[Paper Review] Coherent Distributions and Quantization

M. Grigorescu|arXiv (Cornell University)|Oct 6, 2014

Quantum chaos and dynamical systems25 references3 citations

TL;DR

This paper proposes a mathematical framework linking classical and quantum mechanics through coherent phase-space distributions, specifically Wigner functions and 'action waves,' by treating quantum wave functions as elementary degrees of freedom on a granular phase-space. It demonstrates that quantum behavior arises fundamentally from coherence properties of these distributions, with key results showing that Wigner functions of pure states have zero quantum entropy, while their phase-space entropy remains complex, highlighting a core distinction from classical statistical mechanics.

ABSTRACT

This work presents a selective review of results concerning the mathematical interface between the classical and quantum aspects encountered in problems such as the nuclear mean-field dynamics or quantum Brownian motion. It is shown that the main difference between classical and quantum behavior arises from the coherence properties of the phase-space distributions known as "action waves" and Wigner functions. The quantum wave functions appear as elementary degrees of freedom for the phase space granularity.

Motivation & Objective

• To bridge the conceptual gap between classical and quantum mechanics by analyzing the role of phase-space granularity and coherence in quantum distributions.
• To clarify why quantum fluctuations differ fundamentally from classical thermal fluctuations by examining the coherence structure of Wigner functions and action waves.
• To establish a mathematical interface between Hilbert space formalism and classical phase-space dynamics using coherent states and density operators.
• To investigate how quantum entropy and probability distributions emerge from the geometry of phase-space cells defined by wave functions.
• To demonstrate that the non-negativity of classical probability distributions does not hold for quantum Wigner functions, revealing a fundamental quantum signature.

Proposed method

• Uses Wigner functions as phase-space representations of quantum states, defined via the Weyl transform of density operators.
• Introduces 'quantum cells' as topological subspaces of phase space, with characteristic functions $\chi_{\psi}$ that are separable and correspond to pure states $|\psi\rangle\langle\psi|$.
• Defines a canonical phase-space measure with volume $\mathrm{h}^3$ per quantum cell, linking to the old quantum condition $\oint p\,dq = nh$.
• Applies the trace formula $\mathcal{S}_q(\rho) = -k_B \mathrm{Tr}(\hat{\bar{\rho}} \ln \hat{\bar{\rho}})$ to show that pure states have zero quantum entropy.
• Uses the inner product $ (\chi_\psi, \chi_{\psi'}) = \mathrm{h}^3 |\langle\psi|\psi'\rangle|^2 $ to define orthogonality and probability amplitudes in phase space.
• Analyzes the non-positivity of operators like $\hat{\Delta}_{q_0p_0}$, showing that Wigner functions can take negative values, violating classical probability rules.

Experimental results

Research questions

• RQ1 How do coherent phase-space distributions such as Wigner functions and action waves encode the difference between classical and quantum behavior?
• RQ2 Why do quantum fluctuations not arise from random forces like classical thermal fluctuations, and what is the role of coherence in this distinction?
• RQ3 What is the mathematical structure of quantum phase-space cells, and how do they relate to the old quantum conditions and Hilbert space states?
• RQ4 How does the entropy of a quantum state defined via the Wigner function differ from classical entropy, and what does this imply for the quantum-to-classical transition?
• RQ5 Can the non-negativity of classical probability distributions be preserved in quantum phase-space representations, and what are the consequences of negative Wigner functions?

Key findings

• The Wigner function $\rho_\psi$ of a pure quantum state has zero quantum entropy $\mathcal{S}_q(\rho_\psi) = 0$, indicating maximal quantum coherence.
• The phase-space entropy $\mathcal{S}(\rho_\psi) = -k_B(\rho_\psi, \ln \bar{\rho}_\psi)$ is generally complex, showing that quantum phase-space distributions cannot be interpreted as classical probability densities.
• Negative values in Wigner functions, such as $\rho_\pm(0,0) = \pm 2/\mathrm{h}$, demonstrate that quantum states cannot be represented by non-negative phase-space distributions.
• The inner product $ (\chi_\psi, \chi_{\psi'}) = \mathrm{h}^3 |\langle\psi|\psi'\rangle|^2 $ ensures orthogonality of distinct quantum cells, preserving quantum superposition in phase space.
• For a Gaussian superposition state $\psi_\pm$, the Wigner function exhibits interference terms $\rho_i(x,p) \propto \cos(2dp/\hbar)$, which are responsible for the orthogonality $ (\rho_+, \rho_-) = 0 $.
• The operator $\hat{\Delta}_{00} = 2\hat{\Pi}_0/\mathrm{h}$, representing a delta function in phase space, is not positive-definite, proving that classical phase-space projectors do not extend to quantum mechanics.