Formulations of the Wigner function in Quantum Field Theory (QFT in Phase Space)

Question

I'm studying the phase space formulation of quantum field theory for my final degree project, and I have found two very different ways to construct the Wigner funtion. In the first method, a phase space is constructed as the tensor product of the Hilbert space which represent fields defined in configuration space. More precisely, the Minkowski space is equipped with a cotangent bundle $T^{*}\mathbb{M} \equiv \Gamma$ with coordinates $(q^{\mu},p_{\mu})$, with $\mathcal{H}(\Gamma)$ being the Hilbert space of square-integrable functions $\psi(q,p)$. Then, the operators $Q^{\mu} = q^{\mu}+\frac{i\hbar}{2}\partial_{p_{\mu}}$ and $P_{\mu} = p_{\mu}-\frac{i\hbar}{2}\partial_{q^{\mu}}$ are introduced, which provide $\mathcal{H}(\Gamma)$ with a non-conmutative algebra. Lastly, the Wigner function is defined as $f_{W} = \phi(q,p)\star \phi^{\dagger}(q,p)$, where $\phi(q,p)$ is the solution to the corresponding equation (Klein-Gordon, Dirac, etc.) written in terms of $Q^{\mu}, P_{\mu}$. In summary, this approach uses the phase space as the labels where the different fields are defined, and constructs a "phase space wave function" which, from what I have read, is not very conventional. Of course, I have oversimplyfied the theory and skipped over numerous details which provide mathematical sense. This method was developed by Amorim et al. and is explained in their article Quantum Field Theory in Phase Space link to the article. The second method is the one developed by Mrówczy´nski et al. in their article Wigner Functional Approach to Quantum Field Dynamics, link to the article. In this one, the fields $\Phi(x)$ are defined in each point of the Minkowski space (like a standard field), and not in the cotangent bundle $T^*\mathbb{M}$. However, the Wigner function (or rather, the Wigner functional) is defined as $$W[\Phi(x),\Pi(x)] = \int \mathcal{D}\phi'(x)\ exp\left[-i\int dx \Pi(x)\phi'(x)\right] \langle\Phi(x)+\frac{1}{2}\phi'(x)|\hat{\rho}(t)|\Phi(x)-\frac{1}{2}\phi'(x)\rangle,$$ i.e. the Wigner functional depends on the field and its conjugate momentum, which contitute the "phase space" in this theory. To summarize, in the first method, we have the classical phase space, which constitutes the space in which the fields are defined and, in the second one, the phase space aspect is represented by the field operators and their conjugate momenta. As you can see, these two approaches are very different. My questions are:

• What is the physical meaning behind the difference in how phase space is defined in each theory?
• Are these methods used in different areas of Theoretical Physics?
• Are both equally valid?

I know these topics are not known by many people, but I would appreciate if someone with experience in these field helped me out, since I am a bit lost and I need a little context.

Answer 1

There are some problems in trying to define a quasi-probability distribution in phase space for quantum fields, à la Wigner.

The foremost complication is given by the lack of a universal reference measure in an infinite dimensional vector (e.g. Hilbert) space. For finite dimensional vector spaces, there is the Lebesgue measure, and thus it makes sense to define a probability (or quasi-probability) distribution with respect to the Lebesgue measure. The Wigner function provides such a way, consistently with taking the classical limit (in the limit in fact the Wigner function becomes a true probability distribution).

The second method cited by the OP is trying to do something of this kind, however by using a mathematically ill-defined or at least ambiguous object in the form of the reference path measure "$\mathcal{D}\phi(x)$. What is the measure chosen? The Gaussian (free) one? Some equilibrium measure for an interacting field theory? These measures behave very differently, and are very tricky to define even in the simplest cases. Even if the measure is unambiguously chosen, it would not be good for all quantum states: roughly speaking, the gaussian free measure is not suited to describe interacting states (e.g. the interacting vacua).

There is however a way to define an object similar, at least in spirit, to the Wigner function also for quantum fields. A quantum state (be it of a theory with finitely or infinitely many degrees of freedom) is a noncommutative probability. The Wigner way of characterizing this noncommutativity is by defining a phase space distribution that is not necessarily positive. Another way, that can be generalized more easily to bosonic theories, is to look at Fourier transforms of measures. Let us remain finite dimensional for the moment. The well-known Bochner theorem states that there is a bijection between finite measures and the continuous functions of positive type whose value in zero coincides with the total mass of the measure. The bijection map is given by the Fourier transform (here written directly in phase space terms for convenience): $$ \hat{\mu}(\xi,\eta)= \int_{\mathbb{R}^{2d}} e^{i(\xi\cdot q + \eta\cdot p)} \mathrm{d}\mu(q,p)\;. $$ If one now quantizes the character $e^{i(\xi\cdot q + \eta\cdot p)}$ to an operator, and replaces the integral with a trace w.r.t. a quantum state $\rho_{\hbar}$ (density matrix) obtains a noncommutative Fourier transform: the quantization of the character is the so-called Weyl operator $W_{\hbar}(\xi,\eta)$, thus $$\hat{\rho}_{\hbar}(\xi,\eta)=\mathrm{tr}[\rho_{\hbar} W_{\hbar}(\xi,\eta)]\; .$$ It turns out that the noncommutative Fourier transform is very nice, in particular behaves very similarly to the Fourier transform of a measure: it is continuous, its value in zero is the trace of the state, and it is of almost positive type. With respect to positive type functions, there is a quantum correction in the form of relative phases of the type $e^{i\hbar \sigma(\xi,\eta)}$, where $\sigma(\xi,\eta)$ is the canonical symplectic form associated to the (dual of the) phase space. This is analogous to the non-positivity of the Wigner function, and the appearence of the symplectic form is a clear signature of the phase space structure. Finally, taking suitably the limit $\hbar\to 0$, there exists a classical measure $\mu$ such that the noncommutative Fourier transform converges to the classical one: $$\lim_{\hbar\to 0} \hat{\rho}_{\hbar}(\xi,\eta) = \hat{\mu}(\xi,\eta)\; .$$ Therefore, the quantum description correctly reduces to the classical one in the limit $\hbar\to 0$.

The advantage of this formulation with respect to Wigner's one is that the extension to infinitely many degrees of freedom is possible, without (too many) difficulties. Bosonic quantum fields are indeed defined starting from the Weyl operators, and their canonical commutation relations. In QFTs it is customary to take the complexification of the phase space as the reference space (roughly speaking, taking $z=q+ip$ and $\bar{z}=q-ip$ as canonical variables). In addition, as we have seen the character is defined with respect to the dual space with variables $(\xi,\eta)$ (let us call $\zeta = \xi +i\eta$ its complexification). However, while in finite dimensions any vector space is isomorphic to its dual this is not true for infinite dimensional spaces (in addition, there are several possible dual spaces). To distinguish between the phase space of fields, and its dual space, let us call $z$ the (classical) field and $\zeta$ a test function. So the algebra of Weyl operators, encoding the canonical commutation relations, is given by identifying the space of test functions. Such space shall be infinite dimensional, and endowed with a symplectic form that enters in the CCR (as it entered in the noncommutative Fourier transform). This symplectic form is usually defined starting from the canonical poisson bracket between a field and its momentum. In the complexified version, the standard choice is to define an inner product on the infinite dimensional complex space of test functions, from which the symplectic form is obtained by taking the imaginary part of the inner product (and considering the complex vector space as a real space with "twice many" variables). A standard example is to take the test functions in the Schwartz space of rapidly decreasing functions $\mathscr{S}$ (with inner product given by the standard $L^2$ inner product), and the tempered distributions $\mathscr{S}'$ as the (complexified) phase space of classical fields. This choice amounts to the canonical poisson bracket $$\{\bar{z}(x),z(y)\}= i \delta(x-y)\; .$$

Now, given a bosonic quantum field state $\rho_{\hbar}$, the noncommutative Fourier transform is again defined as $$\hat{\rho}_{\hbar}(\zeta)= \mathrm{tr}[\rho_{\hbar} W_{\hbar}(\zeta)]\;,$$ the same as for finite dimensions. Again, a noncommutative form of the Bochner theorem holds (with some technical restrictions): there is a bijection, given by the noncommutative Fourier transform, between regular$^{1}$ states and complex-valued functionals of test functions that are continuous when restricted to finite dimensional subspaces, whose value in zero coincides with the trace of the state, and that are of almost positive type. An analogous theorem, in the commutative case, holds for so-called cylindrical measures (an object that is more general than measures, and exists only in infinite dimensional spaces; this object is the correct classical counterpart of a quantum state in QFT). So again, the parallel between classical and noncommutative probability is done effectively. Furthermore, as in the finite dimensional case, taking suitably the limit $\hbar\to 0$ there exists a cylindrical measure $m$ such that $$\lim_{\hbar\to 0} \hat{\rho}_{\hbar}(\zeta) = \hat{m}(\zeta)\; .$$

Let me conclude with a small remark on fermionic theories. Since for fermions the Weyl operators (exponentials) are not available, one should use the noncommutative version of the moments of a probability distribution (essentially the trace of arbitrary products of smeared fields). In addition, for fermions there are currently no results available concerning the limit $\hbar\to 0$, and recovering a classical field theory.

References.

Noncommutative Fourier Transform and Bochner Theorem:

1. I. Segal. Foundations of the theory of dyamical systems of infinitely many degrees of freedom. II. Canad. J. Math., 13:1–18, 1961.
2. M. Merkli. The ideal quantum gas, in Open Quantum Systems I. Lecture Notes in Mathematics. 2006

Classical limit of states in bosonic QFTs: limit of Fourier transforms:

1. Z. Ammari, F. Nier. Mean field limit for bosons and infinite dimensional phase-space analysis. Ann. Henri Poincaré, 9(8):1503– 1574, 2008.
2. M. Falconi. Cylindrical Wigner Measures. Documenta Math. 23:1677–1756, 2018.

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1: Regular states are defined rather abstractly, but are essentially the only type of physically relevant quantum states, I won't make a lengthy and probably unnecessary discussion about it here.