Selym Villalba-Chávez

Reinhard Alkofer

Kai Schwenzer

On the connection between Hamilton and Lagrange formalism in Quantum Field Theory

Keywords: numbers: 11.10.Ef, 11.10.Lm, 11.15.Tk Keywords: First order formalism, Functional methods, Dyson-Schwinger equations, Auxiliary fields

The connection between the Hamilton and the standard Lagrange formalism is established for a generic Quantum Field Theory with vanishing vacuum expectation values of the fundamental fields. The Effective Actions in both formalisms are the same if and only if the fundamental fields and the momentum fields are related by the stationarity condition. These momentum fields in general differ from the canonical fields as defined via the Effective Action. By means of functional methods a systematic procedure is presented to identify the full correlation functions, which depend on the momentum fields, as functionals of those usually appearing in the standard Lagrange formalism. Whereas Lagrange correlation functions can be decomposed into tree diagrams the decomposition of Hamilton correlation functions involves loop corrections similar to those arising in n-particle effective actions. To demonstrate the method we derive for theories with linearized interactions the propagators of composite auxiliary fields and the ones of the fundamental degrees of freedom. The formalism is then utilized in the case of Coulomb gauge Yang-Mills theory for which the relations between the two-point correlation functions of the transversal and longitudinal components of the conjugate momentum to the ones of the gauge field are given.

I. INTRODUCTION

The path integral method within the Lagrange formalism is a fundamental tool to formulate Quantum Field Theory. Certainly, the treatment of the same theory within the canonical operator formalism is somewhat more cumbersome. In fact, additionally to the operator-ordering problems of gauge theories [1,2], the formal invariance properties of a generic theory admit additional transformation laws which are not present in the Lagrange formulation. The "momentum fields" in such a case are not related to the fundamental fields by means of the canonical equations, see, e.g., ref. [START_REF] Weinberg | The Quantum theory of fields[END_REF], but constitute separate degrees of freedom with independent transformation properties and thus symmetries. As a consequence the number of variables in the path integral increases in comparison to the Lagrange formalism. This introduces for most theories considerable additional complications in perturbative as well as nonperturbative calculations [START_REF] Watson | [END_REF][5][6][7].

On the other hand, some recent studies indicate that the first order formalism proves to be a successful tool to study the required complete cancellation of the energy divergences [8] that emerge in a perturbative treatment of Coulomb gauge Yang-Mills theory within the standard Lagrange formalism [7,9]. Coulomb gauge Yang-Mills theory has attracted attention since one possible solution to the confinement problem in QCD is provided by the "Gribov-Zwanziger" scenario [8,10] (cf. also ref. [11]). However, the problem of renormalizing Coulomb gauge Yang-Mills theory is still unsolved since these energy divergences cannot be regularized using any of the standard procedures. Recently it was illustrated how these energy divergences cancel at each order in perturbation theory [12]. In order to perform explicit calculations a number of methods have been applied as e.g. introducing a novel method to regularize Feynman integrals in non-covariant gauges [13,14], employing algebraic renormalizability [8], and trying to recover Coulomb gauge Yang-Mills theory as a limit of an interpolating gauge [15]. On the other hand, there are several indications that the canonical or first order formalism is better suited for studying Coulomb gauge Yang-Mills theory [START_REF] Watson | [END_REF][5][6][7][8][15][16][17][18]. Yet, even if such an approach would be successful, due to the dramatically complicated form of the functional equations in the first order formalism, an explicit non-perturbative study is far too involved to be computationally feasible. Therefore, an analysis in the second order formalism would be highly desirable. To this end we provide general connections between the Greens functions in the two formulations that should help to perform the renormalization in the Lagrange framework according to the insight in the renormalization procedure obtained in the Hamilton framework.

In order to establish general relations between dressed correlation functions in the different formulations we exploit the following properties. At vanishing sources associated to the momentum fields the Effective Action (i.e. the Generating Functional of one-particle-irreducible (1PI) Green's functions) of the Hamilton formalism reduces to the one of the Lagrangean approach. This allows to reduce the set of Dyson-Schwinger equations (DSEs) [19,[START_REF] Schwinger | Proc. Nat. Acad. Sci[END_REF] in the first order formalism to the corresponding set derived from the standard path integral representation. An important special Confidential: not for distribution. Submitted to IOP Publishing for peer review 9 November 2009 case is given if the Hamiltonian is quadratic in the momentum fields. For a corresponding Generating Functional the integration over the momentum fields can be performed and any m-point function involving this field as the average of a polylocal function of the quantum canonical momentum fields can be determined. This means that the full correlation functions involving these canonical variables can be found as a functional of those that usually appear in the standard path integral representation. The procedure to find these connections is closely related to the functional method used in the derivation of the DSEs (see e.g. ref. [START_REF] Alkofer | [END_REF]). In this paper we give the explicit form of these relations for a general four-dimensional renormalizable theory. This includes the case that the interaction terms involve the time derivative of the fields. In a final step we resolve the relations between the proper two-point functions in both frames by considering the inverse of the matrix-valued propagators in the Hamiltonian approach.

Moreover, we show that similar connections arise for theories where not the kinetic but the interaction part is linearized. Such a bosonization procedure is an important technique used in many parts of physics ranging from hadronic physics to condensed matter systems. Our results explicitly verify, in agreement with other approaches, that there is no double counting in bosonized theories, but instead a given correlation function in the underlying theory is exactly given by the sum of all possible contributions involving both the fundamental and the composite degrees of freedom in the bosonized theory.

This paper is organized as follows: In Sect. II the functional equations for DSEs and a Symmetry-Related identity (like Ward-Green-Takahashi, resp. Slavnov-Taylor, identities (STIs) of a gauge theory [22][23][24][25][26]) in the phase space formulation are given. In Sect. III we present the first order DSEs of theories which are quadratic in the momentum fields. In addition, we derive a method to determine the correlation functions and the STIs including momentum fields from the respective quantities that usually appear in the standard Lagrange representation. Diagrammatic rules and the general decomposition of the full proper functions in the Lagrange formalism are given in Sect. IV, while in Sect. V and Sect. VI we detail the explicit decomposition of the connected and proper two-point functions in the Hamilton framework, respectively. In Sect. VII we show that the formalism can also be utilized in the case composite of auxiliary fields, and last but not least in Sect. VIII we apply the formalism to the case of Coulomb gauge Yang-Mills theory. In the last section we conclude while essential steps of many calculations have been deferred to several appendices.

II. FUNCTIONAL EQUATIONS

We start our analysis by considering a generic Quantum Field Theory formulated within the first order formalism. The Hamiltonian density H (q m (x), p m (x)) depends on the fundamental fields q m (x) and the conjugate momentum fields p m (x), respectively. As will be discussed in Sect. IV.D, for fermionic fields the relation between the two formalisms is trivial. Therefore we will treat only the case of bosonic fields in this and the next section.

In the case of gauge invariant theories we will suppose that the Hamiltonian H includes the additional terms that arise when the constraints associated to such theories, like e.g. Gauss' law in Quantum Electrodynamics, are localized. In this context each Lagrange multiplier necessary to impose the constraints will be treated as a fundamental field. Certainly the presence of these terms leads to the main differences between gauge theories and the more conventional Hamiltonian systems. In addition, in a fixed gauge, ghost fields appear. The usual path integral representation of these Grassmannian variables is formulated within the first order formalism (see also Sect. IV.C). Thus, in the case of a gauge theory they will be analysed in an independent way (for details see Sect. VIII). In this section, to set up the problem, we will disregard for the moment these potential complications.

Let us suppose, in addition, that the fields p m (x) and q m (x) are coupled to a set of classical sources given by J p m (x) and J q m (x), respectively. Under such conditions the source dependent vacuum-to-vacuum transition amplitude between the asymptotic states |Vac, in and |Vac, out looks like

Z[J] = Vac, out|Vac, in J = D[q]D[p] exp i (I [q, p, J] + i -terms) (1)
where [q] denotes the collection of all fundamental fields, whereas [p] the corresponding momentum fields. For more details see section 9.2 of the Ref. [START_REF] Weinberg | The Quantum theory of fields[END_REF]. Here the argument in the exponential has the structure

I[q, p, J] = I 0 [q, p] + d 4 xJ(x) • φ(x)
where

φ(x) ≡ p m (x) q m (x) and J(x) ≡ J p m (x) J q m (x)
, include both momentum and fundamental fields and sources. The components of the sources in this notation are distinguished by upper labels, whereas

I 0 [q, p] = ∞ -∞ dτ d 3 x p m (x) qm (x) -H (q(x), p(x)) (2)
looks like the classical expression for the Hamiltonian action. Note that the part concerning to the terms in Eq. ( 1) have the function to produce the necessary i s in the denominators of all propagators such that the correct boundary conditions of the fields at asymptotic times, q m (x, ±∞), are implemented (see again Ref. [START_REF] Weinberg | The Quantum theory of fields[END_REF]).

The fact that I 0 [q, p] looks like the action expressed in terms of canonical variables is somewhat misleading since the momentum fields p m (x) are independent variables and therefore not yet related to the fundamental fields q m (x) or their derivatives. In particular, since the path integral is not saturated by its saddle point(s) they are not constrained to obey the equations of motion of classical Hamiltonian dynamics δI 0 /δφ = 0 where

δI 0 δφ ≡   δI0 δpm δI0 δqm   =   qm -∂H ∂pm ṗm -∇ • ∂H ∂(∇qm) + ∂H ∂qm   . ( 3
)
The path integrals, however, contain the information about these equations of motion. Indeed, let us consider the operator version of Eq. ( 3). Its vacuum expectation values in the presence of the external classical sources can be expressed as

δI 0 δφ = D[q]D[p] i δ δφ exp i I [q, p, J] 0 out |0 in J -J. ( 4
)
Assuming the absence of any boundary terms, the integral of such a functional derivative vanishes. Then by substituting each of the elementary objects present in Eq. ( 3) by the derivative with regard to the respective classical sources, the following functional differential equation arises

δI 0 δφ φ(x)→ i δ δJ(x) Z[J] = -J(x)Z[J].
(

It contains all equations of motions fulfilled by all Green's functions.

Introducing the Generating Functional of connected Green's functions, W H [J] = -i ln (Z[J]), allows us to rewrite Eq. ( 5) as

δI 0 δφ φ(x)→ δW H δJ(x) + i δ δJ(x) = -J(x), ( 6
)
where in the last step we made use of the following identity

F δ δJ exp(G(J)) = exp(G(J))F δG(J) δJ + δ δJ (7)
for arbitrary functionals F and G. This set of equations constitutes the DSEs for the connected Green's functions.

In the next step we introduce the vacuum expectation values of the momentum and the fundamental fields in the presence of sources

φ[J] = 1 i δW H δJ(x) ≡   p[J] q[J]   =    1 i δW H δJ p m 1 i δW H δJ q m    . ( 8
)
We assume that it is possible to invert these relations such that the sources are expressed as functionals of the vacuum expectation values of Π(x) and Q(x). Hereby the replacements of variables in Eq. ( 6) becomes

φ(x) → φ[J](x) + i d 4 x D[J](x, x ) δ δ φ(x ) (9)
where

D[J](x, x ) ≡   ∆ pp ml [J](x, x ) ∆ pq ml [J](x, x ) ∆ qp ml [J](x, x ) ∆ qq ml [J](x, x )   (10)
are the source-dependent two-point Green's function where

∆ pq = 1 i δW H [J] δJ p (x)δJ q (x ) = δ ql [J](x ) δJ p (x) (11)
and ∆ pp [J], ∆ qq [J] and ∆ qp [J] are analogously defined. The Effective Action in the first order formalism Γ H [q m , pm ] can be defined by the Legendre transform of W H [J] with respect to the associated "averaged" fields of the theory

Γ H [ φ] def ≡ W H [J] -d 4 xJ(x) • φ(x). ( 12
)
Remember that the compact notation implies a twofold Legendre transformation in the two independent components q and p. By considering the functional derivative of Γ H [q, p] with respect to pi (x) and qi (x) we obtain as expected

δΓ H δ φ(x) = -J(x). ( 13
)
Substituting Eq. ( 13) into Eq. ( 6), however, we obtain with the replacement Eq. ( 9) the desired functional differential equations for Γ H [q, p],

δΓ H δ φ(x) = δI 0 δφ(x) φ(x)→ φ[J](x)+ i R d 4 x D[J](x,x ) δ δ Φ(x ) . ( 14
)
This relation generates all DSEs for the 1PI Green's functions of the first order formalism. The method to obtain such a functional relation will be employed several times in the following. The derivation of the proper propagators is straightforward when keeping in mind that they are embedded in a matrix. In fact, taking a derivative with regard φ(x ) in Eq. ( 14) we obtain

G[φ](x , x ) ≡    δ 2 Γ H δ pm(x )δ pn(x ) δ 2 Γ H δ pm(x )δ qn(x ) δ 2 Γ H δ q m (x )δ pn(x ) δ 2 Γ H δ q m (x )δ qn(x )    . ( 15
)
This expression and Eq. ( 10) are related via the identity

d 4 x D[J](x, x )G[φ](x , x ) = -I, ( 16
)
which can be obtained by taking a derivative with respect to the source J in Eq. (13). Hereby I = δ mn δ 4 (x-x )⊗1 2×2 denotes the identity in the considered space. Any other connected or proper Green's function can be derived by considering higher order derivatives with respect to J and φ in Eq. ( 10) and Eq. ( 15), respectively. In general, these functions are constrained by the symmetry properties of the initial "action". In the case of a gauge theory a corresponding derivation leads to the STIs. For Coulomb gauge Yang-Mills theory it is presented in Sect. VIII. Here we will illustrate the potential complications arising in the first order formalism. For this it sufficient to assume that I 0 is invariant under the simultaneous infinitesimal transformations

δφ[φ] ≡ •   G q m [p, q] F p m [p, q]   (17)
and also assume that they leave the path integration measure invariant. Performing this substitution in Eq. ( 1) we can obtain the following "Symmetry-Related Functional Identity"

0 = D[q]D[p] d 4 xJ(x) • δφ(x) exp i I [φ, J] . ( 18
)
In the next step we substitute the fundamental and momentum fields by the respective derivatives with respect to the classical source. Following a procedure similar to the derivation of the DSEs we rewrite the symmetry-related identity Eq. ( 18) and express it in terms of the full Generating Functional Z[J]:

0 = d 4 xJ(x) • δφ i δ δJ Z[J]. ( 19
)
Analogously, the above relation can be rewritten for connected Green's functions

0 = d 4 xJ(x) • δφ[φ]| φ→ δW H δJ + i δ δJ . ( 20
)
As before, Eq. ( 18) can be reformulated in terms of the Effective Action

0 = d 4 x δΓ H δ φ • δφ[φ] φ→ φ[J]+ i R d 4 yD[J](x,y) δ δ φ(y) . ( 21
)
This identity verifies that the Effective Action in first order formalism preserves the continuous symmetries of the initial canonical quantum action. At this point the DSEs and the symmetry-related identity Eq. ( 18) of a QFT seem to be more cumbersome than the usual one which appear in the standard path integral formulation. However, in the next section we shall show that under certain conditions the above system can be reduced to the latter one.

III. THE CONNECTION BETWEEN LAGRANGE AND HAMILTON FORMALISM

From here on we will denote all the variables a field depends on by a single latin index. For instance in case of a gauge field such indices will indicate the space-time point x, the vectorial index µ, and the adjoint gauge group index a. Repeated indices are summed and integrated over for discrete and continuous variables, respectively. Also all fundamental fields will be denoted by the same letter q. Similarly, p will represent all momentum fields.

Let us now consider the path integral representation of a n-point function involving only momentum fields

∆ p...p i1...in = 0 out T Πi1 . . . Πin 0 in J i 1-n 0 out |0 in J = D[q]D[p]p i1 . . . p in exp i I[q, p, J] i 1-n D[q]D[p] exp i I[q, p, J] .
Here T is the time ordering operator, and Πi ≡ Πm (x, τ) are the Hermitian momentum field operators in the Heisenberg picture. The expression (22) can be written via functional derivatives with respect to J p i as

∆ p...p i1...in = D[q]D[p] i δ δJ p i 1 . . . i δ δJ p in exp i {I[q, p, J]} i 1-n D[q]D[p] exp i {I[q, p, J]} . ( 22
)
In the following we will consider only the important class of Hamiltonians which are at most quadratic in the momentum fields. Note that this case is realized for all renormalizable and most non-renormalizable theories. For later use we write the Hamiltonian in the form

H = 1 2 p i A ij [q]p m + B i [q]p i + C[q], (23)
which defines a real, symmetric, positive and non-singular matrix A as well as the real functionals of q, B i [q] and C[q]. For the considered class of Hamiltonians the Gaussian integration over p can be performed analytically. As shown in appendix A it yields an expression of the form

∆ p...p i1...in = D[q] i δ δJ p i 1 . . . i δ δJ p in exp i S[q, J] i 1-n D[q] exp i S[q, J] . ( 24
)
The general result for the new action arising in the exponential is given in Eq. (A5).

For simplicity we will discuss first the standard Hamiltonian with A ij [q] = δ ij . The general case where A is nontrivial is realized e.g. in the case of Coulomb gauge QCD which will be discussed in Sect. VIII. (For the moment we only note that then the inverse matrix, A -1 [q], appears in the following expressions as a prefactor of J p making in general the action S 0 non-local.) In the considered simpler standard case the new action possesses the structure

S[q, J] = S 0 + 1 2 J p i J p i + J p i δS 0 δ qi + J q i q i (25)
where S 0 is the standard action

S 0 [q] = 1 2 qi qi -qi B i [q] + 1 2 B i [q]B i [q] -C[q] . ( 26
)
The application of Eq. ( 7) on the integrand present in Eq. ( 24) makes it possible to write

∆ p...p i1...in = D[q] Ôp...p i1...in-1 J p in + δS0 δ qin exp i S i 1-n D[q] exp i S . ( 27
)
where the operator (28) only acts on the function inside the curly brackets. Note that the integrand in the numerator is a functional depending on the fields q as well as the sources J q and J p .

Ôp...p i1...in-1 = n-1 l=1 J p i l + δS 0 δ qi l + i δ δJ p i l

A. Alternative form of the DSEs

Analogous to Eq. ( 6) we can write the partially integrated expression Eq. ( 27) in terms of the Generating Functional of connected Green's functions. Let us specialize to the case of one-point Green's functions, i.e., the vacuum expectation values of the momentum fields pi

[J] = δS 0 δ qi q→ δW H δJ q + i δ δJ q + J p i = δW H δJ p i (x, τ) . ( 29
)
Eq. ( 13) allows to express Eq. (29) as

δΓ H δ pi = -p i + δS 0 δ qi q→q[J]+ i ∆ qq [J] δ δ q . ( 30
)
Here the dependence on the average momentum fields is made explicit and the action in the Lagrange formalism appears which depends only on the q-fields. From a functional point of view it is more convenient to express this via Eq. (29) as

δΓ H δ pi = -p i - δS 0 δ qi φ→ φ[J]+ i ∆ φφ [J] δ δφ . ( 31
)
Now, employing the identity

0 = D[q] δ δq m D[p] exp i I[q, p, J] (32)
we perform the p-integration, so that the above expression reads

0 = D[q] δS 0 δq i + J p m δ 2 S 0 δq i δ qm + J q i exp i S[q, J] , ( 33
)
which makes it possible to obtain an analogous form of the field equations

δΓ H δ qi = δS 0 δq i - δΓ H δ pm δ 2 S 0 δq i δ qm φ→ φ[J]+ i ∆ φφ [J] δ δ φ . ( 34
)
Eqs. (31) and (34) represent alternative forms of the first order DSEs (14). This alternative form of the DSEs allows us to find corresponding equations for the fundamental proper Green's functions by taking derivatives with respect to the momentum and/or fundamental fields. This way the proper momentum propagator can be written as

δ 2 Γ H δ pi δ pj = -δ ij + δ δ pi δS 0 δ qj q→q[J]+ i ∆ qq [J] δ δ q , ( 35
)
whereas the corresponding mixed Green's function is

δ 2 Γ H δ qi δ pj = δ δ qi δS 0 δ qj q→q[J]+ i ∆ qq [J] δ δ q . ( 36
)
In a similar way, the other mixed propagator can be written as

δ 2 Γ H δ pi δ qj = δ δ pi δS 0 δq j q→q[J]+ i ∆ qq [J] δ δ q - δ 2 Γ H δ pi δ pm δ 2 S 0 δq j δ qm φ→ φ[J]+ i ∆ φφ [J] δ δ φ , ( 37
)
whereas the fundamental field propagator is given by

δ 2 Γ H δ qi δ qj = δ δ qi δS 0 δq j q→q[J]+ i ∆ qq [J] δ δ q - δ 2 Γ H δ qi δ pm δ 2 S 0 δq j δ qm φj → φ[J]+ i ∆ φφ [J] δ δ φ . ( 38
)
Hereby, in the derivation of Eqs. (37) and (38) we have discarded those terms that vanish when the sources are set to zero. In addition, note that the proliferated occurrence of Green's functions involving the p-field arises from functional derivatives on ∆ qq ij [J]. As will become clear below, despite the different appearance, the structure of Eqs. (37) and ( 38) is related to the one of Eqs. (35) and (36).

B. Pure and mixed momentum correlation functions

In this subsection we derive expressions for first order correlation functions entirely in terms of second order correlation functions. To this end, we start from Eq. (27). Subsequently, we again follow a procedure similar to that used in the derivation of Eq. ( 6) thus arriving at a general expression for arbitrary momentum correlation functions,

∆ p...p i1...in = i n-1 n-1 l=1 J p i l + δS 0 δ qi l + i δ δJ p i l J p in + δS 0 δ qin J p =0 qa→qa[J q ]+ i ∆ qq ab [J q ] δ δ qb (39)
In contrast to the corresponding equation for the one-point function, Eq. ( 29), (which did lead to the alternative set of DSEs) here the connected Generating Functional in the standard Lagrange formalism appears. Note that the latter does not depend on J p anymore. From the above functional equation, any other m-point Green's function with n-external legs associated to the J p 's is obtained by taking m-n derivatives with regard to J q m and setting them to zero. In particular, the quantum average of the momentum fields becomes a functional of the averaged field q,

pi [q] = δS 0 δ qi qm→qi[J q ]+ i ∆ qq ij [J q ] δ δq j . ( 40
)
Already at this point we want to point out that the averaged momentum field is generally not given by the usual definition of a canonical momentum field p can i on the level of the Effective Action pi

[J = 0] = δS δ q = δΓ δ q ≡ p can i . ( 41
)
This will be shown explicitly in subsection VI A.

In addition, the relation between the pp-correlation functions and those that appear in the standard formalism can be expressed as

∆ pp ij = δ ij + i δS 0 δ qi δS 0 δ qj q→q[J q ]+ i ∆ qq [J q ] δ δq , ( 42
)
whereas the corresponding mixed qp-correlation function can be written as

∆ qp ij [J q ] = δ δJ q i δS 0 δ qj qm→ δW δJ q + i δ δJ q . ( 43
)
By using the chain rule δ/δJ q m = δq n /δJ q m δ/δq n , Eq. (43) can also be given in the form

∆ qp ij [J q ] = ∆ qq il δ δ ql δS 0 δ qj q→q[J q ]+ i ∆ qq [J q ] δ δq = ∆ qq il δ pj δ ql . ( 44
)
The other mixed correlator ∆ pq can be immediately inferred from the bosonic nature of the fields, ∆ pq ij = ∆ qp ji . Similarly, it is possible to determine the connected three point correlation functions

∆ qpp ijk [J q ] = ∆ qq il δ δ ql ∆ pp jk , ∆ qqp ijk [J q ] = ∆ qq im ∆ qq jl δ 3 Γ δ ql δ qm δ qn ∆ qp nk + δ 2 pk δ qm δ ql , ( 45
) ∆ ppp ijk [J q ] = iδ ij pk + iδ ik pj + iδ jk pi + i 2 δS 0 δ qi δS 0 δ qj δS 0 δ qk q→q[J q ]+ i ∆ qq [J q ] δ δq ( 46
)
where in the last expression we have used the previous results. Employing Eqs. (44) and Eq. ( 45) the first and second derivatives of p with respect to q can be expressed as functionals of the remaining elements. In this way the vacuum expectation value of p can be expanded. In fact, up to second order in the classical field it reads pm = -

δ 2 Γ δ qj δ ql ∆ qp lm qj - 1 2! δ 3 Γ δ qi δ qj δ ql ∆ qp lm - δ 2 Γ δ qi δ qn δ 2 Γ δ qj δ ql ∆ qqp lnm qi qj + . . . ( 47
)
Since ∆ qp ij can be computed using the procedure detailed above, the expression (47) determines p as a function of the second-order dressed correlation functions.

C. Inverting the matrix propagator D

The determination of the proper functions within the canonical formalism as a functional of those appearing in the standard Lagrange framework is rather cumbersome. This task involves connected tensors of rank larger than two and depends on the possibility to invert the propagator D. Once the individual elements of this propagator are computed, the proper two-point function is completely determined in terms of the elements of the Lagrange formalism. In case the external sources associated to the p-fields vanish, the inverse of -∆ qq is the proper Green's function that arises in the standard path integral representation.

By direct inversion of the 2 × 2 block matrix one obtains

G =   Γ pp ij Γ pp il ∆ pq lm Γ qq mj Γ qq il ∆ qp lm Γ pp mj Γ qq ij H   (48)
where

Γ pp ij = -(∆ pp + ∆ pq Γ qq ∆ qp ) -1 ij and Γ qq ij H = Γ qq ij + Γ qq il ∆ qp lk Γ pp km ∆ pq mn Γ qq nj . ( 49
)
According to Eq. ( 48) there are several equivalent representations of the latter expression, e.g.

Γ qq ij H = (δ im + Γ qp il ∆ pq lm ) Γ qq mj .
However, in what follows we will consider the most simple form given by

Γ qq ij H = Γ qq ij + Γ qp il (Γ pp lm ) -1 Γ pq mj , ( 50
)
where we have introduced a unity in the form I = (Γ pp ) -1 Γ pp in the last expression of Eq. ( 49), and furthermore the expressions for the off-diagonal elements of Eq. ( 48) have been used.

The method to obtain G can be generalized to any proper m-point function. For instance, let us suppose that we want to compute the proper three-point function. We denote it as G 3 and the correspondingly connected version as D (3) . By considering the action of the symbolic functional derivative δ/δJ on Eq. ( 16) we get the equation D (3) G + DDG 3 = 0. By inversion of D in the second term we obtain the desired form for the proper 3-point Green's function G 3 = -GGD (3) G. Proceeding in an analogous way it is possible to express the proper four-point Green's function G 4 , and so on. Thereby, the number of variables within the first order formalism can be reduced. From that point of view, the initially cumbersome problem becomes simpler.

D. Connecting the Symmetry-Related identities

Let us now return to the symmetry identity. To this end we write Eq. ( 18) in the following form

0 = D[q] J q m δq m i δ δJ p , q + J p m δp m i δ δJ p , q D[p] exp i I[φ, J] . ( 51
)
In order to simplify the following analysis let us consider theories where the transformations are linear in the momentum fields. Under this condition and restricting to the class of Hamiltonians analyzed so far we obtain

0 = D[q] J q m δq m J p + δS 0 δ q , q + J p m δp m J p + δS 0 δ q , q exp i S[q, J] . ( 52
)
In particular, if J p = 0 we find that the action S 0 is invariant under a symmetry transformation δq m which is a functional of q only, i.e. a symmetry transformation δq m δS0 δ q , q → δq m [q]. The structure of Eq. ( 52) allows to write the symmetry identities in a similar form to Eq. ( 20) and Eq. ( 21),

0 = d 4 x δΓ H δ pm δp m δS δ q - δΓ H δ p , q + δΓ H δ qm δq m δS 0 δ q - δΓ H δ p , q φ→ φ[J]+ i ∆ φφ [J] δ δ φ .
This concludes the formal discussion of the functional symmetry identities. A complete derivation, especially for constrained systems as Coulomb gauge Yang-Mills theories, is presented below in Sect. VIII.

IV. DECOMPOSITION OF PROPER LAGRANGE CORRELATION FUNCTIONS

In this section we will show how a general proper correlation function in the Lagrange formalism can be decomposed into correlation functions in the Hamilton approach.

A. Relations between the bare elements

So far we have presented the formalism for a general field theory. In the present and forthcoming sections we will restrict ourselves to the most important class of renormalizable field theories. In four dimensions the most general renormalizable "canonical action" for a pure bosonic theory can be expressed as a functional Taylor expansion,

I 0 [q, p] = I qp 0ji p i q j + 1 2 I pp 0ij p i p j + 1 2 I pqq 0ijk p i q j q k + 1 2 I qq 0ij q i q j + 1 3! I qqq 0ijk q i q j q k + 1 4! I qqqq 0ijkl q i q j q k q l . ( 53
)
Here

I qp 0ji = ∂ τi δ ij is such that I qp 0ji q j = qi . ( 54
)
Clearly, the coefficients I φ... 0i... are field independent. They are given by the functional derivatives of I 0 evaluated at φ = 0, namely

I φφ 0ij... ≡ δ δφ i δ δφ j δ δφ k . . . I 0 φ=0 .
We remark that I pqq 0ijk as well I qqqq 0ijkl are dimensionless tensor couplings whereas I qq 0ij and I qqq 0ijk have mass dimension 2 and 1, respectively, and do not involve time derivatives of the fields. Nevertheless, depending on the assumed theory the latter two might depend on ∇ 2 and ∇. We are considering bosonic theories, so that all coefficients are symmetric. In particular, the first term in Eq. ( 2) can be written as I pq 0ji p j q i where I pq 0ji = -∂ τi δ ij which leads to the functional relation

∂ τj δ ji = -∂ τi δ ij .
We can identify the coefficients present in the Hamiltonian density Eq. ( 23) as

A ij = -I pp 0ij = δ ij , B i = - 1 2 I pqq 0ijk q j q k and C = - 1 2 I qq 0ij q i q j - 1 3! I qqq 0ijk q i q j q k - 1 4! I qqqq 0ijkl q i q j q k q l . ( 55
)
Substituting Eqs. (55) in Eq. ( 26) and collecting the terms of the same order in q we get that the action S 0 can be written as a polynomial functional,

S 0 = 1 2 S 0ij q i q j + 1 3! S 0ijk q i q j q k + 1 4! S 0ijkl q i q j q k q l . ( 56
)
In this context the following relations between the bare coefficients arise

S 0ij ≡ δ 2 S 0 δq i δq j q=0 = I qp 0il I pq 0lj + I qq 0ij , ( 57
)
S 0ijk ≡ δ 3 S 0 δ qi δ qj δ qk q=0 = I qqp 0ikn I pq 0nj + qk ↔ qj permut. + qi ↔ qj permut. + I qqq 0ijk , ( 58
)
S 0ijkl ≡ δ 4 S 0 δq i δq j δq k δq l q=0 = I qqp 0ikn I pqq 0njl + qi ↔ ql perm. + qk ↔ ql permut. + I qqqq 0ijkl ( 59
)
Substituting the above equations in Eq. ( 56) and considering the relation given by Eq. ( 54) allows to find additional relations. In fact, taking the second, third and fourth functional derivatives of S 0 and evaluating at q = 0, respectively, we obtain

S q q 0ij ≡ δ 2 S 0 δq i δ qj q=0 = I qp 0ij = ∂ τj δ ji , S qq 0ji ≡ δ 2 S 0 δ qj δq i q=0 = I pq 0ij = -∂ τj δ ji , S qq q 0ijk ≡ δ 3 S 0 δq i δq j δ qk q=0 = I qqp 0ijk , ( 60
)
δ3 S 0 δq i δq j δq k q=0 = I qqq 0ijk and δ4 S 0 δq i δq j δq k δq l q=0 = I qqqq 0ijkl .

(61)

In the last two relations we have introduced δ denoting a partial functional differentiation acting just on those terms in the action that do not involve the time derivative of the field.

To complete our analysis we point out that from Eq. (A7) the general form of the quantum canonical momentum fields in four-dimensional renormalizable theories is given by

p i = S q q 0ji q j + 1 2 S qq q 0kji q j q k . ( 62
)
As these polynomial representations allow to identify a priori the bare elements, the expansions given above prove to be very convenient to derive the DSEs for a general bosonic theory in both formulations.

B. Diagrammatic representation

The relations between the correlation functions in both formulations will be given in the following via explicit diagrammatic expressions. To enable this, we will first introduce a graphical representation in terms of the fundamental objects that characterize the theory in both formulations. I) As before we consider fields that involve all the irreducible representations in the theory. The fundamental fields are represented by solid lines whereas the corresponding momentum fields are denoted by zigzag lines. IV) In our analysis it will become useful to introduce the inverse proper momentum 2-point function

D pp ij ≡ - δ 2 Γ H δ pi δ pj -1 (63)
which will be represented by a thick dotted line. Since it has the form of an alternative momentum propagator it can connect as an internal momentum line to proper vertices. Similarly all connected correlators which are one-particle-irreducible (1PI) in the fields but merely connected via the "propagator" D pp are called p-connected and are denoted by a blob labeled by a P.

P D pp p -connected

C. Diagrammatic decomposition

To express proper functions in the Lagrange formalism in terms of those of the Hamilton formalism, we exploit the underlying equivalence between the first and second order formalism. Due to the equivalence of the Generating Functionals of connected Green's functions at vanishing sources J p , the Effective Actions in the two formalisms are identical when p is a functional of q which in turn is implicitly given by the stationarity in p,

Γ[q] ≡ Γ H [p(q), q] whenever δΓ H δ p = -J p = 0. ( 64
)
Here Γ H still depends explicitly on q and p, where the quantum average of the momentum fields is in general a complicated functional of the fundamental field. Any proper n-point function in the standard formalism can be determined by taking n derivatives with respect to the fields in Eq. ( 64) and evaluated at the vacuum expectation value. In particular applying the chain rule and Eq. (64) the first derivative reads

δΓ δ qi = δΓ H δ qi + δΓ H δ pj δ pj δ qi = δΓ H δ qi . ( 65
)
In the above defined graphical representation this equation reads

= .
As usual in the standard formalism, a differentiation with respect to a field is equivalent to attaching an external leg in the graphical representation (cf., e.g., ref. [27,35]). Next we consider the second derivative of the action given by

δ 2 Γ δ qi δ qj = δ pn δ qi δ 2 Γ H δ pn δ qj + δ 2 Γ H δ qi δ qj . ( 66
)
The field derivative of the momentum field can alternatively to the previous result (47) also be obtained from a field derivative of the constraint equation in Eq. ( 64),

δ 2 Γ H δ qi δ pj = - δ pn δ qi δ 2 Γ H δ pn δ pj ⇒ δ pn δ qi = - δ 2 Γ H δ qi δ pj δ 2 Γ H δ pn δ pj -1 . ( 67
)
Inserting this in Eq. ( 66) it is expressed entirely in terms of proper first order Green's functions and takes the symmetric form

δ 2 Γ δ qi δ qj = δ 2 Γ H δ qi δ qj - δ 2 Γ H δ qi δ pj δ 2 Γ H δ pn δ pj -1 δ 2 Γ H δ pn δ qj (68)
in accordance with Eq. (50). The above equation yields the graphical representation of the decomposition of the proper two-point Green's function

= + .
Interestingly, in terms of the first order correlation functions there is in addition to the proper part also a connected contribution due to the fact that the fundamental field mixes with the corresponding momentum field. This is evident from the off-diagonal elements Eq. ( 43) in the propagator matrix in the first order formalism. Due to the mixing the arising propagator in Eq. ( 67) is not the elementary p-propagator but the propagator for a collective mode described by the inverse of the proper two-point momentum correlation function D pp which is represented by the dotted line and related to the actual momentum propagator via Eq. ( 49).

The result for the fundamental field derivative of a proper correlation function in the Hamilton formalism yields the replacement rule + → .

In the next step a field derivative can also act on the propagator D pp . Its derivative is obtained from the derivative of the inverse of an operator as This yields directly a symmetric result, whereas the computation without the replacement Eq. ( 67) in each step would produce an asymmetric result involving higher derivatives of p. To derive the decomposition of higher order Lagrange correlation functions it is useful to note that both of the above replacement rules involve the external legs in the form of the composite expression ≡ + denoted by a dashed line. By introducing the composite external leg into the graphical representation, the decomposition of the 3-point function is given by a single graph with three of these new external legs. Via the previous rules it is easy to obtain a corresponding replacement rule for the composite external leg. Thereby the extension of a general n-point function by an additional leg can be obtained from the simplified set of rules in Fig. 1. Starting from the 3-point function these rules allow to derive the decomposition of arbitrary proper n-point functions in the Lagrange formalism. In the appendix we show that the graphs generated by these replacement rules have a very simple structure that can be summarized by the following general statement:

δD pp ij δ qk = - δ δ qk δ 2 Γ H δ pi δ pj -1 = δ 2 Γ H δ pi δ pm -1 δ 3 Γ H δ pm δ pn δ qk δ 2 Γ H δ pn δ pj -1 = D pp im Γ ppq mnk D pp
A proper n-point function in the Lagrange formalism can be decomposed into the sum of all p-connected n-point functions with composite external legs in the Hamilton framework. This is shown in graphical form in Fig. 2.

The graphical representation manifestly shows that the propagation, decay and dispersion of particles are more complicated when analyzed in the Hamilton framework. At the tree level the arising propagators D pp are constant and correspondingly the dotted internal lines between the proper vertices represent merely contact terms. Similarly the external leg corrections involve no additional poles at tree level and the explicit energy dependence arising from the mixed correlator in Eq. ( 67) just cancels the one from the momentum field derivative of the initial vertex. In the fully dressed action, however, there could arise additional poles due to the actual propagation of the momentum fields via higher order kinetic terms, as well as an explicit energy dependence of the terms that cancels only after summing over all contributions.

Via graphical rules, one can therefore generate in a systematical way the connection between any proper Green's function in the standard second-order formalism to the appropriate correlation functions in the first-order formalism. We point out already here that the inverse relations for Green's functions in the latter context are more complicated and can involve loops related to connected correlation functions with both mixed and pure momentum fields. The method used in the last section allows us to represent them in terms of those appearing in the Lagrange formalism.

D. Inclusion of Grassmannian fields

As already mentioned we will consider for completeness also quantum field theories involving Grassmannian fields c i and c † j fulfilling the anti-commutation rules

c i , c † j = δ ij , ( 69
)
c † i , c † j = {c i , c j } = 0. ( 70
)
The Grassmannian action for a renormalizable theory in four dimensions has the general form

I 0 = iκ ij c † i ċj + iλ ijk q i c † j c k + iα ij c † i c j . ( 71
)
The quantum canonical momentum fields associated to the c i are given by p i = δI0 δR ċi = iκ ij c † j , where the suffix R denotes differentiation from the right. It is clear that the usual path integral representation for Grassmannian field is already of first-order form since the momentum fields are treated as independent variables. Purely fermionic correlation functions are therefore trivially identical in the two formulations.

Let us now study mixed correlation functions involving bosonic fields represented by the q i in Eq. ( 71) when the bosonic path integral is written in the canonical form as well. Generally the tensors κ ij , α ij and λ ijk are real and field independent. The kinetic tensor reads in particular

κ ij =    δ ij with c, c † ∈ Fermions δ ij σ∂ 0 with c, c † ∈ Fadeev -Popov Ghosts
where σ ∈ R. Since it is a bosonic variable, p is a functional of fermionic bilinears ci † Γ ij cj and the field q. As a consequence the second term in Eq. (66) vanishes identically, which confirms the equality of the propagators in both formulations

δ 2 Γ δc i δc † j = δ 2 Γ H δc i δc † j . ( 72
)
Analogous to the bosonic case the three-point vertex can be decomposed as

δ 3 Γ δ qi δc j δc † k = δ pn δ qi δ 3 Γ H δ pn δc j δc † k + δ 3 Γ H δ qi δc j δc † k . ( 73
)
Although the propagators are the same in both formulations the vertices can be different. The above chain rule again results in the appearance of composite legs for all bosons in the graphical representation. The decomposition of a general n-point correlation function is then again given by all p-connected graphs where only the bosonic propagators D pp are involved and only bosonic external legs are composite.

V. DECOMPOSITION OF CONNECTED HAMILTON CORRELATION FUNCTIONS

So far we gave in subsection III B general expressions for correlation functions in the Hamilton formalism. The goal in this section is to evaluate the general decomposition of the two-point functions in the Hamilton framework in case of a generic four-dimensional renormalizable quantum field theory in terms of Lagrange correlation functions. To do this, we start by computing the elements of ∆ qp ij .

A. The mixed connected 2-point function

By considering Eq. (44) and Eq. (62) the mixed connected two-point function in the first order formalism can be written as (from now on we skip the explicit factor )

∆ qp ij = ∆ qq il S q q 0lj - i 2 S qq q 0ukj δ δ ql ∆ qq uk . ( 74
)
Using partial differentiation for the identity ∆ qq il Γ qq lj = -δ ij we obtain

δ δ ql ∆ qq uk = ∆ qq um Γ qqq mln ∆ qq nk , (75)
and therefore in configuration space

∆ qp ij = ∆ qq il I qp 0lj - i 2 I qqp 0ukj ∆ qq um Γ qqq mln ∆ qq nk , ( 76
)
where Eq. ( 60) has been used. According to the diagrammatic representation the above function has the decomposition given in Fig. 3.

= -i 2 FIG. 3:
The decomposition of the mixed Hamilton propagator in terms of Lagrange correlation functions.

Via the bosonic symmetry of the propagator it is easy to see that

∆ pq ij = I qp 0li - i 2 I qqp 0uki ∆ qq um Γ qqq mnl ∆ qq nk ∆ qq lj . ( 77
)
Clearly, Eq. ( 76) and Eq. ( 77) fulfill the condition ∆ qp ij = ∆ pq ji . Our convention for the Fourier transform of a general two-point function (connected or proper) obeying translational invariance is 4 . As a consequence of this convention and the equivalence ∆ qp ij = ∆ pq ji we obtain the relation

∆ ΦΦ ij ≡ ∆ ΦΦ ij (x i -x j ) = dk∆ ΦΦ ij (k)e -ik(xi-xj ) , ( 78
) with k(x i -x j ) = k 0 (x 0i -x 0j ) -k • ( x i -x j ) and dk ≡ d 4 k/(2π)
∆ qp ij (k) = ∆ pq ji (-k) ( 79
)
where i and j represent the remaining internal indices. This yields the corresponding equation in momentum space

∆ qp ij (k) = ∆ qq il (k) ik 0 δ lj - i 2 dωI qqp 0ukj (k -ω, ω, -k)∆ qq um (ω -k)Γ qqq mln (ω -k, k, -ω)∆ qq nk (ω) . ( 80
)
In the derivation of the latter equation we have taken into account the Fourier transformation of the proper 3-point function

Γ ΦΦΦ αβγ = dk α dk β dk γ (2π) 4 δ (4) (k α + k β + k γ )Γ ΦΦΦ αβγ (k α , k β , k γ )e -ikαxα-ik β x β -ikγ xγ (81)
where the δ-function expresses the momentum conservation.

B. The momentum propagator

The pure pp-correlator can be computed using Eqs. ( 42) and (62). Neglecting those terms that will eventually vanish when the sources are set to zero, it reads

∆ pp ij = δ ij + S q q 0li S q q 0mj ∆ qq lm - i 2 S q q 0li S qq q 0nmj ∆ qq lu δ δ qu ∆ qq nm - i 2 S q q 0mj S qq q 0kli ∆ qq ku δ δ qu ∆ qq lm - i 2 S qq q 0kli S qq q 0nmj ∆ qq km ∆ qq ln - 1 4 S qq q 0kli S qq q 0nmj ∆ qq ku δ δ qu ∆ qq lx δ δ qx ∆ qq nm . ( 82
)
By iterated application of Eq. ( 75) and considering Eq. ( 60) this expression yields

= -i 2 + -1 4 -1 4 -1 2 -i 2 -i 2 FIG. 4:
The decomposition of the momentum propagator in terms of Lagrange correlation functions.

∆ pp ij = δ ij - i 2 I pqq 0ikl ∆
) 83
Here we have already suppressed disconnected expressions involving tadpoles that cancel since Eq. (30) gives

δΓ H δ pi q, p=0 ∼ I qqp 0mni ∆ qq mn = 0 . ( 84
)
The graphical representation of the decomposition of the momentum propagator in the Hamilton framework is shown in Fig. 4. Note that the loop graphs in the first line represent precisely the vacuum graphs of the n-particle irreducible (nPI) actions in the Lagrange formalism of order one and two with attached external p-legs. In addition to these proper contributions this result involves again connected graphs in the second line. These graphs correspond to the second and third line in Eq. ( 83) which can alternatively be expressed by the known mixed correlation functions derived in the last subsection as -∆ pq im Γ qq ml ∆ qp lj and correspondingly represented as a single graph. After Fourier transformation of Eq. ( 83) we find the representation of ∆ pp in momentum space

∆ pp ij (k) = δ ij -∆ pq im (k)Γ qq ml (k)∆ qp lj (k) - i 2 dωI pqq 0ikl (k, ω -k, -ω)I qqp 0nmj (ω, k -ω, -k)∆ qq km (ω -k)∆ qq ln (ω) - 1 4 dωdµI pqq 0ikl (k, µ -k, -µ)I qqp 0nmj (-ω, ω + k, -k)∆ qq ku (µ -k)∆ qq lx (-µ)∆ qq ny (-ω)Γ qqqq yxuz (ω, µ, k -µ, -ω -k) × ∆ qq zm (-ω -k) - 1 2 dωdµI pqq 0ikl (k, ω -k, -ω)I qqp 0nmj (µ + k, -µ, -k)∆ qq ku (ω -k)Γ qqq puq (-k -µ, k -ω, ω + µ) × ∆ qq lx (-ω)∆ qq np (µ + k)∆ qq qy (ω + µ)Γ qqq yxz (-ω -µ, ω, µ)∆ qq zm (µ). ( 85
)
The summation over all field components indicated by the subindices in Eq. (80) (see also Eq. ( 83)) leads to multiple possibilities. Yet many of these might not be allowed due to the symmetries of the theory. The decomposition of higher order functions involving one or two external momentum fields is simply obtained by further derivatives of these generating equations. Via the chain rule a derivative w.r.t. to the source can be transformed into a derivative with respect to the averaged fields yielding in addition external propagators. Correspondingly the graphical decomposition of such correlation functions can be obtained recursively via the graphical replacement rules in the Lagrange framework given in [27,35]. On the other hand, from Eq. (39) it is clear that higher order correlation functions involving n momentum fields involve graphs of loop order n. Therefore it is not possible to give a close result for all n-point functions.

To conclude this section we remark that the multiplication of Γ qq on the left hand side amputates the Green's function ∆ qq . According to Eq. (47), this operation allows to derive

δ pi δ qj = I qp 0ji - i 2 I qqp 0uki ∆ qq um Γ qqq mjn ∆ qq nk . ( 86
)
Considering this expression and the diagrammatic rules in the standard path integral representation we get

δ pi δ qj δ qk = I qqp 0jki - i 2 I qqp 0mui ∆ qq mn Γ qqq nkx ∆ qq xy Γ qqq yjt ∆ qq tu + j ↔ k - i 2 I pqq 0imu ∆ qq mn Γ qqqq njkl ∆ qq lu , ( 87
)
which can be represented graphically as

= -i 2 -i 2 -i 2
We note in passing, that similar to the composite collective propagator and composite leg in the last section this "vertex" allows to write the graphs in the first line by a single loop graph. Yet, here this is then surely no explicit decomposition in terms of second order correlation functions anymore.

VI. DECOMPOSITION OF PROPER HAMILTON CORRELATION FUNCTIONS

While in the context of the canonical formulation it is entirely possible to deduce the complete set of DSEs directly from Eqs. (35-37) we will follow a slightly less obvious path here. We proceed to give the diagrammatic and analytic expressions for the proper propagators, typically the ones of the first order formalism. Subsequently, we will show that such representations can be encoded into the usual ones, i.e. Eqs. (35)(36)(37), which then completes the proof of equivalence between both types of derivations.

A. The inverse propagators

The result given by Eq. ( 48) allows to analyze the structure of the inverse propagator. By considering the Fourier transformation of this equation we find that

Γ pp ij (k) = -(∆ pp (k) + ∆ pq (k)Γ qq (k)∆ qp (k)) -1 ij , Γ pq ij (k) = Γ pp il (k)∆ pq lm (k)Γ qq mj (k), (88)
Γ Hqq ij (k) = Γ qq ij (k)+Γ qq il (k)∆ qp lk (k)Γ pp km (k)∆ pq mn (k)Γ qq nj (k). ( 89
)
For a field theory without a three-point interaction vertex involving the time derivative of the fields, one has Γ pp = -I, and Eq. ( 48) reduces to

G =   -I -ik 0 I ik 0 I Γ qq -k 2 0 I   . ( 90
)
This simplified expression holds for theories like QED and/or self-interacting φ 4 -theory, where the vacuum expectation value of the momentum field is completely determined in terms of q, pi = qi . Actually, the expressions for Γ pp , Γ qp and Γ pq in Eq. (90) encode a more general result because they could be obtained using Eq. ( 35) and Eq. (36) without setting variables to zero. As a consequence they show that dressed proper functions involving more than two external momentum legs are not present in such theories. This means that the mass and the coupling constant receive no contribution coming from the higher order corrections besides the usual ones given in the standard framework. On the other hand the absence of quantum corrections to Γ pp , Γ qp , and Γ pq means that only wave-function renormalization contributes to these kinetic terms.

As in the present framework the quantum corrections to the propagators and vertices depend in general on temporal derivatives this shows that within the standard formalism there are several pieces in the Effective Action that depend on the time derivative of the averaged field. As a consequence, the canonical momentum fields defined on the level of the Effective Action differ from those given by the quantum average of p since the involved limiting processes do not commute.

B. Recovering the first order DSEs

Although the decomposition of proper Hamilton functions given above involves inversions that cannot be omitted in terms of second order correlation functions it is possible to transform these equations into a form where such matrix inversions are eliminated. This is done by explicitly introducing first order correlation functions on the right hand side again. As we will show in this subsection, this leads precisely to the first order DSEs. We start from Eq. (87) and use it to express Γ pp as

Γ pp ij = -δ ij - i 2 I pqq 0ikl ∆ qq km ∆ qq ln δ pj δ qn δ qm -1 . ( 91
)
The multiplication of -Γ pp -1 from the left hand side allows to write the above equation as

δ is - i 2 I pqq 0ikl (∆ qqp kls -∆ qq km ∆ qq ln Γ qqq umn ∆ qp us ) Γ pp sj = -δ ij ,
where Eq. ( 45) has been taken into account. The introduction of a I as -∆ qq Γ qq in the last term in the bracket and the use of Eq. ( 45) makes it possible to express the above relation in the following form

Γ pp ij = -δ ij + i 2 I pqq 0ikl ∆ qqp kls Γ pp sj + ∆ qqq kls Γ qp sj , ( 92
)
which can be translated to

Γ pp ij = -δ ij - i 2 I pqq 0ikl ∆ qφ kn Γ φpφ njm ∆ φq ml . ( 93
)
This presents the DSE for the proper momentum 2-point correlation function.

It is possible to obtain a similar equation for the pq-propagator:

Γ pq ij = δ il - i 2 I pqq 0ikt ∆ qq km ∆ qq tn δ 2 pl δ qn δ qm -1 I pq 0lj - i 2 I pqq 0luk ∆ qq um Γ qqq mjn ∆ qq nk . ( 94
)
The multiplication of -Γ pp -1 from the right hand side allows to rewrite Eq. ( 94) as

Γ pq ij = I pq 0ij - i 2 I pqq 0iuk ∆ qq um Γ qqq mnj ∆ qq nk + i 2 I pqq 0iuk ∆ qq um ∆ qq nk δ pl δ qm δ qn Γ pq lj . ( 95
)
This can be rewritten in the following form

Γ pq ij = I pq 0ij - i 2 I pqq 0iuk ∆ qq um Γ qqq mnj ∆ qq nk + ∆ qq um ∆ qq nk Γ qqq mnl ∆ qp lv Γ pq vj -∆ qqp ukl Γ pq lj . ( 96
)
The introduction of a I as -∆ qq Γ qq in the first and second term inside the brackets allows to write this as

Γ pq ij = I pq 0ij + i 2 I pqq 0iuk ∆
Γ pq ij = I pq 0ij - i 2 I pqq 0ikl ∆ qφ kn Γ φqφ njm ∆ φq ml . ( 99
)
We remark that the last terms of Eqs. ( 93) and (99) include several combinations of fundamental and momentum fields. The relations given by Eq. (93) and Eq. ( 99) are the DSEs of the proper ppand pq-propagators expressed in terms of the usual elements of the canonical formulation.

It is straightforward to prove that Eq. ( 93) coincides with the corresponding equation derived using Eq. ( 35) and Eq. (62). This is different for Eq. (99). In fact, as discussed previously, we could in principle calculate these propagators from Eq. (37), nevertheless by considering the action Eq. ( 56) and the relations between the bare elements we find that the structure is more cumbersome than that given in Eq. (99). Instead, the latter one is in correspondence with Eq. (36) via the symmetry relation Γ pq ij = Γ qp ji in case both q and p are bosonic fields. The fact that we recover the standard first order propagator DSEs from the decomposition of proper Hamilton correlation functions has its origin in the equivalence between the canonical and Lagrange equations of motion at the quantum level. In appendix C it is shown that whenever they describe the same dynamical processes, the DSEs derived from them will be equivalent too. However, as we argued in the last subsection, the classical canonical momentum fields defined from the Effective Action and those given by the quantum average of p are not the same. This means that the equations expressed in term of p and q do not correspond to the classical canonical ones.

Based on this statement, the derivation of Γ qq ij H is considerably simpler using Eq. (38) than via the procedure performed in the last two cases. Indeed, by considering Eqs. ( 62) and (56) as well as the relations between the bare elements, we arrive at

δΓ H δ qi = I qp 0il p l + I qqp 0ijk q j p k + I qq 0ij q j + 1 2 I qqq ijk q j q k + 1 3! I qqqq 0ijkl q j q k q l q→q+ i ∆ qφ δ δ φ . (100)
Now, in order to give the explicit form of Γ qq ij H , we rewrite Eq. ( 100) as

δΓ H δ qi = I qp 0ij pj + I qq 0ij qj + I qqp 0ijk qj pk + 1 2 I qqq 0ijk qj qk + 1 3! I qqqq 0ijkl qj qk ql -iI qqp 0ijk ∆ qp jk [J] - i 2 I qqq 0ijk ∆ qq jk [J] - 1 6 I qqqq 0ijkl ∆ qφ jm [J]∆ qφ ku [J]Γ φφφ mun [q, p]∆ φq nl [J q ] - i 2 I qqqq 0ijkl qj ∆ qq kl [J]. ( 101
)
Taking the functional derivative with respect to q and setting the vacuum expectation values of the fundamental fields to zero we get

Γ H ij = I qq 0ij - i 2 I qqqq ijkl ∆ qq kl -iI qqp 0jlm ∆ qφ lu Γ φqφ uin ∆ φp nm - i 2 I qqq 0jlm ∆ qφ lu Γ φqφ uin ∆ φq nm - 1 2 I qqqq jlmn ∆ qφ lh Γ φφφ hf k ∆ φφ f y ∆ qφ mu × Γ φqφ uiy ∆ φq kn - 1 6 I qqqq jlmn ∆ qφ lh ∆ qφ mf Γ φφqφ hf ik ∆ φq kn . ( 102
)
The Fourier transformation of Eqs. ( 93), ( 99) and ( 102) yields finally the corresponding DSEs in momentum space

Γ pp ij (k) = -δ ij - i 2 dωI pqq 0ikl (k, ω -k, -ω)∆ qφ kn (k -ω)Γ φpφ njm (k -ω, -k, ω)∆ φq ml (-ω) (103) Γ pq ij (k) = -ik 0 δ ij - i 2 dωI pqq 0ikl (k, ω -k, -ω)∆ qφ kn (k -ω)Γ φpφ njm (k -ω, -k, -ω)∆ φq ml (-ω) (104) Γ qq ij H (k) = I qq 0ij (k) - i 2 dωI qqqq 0ijkl (k, -k, -ω, ω)∆ qq kl (ω) -i dωI qqp 0ilm (k, ω -k, -ω)∆ qφ lu (k -ω)Γ φqφ ujn (k -ω, -k, ω) × ∆ φp nm (-ω) - i 2 dωI qqq 0ilm (k, ω -k, -ω)∆ qφ lu (k -ω)Γ φqφ ujm (k -ω, -k, ω)∆ φq nm (-ω) - 1 2 dωdµI qqqq 0ilmn (k, -µ -ω, ω -k, µ)∆ qφ lh (ω + µ)Γ φφφ hf k (ω + µ, -ω, -µ)∆ φφ f y (ω)∆ qφ mu (k -ω) × Γ φqφ ujy (k -ω, -k, ω)∆ φq kn (µ) - 1 6 dωdµI qqqq 0ilmn (k, ω -k -µ, -ω, µ)∆ qφ lh (k + µ -ω)∆ qφ mf (ω) × Γ φφqφ hf jk (k + µ -ω, ω, -k, -µ)∆ φq kn (µ) (105)
We show the graphical representation of the complete set of DSEs in Fig. 5 where for conciseness we use the matrix propagator which yields all possible loop graphs involving physical vertices in accordance with the symmetries of the action as a consequence of the implied summation over repeated indices.

VII. THEORIES WITH AUXILIARY FIELDS

Before we come to the main application of the developed formalism in the context of Coulomb gauge QCD, we will show in this section that the above derived results also apply to theories involving auxiliary fields, i.e. the typical treatment of such theories represent a special case of the discussion presented here. However, it is not the kinetic but the interaction terms which are linearized. In the case of fermionic theories this is also referred to as bosonization [28]. The action is by construction only quadratic in the auxiliary fields and lacks kinetic terms for them. Therefore, as we detail below the above described analysis directly applies and gives general relations between correlation functions in the fundamental theory and the linearized form involving auxiliary fields.

= -i 2 = -i 2 , = -i 2 -i 2 -1 2 -1 6
FIG. 5: Diagrammatical representation of the coupled system of DSEs within the first order formalism. The double lines represent the matrix propagator so that all possible graphs involving the individual propagators arise which are compatible with the symmetries and the restriction that the pqq-vertex is the only bare vertex involving momentum lines. In this form the formal equivalence to the second order equation is entirely manifest.

We will illustrate this in the case of fermionic theories with non-renormalizable, quartic interactions. Examples for this class of theories are the Nambu-Jona-Lasinio model [29] or the BCS theory of superconductivity [30]. The functional integral is given by

Dψ † Dψ exp i d 4 x L ψ +J ψ (x) ψ(x)+J ψ (x)ψ(x) ( 106
)
with the Lagrangian containing local quartic interactions

L ψ = ψ(x) (i ∂ -m ψ ) ψ(x) - i g i ψ ψ(x)Γ i ψ(x) 2 . ( 107
)
Here the Γ i are Dirac matrices. The linearization of the fermionic interaction can be performed by formally introducing a one of the form

I = i Dη i Dσ i exp i d 4 xσ i (x) η i (x)-ψ(x)Γ i ψ(x) (108)
into the fermionic path integral, where this path integral over σ i enforces a functional δ-function that allows to rewrite the non-linear fermionic interaction in terms of η i . Integrating then over the η i yields the path integral of the corresponding linear sigma model

Dψ † DψDσ exp i d 4 x L σ +J ψ(x) ψ(x)+J ψ (x)ψ(x)+J i σ (x)σ i (x)
where we have introduced additional sources for the auxiliary fields σ i . After this bosonization procedure the Lagrangian of the corresponding "linear σ-model" reads

L σ = ψ(x) i ∂ -m ψ + g i σ Γ i σ i (x) ψ(x) + m 2 σ 2 σ i (x) 2 (109)
where g ψ = g 2 σ /(2m 2 σ ). Here, there is, in contrast to the first order formalism, by construction no mixing between the fundamental and the auxiliary fields.

Analogous to Eq. ( 64) the Effective Actions of the two theories are again identical at vanishing sources

J i σ = 0 Γ ψ [ ψ † , ψ] = Γ σ [ ψ † , ψ, σi [ ψ † , ψ]] ( 110
)
where the auxiliary fields are implicitly given by

δΓ σ δσ i = -J i σ = 0. ( 111
)
Via the chain rule of functional differentiation we obtain analogous expressions for Eqs. ( 65)-( 69) with pi replaced by σi , respectively. Correspondingly, the same graphical rules apply. With the simplification that there can be no mixing due to the different statistics of the fields, the external legs are not composite and the arising inverse 2-point function is the ordinary σ-propagator. Denoting as before connected correlation functions that are 1PI with respect to the fundamental fields and connected in the auxiliary fields as σ-connected, this leads to the analogous general result:

A proper n-point function in the fundamental theory can be decomposed into the sum of all σ-connected n-point functions in the linearized theory.

In particular the decomposition of the proper 4-fermion vertex in the fundamental theory Eq. (107) reads

= + + +
where we represent in an analogous way the proper vertices in the fundamental and linearized theory by large and small blobs and the ordinary σ-propagator by the dotted line. This result confirms a well-known fact, namely, that no double-counting occurs in the linearized theory although the original fermion field and the auxiliary bosonic field are employed both. This redundancy in the description can also be prevented from the outset by partial re-bosonization [31] in the context of the functional renormalization group. In this approach the contribution of the fundamental degrees of freedeom is entirely absorbed into the bosonized interactions even at the level of the effective action.

Since the Lagrangian of the linearized theory is at most quadratic and lacks kinetic terms for the σ-fields by construction, these fields can be trivially integrated out retaining their sources at this point. This leads to an analogous expression to Eq. ( 39) for the correlation function of n auxiliary fields σ i1 (x 1 ) • • • σ in (x n ) after the sources are set to their vacuum expectation value. Due to the absence of 3-point interactions and mixing, the decomposition of the auxiliary σ-propagator is again simplified compared to the result given in Fig. 4,

= +i -
where the different prefactors of the loop correction arise due to the fermionic nature of the fields. In contrast to the decomposition of the Hamilton propagators there is no difference between proper and connected 2-point functions in the case of the auxiliary field due to the absence of mixing, and the proper correlation functions is simply the inverse of the above equation. This concludes the demonstration of the developed formalism to the case of theories with auxiliary fields.

VIII. COULOMB GAUGE YANG-MILLS THEORY

A. The canonical action and the momentum correlation functions

As detailed in the Introduction the main motivation for developing the presented formalism is given by the fact that the first order formalism might be better suited for non-perturbative studies of Coulomb gauge Yang-Mills theory. As a first step into this direction we will apply the formalism developed so far to give the explicit relations between the two-point correlation functions of the transversal and longitudinal components of the conjugate momentum to the ones of the gauge field.

The starting point is the "canonical action" of Coulomb gauge Yang-Mills theory whose structure has been derived in [START_REF] Watson | [END_REF]8]

I 0 = d 4 x p a • Ȧa - 1 2 (p a • p a + B a • B a ) + p a • D ab σ b -λ a ∇ • A a -ca ∇ • D ab c b . ( 112
)
Here, p is the conjugate momentum of the gauge field A µ,a ≡ (A a , σ a ), c and c are the Grassmann-valued Faddeev-Popov ghost fields introduced by fixing the gauge, and λ a is a "colored" Lagrange multiplier field. Above

B a i = ijk ∇ j A a k -1 2 gf abc A b j A c
k represents the chromomagnetic field, σ b the time-component of the gluon field, whereas D ac = ∇δ acgf abc A b is the covariant derivative in the adjoint representation, with the structure constants f abc of the color-group SU (3), and the gauge coupling g. The last two terms constitute the gauge fixing pieces introduced via the Faddeev-Popov procedure. However, as was originally pointed out by Gribov [32], the latter is plagued by the existence of equivalent gauge field configurations that are related by finite gauge transformations. A further condition must be imposed on the configuration space of gauge fields which restricts it to the so called "Gribov region":

Ω ≡ A : ∇ • A = 0 | -∇ • D ≥ 0 . ( 113
)
This simple domain is still not totally free of Gribov copies. Instead one should consider the fundamental modular region Λ [33]. It turns out though that functional integrals are dominated by configurations on the common boundary of Ω and Λ [START_REF] Watson | [END_REF] so that, in practice, it is enough to consider the domain defined by Eq. ( 113). Now, the term p • Ȧ in Eq. ( 112) is equivalent to p q in Eq. ( 2), and the remaining part can be identified as a classical "Hamiltonian" density of Coulomb gauge Yang-Mills theory:

H Coul = 1 2 (p a • p a + B a • B a ) -p a • D ab σ b + λ a ∇ • A a + ca ∇ • D ab c b . ( 114
)
Actually H Coul is not the Coulomb gauge Hamiltonian density derived by Christ and Lee [2]. The derivation of the latter through Eq. ( 112) and the corresponding path integral representation requires further steps detailed e.g. in Ref. [34] and references therein. Although the Christ and Lee Hamiltonian has the desirable property that it involves only the physical degrees of freedom, the non-local character of the Color-Coulomb potential prevents an effective analysis of renormalizability in this framework. On the contrary, the canonical action given in Eq. ( 112) proved useful to study the renormalization properties of Coulomb gauge Yang-Mills theory [START_REF] Watson | [END_REF]5,8,18]. It is noteworthy that within the context of Eq. ( 112) the ghost sector is fully disconnected from p a which allows to write the "canonical action" as a functional Taylor expansion involving a pure bosonic piece like Eq. ( 53) and one containing the ghost field via Eq. ( 71). Therefore, we can identify q = (A a , σ a , λ a ) and employ the general expressions derived so far.

Except for the inverse, bare ghost two-point function

I ab(cc) 0 = -k 2 δ ab , ( 115
)
the remaining inverse tree level propagators of the theory are presented in Table I. Note that I 0 involves four-vertices that have the following form in momentum space

I abc(pAσ) 0ij = -gf abc δ ij , I abc(ccA) 0i = igf abc k ci , ( 116
)
I abc(AAA) 0ijl = igf abc [δ ij (k a -k b ) l + δ jl (k b -k c ) i + δ li (k c -k a ) j ] , (117)
I abcd(AAAA) 0ijlm = -g 2 δ ij δ lm f ace f bde -f ade f cbe + δ il δ jm f abe f cde -f ade f bce + δ im δ jl f ace f dbe -f abe f cde , (118)
with all momenta defined as incoming.

The decomposition of the conjugate momentum into transverse and longitudinal parts, p a = π a -∇ • Ω a (note that Ω a must not be confused with the Gribov region), makes it convenient to study the required complete cancellation of the energy divergences [START_REF] Watson | [END_REF]5,8] that emerge in a perturbative treatment of Coulomb gauge Yang-Mills theory. Certainly, such a decomposition increases the number of fields of the theory and leads to

Ĩ0 = d 4 x π a • Ȧa -∇Ω a • Ȧa + 1 2 Ω a ∇ 2 Ω a -H Coul (p → π) -τ a ∇ • π a -∇Ω a • D ab σ b , ( 119
)
where τ is a colored Langrange multiplier field which appears via the transversality condition of π. In this context, the general structure of the connected two-point functions is given in Table II. The latter may be found independent of any approximation but based solely on the principles of BRST invariance, spatial and time-reversal symmetries and transversality properties of the vector propagators. Each dressing function Λ φφ is a dimensionless scalar function of k 2 0 and k 2 except the ghost propagator

∆ ab(cc) = δ ab Λ c (k 2 ) k 2 (120)
depending only on k 2 . The tree-level propagators are obtained when

Λ ΩΩ = Λ Ωλ = 0, Λ AA = Λ Aπ = Λ ππ = Λ σσ = Λ σΩ = Λ σλ = Λ c = 1. (121)
For a complete description, the reader is referred to ref. [START_REF] Watson | [END_REF].

The general results of the preceeding sections can in principle directly applied to Coulomb gauge QCD as well by decomposing the momentum field into its individual components. However, since the longitudinal momentum field features more complicated bare correlation functions that involve additional derivative operators we derive the corresponding expressions of subsection III B once more taking now the general action (A5) into account. To this end we first have to relate the bare momentum correlation functions with the corresponding ones for the individual momentum components. We formally express the decomposition via a the operator X

I0 Aj pj σ λ Ai -Tk 2 ik0I -k0k -ik pi -ik0I -I -ik 0 σ 0 ik 0 0 λ ik 0 0 0
p i = π i + ∂ i Ω = π i + X ij Ω j and X ij = δp i δΩ j = -∇ i δ ij . ( 122
)
Expanded in a local series in terms of the individual momentum components the general canonical action takes the form

I 0 [q, p] = I qp 0ji π i q j + 1 2 I pp 0ij π i π j + 1 2 I pqq 0ijk π i q j q k + I qp 0ji X il Ω l q j + 1 2 X il I pp 0ij X jk Ω l Ω k + 1 2 I pqq 0ijk X il Ω l q j q k + 1 2 I qq 0ij q i q j + 1 3! I qqq 0ijk q i q j q k + 1 4! I qqqq 0ijkl q i q j q k q l with I pp 0ij = -δ ij .
The above expression yields for the transverse tree level correlators

I πq 0ij = I pq 0ij , I ππ 0ij = I pp 0ij and I πqq 0ijk = δ δπ i δ δq j δ δq k I 0 π,Ω,q=0 = I pqq 0ijk , (123)
as well as analogously for the longitudinal expressions

I Ωq 0ij = I qp 0jl X li , I ΩΩ 0ij = X li I pp 0lk X kj = -X li δ lk X kj = -∇ 2 j δ ji (124)
and

I Ωqq 0ijk = δ δΩ i δ δq j δ δq k I 0 π,Ω,q=0 = -I pqq 0ijk ∇ i δ ii . ( 125
)
The Gaussian integration over π yields

S[q, Ω, J π , J Ω , J q ] = S 0 + 1 2 J π i J π i + J π i δS 0 δ qi + I qΩ 0ji Ω i q j + 1 2 Ω j I ΩΩ oij Ω i + 1 2 I Ωqq 0ijk Ω i q j q k + J Ω i Ω i + J q i q i . ( 126
)
where S 0 is given by Eq. (62) and J i are the corresponding sources associated to the momentum fields. Similarly, the subsequent Gaussian integration over Ω gives

S[q, J π , J Ω , J q ] = S 0 + 1 2 J π i J π i + J π i δS 0 δ qi + J q i q i - 1 2 J Ω i se I ΩΩ 0 -1 ij J Ω i -J Ω i I ΩΩ 0 -1 ij I qΩ 0uj q u + 1 2 I Ωqq 0juk q u q k - 1 2 I qΩ 0ji I ΩΩ 0 -1 il I qΩ 0ml q j q m - 1 2 I qΩ 0ji I ΩΩ 0 -1 il I Ωqq 0lmu q j q m q u - 1 8 I Ωqq 0imu I ΩΩ 0 -1 ij I Ωqq 0jkl q m q u q l q k . ( 127
)
By collecting the terms of the same order in q we obtain the bare vertices in the second order formalism in terms of those that arise in the first order formalism. Thereby, the master equation for the momentum propagator Eq. ( 42) becomes in the case of the Ω-field

∆ ΩΩ ij = I ΩΩ 0 -1 il δ lm + i I qΩ 0ul q u + 1 2 I Ωqq 0lku q k q u I qΩ 0sm q s + 1 2 I Ωqq 0msd q s q d I ΩΩ 0 -1 mj
whereas the corresponding mixed version Eq. ( 44) is given by 6: Decomposition of the proper 2-point functions of Coulomb gauge QCD in the first order formalism in terms of the corresponding correlation functions of the second order representation. The spatial (A) and temporal (σ) gauge fields are represented by solid respectively dotted lines whereas the corresponding transverse (π) andse longitudinal (Ω) momenta by zigzag respectively wavy lines. The equations for the longitudinal momentum propagator are identical to the those for the transverse component and given by replacing zigzag by wavy lines. Here the q fields in Eqs. ( 128) and (128) again have to be replaced by

∆ qΩ ij = -∆ qq il δ δq l I qΩ 0um q u + 1 2 I Ωqq mku q k q u I ΩΩ 0 -1 mj . (128) = -i , = -i = -i - -i -i - - - - - + - FIG.
W H Aj πj σ Ω λ τ Ai T -Λ AA (k 2 0 -k 2 ) T -ik 0 Λ Aπ (k 2 0 -k 2 ) 0 0 -ik i k 2 0 πi T ik 0 Λ Aπ (k 2 0 -k 2 ) T -k 2 Λππ (k 2 0 -k 2 ) 0 0 0 -ik i k 2 σ 0 0 -Λσσ k 2 Λ σΩ k 2 -ik 0 Λ σλ k 2 0 Ω 0 0 Λ σΩ k 2 Λ ΩΩ k 2 -ik 0 Λ λ k 2 -1 k 2 λ ik j k 2 0 ik 0 Λ σλ k 2 ik 0 Λ Ωλ k 2 0 0 τ 0 ik j k 2 0 -1 k 2 0 0
q → q[J q ] + i ∆ qq [J q ] δ δ q . ( 129
)
The corresponding equations involving the π field are identical to Eqs. ( 42) and (44). Performing the same algebraic steps as in sect. V yields the corresponding decomposition of the Hamilton propagators in Coulomb gauge QCD. The result is displayed in diagrammatic form in Fig. 6, it constitutes the main result of this section. We have also checked the structure of these expressions by an explicit projection of the general equations on the corresponding momentum components in appendix D. Next we will consider the other direction of the connection discussed in Sect. IV, the one which expresses proper Lagrange correlators in terms of Hamiltonian ones. This is interesting in the context of Coulomb gauge QCD, since in the Hamilton framework a complete proof of the renormalizability of the theory seems possible due to explicit cancellations ensured by powerful Ward identities [8]. Here again it is the particular definition of the longitudinal momentum that complicates the issue and does not allow to give these equations explicitly. Nevertheless, one can easily convince oneself that these equations are given entirely by tree graphs that introduce no additional divergences. Therefore, once the renormalizability of the theory in the first order formalism is established, the corresponding connection immediately implies renormalizability of the theory in the Lagrange framework. Yet the cancellation mechanism of arising divergences might be far from obvious in the latter framework and could nevertheless prevent simple truncation schemes.

B. Connection between the renormalizability in Hamilton & Lagrange Coulomb gauge QCD

In the previous subsection we have derived general connections that give Coulomb gauge Greens functions in the first order formalism in terms of those in the second order formalism and vice versa. These general connections will allow to show the following statement:

Coulomb gauge is renormalizable in both formalisms if it is renormalizable in either one of them.

The renormalizability in the second order formalism is trivial if the first order formalism proves renormalizable since there are no loop graphs in the connection, as has already been noted.

In order to prove the other direction we assume that the Green's functions of the second order theory have been properly renormalized. As the following statements hold for both π a and Ω a we adopt the notation where p stands for both of these fields. We will show that the explicit expressions for the first order Greens functions given by the connections Fig. 6 and the respective ones for higher order Greens functions are likewise renormalizable. As a first step we show that there are no energy divergences in these connections. Energy divergences arise from loop graphs whose integrands are energy independent. In the case of the propagators given in Fig. 6, all loop graphs involve dressed spatial gluon propagators that according to table II are energy dependent. Without cancellations of the corresponding propagators these integrals therefore do not feature energy divergences. Actually this holds not only for the propagators but for the corresponding connection of arbitrary correlation functions in the first order formalism. To see this it is useful to consider the general structure of the loop graphs arising in these connection equations. The loop structure of these equations is determined by Eq. (39) which describes arbitrary pure momentum correlation functions. Mixed correlation functions are obtained from this equations by further derivatives with respect to the fields which correspond to attaching additional external legs to the graphs but do not alter the loop structure. The terms involving the momentum sources do not involve any summation and only yield the tree level correlators on the classical action. In Coulomb gauge Yang-Mills theory, the arising derivative of the bare action takes the form given in Eq. (62), i.e. it involves bare qpand qqp-vertices where the external momentum leg is attached. The subsequent replacement of Eq. (129) in Eq. (39) produces a series of graphs. This latter replacement term in particular involves a summation (over the index b in Eq. ( 39)) that can lead to loop integrals.

As discussed in detail in (author?) [35] the field derivatives can either act on fields, propagators, or dressed vertices that were created before. Acting on a field simply attaches the associated propagator to the corresponding leg of the respective bare vertex. Acting on a dressed propagator splits it in two, inserts a dressed 3-point vertex and attaches the associated propagator to it. Finally acting on a dressed vertex attaches an additional leg and connects the associated propagator to it. Correspondingly, these individual rules effect merely that the dressed propagators in Eq. (39) are connected in all possible ways to the bare vertices or to each other via dressed vertices. All disconnected contributions vanish analogous to Eq. (84). With this graphical picture in mind it is easy to establish a few general properties of arbitrary connection graphs. Each external momentum leg is either attached indirectly via a tree level pq-mixing term and a field propagator to the residual graph or directly to a bare pσA-vertex. In case the momentum leg is indirectly attached to a loop graph this is done via a dressed field-vertex. Loop graphs arise if a propagator is connected to the bare vertex it started from or if a chain of connected propagators returns to its starting point. Therefore, the number of loops in a graph is at most the number of its external momentum legs. Moreover, each loop graph includes at least one bare pσA-vertex and correspondingly each loop graph also contains at least one spatial gluon propagator which according to table II is manifestly energy dependent. Now let use exclude the possibility of cancellations. Since energy divergences arise from the unsuppressed integration over all modes they arise from the UV part of the loop integral. In the UV limit, however, the dressed proper correlators arising in the loop graphs take due to asymptotic freedom up to possible logarithmic corrections their bare form. Therefore, the perturbative pole of the transverse gluon propagator had to be cancelled by a corresponding factor in a vertex to remove the energy dependence of the propagator and render the loop energy divergent. Yet, contributions from the UV regime of graphs involving other than primitively divergent vertices vanish, whereas the bare form of the primitively divergent vertices, given in Eqs. ( 116) -(118), does not involve a corresponding factor that could cancel the bare propagator pole. Correspondingly it is clear that there cannot be cancellations that would remove the energy dependence of the integrand in the UV regime and so the loop integrals do not involve any explicit energy divergences. All loop integrals are thereby -aside from usual UV divergences that have to be renormalizedfinite and well defined.

On dimensional grounds only propagator and 3-point functions involving momentum fields can feature ordinary UV divergences. These primitively divergent correlators are the qpand pp-propagator in Fig. 6 and the qqp-vertices. Note first that they do not feature power-law divergences. This is trivial for graphs where the external momentum fields are all directly attached to the loop via a bare vertex, since the corresponding bare vertices are momentum independent. According to the above rules there is also the possibility that exactly one external momentum field is attached indirectly via a dressed 3-gluon vertex to a loop graph in the connection of one of the primitively divergent correlation functions. If the corresponding external momentum is transverse the 3-gluon vertex is proportional to the energy and the linear divergent term vanishes in the symmetric integration. Similarly if the external momentum is longitudinal the corresponding 3-gluon vertex is proportional to the momentum and the transversality of the gluon propagator cancels this contribution. Correspondingly there is no linear divergence as should be the case for a gauge theory. The loop integrals appearing in the expressions for the primitively divergent momentum propagators can surely feature ordinary logarithmic UV divergences. Moreover, there could in principle be overlapping logarithmic divergences in these up to 3-loop graphs. Yet, the important point is that there are dedicated additional counterterms in the first order Lagrangian that allow to cancel any divergences explicitly by defining renormalized quantities. Finally there could in principle be overlapping divergences in subgraphs of higher order n-point functions, but due to the absence of energy divergences and the explicit renormalizability of all primitively divergent correlation functions we do not expect this to be a real issue. Clearly, a rigorous analysis of this problem requests to study the associated Slavnov-Taylor identities (derived in the last subsection). We finally note that since all connected Greens functions are finite after the renormalization process the corresponding proper Greens functions related via matrix inversion are likewise finite.

C. First order formalism and BRS-invariance

Next we consider the functional symmetry identity. The canonical action I 0 is BRST-invariant, i.e. invariant under

δA a = 1 g D ac c c δλ, δσ a = - 1 g D 0ac c c δλ, δc a = 1 g λ a δλ, δc a = - 1 2 f abc c b c c δλ, ( 130
)
δp a = f abc c b δλ [(1 -α)p c -αE c ] , δλ a = 0.
Here δλ is a Grassmannian infinitesimal parameter, whereas D 0ac = δ ac ∂ 0 + gf abc σ b . On the other hand E a = -∇σ a -D 0ac A c is the chromoelectric field. Note in addition that α is some color-singlet constant which in general could be some function of position. By considering the above transformation the Slavnov-Taylor identity reads

0 = D[φ] d 4 x - 1 g ρ a D 0ab c b + 1 g j a • D ab c b - 1 g λ a η a - 1 2 f abc ηa c b c c + f abc c b [(1 -α)p c + αE c ] • J p × exp {iI 0 + iI s } , (131)
where

I s = d 4 x ρ a σ a + j a • A a + ca η a + ηa c a + p a • J a p . ( 132
)
Employing a procedure analogous to the one used in Sect. III D we obtain

0 = D[ φ] d 4 x - 1 g ρ a D 0ab c b + 1 g j a • D ab c b - 1 g λ a η a - 1 2 f abc ηa c b c c + f abc c b (1 -α)J c p -E c • J p × exp {iS 0 + iS s } , (133)
where D[ φ] denotes the remaining integration measure of the fields c, c, A and σ. Expressed in the field strength tensor

F a µν = ∂ µ A a ν -∂ ν A a µ + gf abc A b µ A c ν one has S 0 = d 4 x - 1 4 F aµν F a µν -λ a ∇ • A a -ca ∇ • D ab c b (134) S s = d 4 x ρ a σ a + j a • A a + ca η a + ηa c a -J a p • E a + 1 2 J a p • J a p . ( 135
)
In addition we decompose the source J p a = J π a -∇ -∇ 2 J Ω a into transversal and longitudinal components. Employing this decomposition the Slavnov-Taylor identity can be written as 0

= D[ φ] d 4 x - 1 g ρ a D 0ab c b + 1 g j a • D ab c b - 1 g λ a η a (136) - 1 2 f abc ηa c b c c + f abc c b (1 -α) J π c • J π a + ∇ -∇ 2 J Ω c ∇ -∇ 2 J Ω a -f abc c b E c T • J π a -E c L ∇ -∇ 2 J Ω a exp {iS 0 + iS s } ,
where

E ci T = T ij E ic and E ci L = ∂ i ∂ j ∇ 2 E jc . ( 137
)
This completes the application of the developed formalism to Coulomb gauge Yang-Mills theory.

IX. SUMMARY AND OUTLOOK

We conclude with our main result that given a quantum field theory in the context of the first order formalism it is possible to decompose all Green's functions in terms of those obtained from the second order formalism and vice versa. Whereas proper Lagrange correlation functions are given explicitly to all orders and involve only tree graphs involving dressed Hamilton correlation functions, the decomposition of Hamilton n-point functions involves loop graphs of loop order n. Although the structure of the latter equations seems to be somewhat cumbersome, they are still more compact and simple than the usual DSEs within the Hamilton formalism. We have discussed the connection between the Hamilton and the Lagrange formalism for a general quantum field theory and illustrated the detailed structure of the arising relations in the important case of a generic four-dimensional renormalizable field theory.

In accordance with the obtained equations we have argued that in theories where the quantum average of the momentum fields is completely determined as p = q the proper p-propagators receives quantum corrections only via wave-function renormalization. Additionally, it has been shown that the canonical momentum fields which can be defined from the Effective Action and those given by the quantum average of p are in general different.

As a demonstration of the general nature of the presented formalism we also showed that the results obtained in this paper can also be applied to the case of theories involving auxiliary fields from a linearization of the interaction part of the action. This yields general relations between correlation functions in the fundamental theory and the linearized form involving auxiliary fields. A major difference is that in this case the fundamental and the auxiliary fields do not mix which simplifies the connection considerably.

Clearly, the determination of Green's functions of Yang-Mills theories in the first order formalism is considerably more complicated compared to scalar or Abelian gauge theories. The presence of a coupling between the gauge fields and their time derivative as well as the subtleties of gauge fixing present major complications. In particular the renormalizability still poses a major challenge. The presented general connection of arbitrary Green functions in the two formulations allows to relate the strucure of arising divergences in the different formulations. In particular, due to this relation the following important statement has been derived: Coulomb gauge is renormalizable in both formalisms if it is renormalizable in either one of them. This is useful since a proof of the renormalizability of the theory seems more feasible in the first order formalism where energy divergences explicitly cancel. The obtained connections should then help to understand the more intricate cancellation mechanism and thereby how to explicitly renormalize the theory in the second order formalism, where actual analytic or numerical calculations are considerably simpler. A detailed analysis of the renormalizability in Coulomb gauge Yang-Mills theory will be presented in a forthcoming publication. Here the connected cluster with n + 1 legs in the first class of graphs is labeled by the additional index R (for reduced) since one of the legs is connected to the p-propagator instead to a composite external leg. The derivative of this propagator is explicitly present via the second class of graphs. When the new proper vertex is absorbed in the connected cluster, according to the induction assumption (m i + 1 < n) these two terms together precisely yield the full p-connected vertex with m + 1 external legs and without the index R. Since each connected cluster had before already at least two external legs, together with the new one there are now at least three. Terms where besides the new external leg there is only one other external leg at a connected cluster are explicitly given by the fourth class of graphs involving a new "connected cluster" that consists only of the proper 3-point vertex. Finally, the general expression at order n + 1 contains also graphs where the new external leg is attached to the proper vertex itself, given by the third class of graphs. Altogether, the sum of the different classes of graphs precisely yields all necessary graphs at order n + 1, which completes the proof.

APPENDIX C: FIRST ORDER DYSON-SCHWINGER EQUATIONS

In this appendix we derive the DSEs in the first order formalism directly. In order to do this we compute Eq. ( 14) with the canonical action given in Eq. ( 53) δΓ H δ pi = δI 0 δp i φ→ φ+ i ∆ φφ δ δ φ = -p i + I qp 0li q l + 1 2 I qqp 0lki q l q k φ→ φ+ i ∆ φφ δ δ φ .

(C1)

We can identify the sum of the last two terms inside the bracket as the quantum canonical momentum fields, which allows to write the above relation as Eq. (30). The substitution of Eq. (C1) in Eq. ( 34), allows to write the latter as

δΓ H δ qi = δS 0 δq i + p m - δS 0 δ qm δ 2 S 0 δq i δ qm φ→ φ+ i ∆ φφ δ δ φ (C2)
where δS 0 δq i = S qq 0ij q j + 1 2 S qqq 0ijk q j q k + 1 3! S qqqq 0ijkl q j q k q l + I qp 0jl I pq 0li q j + I qq 0ij q j + I qqp 0ijm I pq 0mk q j q k + 1 2 I qqp 0jkm I pq 0mi q j q k + 1 2 I qqq 0ijk q j q k + 1 3! I qqqq 0ijkl q j q k q l + 1 2 I qqp 0ijm I ppqq 0mkl q j q k q l (C3) and p m -δS 0 δ qm δ 2 S 0 δq i δ qm = I qp 0im p m + I qqp 0ijm p m q j -I qp 0jl I pq 0li q j -I qqp 0ijm I pq 0mk q j q k -1 2 I qqp 0jkm I pq 0mi q j q k -1 2 I qqp 0ijm × I ppq 0mkl q j q k q l . (C4)

In the last two equations we have used the elementary decomposition of the bare elements and Eq. (62). Plugging Eqs. (C3) and (C4) into Eq. (C2) we arrive at the DSE (100).

APPENDIX D: EXPLICIT PROJECTION ON THE INDIVIDUAL MOMENTUM COMPONENTS

Defining the transverse projector T ij = δ ij -∂ i ∂ j /∇ 2 one has π a i = T ij p a j and Ω a = ∇ -∇ 2 • p a . As no time derivative appears one can obtain ∆ Aπ ij , ∆ ππ ij , ∆ σΩ ij , and ∆ ΩΩ ij by projecting p in the two-point functions ∆ Ap ij , ∆ σp ij and ∆ pp ij . For instance to obtain ∆ Aπ ij we decode the result given in Eq. ( 76). Let us first identify the fundamental field q with index i with A a i . We then expand the remaining sums over the fields involved in the q's considering all possible combinations which generate the bare vertex I Obviously the case corresponding to ∆ σΩ ij is obtained by replacing in Eq. (D1) A → σ where appropriate and considering the relation

∆ σΩ ij = ∂ j ∇ 2 ∆ σp ij = ∂ j ∇ 2 ∆ σσ il I σp 0lj -iI σAp 0ukj ∆ σσ um Γ σσA mln ∆ AA nk . ( D3
)
Note that the sum over r in Eq. (D2) and over j in Eq. (D3) must be understood over the discrete spatial indices only.

Clearly, by demanding a similar procedure we obtain that Note that the sum over all field components q i in the internal progators leads to multiple possibilities. The Fourier transformation of these expressions read therefore The bare Greens functions involving the full momentum field can now be expressed by the corresponding expression for the individual components via the explicit expressions given in section VIII which results in the equations given in diagrammatic form in Fig. 6.

∆ ππ ij = T ii T jj ∆ pp i j and ∆ ΩΩ ij = ∂ i ∇ 2 ∂ j ∇ 2 ∆ pp i j (D4) with ∆ pp ij = δ ij -

  propagators are denoted by thick, whereas bare propagators and external lines by thin lines, respectively. Off-diagonal propagator components are represented by a thick, half solid and half zigzag line. dressed bare, external To keep the representation of the DSEs in the first order formalism concise we also include the matrix propagator Eq. (10) represented by a double line. D III) All proper correlation functions (including proper 2-point functions) in the first and second order formalism are denoted by small and large filled blobs, proper, Lagrange proper, Hamilton whereas bare vertex functions are represented by open blobs. bare, Lagrange bare, Hamilton

FIG. 1 :FIG. 2 :

 12 FIG.1:The replacement rules that create the general decomposition of a proper correlation function in the Lagrange formalism in terms of Hamilton correlation functions.

  bosonic symmetry of the latter allows to write∆ Ap ij = ∆ AA il I Ap 0lj -iI σAp 0ukj ∆ σσ um Γ σAA mln ∆ AA nk . (D1)Here no internal propagator like ∆ Aλ ij appears since no proper vertex functions with λ-derivative exist. The relation between this function and ∆ Aπ ij is given by∆ Aπ ij = T jr ∆ Ap ir = T rj ∆ AA il I Ap 0lr -iI σAp 0ukr ∆ σσ um Γ σAA mln ∆ AA nk .

k i k j k 4

 4 ik 0 T lj (k)δ rb -iT jv (k) dωI cdb(σAp) 0uv (k -ω, ω, -k)∆ de(AA) um (ω -k)Γ erh(AAσ) ml (ω -k, k, -ω) × ∆ hc(σσ) (ω) , (D6) ∆ ab(σΩ) (k) = ∆ al(σσ) (k) -δ lbk j k 2 dωI cdb(σAp) 0uj (k -ω, ω, -k)∆ de(AA) um (ω -k)Γ elh(Aσσ) m (ω -k, k, -ω) × ∆ hc(σσ) (ω) , -T ii (k)T ij (k) i dωI acd(pAσ) 0i k (k, ω -k, -ω) × ∆ cf (AA) km (ω -k)∆ de(σσ) (ω)I ef b(σAp) 0mj (ω, k -ω, -k) + dω dµI acd(pAσ) 0i k (k, µ -k, -µ)∆ cn(AA) ku (µ -k) × ∆ df (σσ) (-µ)∆ ey(σσ) (-ω)Γ yf nl(σσAA) uz (ω, µ, k -µ, -ω -k)∆ lf (AA) zm (-ω -k)I ef b(σAp) 0mj (-ω, ω + k, -k) ω -k, -ω)∆ cl(AA) ku (ω -k)Γ vlh(σAq) up (-k -µ, k -ω, ω + µ)∆ dx(σσ) (-ω) × ∆ ev(σσ) (µ + k)∆ hs(qq) py (ω + µ)Γ sxt(qσA) yz (-ω -µ, ω, µ)∆ tf (AA) zm (µ)I ef b(σAp) 0mj (µ + k, -µ, -k) + dω dµI acd(pAσ) 0i k (k, ω -k, -ω)∆ cl(AA) ku (ω -k)Γ vlh(AAq) zup (-k -µ, k -ω, ω + µ)∆ ev(AA) zm (µ + k) × ∆ dx(σσ) (-ω)∆ hs(qq) py (ω + µ)Γ sxt(qσσ) y (-ω -µ, ω, µ)∆ tf (σσ) (µ)I ef b(σAp) 0mj (µ + k, -µ, -k), i dωI acd(pAσ) 0ik (k, ω -k, -ω) × ∆ cf (AA) km (ω -k)∆ de(σσ) (ω)I ef b(σAp) 0mj (ω, k -ω, -k) + dω dµI acd(pAσ) 0ik (k, µ -k, -µ)∆ cn(AA) ku (µ -k) × ∆ df (σσ) (-µ)∆ ey(σσ) (-ω)Γ yf nl(σσAA) uz (ω, µ, k -µ, -ω -k)∆ lf (AA) zm (-ω -k)I ef b(σAp) 0mj (-ω, ω + k, -k) + dω dµI acd(pAσ) 0ik (k, ω -k, -ω)∆ cl(AA) ku (ω -k)Γ vlh(σAq) up (-k -µ, k -ω, ω + µ)∆ dx(σσ) (-ω) × ∆ ev(σσ) (µ + k)∆ hs(qq) py (ω + µ)Γ sxt(qσA) yz (-ω -µ, ω, µ)∆ tf (AA) ω -k, -ω)∆ cl(AA) ku (ω -k)Γ vlh(AAq) zup (-k -µ, k -ω, ω + µ)∆ ev(AA) zm (µ + k) × ∆ dx(σσ) (-ω)∆ hs(qq) py (ω + µ)Γ sxt(qσσ) y (-ω -µ, ω, µ)∆ tf (σσ) (µ)I ef b(σAp) 0mj (µ + k, -µ, -k) . (D9)

TABLE I :

 I Tree level proper two point functions (without color factors) in momentum space.

TABLE II :

 II General form of propagators in momentum space. The global color factor δ ab has been extracted. All unknown functions Λ φφ are dimensionless, scalar functions of k 2 0 and k 2 . Here Tij = Iij -kikj /k 2 is the transverse projector in momentum space.

Acknowledgments

It is our pleasure to thank Peter Watson for helpful discussions. S. V.-C. is supported by the Doktoratskolleg "Hadrons in Vacuum, Nuclei and Stars" of the Austrian science fund (FWF) under contract W1203-N08. K. S. acknowledges support from the FWF under contract M979-N16, and R. A. from the German research foundation (DFG) under contract AL 279/5-2.

APPENDIX A: p-INTEGRATION OF Z[J].

In this appendix we perform the integration over the momentum fields in the vacuum-vacuum transition amplitude in presence of the classical sources. In order to do this, we assume a general quadratic Hamiltonian density given by Eq. (23). In order to evaluate the Gaussian functional integral over the momentum fields we first consider a finite dimensional generating functional

with the momentum field dependent part of the action

By completing the squares in the above expression we arrive at

where

By considering the change of the integration variables from p i to π i we obtain

and

Here, the action S 0 can be obtained by considering the limit of S[q, J] when J → 0. Explicitly it reads

where

The canonical momentum fields defined as

allow us to identify and substitute the last term in S[q, J] by its respective definition, and to eventually obtain the desired form of Eq. ( 25).

APPENDIX B: PROOF OF THE GENERAL FORM OF THE DECOMPOSITION OF PROPER LAGRANGE CORRELATION FUNCTIONS

In this appendix we prove the statement made in subsection IV C that the replacement rules precisely generate the p-connected correlators. As usual for a statement over the integers this is done by induction. The statement is trivially fulfilled in the case n = 3 given explicitly before. Now let us assume it is fulfilled for all integers ≤ n and show that this implies its validity for n + 1. In the p-connected diagrams with n legs we single out the proper vertex we started the iteration with. This vertex can be connected to other connected clusters so that all arising graphs have the general form

Here we suppress all external legs and only give their number next to the corresponding vertex. The sum over s counts the connected clusters attached to the considered proper vertex which in addition has n -m external legs and the implicitly given sum labeled D({m i }) runs over all ways to distribute the remaining m external legs to the s indistinguishable connected vertices (taking into account that a vertex has at least 3 legs).

Next we consider the attachment of an additional external leg via the rules given in Fig. 1, applied in all possible ways. The above representation then goes over to a n + 1-point function of the following form