Functional Quantum Field Theory in Phase Space

Abstract

The formulation of Quantum Field Theory (QFT) in phase space offers a unique alternative to operator and path-integral paradigms, providing distinct conceptual advantages for semiclassical expansions. In this work, we present a systematic and self-consistent functional framework that maps stationary Schrodinger functional equations directly onto phase-space star-eigenvalue equations across different spin statistics. Operating within a non-manifestly covariant equal-time formalism, we derive explicit vacuum Wigner functionals for scalar, gauge, and fermionic fields, establishing the rigorous theoretical consistency of the formalism from first principles prior to phenomenological applications. We analyze how ordering prescriptions and continuous symmetries manifest under the functional star-product, including an explicit phase-space formulation of Noether’s theorem and field regularization. Finally, the framework is extended to interacting systems via a functional Rayleigh–Schrodinger perturbative scheme, illustrated explicitly through the non-trivial first-order Wigner functional correction $W^{\left(1\right)}$ and the vacuum energy correction for a ${\varphi }^4$ self-interacting theory, establishing a solid foundation for evaluating real-time quantum field dynamics.

1. Introduction

The laws of classical mechanics are well established since the seventeenth century. However, although Newton’s laws describe the world correctly within their range of application, they are ineffective when dealing with more complex problems. In the nineteenth century, other formulations emerged, such as those of Lagrange and Hamilton, which constitute a much more elegant description, treating positions and momenta equally, i.e., within the phase space.

Quantum physics emerged at the beginning of the 20th century seeking to describe specific phenomena. Its formulation began to be developed in the 1920s as a description of the subatomic world. In 1926, Schrödinger formulated wave mechanics, where the central concept was the wave function. This wave function represents an element of an infinite-dimensional Hilbert space, which contains the possible states of the system. Its squared modulus represents the probability of finding a particle at a given point. In 1927, Heisenberg enunciated the Principle of Uncertainty, which places a limit on the accuracy of certain observations. Specifically, it ensures that it is not possible to measure the momentum and position of a particle with absolute precision. This seems to prevent the formulation of quantum physics within phase space, since the operators associated with both observables do not commute. However, this is not the case.

The first attempt at defining quantum physics in phase space was made by Weyl in 1927 [1]. In this pioneering work, he establishes a relationship between functions defined in phase space and Weyl ordered operators. Such order is defined as completely symmetric for non-commuting position and momenta operators defined in Hilbert space. Weyl thought that it was the bridge that linked the classical and the quantum, although later it was shown that it was just a change of representation. In 1932, von Neumann postulated the correspondence rules between quantum and classical physics [2]. Also in 1932, Wigner introduced his homonym function in a different context, attempting to calculate quantum corrections to thermodynamic equilibrium [3]. This function can be understood as a generalization of the Liouville density, but it cannot be interpreted as a classical probability distribution, since it can take negative values. Based on it, in the 1940s, Groenewold and Moyal developed a formulation of quantum mechanics in phase space, where a non-commutative product appears, the star product, and the evolution of the Wigner function occurs through a deformation of the Poisson bracket: the Moyal bracket [4,5].

It is important to emphasize that the application of phase-space methods to fields and systems with internal degrees of freedom has a rich history. The quantization of classical systems using anticommuting variables and the formulation of fermionic phase-space dynamics were foundational achievements by Berezin [6] and Marinov [7]. Furthermore, the precise mapping between operator representations and phase-space quasiprobability distributions for bosons and fermions was rigorously established by Cahill and Glauber [8,9], providing the mathematical bedrock for functional expansions. The global mathematical structure of deformation quantization was subsequently established in the landmark work of Bayen et al. [10] and Kontsevich [11], while comprehensive foundational reviews of quantum phase-space distribution functions were provided by Hillery et al. [12]. In parallel, the deformation quantization of gauge theories, field-theoretic transport, and nonequilibrium systems was pioneered by Curtright, Zachos, Carruthers, and others [13,14,15,16,17], highlighting the subtle interplay between gauge constraints and the Moyal ⋆-product.

In recent years, phase-space methods have experienced renewed interest due to their versatility in addressing specialized quantum phenomena beyond standard quantum mechanics. In particular, concrete physical applications demonstrate the adaptability of phase-space techniques across diverse domains, including the analysis of topological phase shifts in the Aharonov–Bohm effect [18], the characterization of Wehrl entropy in entangled harmonic oscillators via the Segal–Bargmann representation [19], and the extension to quantum fields in curved background geometries, such as expanding FLRW spacetimes [20] and black hole horizons [21]. These application-driven developments illustrate the practical utility of phase-space representations for complex physical systems, further motivating a rigorous, unified extension to infinite-dimensional quantum field theory.

From a physical perspective, formulating quantum field theory in phase space offers distinct conceptual advantages over standard operator or wave-functional representations in Hilbert space:

• Transparent Semiclassical Limit: It expresses field dynamics directly in terms of classical field and momentum configurations, providing an intuitive and systematic expansion in powers of ℏ.
• Natural Description of Statistical and Mixed Ensembles: Unlike the Schrödinger wave-functional framework—which is strictly suited for pure states—the Wigner functional natively accommodates mixed quantum states, thermal field configurations, and statistical quantum ensembles without requiring functional density operators.
• Framework for Non-Equilibrium Dynamics: It establishes a natural starting point for deriving field-theoretic quantum transport equations, decoherence processes, and open quantum field systems.

Before exploiting these features in specific phenomenological applications, it is essential to ensure that the foundational theoretical apparatus is fully consistent, transparent, and unified.

Rather than claiming the historical introduction of these individual tools, the primary objective of the present work is to establish the fundamental theoretical consistency and operational coherence of the functional phase-space formalism from first principles. We construct a systematic, unified, and self-consistent functional framework that directly maps the stationary Schrödinger functional equations to their phase-space counterparts across scalar, gauge, and fermionic fields, followed by a consistent perturbative treatment for interacting theories. Furthermore, we explicitly operate within a non-manifestly covariant framework by choosing equal-time hypersurfaces inherent to the functional Schrödinger picture. This allows us to directly compare the structural nuances and ordering prescriptions of different spin statistics under a single functional star-product architecture, bridging the gap between historical foundational math and concrete field-theoretic calculations.

At present, the use of the quantum mechanics formulation in phase space is restricted to very specific areas. On the other hand, its generalization to quantum field theory (QFT) is not straightforward. It requires to make decisions about different assumptions and subtleties. Throughout this work, we will develop the functional formalism of QFT, where the states are represented by functionals of fundamental fields. In particular, we will introduce such formulation for fermions and vector fields.

The manuscript is organized as follows: In Section 2, we introduce some of the key concepts of the Wigner formalism and its functional extension in Quantum Field Theory. The next four sections are devoted to finding the stationary Wigner functionals of the real and scalar field, the massless electromagnetic field, and the fermionic field, respectively. Lastly, we will briefly develop the fundamentals of interacting field theories in phase space, which are necessary in order to calculate transition amplitudes of different processes. The main conclusions are summarized in Section 8.

2. Wigner Functional

The term quantum physics in phase space refers to an alternative representation of quantum mechanics, equivalent to the conventional Hilbert space and the path integral formulation. Its main characteristic is the fact that the state of a system is described simultaneously by the positions and momenta of the particles which conform it. Each quantum state is associated to its Wigner function [3]:

$$W(x,p)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }dy{\psi }^{\ast }\left(x-{\displaystyle \frac{\hslash }{2}}y\right)e^{-ipy}\psi \left(x+{\displaystyle \frac{\hslash }{2}}y\right),$$

(1)

where $\psi \left(x\right)$ is the wave function that describes the quantum state in the Schrödinger position representation. In order to construct a non-commutative algebra, the Moyal star product between functions in phase space is introduced as [4]

$$f(x,p)\star g(x,p)=f(x,p)e^{{\displaystyle \frac{i\hslash }{2}}(\overleftarrow{{\partial }_x}\overrightarrow{{\partial }_p}-\overleftarrow{{\partial }_p}\overrightarrow{{\partial }_x})}g(x,p).$$

(2)

Alternatively, this can be expressed in terms of a Bopp shift [13]:

$$f(x,p)\star g(x,p)=f\left(x+{\displaystyle \frac{i\hslash }{2}}{\partial }_p,p-{\displaystyle \frac{i\hslash }{2}}{\partial }_x\right)g(x,p).$$

(3)

Finally, the dynamics of a quantum state is determined by Moyal’s equation [5]:

$${\displaystyle \frac{\partial W}{\partial t}}=\left\{\{H,W\}\right\},$$

(4)

where the Moyal bracket is defined as $i\hslash \left\{\right\{f,g\left\}\right\}=f\star g-g\star f$. Thus, stationary states must satisfy the eigenvalue equation,

$$H(x,p)\star W(x,p)=W(x,p)\star H(x,p)=EW(x,p).$$

(5)

In order to extend this formulation to quantum field theory, we will use Jackiw’s functional representation [22,23]. According to this, a quantum system is described by a Hamiltonian density $\mathcal{H}(\widehat{\varphi },\widehat{\pi })$ which depends on the field operators $\widehat{\varphi }$ and their canonical conjugate momenta $\widehat{\pi }$. Since we are working in the Schrödinger picture, these operators are time independent, $\widehat{\varphi }=\widehat{\varphi }\left(\mathbf{x}\right)$, $\widehat{\pi }=\widehat{\pi }\left(\mathbf{x}\right)$. In the position representation, a basis of eigenstates of the field operators is constructed:

$$\widehat{\varphi }|\varphi ⟩=\varphi \left(\mathbf{x}\right)|\varphi ⟩.$$

(6)

Thus, we can define the functional associated with a given state $|\Psi ⟩$ as $\Psi \left[\varphi \right]=⟨\varphi |\Psi ⟩$. From this, one can apply a functional Wigner transform and obtain the Wigner functional of that state [24]:

$$\begin{array}{cc}\hfill & W[\varphi ,\pi ]=\int \mathcal{D}\eta {\Psi }^{\ast }\left[\varphi -{\displaystyle \frac{\hslash }{2}}\eta \right]e^{-i\int d^3\mathbf{x}\eta \left(\mathbf{x}\right)\pi \left(\mathbf{x}\right)}\Psi \left[\varphi +{\displaystyle \frac{\hslash }{2}}\eta \right],\hfill \end{array}$$

(7)

where the integration measure is given by $\mathcal{D}\eta ={\prod }_{\mathbf{x}}(d\eta \left(\mathbf{x}\right)/2\pi )$. Similarly to the finite-dimensional case, the Moyal star product is introduced as the equivalent of the operator product:

$$\begin{array}{cc}\hfill & A[\varphi ,\pi ]\star B[\varphi ,\pi ]=A[\varphi ,\pi ]\exp \left({\displaystyle \frac{i\hslash }{2}}\int d^3\mathbf{x}{\displaystyle \frac{\overleftarrow{\delta }}{\delta \varphi \left(\mathbf{x}\right)}}{\displaystyle \frac{\overrightarrow{\delta }}{\delta \pi \left(\mathbf{x}\right)}}-\right.\hfill \\ & \left.{\displaystyle \frac{\overleftarrow{\delta }}{\delta \pi \left(\mathbf{x}\right)}}{\displaystyle \frac{\overrightarrow{\delta }}{\delta \varphi \left(\mathbf{x}\right)}}\right)B[\varphi ,\pi ],\hfill \end{array}$$

(8)

where $\delta /\delta \varphi \left(\mathbf{x}\right)$ denotes the functional derivative with respect to $\varphi \left(\mathbf{x}\right)$. Or, alternatively:

$$\begin{array}{cc}\hfill & A[\varphi ,\pi ]\star B[\varphi ,\pi ]=A\left[\varphi +{\displaystyle \frac{i\hslash }{2}}{\displaystyle \frac{\delta }{\delta \pi \left(\mathbf{x}\right)}},\pi -{\displaystyle \frac{i\hslash }{2}}{\displaystyle \frac{\delta }{\delta \varphi \left(\mathbf{x}\right)}}\right]B[\varphi ,\pi ]=\hfill \\ & A[\varphi ,\pi ]B\left[\varphi -{\displaystyle \frac{i\hslash }{2}}{\displaystyle \frac{\overleftarrow{\delta }}{\delta \pi \left(\mathbf{x}\right)}},\pi +{\displaystyle \frac{i\hslash }{2}}{\displaystyle \frac{\overleftarrow{\delta }}{\delta \varphi \left(\mathbf{x}\right)}}\right].\hfill \end{array}$$

(9)

Conceptually, the functional star-product in Equations (8) and (3) encodes operator non-commutativity directly into field phase space. The functional derivatives act as differential generators that shift field configurations, generating interference structures between classical field profiles $\varphi \left(\mathbf{x}\right)$ and their conjugate momentum fields $\pi \left(\mathbf{x}\right)$.

With this definition, the dynamics of a system is completely determined by Moyal’s equation:

$$\left\{\{H,W\}\right\}={\displaystyle \frac{\partial W}{\partial t}},$$

(10)

where the functional Moyal bracket $i\hslash \left\{\right\{A,B\left\}\right\}=A\star B-B\star A$ replaces the quantum commutator, defining the exact time evolution of the quasiprobability functional in field space.

In the case of a stationary state, we have to solve the eigenvalue equation $H[\varphi ,\pi ]\star W[\varphi ,\pi ]=W[\varphi ,\pi ]\star H[\varphi ,\pi ]=EW[\varphi ,\pi ]$, which is a functional differential equation.

In the following sections, we systematically solve this functional eigenvalue equation for free field theories spanning scalar, vector gauge, and fermionic statistics. Establishing these explicit vacuum and excited Wigner functionals provides a rigorous, self-consistent theoretical baseline before extending the methodology to interacting field theories.

3. Real Scalar Field

For the real scalar field, the Hamiltonian is given by

$$H[\varphi ,\pi ]=\int d^3\mathbf{x}{\displaystyle \frac{1}{2}}\left[{\pi }^2\left(\mathbf{x}\right)+\varphi \left(\mathbf{x}\right)(-{\nabla }^2+m^2)\varphi \left(\mathbf{x}\right)\right].$$

(11)

The real and imaginary parts of the eigenvalue equation for the Wigner functional are obtained by applying the star product with this Hamiltonian:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\int d^3\mathbf{x}\left[{\pi }^2\left(\mathbf{x}\right)-{\displaystyle \frac{{\hslash }^2}{4}}{\displaystyle \frac{\delta }{\delta \pi \left(\mathbf{x}\right)}}(-{\nabla }^2+m^2){\displaystyle \frac{\delta }{\delta \pi \left(\mathbf{x}\right)}}\right.\hfill \\ \hfill & \left.+\varphi \left(\mathbf{x}\right)(-{\nabla }^2+m^2)\varphi \left(\mathbf{x}\right)-{\displaystyle \frac{{\hslash }^2}{4}}{\displaystyle \frac{{\delta }^2}{\delta {\varphi }^2\left(\mathbf{x}\right)}}\right]W[\varphi ,\pi ]\hfill \\ \hfill & =EW[\varphi ,\pi ],\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill & & \int d^3\mathbf{x}\left(\pi \left(\mathbf{x}\right){\displaystyle \frac{\delta }{\delta \varphi \left(\mathbf{x}\right)}}-{\displaystyle \frac{1}{2}}\varphi \left(\mathbf{x}\right)(-{\nabla }^2+m^2){\displaystyle \frac{\delta }{\delta \pi \left(\mathbf{x}\right)}}-\right.\hfill \\ & & \left.{\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta }{\delta \pi \left(\mathbf{x}\right)}}(-{\nabla }^2+m^2)\varphi \left(\mathbf{x}\right)\right)W[\varphi ,\pi ]=0.\hfill \end{array}$$

(13)

Physically, the real part in Equation (12) represents the phase-space constraint on energy density, balanceable only by second-order functional derivatives, whereas the imaginary part in Equation (13) enforces stationary phase-space flow, guaranteeing canonical phase-space continuity.

It is clear that this equation is separable in $\varphi$ and $\pi$. However, solving it directly for an arbitrary state is computationally involved. Therefore, only the ground state (i.e., the vacuum of the Fock space) will be calculated directly. In this case, we assume that the Wigner functional is strictly positive (this can be justified by noting that the free field theory is simply an infinite set of harmonic oscillators, which implies that the Wigner functional describing the ground state is positive). Thus, the vacuum state can be written as

$$W_0[\varphi ,\pi ]=\mathcal{N}\exp \left[-\mathcal{W}[\varphi ,\pi ]\right].$$

(14)

Now, the operators present in the eigenvalue equation suggest that the Wigner functional $\mathcal{W}$ is quadratic in $\varphi$ and $\pi$. Thus, the second derivatives will contribute to the right-hand-side energy term, and the squared first derivatives to the quadratic terms in $\varphi \left(\mathbf{x}\right)$, $\pi \left(\mathbf{x}\right)$. Assuming a standard homogeneous background, the system possesses translational invariance, which implies that the kernel functions will depend only on the coordinate differences, $g_{\varphi ,\pi }(\mathbf{x},\mathbf{y})=g_{\varphi ,\pi }(\mathbf{x}-\mathbf{y})$. With that in mind, it is reasonable to assume that the solution can be written as

$$\begin{array}{cc}\hfill \mathcal{W}[\varphi ,\pi ]=\int d^3\mathbf{x}{d}^3\mathbf{y}& \varphi \left(\mathbf{x}\right)g_{\varphi }(\mathbf{x},\mathbf{y})\varphi \left(\mathbf{y}\right)+\hfill \\ & \pi \left(\mathbf{x}\right)g_{\pi }(\mathbf{x},\mathbf{y})\pi \left(\mathbf{y}\right),\hfill \end{array}$$

(15)

with $g_{\varphi ,\pi }(\mathbf{x},\mathbf{y})$ being symmetric functions of $\mathbf{x},\mathbf{y}$. One can argue that this is not the most general form of a quadratic functional: the functions $g(\mathbf{x},\mathbf{y})$ could be replaced, for example, by a differential operator and integrated only in $\mathbf{x}$. However, when performing these calculations, it is useful to interpret the fields $\varphi \left(\mathbf{x}\right),\pi \left(\mathbf{x}\right)$ as infinite-dimensional vectors, whose components are the values of these fields at each point in space. Under this approach, the differential operators and functions of two variables are bilinear forms (i.e., infinite-dimensional matrices) whose action on the vectors $\varphi ,\pi$ is expressed as integrals like the one in (15). Since these quantities are invariant under unitary transformations, it is always possible to find a change of basis, which would take the form of a change of variables in the integral, such that the differential operators are transformed into functions of two variables.

As a last premise, it will be assumed that the functions $g_{\varphi ,\pi }(\mathbf{x},\mathbf{y})$ are invariant under translations, i.e., $g_{\varphi ,\pi }(\mathbf{x},\mathbf{y})=g_{\varphi ,\pi }(\mathbf{x}-\mathbf{y})$. Therefore, if we perform a Fourier transform on each of the functions:

$$\begin{array}{cc}\hfill & \varphi \left(\mathbf{x}\right)=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^{3/2}}}\tilde{\varphi }\left(\mathbf{k}\right)e^{i\mathbf{k}·\mathbf{x}},\hfill \end{array}$$

(16)

$$\begin{array}{cc}& \pi \left(\mathbf{x}\right)=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^{3/2}}}\tilde{\pi }\left(\mathbf{k}\right)e^{i\mathbf{k}·\mathbf{x}},\hfill \end{array}$$

(17)

$$\begin{array}{cc}& g_{\varphi ,\pi }(\mathbf{x}-\mathbf{y})=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^{3/2}}}{\tilde{g}}_{\varphi ,\pi }\left(\mathbf{k}\right)e^{i\mathbf{k}·(\mathbf{x}-\mathbf{y})},\hfill \end{array}$$

(18)

then the equations for the kernel functions take the form:

$$\begin{array}{c}\hfill {\tilde{g}}_{\varphi }^2\left(\mathbf{k}\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^3{\hslash }^2}}{\omega }_{\mathbf{k}}^2,\end{array}$$

(19)

$$\begin{array}{c}\hfill {\tilde{g}}_{\pi }^2\left(\mathbf{k}\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^3{\hslash }^2{\omega }_{\mathbf{k}}^2}},\end{array}$$

(20)

with ${\omega }_{\mathbf{k}}=\sqrt{{\mathbf{k}}^2+m^2}$. Finally, it follows that

$$\begin{array}{cc}\hfill & W[\tilde{\varphi },\tilde{\pi }]=\hfill \\ \hfill =& \mathcal{N}\exp \left[-{\displaystyle \frac{1}{\hslash }}\int d^3\mathbf{k}\left({\omega }_{\mathbf{k}}\tilde{\varphi }\left(\mathbf{k}\right)\tilde{\varphi }(-\mathbf{k})+{\displaystyle \frac{\tilde{\pi }\left(\mathbf{k}\right)\tilde{\pi }(-\mathbf{k})}{{\omega }_{\mathbf{k}}}}\right)\right].\hfill \end{array}$$

(21)

Note that this solution also satisfies Equation (13). Moreover, this Wigner functional calculated by means of the eigenvalue equation agrees with the Wigner transform of the Schrödinger functional in the position representation, which is given by

$$\begin{array}{cc}\hfill & \Psi \left[\varphi \right]=\tilde{\mathcal{N}}\exp \left[-{\displaystyle \frac{1}{2\hslash }}\int d^3\mathbf{x}\varphi \left(\mathbf{x}\right){(-{\nabla }^2+m^2)}^{1/2}\varphi \left(\mathbf{x}\right)\right]=\hfill \\ & =\tilde{\mathcal{N}}\exp \left[-{\displaystyle \frac{1}{2\hslash }}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{\omega }_{\mathbf{k}}\varphi \left(\mathbf{k}\right)\varphi \left(-\mathbf{k}\right)\right].\hfill \end{array}$$

(22)

Performing the Wigner transform, we obtain a functional integral having the form of a Gaussian with a linear term. Using the well-known result for a Gaussian functional integral [23], we arrive at

$$\begin{array}{cc}\hfill & W_0[\varphi ,\pi ]\propto \exp \left[-{\displaystyle \frac{1}{\hslash }}\left(\int d^3\mathbf{x}(\varphi \left(\mathbf{x}\right){(-{\nabla }^2+m^2)}^{1/2}\varphi \left(\mathbf{x}\right)+\right.\right.\hfill \\ & \left.\left.\pi \left(\mathbf{x}\right){(-{\nabla }^2+m^2)}^{-1/2}\pi (\mathbf{x}))\right)\right],\hfill \end{array}$$

(23)

which agrees with the result obtained using the eigenvalue equation in phase space. It coincides with the one calculated by Curtright and Zachos in reference [24]. To calculate the vacuum energy, we substitute the solution for the functions $g_{\varphi ,\pi }(\mathbf{x}-\mathbf{y})$ into Equation (12):

$$\begin{array}{cc}\hfill & E_0={\displaystyle \frac{{\hslash }^2}{4}}\left[\int d^3\mathbf{x}{g}_{\varphi }(\mathbf{x},\mathbf{x})+\right.\hfill \\ & \left.\int d^3\mathbf{x}{d}^3\mathbf{y}{\delta }^{\left(3\right)}(\mathbf{x}-\mathbf{y})(-{\nabla }_{\mathbf{x}}^2+m^2)g_{\pi }(\mathbf{x},\mathbf{y})\right]=\hfill \\ \hfill =& {\displaystyle \frac{\hslash }{2}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{\omega }_{\mathbf{k}}{\delta }^{\left(3\right)}\left(\mathbf{0}\right).\hfill \end{array}$$

(24)

This divergence in the energy of the vacuum state is consistent with other QFT formalisms. Finally, the normalization constant can be obtained by imposing $⟨0|0⟩=1$, which in terms of the Wigner functional means

$$\int \mathcal{D}\varphi \mathcal{D}\pi W_0[\varphi ,\pi ]=1,$$

(25)

where the integral measures are $\mathcal{D}\varphi ={\prod }_{\mathbf{x}}d\varphi \left(\mathbf{x}\right)$, $\mathcal{D}\pi ={\prod }_{\mathbf{x}}d\pi \left(\mathbf{x}\right)$. In this way, one arrives at

$$\begin{array}{cc}\hfill & 1=\mathcal{N}{\displaystyle \frac{{\left(\sqrt{\pi }\right)}^n}{\sqrt{\det \left(\frac{1}{\hslash }{(-{\nabla }^2+m^2)}^{1/2}\right)}}}\times \hfill \\ & {\displaystyle \frac{{\left(\sqrt{\pi }\right)}^n}{\sqrt{\det \left(\frac{1}{\hslash }{(-{\nabla }^2+m^2)}^{-1/2}\right)}}}=\mathcal{N}{(\hslash \pi )}^n,\hfill \end{array}$$

(26)

with $n\to \infty$. To check this result, the norm of the vacuum state can be rederived by integrating in Fourier space. Since the Jacobian for this change of variables is $1/{\left(2\pi \right)}^{3/2}$ (note that, apart from this factor, the Fourier transform is unitary as it preserves scalar products), it follows that

$$\begin{array}{c}1=\mathcal{N}{\displaystyle \prod _{\mathbf{k}}}{\displaystyle \frac{\sqrt{\hslash \pi }}{{\omega }_{\mathbf{k}}}}{\displaystyle \prod _{\mathbf{k}}}\sqrt{\hslash \pi }{\omega }_{\mathbf{k}}\Rightarrow \mathcal{N}={\displaystyle \prod _{\mathbf{k}}}{\displaystyle \frac{1}{\hslash \pi }}.\end{array}$$

(27)

In conclusion, the complete expression of the Wigner functional for the vacuum state is

$$\begin{array}{ccc}& & W_0[\tilde{\varphi },\tilde{\pi }]=\hfill \\ & & \prod _{\mathbf{k}}{\displaystyle \frac{1}{\hslash \pi }}\exp \left[-{\displaystyle \frac{1}{\hslash }}\left({\omega }_{\mathbf{k}}\tilde{\varphi }\left(\mathbf{k}\right)\tilde{\varphi }(-\mathbf{k})+{\displaystyle \frac{\tilde{\pi }\left(\mathbf{k}\right)\tilde{\pi }(-\mathbf{k})}{{\omega }_{\mathbf{k}}}}\right)\right].\hfill \end{array}$$

(28)

The next step is to calculate the Wigner functional of an excited state of the field. For this purpose, the creation and annihilation operators are introduced (which, in this framework, serve as the phase-space representations of ladder operators, or creation/annihilation functionals):

$$\begin{array}{cc}\hfill & a^{†}\left(\mathbf{k}\right)={\displaystyle \frac{1}{\sqrt{2\hslash }}}\int d^3\mathbf{x}\left(\sqrt{{\omega }_{\mathbf{k}}}\varphi \left(\mathbf{x}\right)-{\displaystyle \frac{i}{\sqrt{{\omega }_{\mathbf{k}}}}}\pi \left(\mathbf{x}\right)\right)e^{-i\mathbf{k}·\mathbf{x}},\hfill \\ \hfill & a\left(\mathbf{k}\right)={\displaystyle \frac{1}{\sqrt{2\hslash }}}\int d^3\mathbf{x}\left(\sqrt{{\omega }_{\mathbf{k}}}\varphi \left(\mathbf{x}\right)+{\displaystyle \frac{i}{\sqrt{{\omega }_{\mathbf{k}}}}}\pi \left(\mathbf{x}\right)\right)e^{i\mathbf{k}·\mathbf{x}},\hfill \end{array}$$

(29)

which obey the commutation relations:

$$\left\{\{a\left(\mathbf{k}\right),a^{†}\left({\mathbf{k}}^{\prime }\right)\}\right\}={\left(2\pi \right)}^3{\delta }^{\left(3\right)}(\mathbf{k}-{\mathbf{k}}^{\prime }).$$

(30)

First, we observe that

$$a\left(\mathbf{k}\right)\star W_0[\varphi ,\pi ]=W_0[\varphi ,\pi ]\star a^{†}\left(\mathbf{k}\right)=0\forall \mathbf{k},$$

(31)

which can be checked by using the expression for the functional star product (8) and the explicit form of the vacuum Wigner functional. On the other hand, if the operators are applied in the opposite direction, the resulting functional has the form:

$$\begin{array}{cc}\hfill & W_1[\varphi ,\pi ]=(2\hslash {\omega }_{{\mathbf{k}}_1})a^{†}\left({\mathbf{k}}_1\right)\star W_0[\varphi ,\pi ]\star a\left({\mathbf{k}}_1\right)=\hfill \\ \hfill & =2{\omega }_{{\mathbf{k}}_1}{W}_0[\varphi ,\pi ]\left[\int d^3\mathbf{x}{d}^3\mathbf{y}\left({\omega }_{{\mathbf{k}}_1}\varphi \left(\mathbf{x}\right)\varphi \left(\mathbf{y}\right)+{\displaystyle \frac{\pi \left(\mathbf{x}\right)\pi \left(\mathbf{y}\right)}{{\omega }_{{\mathbf{k}}_1}}}-\right.\right.\hfill \\ & \left.\left.i(\varphi \left(\mathbf{x}\right)\pi \left(\mathbf{y}\right)-\varphi \left(\mathbf{y}\right)\pi \left(\mathbf{x}\right))-{\displaystyle \frac{\hslash }{2}}{\delta }^{\left(3\right)}(\mathbf{x}-\mathbf{y})\right)e^{i{\mathbf{k}}_1(\mathbf{x}-\mathbf{y})}\right].\hfill \end{array}$$

(32)

Operating via functional Bopp shifts, these ladder phase-space functionals dynamically generate polynomial prefactors in field space without requiring underlying Hilbert-space state manipulation.

To determine the physical meaning of these states, one can see that both the Hamiltonian and the momentum functionals can be expressed in terms of the creation and annihilation operators as

$$\begin{array}{cc}\hfill & H=\int d^3\mathbf{x}{\displaystyle \frac{1}{2}}\left[{\pi }^2\left(\mathbf{x}\right)+\varphi \left(\mathbf{x}\right)(-{\nabla }^2+m^2)\varphi \left(\mathbf{x}\right)\right]=\hfill \\ & {\displaystyle \frac{\hslash }{2}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{\omega }_{\mathbf{k}}\left(a^{†}\left(\mathbf{k}\right)\star a\left(\mathbf{k}\right)+a\left(\mathbf{k}\right)\star a^{†}\left(\mathbf{k}\right)\right),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & P_j=-\int d^3\mathbf{x}\varphi \left(\mathbf{x}\right)\star {\partial }_j\pi \left(\mathbf{x}\right)=\hfill \\ & {\displaystyle \frac{\hslash }{2}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{k}_j\left(a^{†}\left(\mathbf{k}\right)\star a\left(\mathbf{k}\right)+a\left(\mathbf{k}\right)\star a^{†}\left(\mathbf{k}\right)\right).\hfill \end{array}$$

(34)

Expressing these functionals as functions of the ladder ones allows us to easily deduce the following commutation relations:

$$\begin{array}{c}\left\{\{H,a^{†}\left(\mathbf{k}\right)\}\right\}=\left\{\{a\left(\mathbf{k}\right),H\}\right\}=\hslash {\omega }_{\mathbf{k}},\\ \left\{\{P_j,a^{†}\left(\mathbf{k}\right)\}\right\}=\left\{\{a\left(\mathbf{k}\right),P_j\}\right\}=\hslash k_j,\end{array}$$

(35)

Using them, one arrives at the energy and momentum of the state represented by the excited Wigner functional:

$$\begin{array}{cc}\hfill & H\star W_1=W_1\star H=(E_0+\hslash {\omega }_{{\mathbf{k}}_1})W_1=E_1{W}_1,\hfill \\ \hfill & P_j\star W_1=W_1\star P_j=\hslash {\left(k_1\right)}_j{W}_1.\hfill \end{array}$$

(36)

Thus, we can conclude that the functional $W_1[\varphi ,\pi ]$ represents the state of a particle having momentum and energy given by ${\mathbf{k}}_1$, ${\omega }_{{\mathbf{k}}_1}$. In a completely analogous way, functionals corresponding to multi-particle states are constructed by repeatedly applying the creation operator to the vacuum state.

4. Complex Scalar Field

The procedure for the complex scalar field is very similar to the real one. This time, the Hamiltonian is

$$H[\varphi ,\pi ]=\int d^3\mathbf{x}{\pi }^{†}\left(\mathbf{x}\right)\pi \left(\mathbf{x}\right)+{\varphi }^{†}(-{\nabla }^2+m^2)\varphi \left(\mathbf{x}\right).$$

(37)

Since the fields are now complex, one must be careful when applying the star product with conjugate fields. If the fields are expressed as $\varphi ={\varphi }_1+i{\varphi }_2$, $\pi ={\pi }_1+i{\pi }_2$, then

$$\begin{array}{c}{\varphi }_{1,2}\left(\mathbf{x}\right)\star ={\varphi }_{1,2}+{\displaystyle \frac{i\hslash }{2}}{\displaystyle \frac{\delta }{\delta {\pi }_{1,2}\left(\mathbf{x}\right)}},\\ {\pi }_{1,2}\left(\mathbf{x}\right)\star ={\pi }_{1,2}-{\displaystyle \frac{i\hslash }{2}}{\displaystyle \frac{\delta }{\delta {\varphi }_{1,2}\left(\mathbf{x}\right)}}.\end{array}$$

(38)

Therefore, in terms of the field $\varphi$ and its complex conjugate:

$$\begin{array}{cc}\hfill & \varphi \left(\mathbf{x}\right)\star =({\varphi }_1\left(\mathbf{x}\right)\star )+i({\varphi }_2\left(\mathbf{x}\right)\star )=\varphi \left(\mathbf{x}\right)+{\displaystyle \frac{i\hslash }{2}}{\displaystyle \frac{\delta }{\delta {\pi }^{†}\left(\mathbf{x}\right)}},\hfill \end{array}$$

$$\begin{array}{cc}& {\varphi }^{†}\left(\mathbf{x}\right)\star =({\varphi }_1\left(\mathbf{x}\right)\star )-i({\varphi }_2\left(\mathbf{x}\right)\star )={\varphi }^{†}\left(\mathbf{x}\right)+{\displaystyle \frac{i\hslash }{2}}{\displaystyle \frac{\delta }{\delta \pi \left(\mathbf{x}\right)}}.\hfill \\ \hfill & \pi \left(\mathbf{x}\right)\star =({\pi }_1\left(\mathbf{x}\right)\star )+i({\pi }_2\left(\mathbf{x}\right)\star )=\pi \left(\mathbf{x}\right)-{\displaystyle \frac{i\hslash }{2}}{\displaystyle \frac{\delta }{\delta {\varphi }^{†}\left(\mathbf{x}\right)}},\hfill \end{array}$$

(39)

$$\begin{array}{cc}& {\pi }^{†}\left(\mathbf{x}\right)\star =({\pi }_1\left(\mathbf{x}\right)\star )-i({\pi }_2\left(\mathbf{x}\right)\star )={\pi }^{†}\left(\mathbf{x}\right)-{\displaystyle \frac{i\hslash }{2}}{\displaystyle \frac{\delta }{\delta \varphi \left(\mathbf{x}\right)}}.\hfill \end{array}$$

(40)

The asymmetric appearance of functional derivatives in $\varphi \star$ versus ${\varphi }^{†}\star$ reflects the non-commutative phase-space algebra inherited from complex canonical conjugate pairs.

The relations between $\varphi$ and ${\varphi }^{†}$ (resp. $\pi$ and ${\pi }^{†}$) are not complex conjugates of each other. Recall that, when performing the adjoint of a product of operators in Hilbert space, the order of the factors is reversed.

In a completely analogous way to the real scalar field, it can be shown that the solution for the ground state is given by a functional quadratic exponential:

$$\begin{array}{c}\hfill W_0[\tilde{\varphi },\tilde{\pi }]\propto \exp \left[-{\displaystyle \frac{2}{\hslash }}\int d^3\mathbf{k}{\tilde{\varphi }}^{†}\left(\mathbf{k}\right){\omega }_{\mathbf{k}}\tilde{\varphi }\left(\mathbf{k}\right)+{\displaystyle \frac{{\tilde{\pi }}^{†}\left(\mathbf{k}\right)\tilde{\pi }\left(\mathbf{k}\right)}{{\omega }_{\mathbf{k}}}}\right].\end{array}$$

(41)

Here, the factor of 2 in the exponent arises naturally from the standard Gaussian integration over the independent real and imaginary parts of the complex field degrees of freedom, combined with our chosen normalization convention.

In this case, we have to generate two sets of creation/annihilation functionals.

$$\begin{array}{cc}\hfill & a^{†}\left(\mathbf{k}\right)={\displaystyle \frac{1}{\sqrt{2\hslash }}}\int d^3\mathbf{x}\left(\sqrt{{\omega }_{\mathbf{k}}}{\varphi }^{†}\left(\mathbf{x}\right)-{\displaystyle \frac{i}{\sqrt{{\omega }_{\mathbf{k}}}}}\pi \left(\mathbf{x}\right)\right)e^{-i\mathbf{k}·\mathbf{x}},\hfill \\ \hfill & a\left(\mathbf{k}\right)={\displaystyle \frac{1}{\sqrt{2\hslash }}}\int d^3\mathbf{x}\left(\sqrt{{\omega }_{\mathbf{k}}}\varphi \left(\mathbf{x}\right)+{\displaystyle \frac{i}{\sqrt{{\omega }_{\mathbf{k}}}}}{\pi }^{†}\left(\mathbf{x}\right)\right)e^{i\mathbf{k}·\mathbf{x}},\hfill \\ \hfill & b^{†}\left(\mathbf{k}\right)={\displaystyle \frac{1}{\sqrt{2\hslash }}}\int d^3\mathbf{x}\left(\sqrt{{\omega }_{\mathbf{k}}}\varphi \left(\mathbf{x}\right)-{\displaystyle \frac{i}{\sqrt{{\omega }_{\mathbf{k}}}}}{\pi }^{†}\left(\mathbf{x}\right)\right)e^{-i\mathbf{k}·\mathbf{x}},\hfill \\ \hfill & b\left(\mathbf{k}\right)={\displaystyle \frac{1}{\sqrt{2\hslash }}}\int d^3\mathbf{x}\left(\sqrt{{\omega }_{\mathbf{k}}}{\varphi }^{†}\left(\mathbf{x}\right)+{\displaystyle \frac{i}{\sqrt{{\omega }_{\mathbf{k}}}}}\pi \left(\mathbf{x}\right)\right)e^{i\mathbf{k}·\mathbf{x}},\hfill \end{array}$$

(42)

Distinct ladder functionals for a and b decouple particle and antiparticle modes, allowing independent creation/annihilation operations directly within phase space.

Which satisfy the commutation relations:

$$\begin{array}{cc}\hfill & \left\{\{a\left(\mathbf{k}\right),a^{†}\left({\mathbf{k}}^{\prime }\right)\}\right\}=\left\{\{b\left(\mathbf{k}\right),b^{†}\left({\mathbf{k}}^{\prime }\right)\}\right\}\hfill \\ & ={\left(2\pi \right)}^3{\delta }^{\left(3\right)}(\mathbf{k}-{\mathbf{k}}^{\prime }),\hfill \end{array}$$

(43)

with the rest of them being zero. As in the real case, it is found that the Hamiltonian and the linear momentum can be expressed as

$$\begin{array}{cc}\hfill & H={\displaystyle \frac{\hslash }{2}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{\omega }_{\mathbf{k}}(a^{†}\left(\mathbf{k}\right)\star a\left(\mathbf{k}\right)+a\left(\mathbf{k}\right)\star a^{†}\left(\mathbf{k}\right)+\hfill \\ & b^{†}\left(\mathbf{k}\right)\star b\left(\mathbf{k}\right)+b\left(\mathbf{k}\right)\star b^{†}\left(\mathbf{k}\right)),\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & P_j={\displaystyle \frac{\hslash }{2}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{k}_j(a^{†}\left(\mathbf{k}\right)\star a\left(\mathbf{k}\right)+a\left(\mathbf{k}\right)\star a^{†}\left(\mathbf{k}\right)+\hfill \\ & b^{†}\left(\mathbf{k}\right)\star b\left(\mathbf{k}\right)+b\left(\mathbf{k}\right)\star b^{†}\left(\mathbf{k}\right)).\hfill \end{array}$$

(45)

Therefore, the states $a^{†}\left(\mathbf{k}\right)\star W_0\star a\left(\mathbf{k}\right)$ and $b^{†}\left(\mathbf{k}\right)\star W_0\star b\left(\mathbf{k}\right)$ represent states of a particle (antiparticle) with momentum $\mathbf{k}$ and energy ${\omega }_{\mathbf{k}}$. Finally, we consider the quantity:

$$Q=i\int d^3\mathbf{x}({\varphi }^{†}\left(\mathbf{x}\right){\pi }^{†}\left(\mathbf{x}\right)-\varphi \left(\mathbf{x}\right)\pi \left(\mathbf{x}\right)),$$

(46)

which will be denoted as charge. In terms of the ladder functionals, it has the form:

$$Q=\hslash \int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}(a^{†}\left(\mathbf{k}\right)\star a\left(\mathbf{k}\right)-b\left(\mathbf{k}\right)\star b^{†}\left(\mathbf{k}\right)).$$

(47)

Although this has not been done for the previous functionals, in this case, we will make use of normal ordering of the creation and annihilation functionals in order to avoid the vacuum state having a non-zero charge. This can be justified due to the fact that there is some ambiguity in the ordering of the fields in the definition of the charge (46). Therefore, the expression to be used is

$$:Q:=\hslash \int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}(a^{†}\left(\mathbf{k}\right)\star a\left(\mathbf{k}\right)-b^{†}\left(\mathbf{k}\right)\star b\left(\mathbf{k}\right)).$$

(48)

It is easy to show that the charge Q and the Hamiltonian H commute under Moyal’s bracket. Thus, using Moyal’s equation, it follows that Q is conserved:

$${\displaystyle \frac{dQ}{dt}}=\left\{\{Q,H\}\right\}=0.$$

(49)

Consequently, the states obtained by consecutively applying the ladder functionals are eigenstates of the charge functional Q, with eigenvalue $n_a-n_b$, where $n_a$ (resp. $n_b$) is the number of times we have applied the functional $a^{†}\star \left(\right)\star a$ (resp. $b^{†}\star \left(\right)\star b$) to the vacuum state. Therefore, they can be interpreted as states with $n_a$ particles with charge +1 and $n_b$ antiparticles with charge −1.

Functional Noether’s Theorem in Phase Space

The consistency of our functional phase-space framework can be further substantiated by analyzing how continuous symmetries and conservation laws manifest under the star product. Let us consider an infinitesimal transformation of the fields generated by a continuous symmetry, ${\varphi }_a\left(\mathbf{x}\right)\to {\varphi }_a\left(\mathbf{x}\right)+ϵ\Delta {\varphi }_a\left(\mathbf{x}\right)$. The corresponding classical Noether charge can be lifted to a phase-space functional:

$$Q[\varphi ,\pi ]=\int d^3\mathbf{x}{\pi }_a\left(\mathbf{x}\right)\Delta {\varphi }_a\left(\mathbf{x}\right).$$

(50)

In this Wigner–Moyal functional description, the classical Poisson bracket is replaced by the functional Moyal bracket, defined for any two functionals A and B as $\left\{\{A,B\}\right\}={\displaystyle \frac{1}{i\hslash }}(A\star B-B\star A)$. The quantum time evolution of any state or observable is governed by $\partial W/\partial t=\left\{\right\{H,W\left\}\right\}$. Consequently, Noether’s theorem generalizes naturally: a charge functional is conserved, $dQ/dt=0$, if and only if it star-commutes with the Hamiltonian, ${[Q,H]}_{\star }=Q\star H-H\star Q=0$. Furthermore, the infinitesimal symmetry transformations of the fundamental variables are directly generated via $\delta \varphi \left(\mathbf{x}\right)=\left\{\right\{\varphi \left(\mathbf{x}\right),Q\left\}\right\}$ and $\delta \pi \left(\mathbf{x}\right)=\left\{\right\{\pi \left(\mathbf{x}\right),Q\left\}\right\}$.

As a concrete application, for the global $U\left(1\right)$ symmetry characterizing the complex scalar field, $\varphi \left(\mathbf{x}\right)\to e^{i\alpha }\varphi \left(\mathbf{x}\right)$, the infinitesimal variations are $\Delta \varphi =i\varphi$ and $\Delta {\varphi }^{†}=-i{\varphi }^{†}$. The conserved Noether charge functional in phase space is then strictly expressed as

$$Q=i\int d^3\mathbf{x}\left[{\pi }^{†}\left(\mathbf{x}\right)\star {\varphi }^{†}\left(\mathbf{x}\right)-\varphi \left(\mathbf{x}\right)\star \pi \left(\mathbf{x}\right)\right],$$

(51)

which satisfies $\left\{\right\{Q,H\left\}\right\}=0$, providing a phase-space confirmation of quantum charge conservation. We designate this form as the primary definition of the charge functional; alternative ordering choices or placements of the star product differ from it at most by an additive c-number vacuum charge contribution, which commutes with all observables and drops out of physical expectation values.

5. Massless Electromagnetic Field

To construct the Wigner functionals of the free electromagnetic field, it is necessary to make two new considerations with respect to the scalar case: firstly, the fields have more than one component and, secondly, they have certain degrees of freedom under gauge transformations. With this in mind, the configuration space will be made up by the time-independent $A_{\mu }\left(\mathbf{x}\right)$ fields (recall that we are working in the Schrödinger picture, where the Hilbert space operators do not evolve in time). Because of this, it is very useful to choose the temporal gauge, $A_0=0$. This way, the conjugate momentum of the vector potential, that is, the electric field $\mathbf{E}\left(\mathbf{x}\right)$, will be completely independent of $\mathbf{A}\left(\mathbf{x}\right)$ (which is essential when working in phase space). However, there is still some residual gauge freedom left:

$$\mathbf{A}\left(\mathbf{x}\right)\to \mathbf{A}\left(\mathbf{x}\right)-\nabla \Lambda \left(\mathbf{x}\right).$$

(52)

We could impose some other condition on the fields, for example, the Coulomb gauge $\nabla ·\mathbf{A}\left(\mathbf{x}\right)=0$. However, this implies that we would have to modify the canonical commutation rules to meet this condition, replacing the conventional Dirac delta by a transverse Dirac delta. Since we do not want to change the definition of the star product, we will choose not to impose any further conditions on the fields.

On the other hand, the fact that the fields are time-independent and that $A_0=0$ eliminates the presence of Gauss’s law $\nabla ·\mathbf{E}\left(\mathbf{x}\right)=0$. To solve this, following the example of Hatfield [23], the following constraint will be added as a condition on the states:

$$\begin{array}{cc}\hfill & \nabla ·\mathbf{E}\left(\mathbf{x}\right)\star W[\mathbf{A},\mathbf{E}]=W[\mathbf{A},\mathbf{E}]\star \nabla ·\mathbf{E}\left(\mathbf{x}\right)=0\Rightarrow \hfill \\ & \nabla ·{\displaystyle \frac{\delta W}{\delta \mathbf{A}\left(\mathbf{x}\right)}}=0,\nabla ·\mathbf{E}\left(\mathbf{x}\right)=0.\hfill \end{array}$$

(53)

In phase space, Gauss’s law manifests as a dual constraint enforcing transverse functional derivatives with respect to $A_i\left(\mathbf{x}\right)$, guaranteeing physical state invariance along gauge orbits.

It can be seen that this guarantees that the Wigner functional is invariant under residual gauge transformations. Indeed, if $\delta \mathbf{A}\left(\mathbf{x}\right)=-\nabla \Lambda \left(\mathbf{x}\right)$, then

$$\begin{array}{cc}\hfill & \delta W[\mathbf{A},\mathbf{E}]=\int d^3\mathbf{x}{\displaystyle \frac{\delta W}{\delta \mathbf{A}\left(\mathbf{x}\right)}}·\delta \mathbf{A}\left(\mathbf{x}\right)=\hfill \\ & -\int d^3\mathbf{x}{\displaystyle \frac{\delta W}{\delta \mathbf{A}\left(\mathbf{x}\right)}}·\nabla \Lambda \left(\mathbf{x}\right)=0,\hfill \end{array}$$

(54)

where we have integrated by parts. Equivalently, one can interpret that the action of ${\displaystyle \frac{\delta }{\delta \mathbf{A}\left(\mathbf{x}\right)}}$ on the Wigner functional ‘extracts’ the factors $\mathbf{A}\left(\mathbf{x}\right)$ appearing in $W[\mathbf{A},\mathbf{E}]$. Thus, the condition (53) implies that all components of $\mathbf{A}\left(\mathbf{x}\right)$ that are present in the Wigner functional are transverse, while the longitudinal ones are eliminated. Since the transverse components ${\mathbf{A}}_T$ are gauge invariant (while the longitudinal components ${\mathbf{A}}_L$ are not), it follows that $W[\mathbf{A},\mathbf{E}]$ is also gauge invariant.

On the other hand, one should not forget the other condition that we imposed, that is, $\nabla ·\mathbf{E}\left(\mathbf{x}\right)=0$. In contrast to the vector potential, where this requisite is defined by its action on the functionals, the electric field is directly restricted, i.e., only the transverse components of the electric field are considered. We will have to take this into account at all times when calculating Wigner functionals, since it will not be denoted explicitly.

One problem with this choice of gauge is that, when calculating expected values, unwanted divergences appear. This is because any physical quantity $\mathcal{O}[\mathbf{A},\mathbf{E}]$ must be gauge invariant, which implies that it only depends, like the Wigner functional, on the transverse components of the vector potential. Therefore, the expectation value:

$$⟨\mathcal{O}⟩={\displaystyle \frac{\int \mathcal{D}\mathbf{A}\mathcal{D}\mathbf{E}\mathcal{O}[{\mathbf{A}}_T,\mathbf{E}]W[{\mathbf{A}}_T,\mathbf{E}]}{\int \mathcal{D}\mathbf{A}\mathcal{D}\mathbf{E}W[{\mathbf{A}}_T,\mathbf{E}]}},$$

(55)

adds the values of $\mathcal{O}[{\mathbf{A}}_T,{\mathbf{E}}_T]$ and $W[{\mathbf{A}}_T,{\mathbf{E}}_T]$ infinitely many times, since the integration measures can be factored into $\mathcal{D}\mathbf{A}=\mathcal{D}{\mathbf{A}}_T\mathcal{D}{\mathbf{A}}_L$ (this does not apply to the electric field because, by definition, all possible values of the field are transverse). However, for abelian gauge theories such as the massless electromagnetic field (where the group of gauge transformations is $U\left(1\right)$), the longitudinal and transverse components are easy to separate, so that their contributions in the numerator and denominator of (55) cancel.

Once all these details have been taken into account, one can proceed to calculate the Wigner functionals. In the temporal gauge, the Hamiltonian of the theory is given by

$$H={\displaystyle \frac{1}{2}}\int d^3\mathbf{x}\left[{\mathbf{E}}^2\left(\mathbf{x}\right)+{\mathbf{B}}^2\left(\mathbf{x}\right)\right]={\displaystyle \frac{1}{2}}\int d^3\mathbf{x}\left[{\mathbf{E}}^2\left(\mathbf{x}\right)+{(\nabla \times \mathbf{A}\left(\mathbf{x}\right))}^2\right].$$

(56)

This leads, again, to an eigenvalue equation imposed on the stationary states. In order to calculate the vacuum Wigner functional, this equation will be expressed directly in the Fourier space of the fields. To ensure mathematical rigor regarding the functional Dirac delta ${\delta }^{\left(3\right)}\left(\mathbf{0}\right)$ and the infinite volume divergences, a standard box normalization is implicitly assumed to render the functional integrations well-defined. We define the expansions as

$$\begin{array}{cc}\hfill & \mathbf{A}\left(\mathbf{x}\right)=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^{3/2}}}\tilde{\mathbf{A}}\left(\mathbf{k}\right)e^{i\mathbf{k}·\mathbf{x}},\hfill \\ \hfill & \mathbf{E}\left(\mathbf{x}\right)=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^{3/2}}}\tilde{\mathbf{E}}\left(\mathbf{k}\right)e^{i\mathbf{k}·\mathbf{x}},\hfill \\ \hfill & {\displaystyle \frac{\delta }{\delta \mathbf{A}\left(\mathbf{x}\right)}}=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^{3/2}}}{\displaystyle \frac{\delta }{\delta \tilde{\mathbf{A}}\left(\mathbf{k}\right)}}{e}^{-i\mathbf{k}·\mathbf{x}},\hfill \\ \hfill & {\displaystyle \frac{\delta }{\delta \mathbf{E}\left(\mathbf{x}\right)}}=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^{3/2}}}{\displaystyle \frac{\delta }{\delta \tilde{\mathbf{E}}\left(\mathbf{k}\right)}}{e}^{-i\mathbf{k}·\mathbf{x}},\hfill \end{array}$$

(57)

where the multicomponent functional Dirac delta acts in the following way:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\delta \widetilde{A_i}\left(\mathbf{k}\right)}{\delta \widetilde{A_j}\left({\mathbf{k}}^{\prime }\right)}}={\delta }_{ij}{\delta }^{\left(3\right)}(\mathbf{k}-{\mathbf{k}}^{\prime }),\hfill \\ & {\displaystyle \frac{\delta \widetilde{E_i}\left(\mathbf{k}\right)}{\delta \widetilde{E_j}\left({\mathbf{k}}^{\prime }\right)}}={\delta }_{ij}{\delta }^{\left(3\right)}(\mathbf{k}-{\mathbf{k}}^{\prime }).\hfill \end{array}$$

(58)

The condition that the electric field is purely transverse translates into $\mathbf{k}·\tilde{\mathbf{E}}\left(\mathbf{k}\right)=0\forall \mathbf{k}$. Based on these expressions, it can be seen that the equation that determines the vacuum state takes the form:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\int d^3\mathbf{k}\left[\tilde{\mathbf{E}}\left(\mathbf{k}\right)·\tilde{\mathbf{E}}(-\mathbf{k})-{\displaystyle \frac{{\hslash }^2}{4}}\left(\mathbf{k}\times {\displaystyle \frac{\delta }{\delta \tilde{\mathbf{E}}\left(\mathbf{k}\right)}}\right)·\right.\hfill \\ \hfill & \left(\mathbf{k}\times {\displaystyle \frac{\delta }{\delta \tilde{\mathbf{E}}(-\mathbf{k})}}\right)+(\mathbf{k}\times \tilde{\mathbf{A}}\left(\mathbf{k}\right))·(\mathbf{k}\times \tilde{\mathbf{A}}(-\mathbf{k}))-\hfill \\ \hfill & \left.{\displaystyle \frac{{\hslash }^2}{4}}{\displaystyle \frac{\delta }{\delta \tilde{\mathbf{A}}\left(\mathbf{k}\right)}}·{\displaystyle \frac{\delta }{\delta \tilde{\mathbf{A}}(-\mathbf{k})}}\right]W_0[\tilde{\mathbf{A}},\tilde{\mathbf{E}}]=E_0{W}_0[\tilde{\mathbf{A}},\tilde{\mathbf{E}}].\hfill \end{array}$$

(59)

As in the previous cases, we assume that the Wigner functional that represents the fundamental state is an exponential of the fields. If $\mathcal{W}$ is taken as a quadratic functional in $\mathbf{A},\mathbf{E}$, as suggested by the presence of the functional derivatives and the quadratic terms and constants, the equation splits into two parts:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\hslash }^2}{8}}\int d^3\mathbf{k}\left[-\left(\mathbf{k}·{\displaystyle \frac{\delta }{\delta \tilde{\mathbf{E}}\left(\mathbf{k}\right)}}\right)\left(\mathbf{k}·{\displaystyle \frac{\delta }{\delta \tilde{\mathbf{E}}(-\mathbf{k})}}\right)\mathcal{W}\right.\hfill \\ & \left.+{\mathbf{k}}^2{\displaystyle \frac{{\delta }^2\mathcal{W}}{\delta \tilde{\mathbf{E}}\left(\mathbf{k}\right)·\delta \tilde{\mathbf{E}}(-\mathbf{k})}}+{\displaystyle \frac{{\delta }^2\mathcal{W}}{\delta \tilde{\mathbf{A}}\left(\mathbf{k}\right)\delta \tilde{\mathbf{A}}(-\mathbf{k})}}\right]=E_0,\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\hslash }^2}{4}}\int d^3\mathbf{k}\left(\mathbf{k}\times {\displaystyle \frac{\delta \mathcal{W}}{\delta \tilde{\mathbf{E}}\left(\mathbf{k}\right)}}\right)·\left(\mathbf{k}\times {\displaystyle \frac{\delta \mathcal{W}}{\delta \tilde{\mathbf{E}}(-\mathbf{k})}}\right)+\hfill \\ & {\displaystyle \frac{\delta \mathcal{W}}{\delta \tilde{\mathbf{A}}\left(\mathbf{k}\right)}}·{\displaystyle \frac{\delta \mathcal{W}}{\delta \tilde{\mathbf{A}}(-\mathbf{k})}}=\int d^3\mathbf{k}\tilde{\mathbf{E}}\left(\mathbf{k}\right)·\tilde{\mathbf{E}}(-\mathbf{k})+\hfill \\ & (\mathbf{k}\times \tilde{\mathbf{A}}\left(\mathbf{k}\right))·(\mathbf{k}\times \tilde{\mathbf{A}}(-\mathbf{k})),\hfill \end{array}$$

(61)

where the functional Laplacian is defined as

$${\displaystyle \frac{{\delta }^2\mathcal{W}}{\delta \tilde{\mathbf{E}}\left(\mathbf{k}\right)·\delta \tilde{\mathbf{E}}(-\mathbf{k})}}=\sum _i{\displaystyle \frac{{\delta }^2\mathcal{W}}{\delta \widetilde{E_i}\left(\mathbf{k}\right)\delta \widetilde{E_i}(-\mathbf{k})}}.$$

(62)

Contrary to the previous fields, a solution of the form:

$$\begin{array}{cc}\hfill & \mathcal{W}[\tilde{\mathbf{A}},\tilde{\mathbf{E}}]=\hfill \\ & \int d^3\mathbf{k}{\tilde{g}}_{\mathbf{A}}\left(\mathbf{k}\right)\tilde{\mathbf{A}}\left(\mathbf{k}\right)·\tilde{\mathbf{A}}(-\mathbf{k})+{\tilde{g}}_{\mathbf{E}}\left(\mathbf{k}\right)\tilde{\mathbf{E}}\left(\mathbf{k}\right)·\tilde{\mathbf{E}}(-\mathbf{k}),\hfill \end{array}$$

(63)

does not satisfy the Gauss condition (53) which, in Fourier space, can be expressed as $\mathbf{k}·\delta W/\delta \tilde{\mathbf{A}}\left(\mathbf{k}\right)=0$. Since, when applying these operators on the Wigner vacuum functional, terms proportional to $\mathbf{k}·\tilde{\mathbf{A}}\left(\mathbf{k}\right)$ appear, one can guarantee (53) if the following substitution is made in the initial quadratic expression (see Hatfield [23]):

$$\tilde{\mathbf{A}}\left(\mathbf{k}\right)\to \tilde{\mathbf{A}}\left(\mathbf{k}\right)-{\displaystyle \frac{(\mathbf{k}·\tilde{\mathbf{A}}\left(\mathbf{k}\right))}{{\mathbf{k}}^2}}={\tilde{\mathbf{A}}}_T\left(\mathbf{k}\right).$$

(64)

It is easy to verify that the restrictions imposed on the states are now fulfilled. In order to calculate the functions ${\tilde{g}}_{\mathbf{A},\mathbf{E}}$, we return to the eigenvalue equations. Assuming that $g_{\mathbf{A},\mathbf{E}}\left(\mathbf{x}\right)$ are real functions, that is, ${\tilde{g}}_{\mathbf{A},\mathbf{E}}\left(\mathbf{k}\right)={\tilde{g}}_{\mathbf{A},\mathbf{E}}(-\mathbf{k})$, and taking into account the vector relation $(\mathbf{a}\times \mathbf{b})·(\mathbf{c}\times \mathbf{d})=(\mathbf{a}·\mathbf{c})(\mathbf{b}·\mathbf{d})-(\mathbf{a}·\mathbf{d})(\mathbf{b}·\mathbf{c})$, it follows that the functions ${\tilde{g}}_{\mathbf{A},\mathbf{E}}\left(\mathbf{k}\right)$ must satisfy

$${\hslash }^2{\tilde{g}}_{\mathbf{E}}^2\left(\mathbf{k}\right){\mathbf{k}}^2=1,{\hslash }^2{\tilde{g}}_{\mathbf{A}}^2\left(\mathbf{k}\right)={\mathbf{k}}^2.$$

(65)

The final expression of the Wigner functional for the fundamental state of the electromagnetic field is

$$\begin{array}{cc}\hfill & W[\tilde{\mathbf{A}},\tilde{\mathbf{E}}]=\hfill \\ & \mathcal{N}\prod _{\mathbf{k}}\exp \left[-\left({\omega }_k{\tilde{\mathbf{A}}}_T\left(\mathbf{k}\right)·{\tilde{\mathbf{A}}}_T(-\mathbf{k})+{\displaystyle \frac{\tilde{\mathbf{E}}\left(\mathbf{k}\right)·\tilde{\mathbf{E}}(-\mathbf{k})}{{\omega }_{\mathbf{k}}}}\right)\right],\hfill \end{array}$$

(66)

where $\mathcal{N}$ represents a formal normalization constant. As is standard in functional methods, this overall factor is left implicit since it systematically cancels out during the computation of physical expectation values and correlation functions. Here, ${\omega }_{\mathbf{k}}=\left|\mathbf{k}\right|$. We emphasize again that $\tilde{\mathbf{E}}\left(\mathbf{k}\right)$ fields are transverse by definition, which is why it is not explicitly stated. One can directly check that this expression coincides with the Wigner transform of the Schrödinger functional, which is given by [23]

$$\Psi \left[\tilde{\mathbf{A}}\right]=\tilde{\mathcal{N}}\exp \left[-{\displaystyle \frac{1}{2\hslash }}\int d^3\mathbf{k}{\omega }_k{\tilde{\mathbf{A}}}_T\left(\mathbf{k}\right)·{\tilde{\mathbf{A}}}_T(-\mathbf{k})\right].$$

(67)

If we use this expression in the formula for the Wigner transform (7), and integrate only over the transverse components of the vector potential (otherwise, there would be an infinite factor in front of the integral), we obtain the exact same result as with the eigenvalue equation.

Now, in order to determine the energy of the vacuum state, the second functional derivatives of the solution are calculated:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\delta }^2\mathcal{W}}{\delta {\tilde{E}}_i\left(\mathbf{k}\right)\delta {\tilde{E}}_i(-\mathbf{k})}}={\displaystyle \frac{2}{{\omega }_{\mathbf{k}}\hslash }}{\delta }^{\left(3\right)}\left(\mathbf{0}\right),\hfill \\ & {\displaystyle \frac{{\delta }^2\mathcal{W}}{\delta {\tilde{A}}_i\left(\mathbf{k}\right)\delta {\tilde{A}}_i(-\mathbf{k})}}={\displaystyle \frac{2{\omega }_{\mathbf{k}}}{\hslash }}{\delta }^{\left(3\right)}\left(\mathbf{0}\right)\left(1-{\displaystyle \frac{k_i^2}{{\mathbf{k}}^2}}\right).\hfill \end{array}$$

(68)

Recalling that the vectors $\tilde{\mathbf{E}}\left(\mathbf{k}\right)$, which are transverse, have only two nonzero components, we arrive at

$$\begin{array}{cc}\hfill & E_0={\displaystyle \frac{{\hslash }^2}{8}}\int d^3\mathbf{k}\left[-\left(\mathbf{k}·{\displaystyle \frac{\delta }{\delta \tilde{\mathbf{E}}\left(\mathbf{k}\right)}}\right)\left(\mathbf{k}·{\displaystyle \frac{\delta }{\delta \tilde{\mathbf{E}}(-\mathbf{k})}}\right)\mathcal{W}+\right.\hfill \\ & \left.{\mathbf{k}}^2{\displaystyle \frac{{\delta }^2\mathcal{W}}{\delta \tilde{\mathbf{E}}\left(\mathbf{k}\right)·\delta \tilde{\mathbf{E}}(-\mathbf{k})}}+{\displaystyle \frac{{\delta }^2\mathcal{W}}{\delta \tilde{\mathbf{A}}\left(\mathbf{k}\right)\delta \tilde{\mathbf{A}}(-\mathbf{k})}}\right]=\hfill \\ \hfill & =\int d^3\mathbf{k}\hslash {\omega }_{\mathbf{k}}{\delta }^{\left(3\right)}\left(\mathbf{0}\right).\hfill \end{array}$$

(69)

Next, the creation and annihilation functionals will be constructed as

$$\begin{array}{cc}\hfill & a^{†}(\mathbf{k},\lambda )={\displaystyle \frac{1}{\sqrt{2\hslash }}}\int d^3\mathbf{x}\mathit{ϵ}(\mathbf{k},\lambda )·\hfill \\ & \left(\sqrt{{\omega }_{\mathbf{k}}}\mathbf{A}\left(\mathbf{x}\right)-{\displaystyle \frac{i}{\sqrt{{\omega }_{\mathbf{k}}}}}\mathbf{E}\left(\mathbf{x}\right)\right)e^{-i\mathbf{k}·\mathbf{x}}\hfill \\ \hfill & a(\mathbf{k},\lambda )={\displaystyle \frac{1}{\sqrt{2\hslash }}}\int d^3\mathbf{x}{\mathit{ϵ}}^{\ast }(\mathbf{k},\lambda )·\hfill \\ & \left(\sqrt{{\omega }_{\mathbf{k}}}\mathbf{A}\left(\mathbf{x}\right)+{\displaystyle \frac{i}{\sqrt{{\omega }_{\mathbf{k}}}}}\mathbf{E}\left(\mathbf{x}\right)\right)e^{i\mathbf{k}·\mathbf{x}},\hfill \end{array}$$

(70)

where $\mathit{ϵ}(\mathbf{k},\lambda )$ are orthonormal vectors (although they need not be real) associated with each transverse polarization ($\lambda =1,2$), that is, $\mathit{ϵ}(\mathbf{k},\lambda )·\mathbf{k}=0$. Actually, the choice of the temporal gauge does not prevent us from defining longitudinal creation and annihilation functionals corresponding to $\mathit{ϵ}(\mathbf{k},\lambda )\parallel \mathbf{k}$. However, it can be easily checked that any state formed from the action of these longitudinal functionals would not satisfy the condition (53). Hence, it makes no sense to define them.

Consequently, the functional corresponding to a photon with momentum ${\mathbf{k}}_1$ and polarization $\lambda$ is calculated as

$$\begin{array}{cc}\hfill & W_1[\tilde{\mathbf{A}},\tilde{\mathbf{E}}]=a^{†}({\mathbf{k}}_1,\lambda )\star W_0[\mathbf{A},\mathbf{E}]\star a({\mathbf{k}}_1,\lambda )=\hfill \\ & 2{\left(2\pi \right)}^3{\omega }_{{\mathbf{k}}_1}[{\omega }_{{\mathbf{k}}_1}(\mathit{ϵ}({\mathbf{k}}_1,\lambda )·{\tilde{\mathbf{A}}}_T\left({\mathbf{k}}_1\right))({\mathit{ϵ}}^{\ast }({\mathbf{k}}_1,\lambda )·{\tilde{\mathbf{A}}}_T(-{\mathbf{k}}_1))+\hfill \\ & {\displaystyle \frac{1}{{\omega }_{{\mathbf{k}}_1}}}(\mathit{ϵ}({\mathbf{k}}_1,\lambda )·\tilde{\mathbf{E}}\left({\mathbf{k}}_1\right))({\mathit{ϵ}}^{\ast }({\mathbf{k}}_1,\lambda )·\tilde{\mathbf{E}}(-{\mathbf{k}}_1))-{\displaystyle \frac{\hslash }{2}}{\delta }^{\left(3\right)}\left(\mathbf{0}\right)+\hfill \\ \hfill & i\left((\mathit{ϵ}({\mathbf{k}}_1,\lambda )·{\tilde{\mathbf{A}}}_T\left({\mathbf{k}}_1\right))({\mathit{ϵ}}^{\ast }({\mathbf{k}}_1,\lambda )·\tilde{\mathbf{E}}(-{\mathbf{k}}_1))-\right.\hfill \\ & \left.(\mathit{ϵ}({\mathbf{k}}_1,\lambda )·\tilde{\mathbf{E}}\left({\mathbf{k}}_1\right))({\mathit{ϵ}}^{\ast }({\mathbf{k}}_1,\lambda )·{\tilde{\mathbf{A}}}_T(-{\mathbf{k}}_1))\right)]W_0[\tilde{\mathbf{A}},\tilde{\mathbf{E}}].\hfill \end{array}$$

(71)

To show that this state represents a photon of momentum ${\mathbf{k}}_1$ and energy ${\omega }_{{\mathbf{k}}_1}$, it is noted that the Hamiltonian and the momentum operator can be written as a function of $a^{†}$ and a as

$$\begin{array}{cc}\hfill & H={\displaystyle \frac{1}{2}}\int d^3\mathbf{x}\left[{\mathbf{E}}^2\left(\mathbf{x}\right)+{(\nabla \times \mathbf{A}\left(\mathbf{x}\right))}^2\right]=\hfill \\ \hfill & {\displaystyle \frac{\hslash }{2}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{\omega }_{\mathbf{k}}\sum _{\lambda }\left[a^{†}(\mathbf{k},\lambda )\star a(\mathbf{k},\lambda )+a(\mathbf{k},\lambda )\star a^{†}(\mathbf{k},\lambda )\right]\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill & P_j=\int d^3\mathbf{x}{\left[\mathbf{E}\left(\mathbf{x}\right)\times (\nabla \times \mathbf{A}(\mathbf{x}\left)\right)\right]}_j=\hfill \\ \hfill & {\displaystyle \frac{\hslash }{2}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{k}_j\sum _{\lambda }\left[a^{†}(\mathbf{k},\lambda )\star a(\mathbf{k},\lambda )+a(\mathbf{k},\lambda )\star a^{†}(\mathbf{k},\lambda )\right],\hfill \end{array}$$

(73)

where we have taken into account the completeness relation of the set of vectors $\left\{\mathit{ϵ}(\mathbf{k},1),\mathit{ϵ}(\mathbf{k},2),\mathbf{k}\right\}$:

$$\sum _{\lambda }{ϵ}_i(\mathbf{k},\lambda ){ϵ}_j^{\ast }(\mathbf{k},\lambda )={\delta }_{ij}-{\displaystyle \frac{k_i{k}_j}{{\mathbf{k}}^2}},$$

(74)

Therefore, in a completely analogous way as in the previous cases, we see that the functional $W_1[\mathbf{A},\mathbf{E}]$ defined in (71) corresponds to a photon with momentum ${\mathbf{k}}_1$ and energy ${\omega }_{{\mathbf{k}}_1}=\left|\mathbf{k}\right|$.

A final important quantity when analyzing the electromagnetic field is helicity:

$$h={\displaystyle \frac{\mathbf{S}·\mathbf{P}}{\left|\mathbf{P}\right|}},$$

(75)

where the spin $\mathbf{S}$ is given by $\mathbf{S}=\int d^3\mathbf{x}\mathbf{E}\left(\mathbf{x}\right)\times \mathbf{A}\left(\mathbf{x}\right)$. Expressing this functional in terms of the creation and annihilation functionals is an algebraically arduous task, mainly because the invariance under changes of basis of the polarization vectors is not as trivial as in the case of the Hamiltonian or the momentum. However, if one expresses the fields $\mathbf{A}$ and $\mathbf{E}$ as a function of the creation and annihilation functionals by inverting the relations (70), it is not difficult to show that linearly polarized states (i.e., states which are created by applying ladder functionals associated with real unit vectors $\mathit{ϵ}\left(\mathbf{k}\right)\perp \mathbf{k}$) are not eigenstates of the helicity functional. However, circularly polarized states,

$$\begin{array}{c}\hfill W_{\pm }=a^{†}(\mathbf{k},\pm )\star W_0\star a(\mathbf{k},\pm ),\end{array}$$

(76)

where $a^{†}(\mathbf{k},\pm )={\displaystyle \frac{1}{\sqrt{2}}}(a^{†}(\mathbf{k},1)\pm i{a}^{†}(\mathbf{k},2))$, and $a(\mathbf{k},\pm )={\displaystyle \frac{1}{\sqrt{2}}}(a(\mathbf{k},1)\mp ia(\mathbf{k},2))$, can be shown to be eigenstates of h with eigenvalue $\pm 1$. This confirms that phase-space Wigner functionals naturally disentangle intrinsic spin-1 helicity modes through complex polarization combinations without requiring explicitly covariant gauge-fixing terms. In conclusion, one can interpret the state $W_{\pm }$ as a photon with momentum, energy, and helicity given by $\mathbf{k}$, ${\omega }_{\mathbf{k}}$, and $h=\pm 1$.

6. Free Fermionic Field

The Wigner formalism for fermionic fields presents an obvious added difficulty, namely, the canonical commutation rules that characterized the scalar and electromagnetic fields become anti-commutation relations. In the Fock space formulation, this does not present a problem, since one simply postulates these relations on the field operators and performs calculations in the standard manner. However, when working with functionals, it is necessary to redefine the nature of the variables and the form of the star product.

This is immediately apparent when imposing the anti-commutation relations of the field operators. Since $i{\widehat{\psi }}^{†}$ is the conjugate momentum of $\widehat{\psi }$, it follows that

$$\begin{array}{cc}\hfill & \{{\widehat{\psi }}_a\left(\mathbf{x}\right),{\widehat{\psi }}_b\left(\mathbf{y}\right)\}=\{{\widehat{\psi }}_a^{†}\left(\mathbf{x}\right),{\widehat{\psi }}_b^{†}\left(\mathbf{y}\right)\}=0,\hfill \\ \hfill & \{{\widehat{\psi }}_a\left(\mathbf{x}\right),{\widehat{\psi }}_b^{†}\left(\mathbf{y}\right)\}=\hslash {\delta }_{ab}{\delta }^{\left(3\right)}(\mathbf{x}-\mathbf{y}).\hfill \end{array}$$

(77)

Consequently, the eigenvalues of such operators, which are functions of position $\mathbf{x}$, must also satisfy them. Defining the Moyal anti-bracket as

$$\begin{array}{cc}\hfill & {\left\{\{A[\psi ,{\psi }^{†}],B[\psi ,{\psi }^{†}]\}\right\}}_{anti}=\hfill \\ \hfill & A[\psi ,{\psi }^{†}]\star B[\psi ,{\psi }^{†}]+B[\psi ,{\psi }^{†}]\star A[\psi ,{\psi }^{†}],\hfill \end{array}$$

(78)

it follows that $\psi$ and ${\psi }^{†}$ obey the relations (77) under the Moyal anti-bracket. In order to satisfy them, we have to redefine the action of the star product:

$$\begin{array}{cc}\hfill & {\psi }_a\left(\mathbf{x}\right)\star ={\psi }_a\left(\mathbf{x}\right)+{\displaystyle \frac{\hslash }{2}}{\displaystyle \frac{\delta }{\delta {\psi }_a^{†}\left(\mathbf{x}\right)}},\hfill \\ \hfill & {\psi }_a^{†}\left(\mathbf{x}\right)\star ={\psi }_a^{†}\left(\mathbf{x}\right)+{\displaystyle \frac{\hslash }{2}}{\displaystyle \frac{\delta }{\delta {\psi }_a\left(\mathbf{x}\right)}}.\hfill \end{array}$$

(79)

However, this is not enough. If, with this definition, the first Moyal anti-bracket is calculated, one arrives at

$$\begin{array}{cc}\hfill & {\psi }_a\left(\mathbf{x}\right){\psi }_b\left(\mathbf{y}\right)+{\psi }_b\left(\mathbf{y}\right){\psi }_a\left(\mathbf{x}\right)=0,\hfill \\ \hfill & {\psi }_a^{†}\left(\mathbf{x}\right){\psi }_b^{†}\left(\mathbf{y}\right)+{\psi }_b^{†}\left(\mathbf{y}\right){\psi }_a^{†}\left(\mathbf{x}\right)=0.\hfill \end{array}$$

(80)

From this, we conclude that the functions $\psi \left(\mathbf{x}\right)$ and ${\psi }^{†}\left(\mathbf{x}\right)$ cannot be usual c-number variables that commute. Instead, they must be interpreted as Grassmann variables which, by definition, obey anti-commutation relations. That is, if ${\theta }_1,{\theta }_2$ are two Grassmann variables, then ${\theta }_1{\theta }_2+{\theta }_2{\theta }_1=0$. Therefore, from now on, all calculations will be performed taking into account the properties of this type of variables [25].

The fact that the fields anti-commute allows us to define the fermionic functional in phase space. Indeed, if the eigenvalues of the field operators were c-numbers, then ${\psi }^{†}\left(\mathbf{x}\right)$ would correspond to the adjoint spinor of $\psi \left(\mathbf{x}\right)$. Therefore, the conjugate momentum of the field would be completely determined by the field itself. This would render the definition of phase space meaningless, since the two variables are not independent. However, since the complex conjugate operation is not completely defined for Grassmann variables (more precisely, it is defined as any antilinear and idempotent application), the conjugate momenta are not determined by the fields and, consequently, can be treated as independent. In phase space, this non-commuting Grassmannian structure lifts the algebraic constraints imposed by first-order field equations, establishing true kinematic independence between the field and its momentum conjugate.

The Hamiltonian of fermionic field theory is the Dirac Hamiltonian:

$$H=\int d^3\mathbf{x}{\psi }^{†}\star \left(\mathbf{x}\right)(-i\mathit{\alpha }·\nabla +\beta m)\psi \left(\mathbf{x}\right),$$

(81)

where the matrices ${\alpha }_i$ ($i=1,2,3$) and $\beta$ can be taken as $\beta ={\gamma }_0$, ${\alpha }_i={\gamma }_0{\gamma }_i$, with ${\gamma }_{\mu }$ ($\mu =0,1,2,3$) being the Dirac matrices. In order to solve the eigenvalue equation, we will work directly in the Fourier space of the fields, i.e., expressing the Hamiltonian in terms of the creation and annihilation operators to avoid explicit manipulation of the Dirac matrices. Thus, the fields are expanded as

$$\begin{array}{cc}\hfill & {\psi }_a\left(\mathbf{x}\right)={\displaystyle \sum _{s=1}^4}\int {\displaystyle \frac{d^3\mathbf{p}}{{\left(2\pi \right)}^{3/2}}}{\displaystyle \frac{1}{\sqrt{{\omega }_{\mathbf{p}}}}}b(\mathbf{p},s)u_a(\mathbf{p},s)e^{i\mathbf{p}·\mathbf{x}}\hfill \\ \hfill & {\psi }_a^{†}\left(\mathbf{x}\right)={\displaystyle \sum _{s=1}^4}\int {\displaystyle \frac{d^3\mathbf{p}}{{\left(2\pi \right)}^{3/2}}}{\displaystyle \frac{1}{\sqrt{{\omega }_{\mathbf{p}}}}}{b}^{†}(\mathbf{p},s)u_a^{†}(\mathbf{p},s)e^{-i\mathbf{p}·\mathbf{x}},\hfill \end{array}$$

(82)

where the spinors $u(\mathbf{p},s)$, $s=1,\dots ,4$ satisfy

$$\begin{array}{cc}\hfill & (\cancel{p}-m)u(\mathbf{p},s)=0\Rightarrow (\mathbf{p}·\overrightarrow{\gamma }+m)u(\mathbf{p},s)=p_0{\gamma }^{0}u(\mathbf{p},s)={\omega }_{\mathbf{p}}{\gamma }^{0}u(\mathbf{p},s),\hfill \\ \hfill & u^{†}(\mathbf{p},s)u(\mathbf{p},r)={\omega }_{\mathbf{p}}{\delta }_{sr},\hfill \\ \hfill & \overline{u}(\mathbf{p},s)u(\mathbf{p},r)={\sigma }_s{\omega }_{\mathbf{p}}{\delta }_{sr}.\hfill \end{array}$$

(83)

Here, ${\sigma }_s=1$ if $s=1,2$ and ${\sigma }_s=-1$ if $s=3,4$. By directly substituting the field mode expansions into the classical phase-space Dirac Hamiltonian functional (81) and applying the corresponding Moyal star product rules, the expression simplifies to

$$H=\hslash {\displaystyle \sum _{s=1}^4}\int d^3\mathbf{p}{\sigma }_s{\omega }_{\mathbf{p}}{b}^{†}(\mathbf{p},s)\star b(\mathbf{p},s).$$

(84)

Also, the star product can be expressed in terms of $b,b^{†}$:

$$\begin{array}{cc}\hfill & b(\mathbf{p},s)\star =b(\mathbf{p},s)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta }{\delta b^{†}(\mathbf{p},s)}},\hfill \\ \hfill & b^{†}(\mathbf{p},s)\star =b^{†}(\mathbf{p},s)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta }{\delta b(\mathbf{p},s)}},\hfill \end{array}$$

(85)

which means that the anticommutation relations are now

$$\begin{array}{cc}\hfill & {\left\{\{b(\mathbf{p},s),b({\mathbf{p}}^{\prime },r)\}\right\}}_{anti}={\left\{\{b^{†}(\mathbf{p},s),b^{†}({\mathbf{p}}^{\prime },r)\}\right\}}_{anti}=0,\hfill \\ \hfill & {\left\{\{b(\mathbf{p},s),b^{†}({\mathbf{p}}^{\prime },r)\}\right\}}_{anti}={\delta }_{rs}{\delta }^{\left(3\right)}(\mathbf{p}-{\mathbf{p}}^{\prime }).\hfill \end{array}$$

(86)

It can be easily seen that the functionals $b(\mathbf{p},s=1,2)$ correspond to particles of positive energy, while $b(\mathbf{p},s=3,4)$ correspond to particles of negative energy. The label s is related to the spin of the particle.

With this information, we are ready to calculate the fundamental state of the free fermionic field. We start by noting that this state is not empty; instead, all negative energy states are occupied, in accordance with the Dirac sea picture. Therefore, one must construct a Wigner functional in which all states with positive energy are empty (vacuum) and all states with negative energy are filled. These conditions translate into the following relations:

$$\begin{array}{c}b(\mathbf{p},s=1,2)\star W_0[b,b^{†}]=W_0[b,b^{†}]\star b^{†}(\mathbf{p},s=1,2)=0,\\ b^{†}(\mathbf{p},s=3,4)\star W_0[b,b^{†}]=W_0[b,b^{†}]\star b(\mathbf{p},s=3,4)=0.\end{array}$$

(87)

Indeed, since the states with positive energy are empty, annihilating any of them cancels the functional. In the same way, since the states with negative energy are filled, creating any of them also cancels the vacuum functional, because two fermions cannot occupy the same state.

To simplify the calculation, let us assume that only one possible state exists. If it is empty, then the first of the conditions (87) must be satisfied, that is:

$$\begin{array}{cc}\hfill & b{W}_e[b,b^{†}]=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta W_e}{\delta b^{†}}},\hfill \\ \hfill & W_e[b,b^{†}]b^{†}=-{\displaystyle \frac{1}{2}}{W}_e[b,b^{†}]{\displaystyle \frac{\overleftarrow{\delta }}{\delta b}}.\hfill \end{array}$$

(88)

The solution to these equations is given by

$$W_e[b,b^{†}]=1+2b{b}^{†}=\exp \left(2b{b}^{†}\right),$$

(89)

where, of course, b and $b^{†}$ are Grassmann variables. On the other hand, if the state is occupied, then

$$\begin{array}{cc}\hfill & b^{†}{W}_o[b,b^{†}]=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta W_o}{\delta b}},\hfill \\ \hfill & W_o[b,b^{†}]b=-{\displaystyle \frac{1}{2}}{W}_o[b,b^{†}]{\displaystyle \frac{\overleftarrow{\delta }}{\delta b^{†}}}.\hfill \end{array}$$

(90)

In this case, the Wigner functional is

$$W_o[b,b^{†}]=1+2b^{†}b=\exp \left(2b^{†}b\right).$$

(91)

The swap in the ordering of b and $b^{†}$ between empty ($2b{b}^{†}$) and occupied ($2b^{†}b$) exponent factors directly manifests the Pauli exclusion principle within phase space. As can be seen, the difference between an occupied and an empty state is simply the order in which b and $b^{†}$ appear in the Wigner functional. Extrapolating from this result, it follows that the Wigner functional which represents the vacuum state for the free fermionic field is

$$\begin{array}{cc}\hfill & W_0[\psi ,{\psi }^{†}]=\mathcal{N}\exp \left(2\int d^3\mathbf{p}\sum _{s=1,2}b(\mathbf{p},s)b^{†}(\mathbf{p},s)+\right.\hfill \\ \hfill & \left.\sum _{r=3,4}{b}^{†}(\mathbf{p},r)b(\mathbf{p},r)\right).\hfill \end{array}$$

(92)

This explicit closed-form expression for the fermionic vacuum Wigner functional constitutes one of the central original results of this framework. By unifying the Grassmann degrees of freedom with the functional star-product mechanics, this functional establishes a rigorous cornerstone for constructing multiparticle excited states and evaluating phase-space correlation functions in field theories with fermions without resorting to standard operator-based second quantization a priori.

The energy of this state can be calculated using the expression for the Hamiltonian (84):

$$\begin{array}{c}\hfill H\star W_0=E_0{W}_0\Rightarrow E_0=-2\int d^3\mathbf{p}{\omega }_{\mathbf{p}}{\delta }^{\left(3\right)}\left(\mathbf{0}\right).\end{array}$$

(93)

This is consistent with the idea that the Dirac sea is occupied by all possible particles with negative energy.

Next, in order to create states with positive energy particles (which are the ones that can be observed experimentally), we simply apply the creation/annihilation functionals. For example, the state of a fermion with momentum $\mathbf{q}$ and spin $+1/2$ is represented by

$$\begin{array}{cc}\hfill & b^{†}(\mathbf{q},1)\star W_0[b,b^{†}]\star b(\mathbf{q},1)=\mathcal{N}\exp \left(2b^{†}(\mathbf{q},1)b(\mathbf{q},1)\right)\hfill \\ \hfill & \exp \left(2{\int }_{(\mathbf{p},s)\ne (\mathbf{q},1)}{d}^3\mathbf{p}\sum _{s=1,2}b(\mathbf{p},s)b^{†}(\mathbf{p},s)+\right.\hfill \\ \hfill & \left.\sum _{r=3,4}{b}^{†}(\mathbf{p},r)b(\mathbf{p},r)\right).\hfill \end{array}$$

(94)

That is, one must change the order of the product $b^{†}b$ for that particular momentum and spin. On the contrary, to create an antifermion with momentum $-\mathbf{q}$ and spin $-1/2$, it is necessary to destroy one of the particle states with negative energy (namely, the particle state with momentum $\mathbf{q}$ and spin $+1/2$), since that gap is interpreted as the presence of a positive energy antiparticle. Thus, the Wigner functional for this type of state is given by

$$\begin{array}{cc}\hfill & b(\mathbf{q},3)\star W_0[b,b^{†}]\star b^{†}(\mathbf{q},3)=\mathcal{N}\exp \left(2b(\mathbf{q},3)b^{†}(\mathbf{q},3)\right)\hfill \\ \hfill & \exp \left(2{\int }_{(\mathbf{p},s)\ne (\mathbf{q},3)}{d}^3\mathbf{p}\sum _{s=1,2}b(\mathbf{p},s)b^{†}(\mathbf{p},s)+\right.\hfill \\ \hfill & \left.\sum _{r=3,4}{b}^{†}(\mathbf{p},r)b(\mathbf{p},r)\right).\hfill \end{array}$$

(95)

The next step is to verify that these states represent particles with positive energy. Applying the Hamiltonian, we obtain

$$\begin{array}{cc}\hfill & H\star b^{†}(\mathbf{q},1)\star W_0\star b(\mathbf{q},1)=\hfill \\ \hfill & (E_0+\hslash {\omega }_{\mathbf{q}})b^{†}(\mathbf{q},1)\star W_0\star b(\mathbf{q},1),\hfill \end{array}$$

(96)

$$\begin{array}{cc}\hfill & H\star b(\mathbf{q},3)\star W_0\star b^{†}(\mathbf{q},3)=\hfill \\ \hfill & (E_0+\hslash {\omega }_{\mathbf{q}})b(\mathbf{q},3)\star W_0\star b^{†}(\mathbf{q},3),\hfill \end{array}$$

(97)

where we have used anti-commutation rules for the creation and annihilation functionals. As expected, the states that have been characterized as fermions and antifermions contribute with a positive energy to the field. To show that they correspond to particle and antiparticle, respectively, one can calculate the charge functional, which in fermionic field theory is given by

$$\begin{array}{c}Q=\int d^3\mathbf{x}{\psi }^{†}\left(\mathbf{x}\right)\star \psi \left(\mathbf{x}\right)={\displaystyle \sum _{s=1}^4}\int d^3\mathbf{p}{b}^{†}(\mathbf{p},s)\star b(\mathbf{p},s).\end{array}$$

(98)

Using this expression, a direct calculation shows that the fermion and antifermion states are eigenstates of Q, with eigenvalues $Q_0+1$ and $Q_0-1$, respectively. Here, $Q_0=2\int d^3\mathbf{p}{\delta }^{\left(3\right)}\left(\mathbf{0}\right)$ is the charge of the fundamental state. It is thus evident that the particle states have charge $+1$, while the antiparticle states have charge $-1$. We can also confirm that these states are associated with the corresponding momentum and spin (the computations are completely analogous to those performed for this and the previous fields). In this way, the free fermionic field has been completely characterized in phase space.

Note that the interpretation of the Dirac sea used for the calculation of the Wigner functional is just that, an interpretation. However, functionals characterized in this way perfectly describe well-defined states of particles and antiparticles of positive energy. If we wanted to eliminate the energy and charge of the “vacuum”, we would simply have to renormalize using, for example, normal ordering in the functionals.

7. Interaction Theories

Once the Wigner formalism for free fields has been established, it can be extended to interacting systems, such as the self-interacting scalar ${\varphi }^4$ theory or quantum electrodynamics (QED). The procedure mirrors the Rayleigh–Schrödinger perturbation method adapted to the functional phase space. The total Hamiltonian functional is decomposed into a free part and an interacting piece, $H=H^0+\lambda H^1$, where $\lambda \ll 1$ is the coupling constant. The stationary Wigner functionals and their corresponding energy eigenvalues are expanded in powers of $\lambda$:

$$\begin{array}{cc}\hfill & W_n[\varphi ,\pi ]=W_n^0[\varphi ,\pi ]+\lambda W_n^1[\varphi ,\pi ]+{\lambda }^2{W}_n^2[\varphi ,\pi ]+\dots ,\hfill \\ \hfill & E_n=E_n^0+\lambda E_n^1+{\lambda }^2{E}_n^2+\dots \hfill \end{array}$$

(99)

By inserting these expansions into the stationary functional star-eigenvalue equations, $H\star W_n=W_n\star H=E_n{W}_n$, and matching identical powers of $\lambda$, we obtain the hierarchical perturbative equations:

$$\begin{array}{cc}\hfill & E_n^0={\displaystyle \frac{\int \mathcal{D}\varphi \mathcal{D}\pi H^0[\varphi ,\pi ]W_n^0[\varphi ,\pi ]}{\int \mathcal{D}\varphi \mathcal{D}\pi W_n^0[\varphi ,\pi ]}},\hfill \end{array}$$

(100)

$$\begin{array}{cc}\hfill & E_n^1={\displaystyle \frac{\int \mathcal{D}\varphi \mathcal{D}\pi H^1[\varphi ,\pi ]W_n^0[\varphi ,\pi ]}{\int \mathcal{D}\varphi \mathcal{D}\pi W_n^0[\varphi ,\pi ]}},\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill & W_n^1[\varphi ,\pi ]=\sum _{m\ne n}{\displaystyle \frac{1}{E_n^0-E_m^0}}{W}_m^0[\varphi ,\pi ]{\displaystyle \frac{\int \mathcal{D}\varphi \mathcal{D}\pi H^1[\varphi ,\pi ]W_n^0[\varphi ,\pi ]}{\int \mathcal{D}\varphi \mathcal{D}\pi W_n^0[\varphi ,\pi ]}}+\dots \hfill \end{array}$$

(102)

To illustrate the explicit application of this perturbative phase-space formalism, let us consider a real scalar field with a $\lambda {\varphi }^4$ self-interaction, where the perturbation Hamiltonian is $H^1\left[\varphi \right]={\displaystyle \frac{1}{4!}}\int d^3\mathbf{x}{\varphi }^4\left(\mathbf{x}\right)$. To evaluate the first-order energy correction to the vacuum state, $E_0^1$, we employ the unperturbed Gaussian vacuum Wigner functional $W_0^0[\varphi ,\pi ]$ derived above. The first-order shift is given by the phase-space functional expectation value:

$$E_0^1={⟨H^1⟩}_0={\displaystyle \frac{1}{4!}}\int d^3\mathbf{x}{\displaystyle \frac{\int \mathcal{D}\varphi \mathcal{D}\pi {\varphi }^4\left(\mathbf{x}\right)W_0^0[\varphi ,\pi ]}{\int \mathcal{D}\varphi \mathcal{D}\pi W_0^0[\varphi ,\pi ]}}.$$

(103)

Since the integration over the conjugate momentum $\pi \left(\mathbf{x}\right)$ cancels out identically between the numerator and denominator, this reduces to a Gaussian functional integral over the configuration field $\varphi \left(\mathbf{x}\right)$. Utilizing Isserlis’ theorem (Wick’s theorem) within the functional integration domain, the four-point correlation function splits into permutations of two-point functions:

$${⟨{\varphi }^4\left(\mathbf{x}\right)⟩}_0=3{\left({⟨\varphi \left(\mathbf{x}\right)\varphi \left(\mathbf{x}\right)⟩}_0\right)}^2=3{\left({\displaystyle \frac{\hslash }{2}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{\sqrt{{\mathbf{k}}^2+m^2}}}\right)}^2.$$

(104)

Consequently, the first-order vacuum energy correction evaluates to

$$E_0^1={\displaystyle \frac{\lambda }{8}}\int d^3\mathbf{x}{\left({\displaystyle \frac{\hslash }{2}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{\sqrt{{\mathbf{k}}^2+m^2}}}\right)}^2.$$

(105)

This result matches perfectly with the standard Rayleigh–Schrödinger vacuum energy shift obtained via conventional canonical quantization in the Schrödinger picture. It should be noted that for the expectation value $E_0^1$ at this order, the Moyal ⋆-product acts trivially and reduces to standard functional phase-space integration. The non-trivial action of the ⋆-product enters explicitly when computing the first-order correction to the Wigner functional $W_n^1[\varphi ,\pi ]$, where field and momentum cross-derivatives survive, or in higher-order perturbative calculations.

To explicitly demonstrate this non-trivial action of the functional ⋆-product, we solve the first-order star-eigenvalue equation for the vacuum Wigner functional correction, $W_0^1[\varphi ,\pi ]$:

$${\displaystyle \frac{1}{2}}\left(H^0\star W_0^1+W_0^1\star H^0\right)-E_0^0{W}_0^1=E_0^1{W}_0^0-{\displaystyle \frac{1}{2}}\left(H^1\star W_0^0+W_0^0\star H^1\right).$$

(106)

The unperturbed Gaussian vacuum functional $W_0^0[\varphi ,\pi ]$ is defined with kernel $\Omega (\mathbf{x},\mathbf{y})=\sqrt{-{\nabla }^2+m^2}{\delta }^{\left(3\right)}(\mathbf{x}-\mathbf{y})$. When $H^1\left[\varphi \right]$ acts via the Moyal ⋆-product, functional momentum derivatives yield ${\displaystyle \frac{\delta }{\delta \pi \left(\mathbf{x}\right)}}{W}_0^0[\varphi ,\pi ]=-{\displaystyle \frac{4}{\hslash }}\left({\Omega }^{-1}\pi \right)\left(\mathbf{x}\right)W_0^0[\varphi ,\pi ]$, generating an effective complex translation in field configuration space, $\varphi \left(\mathbf{x}\right)\to \varphi \left(\mathbf{x}\right)-2i\left({\Omega }^{-1}\pi \right)\left(\mathbf{x}\right)$:

$$H^1\left[\varphi \right]\star W_0^0[\varphi ,\pi ]={\displaystyle \frac{1}{4!}}\int d^3\mathbf{x}{\left[\varphi \left(\mathbf{x}\right)-2i\int d^3\mathbf{y}{\Omega }^{-1}(\mathbf{x},\mathbf{y})\pi \left(\mathbf{y}\right)\right]}^4{W}_0^0[\varphi ,\pi ].$$

(107)

Taking the real, symmetric part required by Equation (106), the ⋆-product explicitly generates non-trivial cross-couplings between the field $\varphi$ and momentum $\pi$:

$${\displaystyle \frac{1}{2}}\left(H^1\star W_0^0+W_0^0\star H^1\right)={\displaystyle \frac{1}{4!}}\int d^3\mathbf{x}\left[{\varphi }^4\left(\mathbf{x}\right)-24{\varphi }^2\left(\mathbf{x}\right){\left({\Omega }^{-1}\pi \right)}^2\left(\mathbf{x}\right)+16{\left({\Omega }^{-1}\pi \right)}^4\left(\mathbf{x}\right)\right]W_0^0[\varphi ,\pi ].$$

(108)

Expanding $W_0^1[\varphi ,\pi ]=\mathcal{Q}[\varphi ,\pi ]W_0^0[\varphi ,\pi ]$ as a quartic polynomial in momentum space modes $\tilde{\varphi }\left(\mathbf{k}\right)$ and $\tilde{\pi }\left(\mathbf{k}\right)$, Equation (106) reduces mode-by-mode to yield the exact first-order non-Gaussian correction:

$$\begin{array}{cc}\hfill W_0^1[\varphi ,\pi ]=& -{\displaystyle \frac{\lambda }{4!}}\int {\displaystyle \prod _{i=1}^4}{\displaystyle \frac{d^3{\mathbf{k}}_i}{{\left(2\pi \right)}^{3/2}}}{\left(2\pi \right)}^3{\delta }^{\left(3\right)}\left({\displaystyle \sum _{i=1}^4}{\mathbf{k}}_i\right){\displaystyle \frac{1}{{\sum }_{i=1}^4{\omega }_{{\mathbf{k}}_i}}}\times \hfill \\ \hfill & \left[\tilde{\varphi }\left({\mathbf{k}}_1\right)\tilde{\varphi }\left({\mathbf{k}}_2\right)\tilde{\varphi }\left({\mathbf{k}}_3\right)\tilde{\varphi }\left({\mathbf{k}}_4\right)-{\displaystyle \frac{6}{{\omega }_{{\mathbf{k}}_3}{\omega }_{{\mathbf{k}}_4}}}\tilde{\varphi }\left({\mathbf{k}}_1\right)\tilde{\varphi }\left({\mathbf{k}}_2\right)\tilde{\pi }\left({\mathbf{k}}_3\right)\tilde{\pi }\left({\mathbf{k}}_4\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{1}{{\omega }_{{\mathbf{k}}_1}{\omega }_{{\mathbf{k}}_2}{\omega }_{{\mathbf{k}}_3}{\omega }_{{\mathbf{k}}_4}}}\tilde{\pi }\left({\mathbf{k}}_1\right)\tilde{\pi }\left({\mathbf{k}}_2\right)\tilde{\pi }\left({\mathbf{k}}_3\right)\tilde{\pi }\left({\mathbf{k}}_4\right)\right]W_0^0[\varphi ,\pi ].\hfill \end{array}$$

(109)

This explicit expression proves that the Moyal product deforms the vacuum Wigner functional by directly entangling spatial field fluctuations with momentum modes.

However, as noted previously, the present evaluation requires computing highly non-trivial multidimensional functional integrals. In more complex gauge frameworks like QED, incorporating interactions between the electromagnetic ladder functionals $a(\mathbf{k},\lambda )$ and the fermionic Grassmann elements $b(\mathbf{p},s)$ significantly complicates the algebra due to the duplication of the phase-space variables $(\varphi ,\pi )$ and the tracking of the star product across anticommuting fields. While structurally coherent, this confirms that the Wigner functional framework is less computationally efficient for calculating standard scattering matrices compared to covariant path-integral formalisms, remaining instead a powerful tool for investigating specialized real-time, non-equilibrium field dynamics and semiclassical expansions.

8. Conclusions

In this work, we have established a systematic and unified framework for functional quantum field theory in phase space, extending the Wigner–Moyal formalism from quantum-mechanical particle systems to infinite-dimensional field theories. By systematically mapping the stationary functional Schrödinger equations onto phase-space star-eigenvalue equations, we have successfully derived the analytical vacuum Wigner functionals for real and complex scalar fields, abelian gauge fields, and fermionic systems.

It is critical to reiterate that our construction operates within a non-manifestly covariant framework. By choosing equal-time hypersurfaces inherent to the functional Schrödinger representation, Lorentz covariance is expected to be maintained implicitly through the appropriate transformation laws of the underlying field configurations, rather than being manifestly displayed. While this formulation is conceptually robust for analyzing static properties and state structures, it is computationally less efficient for calculating standard scattering amplitudes and S-matrix elements when compared to traditional manifestly covariant path-integral formalisms.

Nevertheless, the primary virtue of this work does not lie in the introduction of isolated phase-space mathematical tools, which possess a rich pedigree spanning from the foundational works of Berezin, Marinov, Cahill, and Glauber to deformation quantization developments. Instead, its value rests on providing a self-consistent synthesis. By placing bosons and fermions under the exact same functional star-product architecture, we have elucidated how different spin statistics and ordering prescriptions manifest dynamically. In the gauge sector, this includes handling constraints via the phase-space analogue of Dirac quantization, while in the fermionic sector, it naturally recovers the physical features of the Dirac sea directly from the star-eigenvalue structure without requiring ad hoc charge-conjugation operator definitions. Furthermore, by explicitly solving the perturbative star-eigenvalue equation for $\lambda {\varphi }^4$ theory, we have derived the closed-form expression for the first-order vacuum Wigner functional $W_0^1[\varphi ,\pi ]$, demonstrating how the functional Moyal product acts non-trivially as a complex translation operator in field space to generate non-Gaussian mode entanglements between fields and canonical momenta.

Looking forward, the functional phase-space framework developed herein opens several promising avenues for future research. The perturbative scheme and explicit non-Gaussian Wigner functional solutions developed for interaction theories can be extended to compute higher-order loop corrections in ${\varphi }^4$ theory and QED, providing a novel perspective on renormalization constants directly within phase space. More importantly, the real strength of this formalism lies in its capacity to handle time-dependent, non-equilibrium processes and thermal field environments. Applying these functional Wigner methods to heavy-ion collisions, early-universe cosmology, and intense laser-field physics represents the natural next step in exploiting the unique advantages of phase-space quantum field theory.