arxiv: 2604.20776 · v1 · submitted 2026-04-22 · 🪐 quant-ph · math-ph· math.MP

Recognition: unknown

Path integral formulation of finite-dimensional quantum mechanics in discrete phase space

Andres F. Gomez, Leonardo A. Pachon

Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:31 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords path integraldiscrete phase spaceWigner functionfinite-dimensional quantum mechanicsqutritentanglement dynamicsWeyl transformClifford covariant

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The pith

A path integral in discrete phase space reproduces classical Hamiltonian flow for linear systems at lattice-commensurate times while requiring all fluctuation sectors for entanglement.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a sum-over-paths representation for the propagator of the discrete Wigner function on the finite lattice Z_d x Z_d. It begins with the Weyl transform built from generalized displacement operators and constructs an exact short-time evolution kernel whose composition law is iterated to produce a discrete phase-space action. For Hamiltonians linear in the coordinates the fluctuation sum factorizes and, at times that are integer multiples of the lattice spacing, reduces exactly to a deterministic shift operator. Explicit calculations on one and two qutrits show that the linear entropy of entanglement is reproduced only when every fluctuation sector contributes coherently to the sum.

Core claim

Starting from the discrete Wigner function and its associated Weyl transform on Z_d x Z_d for odd prime d, an exact evolution kernel is derived whose iteration yields a path-integral expression weighted by a discrete phase-space action. For linear Hamiltonians this sum factorizes; in the Clifford-covariant regime the fluctuations collapse to a deterministic lattice shift that is the finite-dimensional counterpart of classical Hamiltonian flow. The two-qutrit example demonstrates that the closed-form linear entropy is obtained only from the coherent superposition of all fluctuation sectors of the action.

What carries the argument

The discrete evolution kernel generated by the Weyl transform on Z_d x Z_d, whose iterated composition produces the phase-space action that weights the sum over paths and factorizes for linear Hamiltonians.

If this is right

Linear Hamiltonians produce a propagator that is exactly a lattice shift at commensurate times.
The linear entropy of two interacting qutrits is given by a closed expression valid at all times only when all fluctuation sectors are retained.
The zero-fluctuation sector alone produces a non-real kernel at finite time steps and a uniform kernel in the continuum limit.
The construction supplies a concrete route to semiclassical simulation of many-body spin systems via discrete paths.
Wigner negativity can be tracked along the retained paths to characterize non-classicality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same path-sum structure may allow controlled truncation of fluctuation sectors for approximate dynamics in larger spin chains.
Because the commensurate-time limit aligns with Clifford operations, the formulation is compatible with discrete error-correcting codes.
Extension to nonlinear Hamiltonians would identify the point at which factorization fails and genuine quantum interference appears.
The discrete action offers a finite-system probe of the quantum-to-classical transition without requiring the large-d limit.

Load-bearing premise

The discrete Wigner function and Weyl transform admit a composition law for the evolution kernel that can be iterated through short-time steps without additional finite-dimensional corrections.

What would settle it

Evaluate the path-integral sum for a linear Hamiltonian at a time step commensurate with the lattice and compare the resulting operator to the exact unitary evolution applied to an initial discrete Wigner function; any mismatch falsifies the claimed collapse to a deterministic shift.

read the original abstract

We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function defined on $\mathbb{Z}_d \times \mathbb{Z}_d$ (with $d$ an odd prime), and the associated Weyl transform built from generalized displacement operators, we derive an exact evolution kernel that propagates the discrete Wigner function in time. By exploiting the composition law of the kernel and iterating the short-time approximation, we obtain a sum-over-paths expression for the propagator weighted by a discrete phase-space action that is the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates, we show that the fluctuation sum factorizes and, at times strictly commensurate with the lattice (the Clifford-covariant regime), collapses to a deterministic shift realizing the discrete analog of classical Hamiltonian flow. The formulation is applied to a single qutrit ($d=3$) under a diagonal Hamiltonian, and to two interacting qutrits, where we show explicitly that the full entanglement dynamics -- captured by a closed-form expression for the linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors of the action. The $\tilde\mu = 0$ sector alone is non-real at finite time step and collapses to a trivial (uniform) kernel in the continuum limit, failing to reproduce the entanglement dynamics in either regime. We discuss the relevance of this construction for semiclassical simulation of many-body spin systems and the characterization of non-classicality through Wigner negativity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers a discrete path integral that correctly requires all fluctuation paths for entanglement in qutrit systems.

read the letter

The punchline is that this work gives a path-integral formulation for finite-dimensional quantum mechanics in discrete phase space, and demonstrates that the full set of fluctuation sectors is necessary to reproduce entanglement dynamics. The construction starts from the discrete Wigner function and Weyl transform on Z_d × Z_d for odd prime d. It derives an exact short-time kernel and then composes it into a sum over paths weighted by a discrete action. For linear Hamiltonians, the fluctuations factorize, and in the Clifford-covariant regime the sum collapses to a deterministic classical-like shift. The qutrit examples are explicit: a single qutrit with diagonal Hamiltonian and two interacting qutrits with a closed-form linear entropy valid for all times. Only the complete sum matches the entanglement; the zero-fluctuation sector alone gives a trivial kernel that fails in both finite and continuum limits. This is new in the specific discrete Marinov-like action and the factorization result. The paper does well by keeping everything within the established discrete phase-space tools without introducing free parameters or fitting. The logic flows from the composition law outward, and the applications confirm the claims for d=3. Soft spots are limited. The iteration of the short-time approximation assumes the kernel composition holds without corrections, which works in the presented cases but could be tested more broadly for nonlinear Hamiltonians. The discussion of many-body spin systems is forward-looking rather than supported by explicit calculations here. No internal contradictions show up. This paper is for people already working with discrete Wigner functions or path integrals in finite dimensions. A reader looking for concrete tools to simulate or analyze entanglement in small spin systems will find usable results. It deserves a serious referee because the central claims are verifiable from the derivations and examples provided. I recommend sending it for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a path integral representation for the dynamics of finite-dimensional quantum systems entirely in discrete phase space. Starting from the discrete Wigner function on Z_d × Z_d (d odd prime) and the associated Weyl transform, it derives an exact evolution kernel, then obtains a sum-over-paths expression for the propagator by iterating the short-time kernel using the composition law. For Hamiltonians linear in the phase-space coordinates the fluctuation sum factorizes; at lattice-commensurate times in the Clifford-covariant regime it reduces to a deterministic shift. The formulation is applied to a single qutrit and to two interacting qutrits, where a closed-form linear entropy valid for all times is given and shown to require coherent contributions from all fluctuation sectors of the action (the μ̃=0 sector alone fails to reproduce entanglement).

Significance. If the central derivation holds, the work supplies a discrete analog of the Feynman path integral that is free of adjustable parameters and directly applicable to spin systems. Notable strengths are the explicit construction of the kernel, the factorization result for linear Hamiltonians, the verification on d=3, and the closed-form entropy expression that isolates the role of fluctuation sectors. These features support potential use in semiclassical simulation of many-body systems and in characterizing non-classicality via Wigner negativity.

minor comments (2)

The abstract states that the discrete action is 'the natural finite-dimensional counterpart of Marinov's functional'; a short explicit comparison (or reference to the relevant continuous expression) in the introduction would assist readers who are not already familiar with that construction.
In the two-qutrit application the closed-form linear entropy is presented as valid for all times; confirming that every algebraic step from the path sum to this expression is written out (rather than summarized) would improve reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, the recognition of its strengths in constructing an exact discrete phase-space path integral, the factorization for linear Hamiltonians, and the explicit demonstration on qutrit systems that all fluctuation sectors are required for entanglement dynamics. We appreciate the recommendation for minor revision. No specific major comments were listed in the report, so we have no individual points to address. We are prepared to make any editorial or minor clarifications requested by the editor.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from established discrete Wigner/Weyl starting point

full rationale

The paper begins from the established discrete Wigner function and associated Weyl transform on Z_d × Z_d (d odd prime), which are prior, externally defined objects. It then derives an exact short-time evolution kernel via the composition law of the Weyl operators and iterates that kernel to obtain the path-sum representation. For linear Hamiltonians the factorization of the fluctuation sum follows algebraically from the kernel composition once the short-time approximation is accepted; the deterministic shift in the Clifford-covariant regime is a direct consequence of that algebra rather than a fitted or self-referential quantity. The explicit verification for d=3 (single qutrit and two-qutrit entanglement) uses the derived closed-form expressions without reducing any final result to a parameter fit or to a self-citation whose content is itself unverified. No step matches any of the enumerated circularity patterns; the central claims remain independent of the inputs once the discrete phase-space representation is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard properties of the discrete Wigner function for odd-prime dimension without introducing new fitted parameters or postulated entities in the abstract.

axioms (1)

domain assumption The discrete Wigner function on Z_d x Z_d (d odd prime) together with the Weyl transform built from generalized displacement operators admits an exact evolution kernel whose composition law can be iterated.
Invoked as the starting point for deriving the path-integral representation.

pith-pipeline@v0.9.0 · 5586 in / 1357 out tokens · 28614 ms · 2026-05-10T00:31:55.538153+00:00 · methodology

discussion (0)

Reference graph

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