Dual representation for the generating functional of the Feynman path-integral

The generating functional for scalar theories admits a representation which is dual with respect to the one introduced by Schwinger, interchanging the role of the free and interacting terms. It maps $\int V(\delta_J)$ and $J\Delta J$ to $\delta_{\phi_c}\Delta\delta_{\phi_c}$ and $\int V(\phi_c)$, respectively, with $\phi_c=\int J\Delta$ and $\Delta$ the Feynman propagator. Comparing the Schwinger representation with its dual version one gets a little known relation that we prove to be a particular case of a more general operatorial relation. We then derive a new representation of the generating functional $T[\phi_c]=W[J]$ expressed in terms of covariant derivatives acting on 1 $$ T[\phi_c] = {N\over N_0} \exp(-U_0[\phi_c])\exp\Big(-\int V({\cal D}_{\phi_c}^-)\Big) \cdot 1 $$ where ${\cal D}_{\phi}^{\pm}(x)=\mp\Delta{\delta\over\delta\phi}(x)+\phi(x)$. The dual representation, which is deeply related to the Hermite polynomials, is the key to express the generating functional associated to a sum of potentials in terms of factorized generating functionals. This is applied to renormalization, leading to a factorization of the counterterms of the interaction. We investigate the structure of the functional generator for normal ordered potentials and derive an infinite set of relations in the case of the potential ${\lambda\over n!}:\phi^n:$. Such relations are explicitly derived by using the Fa\`a di Bruno formula. This also yields the explicit expression of the generating functional of connected Green's functions.

where D ± φ (x) = ∓ δ δφ (x) + φ(x). The dual representation, which is deeply related to the Hermite polynomials, is the key to express the generating functional associated to a sum of potentials in terms of factorized generating functionals. This is applied to renormalization, leading to a factorization of the counterterms of the interaction. We investigate the structure of the functional generator for normal ordered potentials and derive an infinite set of relations in the case of the potential λ n! : φ n : . Such relations are explicitly derived by using the Faà di Bruno formula. This also yields the explicit expression of the generating functional of connected Green's functions.

Introduction and summary
The path-integral is a basic tool in several research fields, quantum mechanics, quantum field theory, statistical mechanics etc. The original idea is due to Dirac who observed a deep analogy between the Hamilton-Jacobi theory and the quantum transition amplitudes, proposing the relation [1] q, t|Q, T ∼ e − ī h t T dtL . (1.1) Subsequently, Feynman developed Dirac's idea, starting from an infinitesimal version of (1.1), another key step toward the path-integral formulation [2]. A relevant progress in the path-integral approach is due to Schwinger who expressed the generating functional in the form where V is the potential, J the external source and exp(−Z 0 [J ]) the generating functional of the free theory. It turns out that the Schwinger representation admits the dual representation As we will see, this leads to represent the path-integral operator as an operator acting by functional derivatives. Namely, for any functional F [φ], it holds This is a consequence of the general relation derived in Sec. 2. We then will derive a new representation of the generating functional that simplifies considerably the computations. Namely, in Sec. 6, we will see that T [φ c ] can be expressed in terms of covariant derivatives acting on 1, that is where (1.8) We will derive the little-known representation (1.3), reported in Fried's book [3], in two different ways. In Sec. 2 Eq. (1.3) is derived using the path-integral and the operator formalism. Then, in Sec. 4, we will derive a more general operatorial relation, Eq. (4.3), that, in turn, is the functional analog of an operatorial relation satisfied by the Hermite polynomials. In particular, Eq. (4.3) implies the operatorial identity (1.9) that, applied to a constant, reproduces the relation implied by the identification of (1.2) with (1.3). We note that the operator exp( 1 2 δ J −1 δ J ) appears also in the context of renormalization, see e.g. [4,5] and references therein.
As we will be clear from the investigation, the dual representation is deeply related to the Hermite polynomials which appear in several contexts. For example, in Sec. 2, it is shown that the dual representation is the functional generalization of the Weierstrass representation of the Hermite polynomials. Furthermore, in subsec. 2.4, it is shown how the Hermite polynomials arise in the Schwinger-Dyson equation expressed in terms of the dual representation of the generating functional. Another result concerns the following expression of the Schwinger-Dyson equation for the normal ordered potential : V : (1.10) Even such an equation, anticipated in subsec. 2.4, follows by the operatorial relation (4.3). In Sec. 2 we will also show that the dual representation is naturally related to the S-operator. In particular, it turns out that such an operator is proportional to the normal ordered version of the dual generating functional with φ c replaced by the operator φ , that is The functional structure of T [φ c ] suggests considering the following problem investigated in Sec. 3. Given a potential corresponding to a summation of potentials, , and the normalization constants absorbed in T [φ c ] and T k [φ c ]. Such an investigation, that can be extended to the case of fermions and gauge fields, has several applications. Here we consider the case in which the potential is given by the summation of the original one and the one coming from the counterterms. The outcome is the following relation (see Sec. 3 for the notation) We also show that imposing the associativity condition in the case n = 3 one gets the relation where T jk denotes the generating functional associated to the potential V j + V k . Sec. 4 is devoted to the derivation of the operatorial relation from which follows, as a particular case, Eq. (1.9), which, in turn, implies the identification of (1.2) with (1.3). We then derive, by a different method, the analogous relation in the case of the Hermite polynomials.
In Sec. 5 we consider the functional generator in the case of normal ordered potentials. There is a considerable simplification of the general expression since the normal ordering precisely cancels the action of the operator exp( 1 2 δ φ c δ φ c ) acting on each single potential coming from the expansion. In particular, the functional generator of the connected Green's functions Comparing the expression Eq. (1.17) with the explicit expression of the generating functional for the potentials λ n! : φ n : , obtained in [6], leads to an identity that implies an infinite set of relations. Such relations are explicitly derived by using the Faà di Bruno formula giving the chain rules for higher order derivatives. In particular, we will derive the following explicit expression for the action of the operators exp D jk where the h n,k are multinomials of integrated powers of φ c . By (1.17) this also yields the explicit expression of U [φ c ]. In subsec. 5.4 we show a duality between the field φ and the source φ c . This suggests promoting φ c to a dynamical field, leading to a potential V (φ + φ c ). It turns out that the model is equivalent to the one of a single field φ c with U [φ c ] playing the role of potential. Iterating the construction leads to a rescaling of the kinetic term. This seems a possible alternative with respect to the renormalization procedure where the rescaling of the kinetic term is obtained by rescaling the field itself. In Sec. 6 we derive the new representation (1.7) of the generating functional. We also show how such a representation simplifies the calculations. In particular, the action of the normal ordering operator on a product of functionals simplifies considerable, namely (1.20)

Dual representation of the generating functional
For any even function or distribution h, and any functions or operators f and g, set and .
We will also use the notation and .
Consider the Feynman propagator and its inverse Denote by Z 0 [J ] the functional generator of the free connected Green's function, that is We will focus our investigation on a scalar field in the D-dimensional Euclidean space. The corresponding generating functional is where J is the external source, is the scalar action with potential V (φ), and N = 1/ Dφ exp(−S) the normalization constant. We denote by S 0 [φ] the free action.
To fix the normalization constant in (1.2), note that The correspondence between the Schwinger representation and the operator formalism follows by where | is the vacuum of the interacting theory and | J denotes the vacuum-vacuum amplitude in the presence of the external source J . In particular, denoting by |0 the free vacuum, normalized by 0|0 = 1, we have 0|0 J = 0|T exp( Jφ)|0 . Therefore, Eq. (1.3) has been observed in the case of the exponential interaction, considered as master potential, in [6]. We will prove (1.3) using the Euclidean time-ordering, defined by the analytic continuation from the one in Minkowski space. Excellent references for the analytic continuation and related issues in the axiomatic approach include [7][8][9][10].
Consider the field satisfying the classical equation of motion with V = 0 and external source J (2.14) To prove (1.3), we first consider the shift (2.15) in the generating functional of the interacting theory (2.12), rather than, as usual, on the free one. Dropping the prime in φ , yields which is equivalent to This implies that the path-integral operator is equivalent to the action of a functional derivative. This is Eq. (1.5), namely Applying (2.20) to (2.12), yields Taking the multiple derivative of W [J ] with respect to J or, equivalently, choosing , then exp(− 1 2 δφ δφ) is the normal ordering operator. When the argument of F is not an operator, then T F [φ] = F [φ], and the map . Nevertheless, in the following we will call exp(− 1 2 δ φ δ φ ) normal ordering operator even in this case, and will also interchange the notation on the right hand side of (2.25) with : F [φ] : .
The structure of Eq. (1.3) suggests considering the generating functional In the following we will set (2.29) As shown in Sec. 4, such an identity also follows by an operatorial identity once it acts on a constant. Furthermore, by (2.17) and (2.27), we have We will see that this is the functional generalization of the Weierstrass transform.
Another property of T [φ c ] concerns its relation with the effective action Let us start by showing that the S-operator has a simple expression in terms of T [φ]. To this end note that while the functional derivatives of T [φ c ] with respect to J yield the Green functions, deriving with respect to φ c gives Comparing this with the expansion of the S[φ] operator On the other hand, using exp(φδ χ ) that acts by translating χ by φ , we have Replacing T [χ] by the right hand side of (2.27), we get Using the path-integral representation of T [φ], it follows by (2.38) that such an expression is equivalent to Then, the representation (2.21) of the path-integral yields

Relation with the Weierstrass transform and the Hermite polynomials
The Weierstrass transform can be seen as a particular case of (2.30). This arises by first considering the Laplace transform of the Gaussian and then using is replaced by f (x + y), one recognizes it as a particular case of (2.30). The relation (2.29) can be seen as an extension of the relation between the Hermite polynomials and their Weierstrass representation one gets Note that this also provides the following suggestive "perturbative expansion" In this respect, we recall that the Hermite polynomials can be written explicitly Eq. (2.50) can be in fact used in quantum field theory. An obvious reason is that the dual expression of the generating functional involves exp( 1 2 δ φ c δ φ c ) acting on a functional of φ c . This means that in a perturbative expansion there appear terms such as On the other hand, is the functional version of so that, by (2.50),

Schwinger-Dyson equation in the dual representation
Here we consider the Schwinger-Dyson equation using the dual representation of the generating functional. In particular, here we consider the case of the potential λ n n! : φ n : . This suggests an operatorial relation that will be derived in Sec. 4. As we will see, such a relation implies that for arbitrary normal ordered potentials, the Schwinger-Dyson equation reduces to (2.56) Note that here we used the symbol : V : to denote the normal ordered potential, to distinguish it from the non-normal ordered potential in the square bracket. The standard form of the Schwinger-Dyson equation As we said, later we will prove a more general formula, corresponding to the operatorial extension of (2.59).
The connection with the Hermite polynomials is a consequence of (2.55) and (2.59). To see this, note that Eq. (2.59) implies

Factorization problem
In this section we show that the dual representation is the natural one to investigate the following decomposition problem. Namely, given the summation of potentials find how the generating functional associated to V decomposes in terms of the generating functionals associated to the potentials V k 's. Such an investigation, that may be extended to the case of higher spin fields, may have several applications, for example in considering perturbations with respect to a given background. Other applications concern the analysis of possible symmetries related to a subset of the V k 's. Here we introduce the method and consider the application to the case of renormalization of scalar theories. Other applications will be investigated in a future work. Let us consider the simplest case In the Schwinger representation we have and there is no an obvious way to understand the structure of the decomposition. In the dual representation of the generating functional, we have so that, in this case, we can use a key relation satisfied by exp 1 2 δ φ c δ φ c . Namely, given the functionals F and G we have [3] exp To derive such a relation, first note the identity On the other hand, (3.5) can be expressed in the form Next observe that, since exp μ δ φ c translates a functional of φ c by μ , we have that applied to (3.7) reproduces (3.6). Observe that (3.5) implies the following recursive rule, useful in several computations, e.g. in evaluating the Green's functions (3.10) In this notation, equation (3.5) reads Denote by T k [φ c ] the generating functional associated to V k (φ). For notational reasons we use the same symbol T k [φ c ] to denote T k [φ c ] divided by the normalization factor N k /N 0 , We then have that the factorization of the generating functional (3.4) reads that is where Let us apply Eq. (3.14) to renormalization. Consider the renormalized action where V ct is the counterterm potential and ˆ is the Feynman propagator associated to the full kinetic part of the Lagrangian density It is understood that now the field φ c satisfies the equation of motion We denote by T ren [φ c ] the generating functional associated to S ren , by T [φ c ] the one associated to S ren − V ct and by T ct [φ c ] the one associated to S ren − V . By (3.14), the decomposition of the renormalized generating functional is where θ (1, 2) is given by (3.15) with replaced by ˆ . Note that Eq. (3.19) is non-perturbative, and can be iterated order by order in the loop expansion. We now show that associativity applied to Eq. (3.14) leads to a relation reminiscent of a cocycle condition. Consider the case of the sum of three potentials (3.20) and denote by T jk [φ c ] the generating functional associated to the potential V j (φ) + V k (φ), j, k = 1, 2, 3. The full generating functional T [φ c ] can be derived in the same way of (3.14). One just imposes the associativity condition by identifying (3.14), where V 1 is replaced by V 1 + V 2 and V 2 by V 3 , with the expression obtained replacing V 2 by V 2 + V 3 . This gives that, after a change of notation, reads

Dual representation and normal ordering
A feature of the dual representation is that it provides the explicit factorization of the free part e −U 0 [φ c ] . The remanent part is the inverse of the normal ordering operator acting on e − V (φ c ) . On the other hand, the Schwinger-Dyson equation in the dual representation led us to consider the relation (2.60), making clear how e U 0 [φ c ] and its inverse are related to normal ordering. In this respect, note that, how (2.60) shows, Eq. (2.59) does not hold as an operator relation. In the following, we will prove that the operatorial version of Eq. ( In this section, we first prove (4.3), that implies also (4.1), then we show that this is the functional analog of a relation satisfied by the Hermite polynomials.

Proof of (4.3)
To prove

Operatorial extension of the Weierstrass representation of the Hermite polynomials
Eq. (4.3) is the functional generalization of the identity Acting with (4.9) on a constant yields In the case f (x) = x n , Eq. (4.9) provides the operatorial extension of the Weierstrass representation of the Hermite polynomials. The proof of (4.8) can be done by expressing f in terms of its Laplace transform as done for (4.3). Nevertheless, it is of interest to stress its connection with the Hermite polynomials. We then consider the case f (x) = x n , with n a non-negative integer. We do this using the induction method. Since (4.8) holds for f (x) = x and we should prove the operatorial relation  which is the assumption.

T [φ c ] and normal ordered potentials
Here we first investigate the structure of the generating functional U [φ c ] = − ln T [φ c ] in the case of normal ordered potentials. This is done by specializing the construction in [3]. In particular, the action of the inverse of the Wick operator on each potential, contributing to the expansion, precisely cancels the normal ordering, so that leading to a considerable simplification of the full Comparing the result with the explicit expression of W [J ] in the case of the potential λ n! : φ n : , recently derived in [6], yields an identity that implies an infinite set of relations. We then derive such relations using the Faà di Bruno formula, concerning the chain rules for higher order derivatives. This provides, for all n, the Feynman combinatorics and the explicit form of the full loop expansion of U [φ c ]. We conclude the section by showing a duality between the field φ and the external source φ c . Promoting φ c to a dynamical field leads to a scalar theory with potential V (φ + φ c ), which is described by the theory of a single field φ c with the generating functional U [φ c ] playing the role of potential.

The general case
In the following we consider normal ordered potentials that, for notational reasons, we denote by : V : . Let us set We also define j, k ∈ N + . Let us rewrite (3.5) in this notation , and write Rescaling the potential by a constant μ, and then expanding exp(− ∫ : V :) in (5.2), gives by (5.5) so that There is a considerable simplification since the normal ordering cancels all the e D k s. The first two cases are Q 1 = −e D : V : = − V , (5.8) and Note that, as seen by expanding e D 12 , the effect of the −1 in e D 12 − 1 is to eliminate Feynman diagrams which are not connected by at least one propagator. As in the case of Q 1 and Q 2 , all the operators e D k , k = 1, . . . , n, disappear from the expression of the Q n s and their expression simplify to where the subscript c means that terms non-connected by at least one Feynman propagator should be discarded. It then follows by (5.5) and (5.10) that the generating functionals of connected Green's functions associated to a normal ordered potential is

A combinatorial identity for λ n! : φ n :
In [6] it has been introduced a method to derive the generating functional of scalar potentials starting from the one associated to the exponential potential μ D exp(αφ), seen as a master potential. In particular, it has been shown how this leads to derive the generating functional for normal ordered potentials, absorbing at once all the (0) terms, by using where [a] denotes the integer part of a and 0 ≤ q i ≤ kn − 2p. We have [6] with the m lj 's taking all the values satisfying the conditions 0 ≤ m lj ≤ p and k l>j =1 m lj = p. Furthermore, . . , n. By (5.5), (5.11) and (5.14), we get the non-trivial identity Comparing the coefficients of all powers of λ we will derive the explicit expressions of the action of the product of the e D jk s operators. U [φ c ] generates the connected N -point functions, without the free external legs, of φ. Therefore, In the case of n odd one has, perturbatively, φ(x) = 0. We recall that for N ≥ 2 such correlators coincide with the ones of η(

Feynman combinatorics and Faà di Bruno formula
We now derive the infinitely many relations implied by (5.17), giving the explicit form of the action of the operators e D jk . As we will see, this also implies the following explicit expression for the generating functional U [φ c ], in the case of the potential V = λ n! : φ n : , for all n

φ-φ c duality and U [φ c ] as an effective potential
We now show a duality between the field φ and the source φ c . This naturally leads to promote φ c to a dynamical field, with potential U [φ c ], that is with the action Consider the functional According to (2.17) we have with In spite of the opposite sign of the kinetic term for φ c , it is worth noticing the exchange of roles between the external source φ c and the scalar field φ in (5.36) and (5.37).
Let us consider a slightly modified version of (5.35). Namely, define Note that by (5.5), (5.34) and (5.39), we have and Consider the dual generating functional is the generating functional for two scalar fields, φ and φ c , symmetrically coupled by the potential V (φ +φ c ), with one of the two external currents set to zero. That is Let us consider, for arbitrary I and K = ϕ c −1 , the path integral over φ and φ c in (5.44). This also gives the expression of the path integration over φ c in (5.42). To this end, we set (5.46) Next, defining we get We then set (5.50) According to (5.5) we have which has the same form of the dual generating functional associated to the potential V (φ c ) except for the rescaling by a factor 2, of δ ϕ c δ ϕ c . This is equivalent to rescale the kinetic term by a factor 1/2. To see this, note that (5.54) can be directly obtained by replacing ρ and σ in (5.47), by their rescaled versioñ This leads to . Instead of varying the coefficient of the kinetic term, one may rescale ρ and ϕ c to get the standard normalization of the kinetic term, so that replacing V (ρ + ϕ c ) by V ( √ 2(ρ + ϕ c )). Iterating n-times the procedure mapping T [φ] to T D [ϕ c ], leads to a rescaling of a factor 2 n/2 of the scalar field. Such an iteration may lead to an interpretation of the rescaling of the scalar field in the renormalization procedure.

Generating functional and covariant derivatives
In this section we derive a new representation of the generating functional, expressed in terms of covariant derivatives acting on 1. The key observation is that some of the relations we derived in the previous sections, can be expressed in terms of covariant derivatives. This leads to some new results in the path-integral approach to quantum field theory.
The starting point is to note that, given a functional F , one has the operator identity where D MI (x) denotes the "covariant derivative" Eq. (6.1) is the functional generalization of the operator relation By (2.59), which is not an operator identity, and (6.1), it follows that which is the functional extension of Furthermore, by (4.1) and (6.1) we get the operator relation (6.6) By (6.1) we have where Such operators satisfy the commutation relations and where the action of the Wick operator exp(− 1 2 δ φ c δ φ c ) has been obtained by changing sign to . Since (6.13) it follows that even the Green's functions can be expressed in terms of the covariant derivatives By (2.58) and (6.12) it follows that the Schwinger-Dyson equation reduces to the identity The above representation of the generating functional simplifies the explicit calculations. For example, in the case of V = λ 4! φ 4 , one immediately gets It is interesting to consider the case of normal ordered potentials. This is the way D + φ enters in the formulation. Namely, according to (6.4) we have Note that D ± φ acts as the inverse of (6.20) Since D + φ and D − φ differ only by the sign of (x − y), we can easily get the expression of n k D + φ (x k ) from the one of n k D − φ (x k ). For example, by (6.18), : φ 4 (x) : = φ 4 (x) − 6 (0)φ 2 (x) + 3 2 (0) . (6.21) Another feature of D ± φ concerns the case of a product of functionals of φ. As shown in (3.5), the action of exp ± 1 2 δ φ δ φ on F [φ]G[φ] is rather involved. On the other hand, using D ± φ we have