QED Revisited: Proving Equivalence Between Path Integral and Stochastic Quantization

We perform the stochastic quantization of scalar QED based on a generalization of the stochastic gauge fixing scheme and its geometric interpretation. It is shown that the stochastic quantization scheme exactly agrees with the usual path integral formulation.


Introduction
The stochastic quantization scheme of Parisi and Wu [1] has been applied to QED since many years. Nice agreement with conventional calculations was found in several explicit examples (for reviews see e.g. [2,3]), a general equivalence proof so far was lacking.
The main idea of "stochastic quantization" is to view Euclidean quantum field theory as the equilibrium limit of a statistical system coupled to a thermal reservoir. This system evolves in a new additional time direction which is called stochastic time until it reaches the equilibrium limit for infinite stochastic time. In the equilibrium limit the stochastic averages become identical to ordinary Euclidean vacuum expectation values. Fokker-Planck probability distribution. The equilibrium limit of the probability distribution provides the Euclidean path integral density.
One of the most interesting aspects of this new quantization scheme lies in its rather unconventional treatment of gauge field theories, in specific of Yang-Mills theories. We recall that originally it was formulated by Parisi and Wu [1] without the introduction of gauge fixing terms and without the usual Faddeev-Popov ghost fields; later on a modified approach named stochastic gauge fixing was given by Zwanziger [4]; further generalizations and a globally valid path integral were advocated in [5,6,7,8].
The main difficulty for providing an equivalence proof in the case of QED appears to be a rather nontrivial topological obstruction; all previous attempts failed in the past years to identify the standard -gauge fixed-QED action as a Fokker-Planck equilibrium distribution.
In this paper we introduce new modifications of the original Parisi-Wu stochastic process of QED, yet keeping expectation values of gauge invariant observables unchanged: The modified stochastic process not only has damped flows along the vertical direction [4] but also is modeled on a specific manifold of gauge and matter fields with associated flat connection [5,6,7,8]. It is precisely in this case that the standard -gauge fixed -QED path integral density can indeed be identified with the weak equilibrium limit of the underlying Fokker-Planck probability distribution.
In section 2 the geometrical setting for QED is introduced and the associated bundle structure of the space of gauge potentials and matter fields is summarized. We introduce adapted coordinates, the corresponding vielbeins and metrics.
The generalized stochastic process for QED is presented in section 3, section 4 is devoted to the derivation of the conventional QED path integral density as the equilibrium solution.

The geometrical setting of QED
In this section we present the major geometrical structures of QED. We collect in a somewhat formal style all the necessary ingredients which are needed for a compact and transparent formulation of the stochastic quantization scheme of QED.

Gauge Fields
Let P → M be a principal U(1)-bundle over the n-dimensional boundaryless connected, simply connected and compact Euclidean manifold M. The photon fields are regarded as elements of the affine space A of all connections on P . The gauge group G is given by G = C ∞ (M; U(1)) with Lie algebra Lie G = C ∞ (M; u(1)); here Lie (U(1)) = u(1) = iR.

The action of G on A is defined by
Let us define the subgroup G 0 ⊂ G where G 0 = G/U(1) denotes the group of all gauge transformations reduced by the constant ones. Since G 0 acts freely on A we consider the principal G 0 bundleπ : where △ denotes the invertible Laplacian which is acting on the Lie algebra LieG 0 ;

Matter Fields
In order to discuss scalar matter fields φ we chose a representation ρ of g ∈ G 0 on the vector space V = C ; ρ(g) simply denotes multiplication with g. We consider the associated vector bundles E = P × ρ V on M. Scalar fields are described by appropriately chosen sections of E. In the following we denote by F the space of scalar fields.
The action of G 0 on Φ i := (A, φ) ∈ A × F is given by using the previous construction of ω(A) we obtain a global section σ : The tangent space of the configuration space A × F is given by and α, β ∈ T A, or T F , respectively ; * is the hodge operator on M,ᾱ denotes complex conjugation of α.

Adapted Coordinates
Let the globally defined gauge fixing surface Σ in A × F be defined by where ω is given by (2.2). Note that B and ψ are invariant under the action of G 0 which trivially follows from (2.4) and from (2.3); B satisfies the "gauge fixing condition" We define the adapted coordinates Ψ µ = {B, ψ, g} via the bundle maps χ : and The differentials T χ and T χ −1 are calculated straightforwardly (compare also with [7]) as well as Above we defined a Riemannian structure on A×F in terms of the G 0 invariant metric h. In adapted coordinates this metric is given now as the pullback G = χ * h; explicitly we obtain (2.14) In matrix notation we have G = e † e and G −1 = E E † . The determinant of G is given by det G = △.

Parisi-Wu Stochastic Quantization
For scalar QED we have where D A ϕ = (d − A)ϕ and F = dA. The Parisi-Wu Langevin equations are given by where the Wiener increments fulfill dU dU = 2ds, dV dV = 2ds. (3.4) This can be summarized by Using Ito calculus [9,10] we transform the Parisi-Wu Langevin equations into adapted where The use of adapted coordinates allows to disentangle the complicated dynamics of gauge independent and gauge dependent degrees of freedom; it will be of great value later on.
For completeness we note the Fokker-Planck operator in the original variables as well as in the adapted coordinates We remark that in the case of the Parisi-Wu processes diffusion along the vertical direction takes place and no equilibrium distribution is approached. Thus a Fokker-Planck formulation of the Parisi-Wu stochastic quantization scheme is impossible: The gauge invariance of the action S inv is leading to divergencies along the vertical directions when trying to normalize the Fokker Planck density.

Geometric Obstruction
Our equivalence proof relies on specific allowed modifications of the metric on the field space, which governs the stochastic process. These modifications correspondingly are implying changes of the associated Fokker-Planck operator. We are going to show that this can be achieved in such a way that the resulting Fokker-Planck operator has a positive kernel and is annihilated on its right by the standard gauge fixed QED path integral density. In order for this to be the case, however, a certain integrability condition for the drift term of the considered stochastic process has to be fulfilled. Surprisingly similar as in the pure Yang-Mills theory also in the abelian QED case there appears a violation of this condition; it is only after a nontrivial modification of the underlying stochastic processes (see next subsection) that this obstruction can be overcome.
Proceeding step by step we first note (see Zwanziger [4]) that a damping force along the gauge orbit has to be introduced in order to maintain the probabilistic interpretation of the Fokker-Planck formulation. Although knowing that this additional force will not alter expectation values of gauge invariant quantities it is disappointing to observe that due to its presence the standard -gauge fixed -QED action will never annihilate the Fokker-Planck operator on its right side due to the following reason: We recall that the bundle metric h (A,φ) on the associated fiber bundle A×F → A× G F which is invariant under the corresponding group action gives rise to a natural connection γ, whose horizontal subbundle H is orthogonal to the corresponding group. The horizontal where ξ ∈ C ∞ (M; iR). The orthogonal span with respect to the vertical bundle fulfills which follows from Explicitly we can prove that defines a connection induced by h (A,φ) in the principal bundle A×F → A× G F and is U(1) invariant. Calculating its curvature Ω ((τ 1 , v 1 ), (τ 2 , v 2 )) we find that it is nonvanishing and given by As a consequence [5,6,7,8] there does not exist (even locally) a manifold whose tangent bundle is isomorphic to this horizontal subbundle. Specifically this implies that any vector field along the gauge group cannot be written as a gradient with respect to the metric h (A,φ) . The total drift term -containing the extra vertical force term -thus can never arise as derivative of the standard gauge fixed QED action; the Fokker-Planck operator can never be annihilated on its right by the standard QED path integral density; an equivalence proof presently cannot be given.

The Induced Field Metric with Flat Connection
The crucial observation in [5,6,7,8] is to consider a larger class of modified stochastic processes than considered so far, yet always keeping expectation values of gauge invariant observables unchanged: One introduces not only the extra vertical drift terms as discussed above but one also modifies the Wiener increments by specific extra terms and introduces extra so called Ito-terms, correspondingly.
The idea is to view the new terms multiplying the Wiener increments as vielbeins giving rise to the inverse of a yet not specified metric on the space A×F . The appearance of this metric induces a specific connection with a potentially analogous obstruction as discussed above. A necessary requirement to overcome this obstruction is therefore that the corresponding curvature has to vanish. The question how to find such a metric is reduced to the question how to find a flat connection.
Indeed, there exists a flat connection γ in our bundle. This connection is the pull-back of the Maurer-Cartan form θ on G 0 via the global trivialization χ −1 and pr G where pr * G is the projector Σ × G → G. The projector onto the horizontal subbundle H[A × F ; γ] with respect to γ is given by We see that the horizontal subbundle H is orthogonal to the gauge orbits with respect to the induced field metric; in particular the gauge fixing surface is then orthogonal to the gauge orbits.
In the adapted coordinates the induced field metric is denoted by G = e † e. The just discussed orthogonality condition of the gauge fixing surface and the gauge orbit with respect to the induced field metric is transformed into simply This condition is fulfilled provided E is defined as To complete our discussion we also have to specify the vertical drift term; it is related to the gradient of S G , where we chose where θ U (1) is the Maurer Cartan form on U(1). Note that in the original variables we obtain the standard background-gauge gauge fixing term Summarizing we have where S tot = S inv + S G , and d ζd ζ † = 2 G −1 ds. (4.13)

The Equivalence Proof
It is easy now to prove for QED the equivalence of the stochastic quantization scheme