µF

µν = 4π 〈Jν〉ren.

This corresponds to the constant renormalisation of the permeabilityof free space µ0.

For the stress-energy tensor, we find:

〈Tµν〉ren → 〈Tµν〉ren + Ψµν ln `2, (62)


Renormalisation ambiguities III

Ψµν = α

{1

2

(ξ − 1

6

)[m2 +

(ξ − 1

6

)R

]Rµν +

1

120�Rµν

− 1

2

(ξ2 − 1

3ξ +

1

30

)R;µν −

1

90Rα

µRαν +1

180RαβRαµβν

+1

180Rαβγ

µRαβγν +q2

12Fα

µFνα + gµν

{1

720RαβRαβ

− 1

720RαβγδRαβγδ +

1

2

(ξ2 − 1

3ξ +

1

40

)�R

− 1

8

[m2 +

(ξ − 1

6

)R

]2

+q2

48FαβFαβ

}}. (63)

Curvature terms correspond to higher-order corrections to thegravitational action giving rise to Einstein’s semiclassical equations.Gauge field corrections ∼ to TF

µν so it corresponds to renormalisationof the gravitational constant G .


Trace anomaly

The general expression for the trace anomaly is given by:

〈Tµµ 〉ren = −α

{m2w0 − 3 (ξ − ξc)�w0 − 2V10

}+ gµνΨµν . (64)

This simplifies to:

〈Tµµ 〉ren =

1

4π2

[1

720�R − 1

720RαβRαβ +

1

720RαβγδRαβγδ −

q2

48FαβFαβ

].

(65)

The gauge correction to the trace anomaly depends only on Fµν andalso arises in Minkowski spacetime.

This correction agrees with DeWitt-Schwinger method as welladiabatic regularisation on cosmological spacetimes.

Non-trivial check on the validity of our results and provides strongevidence for the equivalence of these approaches.


Summary and outlook

Generalised the Hadamard procedure for charged scalar fields andexplicitly calculated Hadamard coefficients in D = 4.

Found that the trace anomaly of the RSET is modified by a termproportional to the electromagnetic field strength.

We wish to look at some specific examples, such as a charged scalarfield propagating in Reissner-Nordstrom spacetime.

Equivalence of Hadamard and adiabatic approaches have been provenfor a neutral scalar field - extension to the charged case.

The full work can be accessed via:https://arxiv.org/pdf/1910.03666.pdf

Thanks for listening!Visakan Balakumar (Sheffield) Hadamard renormalisation for charged scalar CRAG 28 / 28

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