For Review OnlyGeneralized brackets, Field theory, Equations of Motion. PACS Nos.: 04.20.Fy,...

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nly

On the equivalence and non-equivalence of Dirac and

Faddeev-Jackiw formalisms for constrained systems

Journal: Canadian Journal of Physics

Manuscript ID cjp-2015-0547.R3

Manuscript Type: Article

Date Submitted by the Author: 16-Aug-2016

Complete List of Authors: Ramos, Jairzinho; Burman University, Science

Keyword: Constrained systems, Hamiltonian and Lagrangian methods, Generalized Brackets, Field Theory, Equations of Motion

https://mc06.manuscriptcentral.com/cjp-pubs

Canadian Journal of Physics

For Review O

nlyOn the equivalence and non-equivalence

of Dirac and Faddeev-Jackiw formalisms

for constrained systems

J. RamosDivision of Science, Burman University.

Lacombe, AB, Canada. T4L [email protected]

November 16, 2016

Abstract

We present a comparative analysis between the Dirac method andthe original Faddeev-Jackiw formalism for constrained systems such asDirac free field, Proca model, electromagnetism coupled to matter andsource-free Maxwell field. We establish the possible differences betweenboth approaches and show that they are not completely equivalent.

Keywords: Constrained systems, Hamiltonian and Lagrangian methods,Generalized brackets, Field theory, Equations of Motion.PACS Nos.: 04.20.Fy, 11.10.Ef, 11.90.+t.

1 Introduction

Constrained systems were studied widely by Dirac [1–3] and Bergmann [4–6]whose works were the pioneers in this treatment. The Dirac formulationis the standard method to study theories with constraints and provides ap-propiate generalized brackets to quantize constrained systems. This methodclassifies the constraints as being first and second class. The first class con-straints give rise to gauge symmetries and the second class constraints leadto a reduction in the number of the phase space variables. When only firstclass constraints are present in the physical system, the quantization pro-cedure is quite straightforward. We implement the constraints as operatorsand make the dynamical coordinates and momenta into operators satisfying

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commutation relations which correspond to the Poisson bracket relations ofthe classical theory. An alternative way [7] to quantize this type of systemis by using a gauge fixing procedure in order to break the gauge symmetriesand bring the first class constraints into the second class constraints. Theexistence of the latter means that there are some non-physical degrees offreedom in our system. In the case, that our theory involves only secondclass constraints we have to remove these degrees of freedom and set upDirac brackets referring only to the physical degrees of freedom. Thus, interms of Dirac brackets we can pass over to the quantum theory.

In general, when a physical system has a number of constraints, some ofthem first class, some second class, we take linear combinations of them insuch a way as to have as many constraints as possible brought into the firstclass. However, there may be some second class constraints left in the systemwhich we cannot bring into the first class by taking linear combinations ofthem. Thus, we obtain a gauge theory with only first class constraints in apartially reduced phase space due to the implementation of Dirac brackets.A separate and additional step to Dirac method is a gauge fixing procedure.

In the 1980s Faddeev and Jackiw [8,9] introduced an alternative methodto analyze constrained systems. The Faddeev-Jackiw (F-J) formalism pur-sues a classical geometric treatment based on the sympletic structure of thephase space and it is only applied to first order Lagrangians, linear in thevelocities. Darboux’s theorem [10] is used to diagonalize the 2-form sym-plectic matrix associated with the reduced Lagrangian. This matrix allowsus to obtain the generalized brackets in the reduced phase space without fol-lowing Dirac’s method step by step. A physical system is constrained if thesymplectic matrix has no inverse otherwise the system is not constrained.In this formalism the constraints are not divided into first and second class.The main feature of the F-J formalism is that it fixes the gauge implicitlyand leads to the physical degrees of freedom only. An interesting discussionon this point is in [11]. It must be stressed that the F-J formalism in thispaper is the original method mentioned in [8]. There are other modifiedversions [12–15] of the original F-J formalism which are not considered inthis work.

We should mention that both methods are technically different. It meansthat the procedures followed by each method to analyze constrained systemsare different. The main difference in procedure between the F-J formalismand Dirac method is that, in the first case, it is a process of reduction ofthe phase space directly to the physical degrees of freedom. In the Diraccase, it also includes the possibility to eliminate superfluous variables. Theimplementation of Dirac brackets allows for the elimination of a variable for

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every second class constraint, yet the Dirac method keeps the superfluousdegrees of freedom due to the presence of the first class constraints whichremain until the end of the procedure. After a gauge fixing process suchdegrees of freedom are removed from the theory.

The equivalence between the Dirac method and the original F-J formal-ism for constrained systems has been discussed in [16] providing argumentsto support the full equivalence. However, in this paper we explore the pos-sibility that the equivalence is restricted to certain conditions and bothmethods are not equivalent. Unlike [16], we applied the Dirac method andF-J formalism to systems with an infinite number of degrees of freedomand avoid constructing artificial systems that are very often used in pointmechanics.

This work is organized as follows. In section 2 and 3 we review theDirac method and F-J formalism respectively. In section 4, Dirac and F-Jformalisms are applied to study some constrained theories such as Diracfree field, Proca theory, the interaction of electromagnetism with matterand free-source electromagnetic theory and compare the results obtained.In section 5, we present our conclusions.

2 Dirac method

We start with the definition of the momentum variables pi (i = 1, 2, ..., N)

pi =∂L

∂qi, (1)

where qi are the generalized coordinates and L = L(q, q) is the Lagrangian.We consider a more general case in which the momenta are dependent func-tions of the velocities qi. In other words, the momenta are not all indepen-dent, but there are, rather, some relations of the type

φm(q, p) = 0, (2)

with m = 1, 2, ...,M , that follow from the definition (1). The conditions (2)are called primary constraints to emphasize that the equations of motionare not used to obtain these relations. The next step in the Dirac methodis to introduce the Hamiltonian H by

H = piqi − L, (3)

which is a function of the Hamiltonian dynamical variables q and p. How-ever, H defined in this way is not uniquely determined because we can add

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any linear combination of the φ′s which are zero. It follows that the formal-ism remains unchanged by taking another Hamiltonian

H∗ = H + cm(q, p)φm, (4)

and our theory cannnot distinguish between H and H∗. From the generalmethod of the calculus of variations with constraints of this type, we obtain

qi =∂H

∂pi+ um

∂φm∂pi

(5)

pi = −∂H∂qi− um

∂φm∂qi

, (6)

where um are unknown coefficients called Lagrange multipliers. With thedefinition of the Poisson bracket, one can rewrite the Hamiltonian equationsof motion (5) and (6). Thus, for any function g of the q’s and p′s, theequations of motion are written concisely in terms of the Poisson bracket asfollows

g = {g,H}+ um{g, φm}. (7)

It is shown in [3] that by extending the definition of the Poisson bracket,the equation of motion (7) is written in a more concise form as

g = {g,H + umφm}. (8)

We can make use of the constraints (2) only after one has worked out allthe Poisson brackets which one is interested in. To remind this rule in theDirac method, the constraints (2) are written in the following way

φm ≈ 0. (9)

We call such equations weak equations. Subject to this rule, Eq.(7) in avery concise form becomes

g ≈ {g,HT }, (10)

with HT , the total Hamiltonian given by

HT = H + umφm. (11)

A basic consistency condition is that the primary constraints φm be pre-served in time. Taking g in Eq.(10) to be one of the φm, we should haveg = φm = 0. This gives rise to the consistency conditions which must beexamined to see what they lead to [3,7]. These conditions can reduce, firstly,to an equation independent of the u′s, thus involving only the q′s and p′s.

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If such an equation is independent of the primary constraints it is calledsecondary constraints. Thus, it is of the form

χ(q, p) = 0.

This means that there is another constraint on the dynamical variables ofour theory, then we impose a new consistency condition χ ≈ 0. One mustagain check whether this condition implies new secondary constraints. If itis so, one has to push the process one stage further. After we have exhaustedall the consistency conditions, we are left with a number of secondary con-straints and a number of restrictions on the u′s. Since the secondary con-straints will be treated on the same footing as the primary constraints, it isconvenient to denote them by

φk ≈ 0, (12)

with k = M + 1, ...,M + K and where K is the total number of secondaryequations. It is useful to be able to write all the constraints together in auniform way as

φj ≈ 0, (13)

where j = 1, ...,M + K = J . Secondly, the consistency conditions canimpose restrictions on the u′s. These restrictions are expressed by

{φj , H}+ um{φj , φm} ≈ 0. (14)

We have in Eq.(14) a set of J nonhomogeneous linear equations in the M un-knowns um, with coefficients that are functions of the q′s and the p′s. Theremust exist solutions to these equations, for otherwise the Euler-Lagrangianequations of motion would be inconsistent. The general solution [3, 7] ofEq.(14) is

um = Um + υaVam, (15)

where Um = Um(q, p) is a particular solution of the inhomogeneous equations(14), Vam = Vam(q, p) with a = 1, ..., A are linearly independent solutionsof the homogeneous equations associated with Eq.(14) and υa are totallyarbitrary coefficients.

Let us substitute Eq.(15) into Eq.(11) and the total Hamiltonian becomes

HT = H ′ + υaφa, (16)

where H ′ = H + Umφm and φa = Vamφm. We have satisfied all the consis-tency conditions of our theory and have explicitly separated that part of u

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that remains arbitrary from the one that is fixed by the consistency condi-tions. We may take the υ′s to be arbitrary functions of the time and we stillsatisfy all the requirements of our theory. As a result of the arbitrarinessin the υ′s, the dynamical variables at future times are not completely de-termined by the initial variables and these arbitray functions appear in thegeneral solution of the equations of motion with given initial conditions.

The distinction between primary and secondary constraints is of littleimportance in the final form of the Dirac method. A different classificationof constraints plays a central role. This is the concept of first-class andsecond-class functions.

We define a function F (q, p) to be first class if its Poisson bracket withevery constraint vanishes weakly

{F, φj} ≈ 0, (17)

otherwise F (q, p) is second-class function. From their definitions, we notethat H ′ and φa are first-class functions. This follows from Eq.(14) and fromthe fact that Vam is any independent solution of the homogeneous equationsassociated with Eq.(14). Any linear combination of the primary constraintsφm is of course another primary constraint. So each φa is a first-class primaryconstraint. Thus, we have that the total Hamiltonian HT is the sum of thefirst-class Hamiltonian H ′ and the first-class primary constraints multipiedby arbitrary functions υa.

Let us try to get a physical understanding of this situation where a so-lution to the equations of motion involves arbitrary functions. The physicalstate of a system is uniquely determined by the set of q′s and p′s and not bythe coefficients υ. The initial state must determine the state at later times.Since the dynamical variables q and p contain the arbitrary functions υ,those variables are not uniquely determined by the initial state. This meansthat the state does not uniquely determine a set of the dynamical variablesq and p. There must be several choices of q’s and p’s which corresponds tothe same physical state.

It is shown in [3,7] that for a general dynamical function g(q, p), with ini-tial value g0, the difference between the values of g at time δt correspondingto two different choices υa and υ′a takes the form

∆g(δt) = εa{g, φa}, (18)

where εa = (υa − υ′a)δt. We can change all the Hamiltonian dynamicalvariables in accordance with Eq.(18) and the new Hamiltonian variables willdescribe the same physical state. A transformation (18) is called a gauge

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transformation because it does not change the physical state of a system.Thus, we conclude that the first-class primary constraints φa generate gaugetransformations. Dirac [3] states that the number of these independentgauge functions is equal to the number of independent primary first-classconstraints. Moreover, these gauge transformation can be found followingthe approach presented in [17] or [18]. Another result is that the Poissonbracket {H ′, φa} is also a generator of gauge transformations. It is notpossible to infer from these considerations that every first-class secondaryconstraint is a gauge generator. Nevertheless, by the Dirac conjecture [7]one postulates, in general, that all first-class constraints, both primary andseconday, generate gauge transformations. Therefore, we should consider amore general equation of motion

g = {g,HE}, (19)

where HE is the extended Hamiltonian and is defined by

HE = HT + υ′a′φa′ = H ′ + υaφa + υ′a′φa′ . (20)

Here, φa, first-class primary constraints, and φa′ , first-class secondary con-straints, are the generators of gauge transformations.

For first-class constraints, the algorithm in the Dirac method ends at thispoint. An additional step not included in the Dirac formalism is a gaugefixing procedure. The problem of quantizing a theory with only first-classconstraints is simple [3, 7].

For a theory that also includes second-class constraints, the Dirac algo-rithm includes more steps. The presence of second-class constraints meansthat there are some non-physical degrees of freedom in our theory [3, 7, 19].We have to remove these degrees of freedom and define new Poisson bracketsreferring only to the physical degrees of freedom. We try to replace all theconstraints φj by taking linear combinations of them so that we can bringas many constraints as possible into the first class. However, some of thesecond-class constraints will remain in our theory and will be denoted byχs, s = 1, ..., S, where S is the number of second-class constraints which aresuch that no linear combination of them is first class. Let us form a matrixC, called Dirac matrix, with the Poisson brackets of each χ with the other

C = (css′) =

0 {χ1, χ2} {χ1, χ3} · · · {χ1, χS}

{χ2, χ1} 0 {χ2, χ3} · · · {χ2, χS}{χ3, χ1} {χ3, χ2} 0 · · · {χ3, χS}

...... . . . · · ·

...{χS , χ1} {χS , χ2} {χS , χ3} · · · 0

. (21)

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A more concise way to represent the matrix C is given in terms of its elements

css′ = {χs, χs′}, s, s′ = 1, ..., S. (22)

We note that the matrix C is antisymmetric and its determinant D does notvanish, not even weakly [3,20], therefore the number of surviving χ′s whichcannot be brought into first class, must be even. Since D does not vanish,we can define the inverse C−1 (css

′) of the matrix C (cs′s′′) by

css′cs′s′′ = css

′{χs′ , χs′′} = δss′′ . (23)

We now introduce new Poisson brackets called Dirac brackets as follows:any two quantities ξ and η have a Dirac bracket {ξ, η}D defined by

{ξ, η}D = {ξ, η} − {ξ, χs}css′{χs′ , η}. (24)

Dirac brackets defined by Eq.(24) satisfy all the properties that Poissonbrackets usually satisfy. We notice that the equations of motion are stillvalid for Dirac brackets. With Eq.(24) and being HT first-class function,the equations of motion become

g ≈ {g,HT }D. (25)

The Dirac bracket of any function ξ(q, p) with one of the χ′s is

{ξ, χs′′} = 0, (26)

therefore we can use χs = 0 before working out Dirac brackets. This meansthat this equation can be considered as strongly zero. In conclusion, whenwe implement Dirac brackets in a theory involving second-class constraints,these constraints are strongly zero. In this way, non-physical degrees offreedom are eliminated from our theory for each second-class constraintand the phase space is reduced partially containing only physical degreesof freedom.

Finally, in terms of Dirac brackets we can pass over to the quantumtheory. The remaining equations, which are weakly zero, are all first-classconstraints and become supplementary conditions on the wave functions.Thus the situation is reduced to the previous situation where there wereonly first-class constraints.

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3 Faddeev-Jackiw Formalism

The F-J formalism is basically a Lagrangian method and applied to first-order Lagrangians [8,9]. We can pass from a Lagrangian L(q, q) to a Hamil-tonian H(q, p) by the Legendre transformation and the definition of momen-tum given in Eq.(1). Therefore, the Hamiltonian becomes

H(q, p) = piqi − L(q, q). (27)

By precisely the same procedure, higher order Lagrangians can be convertedto first-order. To this end, we read Eq.(27) in the opposite direction anddefine the canonical Lagrangian Lc(q, p) by

Lc(q, p) = L(q, q, p) = piqi −H(q, p), (28)

which is of first order (linear in the velocities qi). It is straightforward toverify that Euler-Lagrange equations for the canonical Lagrangian coincidewith the Hamiltonian equations of motion.

Let us introduce the dynamical variables ξi = (qk, pk), i = 1, 2, ..., n, n+1, ..., 2n and k = 1, 2, ..., n, where ξ1 = q1, ..., ξ

n = qn, ξn+1 = p1, ..., ξ

2n =pn. We begin with a general first-order Lagrangian

L = ai(ξ)ξi −H(ξ), (29)

where the first term on the right side becomes the canonical 1-form ai(ξ)ξi =

a(ξ). To introduce the F-J formalism, we consider a particular case where

ai(ξ) =1

2ξjωji, (30)

where the constant matrix ωij = −ωji. The symmetric part of ω merelycontributes an irrelevant total time-derivative to the Lagrangian L and canbe dropped. The Euler-Lagrange equation is

ωij ξj =

∂H(ξ)

∂ξi. (31)

This equation implies two cases. For the first case detωij 6= 0, therefore thematrix ωij has an inverse ωij and there are no constraints. The dynamicalequation for ξi becomes

ξi = ωij∂H(ξ)

∂ξj. (32)

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The generalized brackets in this formalism are defined in such a way thatthe Poisson bracket between ξi and H(ξ) reproduces (32). Hence, we obtain

{ξi, ξj} = ωij . (33)

These generalized brackets satisfy all the usual properties of Poisson brack-ets. For the second case, detωij = 0, and constraints are present. We willdiscuss this point later.

In the most general case, with ai(ξ) being an arbitrary function of ξi,the Euler-Lagrange equation is

fij(ξ)ξj =

∂H(ξ)

∂ξi, (34)

where the matrix fij(ξ) is antisymmetric and is defined by

fij(ξ) =∂aj(ξ)

∂ξi− ∂ai(ξ)

∂ξj. (35)

This matrix gives rise to the canonical two-form f(ξ) = 12fij(ξ)dξ

iξj .We have two cases again. For the first case detfij 6= 0, therefore the ma-

trix fij possesses an inverse f ij and there are no constraints. The dynamicalequation that follows from (34) is

ξi = f ij(ξ)∂H(ξ)

∂ξj, (36)

and the generalized brackets take the form

{ξi, ξj} = f ij(ξ), (37)

which satisfies also the usual properties of Poisson brackets including theJacobi identity since fij(ξ) obeys the Bianchi identity.

We now turn to the second case where fij (or ωij) has no inverse (sin-gular) and the constraints are present. The constraint equations can beobtained by Darboux’s theorem and have the following form [9]

∂H(ξ′, z)

∂za= 0, (38)

here a = 1, 2, .., N ′ = N−N∗, where N∗ is the rank of the matrix ω and N isits dimension. The variables za are the so-called zero modes of ω. Darboux’stheorem allows us to split the N-dimensional space into two subspaces which

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are orthogonal to each other. The first one is described by the variables zaand the second by the coordinates ξi

′, where i′ = 1, 2, .., N∗. In the latter

we assume that the matrix ω(ωi′j′) has an inverse and its dimension N∗ iseven, so we have N = N ′ + 2n. Thus we have reduced the N -dimensionalphase space and solved the problem of singularity of the matrix ω.

The Lagrangian L in the even-dimensional space is

L =1

2ξi

′ωi′j′ ξ

i′ −H(ξ′, z). (39)

The form of the function H(ξ′, z) depends on whether the variable za ispresent linearly in the Hamiltonian. For the za ocurring non-linearly inH(ξ′, z) one can solve Eq.(38). But, when H(ξ′, z) contains a za linearly,Eq.(38) does not allow an evaluation of the corresponding za in terms of ξi

′

since za is absent. Using Eq.(38) we eliminate as many za’s as possible, interms of ξi

′and other za’s. Therefore we are left with a Hamiltonian in the

formH(ξ′, z) = H(ξ′) + zkΦ

k(ξ′) k < a. (40)

Then the Lagrangian L is given as

L =1

2ξi

′ωi′j′ ξ

i′ −H(ξ′)− zkΦk(ξ′). (41)

The last term on the Rigth side of the previous equation contains constraintsin the subspace described by ξi

′, and the z′as play the role of Lagrange

multipliers. The only true constraints in the theory are the coefficients Φk.We solve the constraint equations

Φk(ξ′) = 0, (42)

which establish relations among the ξi′

and permit to reduce the number ofvariables ξi

′below the 2n that are present in (41). It also reproduces a new

canonical one-form a and, of course, new coefficients ai where now i rangesover the reduced space. With this, the Darboux procedure is repeated untilwe find the subspace where the matrix ωi′j′ has an inverse. Thus one is leftwith a completely reduced, unconstrained theory.

4 Comparative Study of the Dirac Method andF-J Formalism

We now apply both formalisms to some physical theories and discuss theresults for each case. Here, we use the metric signature (+,−,−,−).

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4.1 Dirac Free Field

4.1.1 Dirac Method

The Lagrangian density LD for the Dirac free field ψ is given by

LD = i(λ+ 1

2)ψγµ∂µψ + i(

λ− 1

2)(∂µψ)γµψ −mψψ, (43)

where ψ is the complex four-component spinor, its adjoint is ψ = ψ†γ0 and itforms an eight-dimensional Grassmann algebra. The γµ are four-dimensionalDirac matrices and λ is an arbitrary parameter. The reason for introducingλ arises from the observation that sometimes there is a confusion as towhether results derived from a Dirac Lagrangian are dependent on the formone starts with. For λ = 0 and λ = 1, we obtain the symmetrized versionand non-symetrized version of LD, respectively. By using the definitionof conjugate momenta πψ and πψ† to the fields ψ and ψ† respectively, theprimary constraints take the form

φ1 = πψ − i(λ+ 1

2)ψ† = 0, φ2 = πψ† − i(

λ− 1

2)ψ = 0. (44)

The consistency conditions on the primary constraints φ1 and φ2 lead toequations that allow finding the Lagrangian multipliers. Thus, the con-straints in Eq.(44) are the only ones present in our theory. Their Poissonbracket becomes

{φ1, φ2} = −iδ(x− y), (45)

therefore φ1 and φ2 are second-class constraints. With these constraints,we obtain the 2 × 2 Dirac matrix C, Eq.(21), whose elements are cab ={φa(x), φb(y)} where a, b = 1, 2. Since detC 6= 0, the matrix C has aninverse

C−1 =

(0 −ii 0

)δ(x− y). (46)

Then, the constraints can be eliminated (strongly zero) if we substitute thePoisson bracket by the Dirac bracket. This fact allows us to reduce a phasespace with four degrees of freedom: ψ, πψ, ψ

†, πψ† to a phase space with only

two degrees: ψ,ψ†, being πψ, πψ† eliminated. We notice that the physical

degrees of freedom that remain in our theory are ψ and ψ†. Therefore, theDirac brackets in terms of the adjoint field ψ read

{ψ(x), ψ(y)}D = 0

{ψ(x), ψ(y)}D = 0

{ψ(x), ψ(y)}D = −iγ0δ(x− y). (47)

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They are the appropriate brackets to use for the quantum theory.

4.1.2 F-J Method

The Lagrangian density LD, Eq.(43), can be rewritten as

LD = i(λ+ 1)

2ψ†ψ + i

(λ− 1)

2ψ†ψ + i

(λ+ 1)

2ψ†~α.∇ψ

+ i(λ− 1)

2∇ψ†.~αψ −mψ†γ0ψ, (48)

which is of first-order in ψ and ψ† and is already in the appropiate formto apply the F-J formalism. From this equation we identify the dynamicalvariables ξi

ξi = (ψ,ψ†) (49)

with i = 1, 2 (ξ1 = ψ, ξ2 = ψ†) and the coefficients ai(ξ) are given by

a1 = aψ = i(λ+ 1

2)ψ†, a2 = aψ† = i(

λ− 1

2)ψ. (50)

The matrix fij , defined in Eq.(35) has an inverse f ij

f ij(x,y) =

(0 −ii 0

)δ(x− y). (51)

According to Eq.(37), the generalized brackets become

{ψ(x), ψ(y)} = 0

{ψ(x), ψ(y)} = 0

{ψ(x), ψ(y)} = −iγ0δ(x− y), (52)

and they are the appropiate brackets to quantize our theory.Comparing the brackets of Eqs.(47) and Eq.(52), we observe that they

are the same. The F-J formalism is more direct to obtain the generalizedbrackets because it already eliminates all non-physical degrees of freedom.Whereas, in Dirac’s method, all degrees of freedom are present in the theoryuntil the end of the procedure where the non-physical degrees of freedomare removed by using the second class constraints.

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4.2 Proca Model

4.2.1 Dirac Method

The Proca Lagrangian density LP is given by

LP = −1

4FµνF

µν +m2

2AµA

µ. (53)

The dynamical variables are the fields Aµ and their conjugate momenta πµ,defined by

πµ =∂LP∂Aµ

. (54)

Following the Dirac method, we obtain a primary constraint φ1 that takesthe form

φ1 = π0 ≈ 0 (55)

and a secondary constraint φ2 that follows from the consistency conditionsfor φ1

φ2 = m2A0 + ∂iπi ≈ 0. (56)

Imposing the consistency conditions on φ2 we obtain an expression to de-termine the Lagrangian multiplier of our theory.

The Poisson bracket for these constraints is expressed by

{φ1(x), φ2(y)} = −m2δ(x− y), (57)

therefore φ1 and φ2 are second-class constraints. It is straightforward toobtain the 2× 2 Dirac matrix C and since detC 6= 0, its inverse becomes

C−1 =

(0 1

m2

− 1m2 0

)δ(x− y). (58)

With the constraints φ1 and φ2 strongly equal to zero, we reduce the phasespace with degrees of freedom Aµ and πµ, by removing the two non-physicaldegrees of freedom A0 and π0 using Eqs. (55) and (56). Therefore, thephysical degrees of freedom are kept and the Dirac brackets become

{Ai(x), Aj(y)}D = 0

{πi(x), πj(y)}D = 0

{Ai(x), πj(y)}D = δijδ(x− y). (59)

and they are the appropriate brackets to use with the quantum theory.

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4.2.2 F-J Method

The Proca Lagrangian LP , Eq.(53), is rewritten as

LP =1

2A2 + A · ∇A0 +

1

2(∇A0)

2 − 1

2(∇×A)2 +

m2

2A2

0 −m2

2A2. (60)

We notice that LP is of second order in A. To apply the F-J formalism, welinearize (first-order) LP by using the definition of the conjugate momentumπµ, Eq.(54), which yields

π0 = 0, ~π = −(A +∇A0). (61)

With the Legrendre transform, integration by parts, frontier conditions(when x→∞, A0 → 0, ~π → 0) and the definition of canonical Lagrangian,Eq.(28), the original LP becomes linear in A

LP = −~π · A− 1

2[~π2 + (∇×A)2 +m2A2] +A0∇ · ~π +

m2

2A2

0. (62)

By comparing Eq.(62) with Eq.(39), we identify the variable z

z = z1 = A0, (63)

and the dynamical variables ξi = (Ak, πk) with i = 1, 2 (ξ1 = Ak, ξ2 = πk).The constraint equation is given by the equation of motion for A0

δHp

δA0= 0 (64)

where HP is the Hamiltonian obtained from (62). Therefore the resultingequation is

A0 = −∇ · ~πm2

, (65)

that leads to the following form for Lp

LP = −~π · A− 1

2[~π2 + (∇×A)2 +m2A2]− 1

2m2(∇ · ~π)2. (66)

From this equation the coefficients ai(ξ) become ai(ξ) = (aAk, aπk) = (πk, 0).

It is straightforward to obtain the matrix fij from Eq.(35). Since detf 6= 0,its inverse f ij take the form

f ij(x,y) =

(0 δij−δij 0

)δ(x− y). (67)

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Hence, from Eq.(37,) the generalized brackets are

{Ai(x), Aj(y)} = 0

{πi(x), πj(y)} = 0

{Ai(x), πj(y)} = δijδ(x− y). (68)

Thus we are in the reduced phase space where there are no constraints.With these generalized brackets, we can go over to the quantum theory.

The brackets in Eq.(59) and Eq.(68) are the same. Both the Diracmethod and F-J formalism eliminate explicitly and implicitly, respectively,all non-physical degrees of freedom and only the physical ones remain.

4.3 Electromagnetism Coupled to Matter

4.3.1 Dirac Method

The Lagrangian density L for this interaction is given by

L = −1

4FµνF

µν + iψγµ(∂µ + ieAµ)ψ −mψψ. (69)

The dynamical variables are the fields Aµ, ψ and ψ† and their conjugatemomenta πµ, πψ and πψ† , respectively. From their definitions, we obtainthe primary constraints

φ1 = π0 ≈ 0, φ2 = πψ − iψ† ≈ 0, φ3 = πψ† ≈ 0, (70)

and from the consistency conditions we have the secondary constraint

χ = ∂iπi − eρ ≈ 0, (71)

where ρ is the probability density. Calculating Poisson brackets betweenthese constraints, we conclude that φ1 is a first-class constraint and φ2, φ3and χ are second-class constraints.

Let us take linear combinations of constraints to bring as many of themas possible into first-class constraints. We construct a matrix whose elementsare the Poisson brackets among the second-class constraints. This matrixhas an eigenvector 1

ieψieψ†

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with null eigenvalue. This means that there is another first-class constraintϕ, that is a linear combination of the second-class constraints. That is

ϕ = ∂iπi + ie(πψψ + ψ†πψ†) (72)

Following Dirac’s method this ϕ will replace the constraint χ. We are nowleft with two first-class constraints φ1 and ϕ that generate gauge transfor-mations, and two second-class constraints φ2 and φ3 that reduce the phasespace by removing non-physical degrees. It is not possible to bring moreconstraints into first class in this way, therefore we construct the 2×2 Diracmatrix C with the remaining second-class constraints φ2 and φ3. Its inverseis

C−1 =

(0 −ii 0

)δ(x− y). (73)

Since there are two second-class constraints present in our theory, we caneliminate two degress of freedom: πψ and πψ† , by implementing the Diracbrackets. The nonzero Dirac brackets are given by

{Aµ(x), πν(y)}D = δµνδ(x− y)

{ψ(x), ψ†(y)}D = −iδ(x− y)

{ψ(x), πψ(y)}D = δ(x− y) (74)

The gauge freedom which comes from the first-class constraints still remainsin the theory. This means that the gauge has not been fixed yet. A gaugefixing procedure is an additional step for the Dirac method.

4.3.2 F-J Method

The Lagrangian density, Eq.(69), can be written as

L =1

2A2 + A · ∇A0 +

1

2[(∇A0)

2 − (∇×A)2] + (75)

iψ†ψ + iψ†~α · (∇+ ieA)ψ − eρA0 −mψ†γ0ψ,

which is a Lagrangian density of second order in A. We have to convert Lto first order to apply the F-J formalism. With the definition of conjugatemomentum, we obtain the following expressions

π0 = 0, ~π = −(A +∇A0) (76)

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that together with the conditions of frontier and the canonical Lagrangian,transform L into first-order

L = −~π · A + iψ†ψ − 1

2[~π2 + (∇×A)2]−A0(eρ−∇ · ~π) + (77)

iψ†~α · (∇− ieA)ψ −mψ†γ0ψ.

From this Lagrangian, we identify

z = z1 = A0, ξi = (Ak, πk, ψ, ψ†). (78)

The equation of motion for A0 gives rise to the following constraint

∇ · ~π = eρ. (79)

In order to solve the constraint, we decompose both ~π and A into transverseand longitudinal parts

~π = ~πT + ~π‖, A = AT + A‖; (80)

with∇ · ~πT = ∇ ·AT = 0, ∇× ~π‖ = ∇×A‖ = 0. (81)

Then Eqs.(80) and (81) imply

~π‖ =∇√−∇2

π, A‖ =∇√−∇2

A; (82)

where A = ‖A‖. With Eqs.(79) - (82), we obtain

π = ‖~π‖ = − 1√−∇2

eρ, (83)

and for A an analogous equation is obtained. Since the constraint equationhas been solved, therefore all constraints have been eliminated from thetheory. Inserting Eq.(83) for π and A into the Lagrangian density L, we get

L =

∫Ldx (84)

=

∫{−~πT · AT + eρ

1√−∇2

A+ iψ†ψ − 1

2[~π2 + (∇×AT )2 − e2ρ 1

∇2ρ]}dx

−HM [(∇− ieAT − ie∇√−∇2

A)ψ]−∫mψψdx

where HM [(∇− ieA)ψ] =∫iψ†~α · (∇− ieA)ψdx.

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The Darboux transformation replaces ψ −→ exp(i e√−∇2

A)ψ. This will

cancel ρ 1√−∇2

A against a contribution from iψ†ψ and eliminate A from the

Hamiltonian. Then we have the following Lagrangian

L =

∫{−~πT · AT + iψ†ψ − 1

2[~π2 + (∇×AT )2 − e2ρ 1

∇2ρ]}dx (85)

−HM [(∇− ieAT )ψ]−∫mψψdx

From this, the dynamical variables are identified as ξi = (AkT , πkT , ψ, ψ†),

where i = 1, 2, 3, 4, and the coefficients ai(ξ) become

ai(ξ) = (aATk , aπTk , aψ, aψ†) = (−πT , 0, iψ†, 0). (86)

From Eq.(35), with these coefficients it is straightfroward to calculate thematrix fij whose inverse f ij is

f ij(x,y) =

0 δij 0 0−δij 0 0 0

0 0 0 −i0 0 i 0

δ(x− y) (87)

Therefore, the generalized brackets are expressed as

{AiT (x), πjT } = δijδ(x− y)

{ψ(x), ψ†(y)} = −iδ(x− y)

{AiT (x), AjT } = {ψ(x), ψ(y)} = {ψ†(x), ψ†(y)} = 0. (88)

In this way the constraints are removed from our theory and they are theappropiate brackets to go over to the quantum theory.

The brackets in Eqs.(74) and (88) are different. In the Dirac method westill have non-physical degrees of freedom since we have not fixed the gaugeyet. This will require an additional step to the method. Whereas the F-J formalism eliminates all non-physical degrees of freedom from the theory.This means that the gauge (Coulomb gauge) has been fixed implicitly duringthe procedure and only the physical degrees remain.

4.4 Source-Free Electromagnetic Field

4.4.1 Dirac Method

Maxwell’s theory for the electromagnetic field without sources is describedby the Lagrangian density LEM

LEM = −1

4FµνF

µν . (89)

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From the definition of canonical momenta πµ conjugate to the dynamic fieldsAµ, Eq.(54), and from the consistency conditions, we obtain the primary andsecondary constraints

φ1 = π0 ≈ 0, φ2 = ∂iπi ≈ 0. (90)

These define a constraint surface on the phase space and their Poissonbracket is given by

{φ1(x), φ2(y)} = 0. (91)

The functions φ1 and φ2 are first-class constraints and will give cause togauge symmetries in the theory. It is not possible to construct the Diracbrackets. However, the Poisson brackets of the dynamical variables aredetermined to be

{Aµ(x), Aν(y)} = 0

{πµ(x), πν(y)} = 0

{Aµ(x), πν(y)} = δµν δ(x− y), (92)

which are obviously, incompatible with the constraint π0 ≈ 0.We keep following the Dirac method as far as we can. From Eq.(20) the

extended Hamiltonian HE is

HE =

∫(H′ + λφ1 + vφ2)dx (93)

where λ, v are arbitrary functions. Integrating by parts and using conditionsat infinity this Hamiltonian transforms into

HE =

∫[1

2~π2 +

1

2(∇×A)2 + λπ0 + (v −A0)∇ · ~π]dx. (94)

Thus we obtain the equations of motion to be

A0 = λ, Ai = −πi + ∂iA0 − ∂iv, πi = ∂jFji. (95)

We realize that A0 is an arbitrary function and it will be eliminated fromthe equations of motion after a gauge fixing procedure.

4.4.2 F-J Method

The Lagrangian density LEM is expressed as

LEM =1

2A2 + A · ∇A0 +

1

2[(∇A0)

2 − (∇×A)2], (96)

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which is of second order in A. To convert it to a first order Lagragian density,we define the conjugate momenta that leads to the following equations

π0 = 0, ~π = −(A +∇A0). (97)

Integrating by parts, conditions at infinity and the definition of the canonicalLagrangian, LEM is written as

LEM = −~π · A− 1

2[~π2 + (∇×A2)] +A0∇ · ~π. (98)

From this Lagrangian density, we deduce the variable z = z1 = A0 and thedynamical variables ξi = (Ai, πi). The equation of motion for A0 providesthe following constraint equation

∇ · ~π = 0. (99)

Following the same procedure as in part (4.3.2) for ρ = 0, we solve Eq.(99)and obtain

∇2

√−∇2

π = 0, (100)

and a similar equation for A. Inserting these equations for π and A into theLagrangian density LEM , it becomes

LEM = ~πT · AT −1

2[~π2 + (∇×AT )2]. (101)

All constraints have been eliminated and the new dynamical variables areξi = (akT , πkT ) and the coefficients ai(ξ) are ai(ξ) = (aAT

i , aπTi ) = (−πT , 0).Therefore the matrix fij can be calculated and its inverse takes the form

f ij(x,y) =

(0 1−1 0

)δijδ(x− y). (102)

Finally, with this matrix, the generalized brackets are

{AiT (x), πjT (y)} = δijδ(x− y)

{AiT (x), AjT (y)} = {πiT (x), πjT (y)} = 0. (103)

They are the appropiate brackets to go over to the quantum theory.Comparing the results of Eqs.(92) and (103), we realize that the brackets

are different. In the Dirac method, in the first part of the analysis weonly obtain the Poisson brackets and cannot define the Dirac brackets sincethe gauge has not been fixed yet. This will be an additional step in thisprocedure. Whereas the F-J formalism eliminates all non-physical degreesof freedom and allows us to obtain the generalized brackets. The Coulombgauge is fixed implicitly during the F-J procedure.

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5 Conclusions

The main difference between the two formalisms is that in the Dirac methodwe obtain Dirac brackets with some first-class constraints in the reductionprocess. The procedure of imposing the consistency conditions ends whenour theory contains no new constraint. Then, a gauge theory due to thepresence of the first-class constraints is obtained in a partially reduced phasespace. In the F-J formalism, the Lagrangian is reduced by using the con-straint equations and Darboux transformation. The procedure continuesuntil we obtain a non-degenerate matrix fij(ξ) with which the generalizedbrackets are obtained in a reduced phase space with physical degrees offreedom only.

We have analyzed and compared the Dirac method and F-J formalism.This leads to the following conclusions:

1. For first-order Lagrangians, both formalisms are equivalent. Bothmethods eliminate all non-physical degrees of freedom.

2. For second or higher order Lagrangians we have the following cases:

• For Lagrangians with second-class constraints only (no gaugesymmetries in the theory) both methods are equivalent. Theyeliminate all non-physical degrees of freedom. No gauge fixingprocedure is required since there are no gauge variables.

• For Lagrangians with first-class constraints only (the theory con-tains gauge symmetries) both methods are not equivalent. Inthis case the F-J formalism eliminates all gauge variables leavingonly the physical degrees of freedom in the theory. Thus the F-Jformalism fixes the gauges implicitly. Whereas the Dirac methodrequires an additional procedure to eliminate the gauge variables.This procedure is called gauge fixing and it is not part of the Diracmethod.

• For Lagrangian with both first-class and second-class constraintsboth methods are not equivalent. First-class constraints giverise to gauge symmetries and second-class constraints reduce thenumber of the variables in the phase space. In this case, Dirac’smethod eliminates some non-physical variables through second-class constraints, thus reducing partially the phase space, butDirac’s method does not remove the gauge variables from thetheory. However, in the F-J formalism all non-physical variables,

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including gauge variables, are eliminated, only leaving the physi-cal degrees of freedom in the theory. Thus, the F-J method fixesthe gauge implicitly whereas the Dirac method requires an addi-tional gauge fixing procedure.

Acknowledgment

I would like to thank B. M. Pimentel and F. Khanna for their helpful com-ments and fruitful discussions.

References

[1] P.A.M. Dirac. Can. J. Math. 2, 129 (1950).

[2] P.A.M. Dirac. Phys. Rev. 114, 924 (1959).

[3] P.A.M. Dirac. Lectures on Quantum Mechanics. Yeshiva University,New York (1964).

[4] J.L. Anderson and P.G. Bergmann. Phys. Rev. 83, 1018 (1951).

[5] P.G. Bergmann and R. Schiller. Phys. Rev. 89, 4 (1953).

[6] P.G. Bergmann and I. Goldberg. Phys. Rev. 98, 531 (1955).

[7] M. Henneaux and C. Teitelboim. Quantization of Gauge Systems.Princeton University Press, New Jersey (1992).

[8] L. Faddeev and R. Jackiw. Phys. Rev. Lett. 60, 1692 (1988).

[9] R. Jackiw. In Constraint Theory and Quantization Methods: FromRelativistic Particles to Field Theory and General Relativity. Editedby F. Colomo, L. Lusanna, and G. Marmo. 2nd Workshop 1993, Mon-telpuciano, Italy. World Scientific, River Edge, NJ. 1994. p. 163. hep-th/9306075.

[10] P.J. Olver. In Graduate Text in Mathematics: Applications of LieGroups to Differential Equations. 2nd ed. Edited by S. Axler, F. W.Gehring, and K. A. Ribet. Springer-Verlag, New York (1993).

[11] D. J. Toms. Phys. Rev. D, 92, 105026 (2015).

[12] J. Barcelos-Neto and C. Wotzasek. Mod. Phys. Lett. A, 7, 1737 (1992).

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[13] J. Barcelos-Neto and C. Wotzasek. Int. J. Mod. Phys. A, 7, 4981 (1992).

[14] H. Montani and C. Wotzasek. Mod. Phys. Lett. A, 8, 3387 (1993).

[15] L. Liao and Y. C. Huang. Ann. Phys. 322, 2469 (2007).

[16] J. A. Garcia and J. M. Pons. Int. J. Mod. Phys. A, 12, 451 (1997).

[17] L. Castellani. Ann. Phys. 143, 357 (1982).

[18] M. Henneaux, C. Teitelboim and J. Zanelli. Nucl. Phys. B, 332, 169(1990).

[19] K. Sundermeyer. In Lectures Notes in Physics: Constrained Systems,Vol. 169. Edited by H. Araki, J. Ehlers, K. Hepp, R. Kippenhahn, H.A. Weidenmuller, and J. Zittartz. Springer-Verlag, Berlin, Heidelberg(1982).; H. J. Rothe and K. D. Rothe. In World Scientific Lecture Notesin Physics - Vol. 81: Classical and Quantum Dynamics of ConstrainedSystems. World Scientific Publishing Co. Pte. Ltd., Singapore (2010).

[20] E. C. G. Sudarshan and N. Mukunda. Classical Dynamics: A ModernPerspective. John Wiley & Sons, New York (1974).

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