Phase Space Quantum Theory

• Phase space quantum theory is a framework where quantum states and observables are represented on a classical phase space, offering a bridge between quantum and classical dynamics.
• It employs tools like the Wigner function, Weyl symbol, Moyal product, and Bopp operators to systematically incorporate quantum corrections into classical trajectories.
• The approach supports semiclassical methods such as the truncated Wigner approximation, facilitating the simulation of complex many-body systems including cold atom physics.

Phase space quantum theory refers to a formulation of quantum mechanics and quantum dynamics in which states, observables, and the evolution of the system are represented in a classical phase space such as the space of coordinates $(x)$ and momenta $(p)$. This framework offers a direct connection to classical mechanics while simultaneously organizing quantum corrections systematically. The phase space approach is realized through the Wigner–Weyl representation, the Moyal product, the use of Bopp operators, path integral constructions in the Heisenberg representation, and semiclassical approximations such as the truncated Wigner approximation. The methodology finds practical utility in analyzing and simulating interacting bosonic and spin systems, notably in the context of cold atom physics.

1. Phase Space Representation and Fundamental Objects

The central objects of phase space quantum theory are the Wigner function (the Weyl transform of the density matrix) and the Weyl symbol of operators. For any operator $\hat{\Omega}(x, \hat{p}, t)$, its expectation value is given by integrating the product of the Wigner function $W(x, p)$ and the operator’s Weyl symbol $\Omega_W(x, p, t)$: $$\langle \hat{\Omega}(x, \hat{p}, t) \rangle = \int dx\, dp\, W(x, p) \Omega_W(x, p, t)$$ The Weyl symbol provides a map from quantum operators to functions on phase space, enabling an explicit relationship between the operator calculus and classical observables. The product of operators is mapped to the Moyal product, a noncommutative deformation of the pointwise product: $$(A \ast B)(x, p) = A(x, p) \exp\left[ -\frac{i\hbar}{2} \left(\overleftarrow{\partial}_x \overrightarrow{\partial}_p - \overleftarrow{\partial}_p \overrightarrow{\partial}_x\right) \right] B(x, p)$$ In the classical limit $\hbar \to 0$, the Moyal bracket reduces to the usual Poisson bracket, thereby recovering classical Hamiltonian dynamics.

Bopp operators provide a representation for coordinate and momentum operators in the Heisenberg picture as: $$\hat{x}(t) \rightarrow x(t) + \frac{i\hbar}{2} \frac{\partial}{\partial p(t)} \qquad \hat{p}(t) \rightarrow p(t) - \frac{i\hbar}{2} \frac{\partial}{\partial x(t)}$$ These are critical in organizing perturbative expansions in quantum fluctuations.

2. Classical Limits: Corpuscular, Wave, and Bloch

The phase space approach accommodates distinct classical limits via different representations:

• Corpuscular (Newtonian) limit: Phase space is spanned by particle coordinates and momenta. The classical (Hamiltonian or Newtonian) dynamics are recovered, and all quantum corrections vanish as $\hbar \to 0$.
• Wave (Gross–Pitaevskii) limit: Using the coherent state representation, phase space is built from the complex amplitudes $\psi, \psi^\ast$ associated with bosonic field operators. In the mean-field limit, the Gross–Pitaevskii equation

$$i\hbar \frac{d\psi}{dt} = \frac{\partial H_W}{\partial \psi^*}$$

is recovered and quantum corrections are organized, e.g., in $1/N$ (inverse particle number).

• Bloch (spin) limit: Spin coherent states realize a phase space for spin systems, and in the $S \to \infty$ limit (large total spin), quantum spin operators map onto classical vectors with fluctuations of order $1/S$.

In each situation, the unique classical trajectory (saddle-point solution) dominates, and quantum corrections are managed as expansions in small parameters ($\hbar$, $1/N$, $1/S$).

3. Semiclassical Truncated Wigner Approximation and Quantum Corrections

A cornerstone of the phase space approach is the Truncated Wigner Approximation (TWA), a leading-order semiclassical method. In TWA, phase space variables are initialized by sampling from the Wigner function, and their evolution follows the classical Hamiltonian (or Gross–Pitaevskii) equations.

The TWA predicts observables via

$$\langle \hat{\Omega}(x, \hat{p}, t)\rangle \approx \int dx_0\, dp_0\, W_0(x_0, p_0)\, \Omega_W(x_{\text{cl}}(t), p_{\text{cl}}(t), t)$$

where classical trajectories $(x_{\text{cl}}(t), p_{\text{cl}}(t))$ solve the deterministic equations of motion.

Beyond TWA, quantum corrections are systematically included by expanding around classical trajectories, either as nonlinear response terms (higher derivatives acting on observables and potentials) or through the introduction of stochastic quantum jumps (e.g., sudden changes in variables at intermediate times). For instance, quantum corrections to an observable arise as: $$\Omega_W(x, t) \rightarrow \Omega_W(x, t) - \int d\tau\, \frac{1}{3! 2^2} \frac{\hbar^2}{i^2} \frac{\partial^3 V}{\partial x^3} \frac{\partial^3 \Omega_W}{\partial p^3} + \dots$$ or equivalently as a stochastic process where cubic noise, characterized by vanishing lower moments and a unit third moment, is injected.

All corrections are manifestly organized in powers of $\hbar^2$.

4. Path Integral Formulation and the Keldysh Technique

The phase space representation is fundamentally rooted in the path integral approach. The evolution operator is written using the Schwinger–Keldysh contour, duplicating the degrees of freedom as forward and backward paths. Average ("classical") and difference ("quantum") fields produce a natural separation, with quantum corrections emerging from fluctuations about the classical saddle point.

Essential phase space structures arise directly from this formalism:

• The Wigner function is obtained by integrating over initial "quantum" fluctuations.
• The Weyl symbol appears as the Fourier transform of operator matrix elements.
• The Moyal product arises as the mapping of operator products to phase space via an exponential of differential operators.
• Bopp operators emerge upon expanding to first order in quantum variables.

This approach is closely connected to the Keldysh technique, which also involves a doubling of fields and yields classical-quantum kinetic equations. The phase space perspective, however, emphasizes the flow of probability and observables in phase space with explicit quantum corrections.

5. Applications in Interacting and Many-Body Systems

Demonstrations in the paper cover a diverse class of many-body systems:

• Harmonic oscillator: The Wigner function is a Gaussian. Classical motion reproduces exact quantum moments for linear systems.
• Mexican hat (quartic) potential: TWA accurately describes short-time dynamics; quantum corrections become significant for long-time tails, revival phenomena, and tunneling.
• Sine–Gordon model: The discretized model obeys phase space Gross–Pitaevskii-like equations. By tuning the quantum (nonlinearity) parameter $\beta$, the accuracy of the semiclassical expansion can be probed.
• Bose–Hubbard model: For one and two dimensions, TWA (with sampled initial Wigner functions) tracks dynamics of observables such as momentum occupation and energy in the presence of interaction quenches. Systematic quantum corrections improve agreement with exact diagonalization for small lattices and extend tractability to large systems.
• Spin systems / Dicke model / Landau–Zener problem: Using spin coherent states and Schwinger–boson representations, the phase space approach efficiently treats large-$S$ spin problems and mixed spin-boson models, accurately capturing mean values and fluctuations.

The method is particularly powerful for dynamical and non-equilibrium phenomena in cold atom systems, where direct quantum simulations become computationally prohibitive.

6. Summary and Significance

Phase space quantum theory, as developed in the phase space representation of quantum dynamics, provides a rigorous, transparent, and practical connection between quantum and classical physics. By recasting quantum evolution as a flow in phase space governed by Hamiltonian equations with systematically computable quantum corrections (via Bopp operators or stochastic jumps), it produces a controlled expansion in small parameters. The approach is deeply intertwined with the Wigner–Weyl quantization, the path integral formalism, and the Keldysh technique.

This framework facilitates efficient simulation and understanding of dynamics in systems with large numbers of degrees of freedom, particularly in cold atom physics, and enables the quantitative paper of quantum-to-classical correspondence, semiclassicality, and the roles of quantum fluctuations in non-equilibrium processes. The essential phase space methods—Wigner function, Weyl symbol, Moyal product, Bopp operators, and their emergence from first principles—provide both conceptual insight and practical computational tools for quantum dynamics.