Ordering, symbols, and finite-dimensional approximations of path integrals

is infinitesimal time interval determining the accuracy of the approximations. A new class of such approximations is found for both c-number and Grassmannian dynamical variables. The actions determined by the approximations are non-local and have no classical continuum limit except the cases of

pq

- and

qp

-ordeeing. As an explicit example the fermionic oscillator is considered in detail.

Related papers

Ordering, Symbols and Finite-Dimensional Approximations of Path Integrals

T. Kashiwa, S. Sakoda, S. V. Zenkin

We derive general form of finite-dimensional approximations of path integrals for both bosonic and fermionic canonical systems in terms of symbols of operators determined by operator ordering. We argue that for a system with a given quantum Hamiltonian such approximations are independent of the type of symbols up to terms of O(ε), where ε is infinitesimal time interval determining the accuracy of the approximations. A new class of such approximations is found for both c-number and Grassmannian dynamical variables. The actions determined by the approximations are non-local and have no classical continuum limit except the cases of pq- and qp-ordering. As an explicit example the fermionic oscillator is considered in detail.We derive general form of finite-dimensional approximations of path integrals for both bosonic and fermionic canonical systems in terms of symbols of operators determined by operator ordering. We argue that for a system with a given quantum Hamiltonian such approximations are independent of the type of symbols up to terms of O(ε), where ε is infinitesimal time interval determining the accuracy of the approximations. A new class of such approximations is found for both c-number and Grassmannian dynamical variables. The actions determined by the approximations are non-local and have no classical continuum limit except the cases of pq- and qp-ordering. As an explicit example the fermionic oscillator is considered in detail.

arXiv:hep-th/9401063Progress of Theoretical PhysicsSep 1, 19942 citations

Quantum mechanics (QM): Path integrals in phase space

Jean Zinn‐Justin

Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.

Apr 15, 20210 citations

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

Leonardo A. Pachon, Andres D. Gomez

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 on ${\mathbb{Z}_d} \times {\mathbb{Z}_d}$ ( $d$ an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits ( $d=3$ ), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.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 on ${\mathbb{Z}_d} \times {\mathbb{Z}_d}$ ($d$ an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits ($d=3$), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.

quant-pharXiv:2604.20776Apr 23, 20260 citations

Grassmann Variables

I. J. R. Aitchison, Anthony J.G. Hey

In the path integral representation of quantum amplitudes ( chapter 16 ) the fields are regarded as classical functions. Matrix elements of time-ordered products of bosonic operators could be satisfactorily represented (see the discussion following ( 16.79 )). But something new is needed to represent, for example, the time-ordered product of two fermionic operators: there must be a sign difference between the two orderings, since the fermionic operators anticommute . Thus it seems that to represent amplitudes involving fermionic operators by path integrals we must think in terms of ‘classical’ anticommuting variables.In the path integral representation of quantum amplitudes ( chapter 16 ) the fields are regarded as classical functions. Matrix elements of time-ordered products of bosonic operators could be satisfactorily represented (see the discussion following ( 16.79 )). But something new is needed to represent, for example, the time-ordered product of two fermionic operators: there must be a sign difference between the two orderings, since the fermionic operators anticommute . Thus it seems that to represent amplitudes involving fermionic operators by path integrals we must think in terms of ‘classical’ anticommuting variables.

CRC Press eBooksApr 19, 20240 citations

A note on the limiting procedures for path integrals

Valery V Smirnov

Limiting procedures for the Feynman type path integral are considered. The evolution operator is approximated with operators corresponding to the exponential of the Hamiltonian's symbol. The proof of convergence uses Chernoff's theory and its extension on the class of stable operators. The approach is especially adequate for the generalized coherent states path integral, where decomposition of a Hamiltonian into the sums used in the approach based on Trotter's formula may be unnatural.Limiting procedures for the Feynman type path integral are considered. The evolution operator is approximated with operators corresponding to the exponential of the Hamiltonian's symbol. The proof of convergence uses Chernoff's theory and its extension on the class of stable operators. The approach is especially adequate for the generalized coherent states path integral, where decomposition of a Hamiltonian into the sums used in the approach based on Trotter's formula may be unnatural.

Journal of Physics A Mathematical and TheoreticalJan 4, 200810 citations

Quantum path integrals and reduced fermion density matrices: One-dimensional noninteracting systems

J. C. Light, Ji-Bing Yuan

A new approach has been used to determine directly the reduced single particle density matrix for noninteracting fermions in a one-dimensional potential. A time evolution operator closely related to the classical limit of the Feynman path integral propagator is derived which has superior behavior in turning point and nonclassical regions. This is used to determine the reduced density matrix in terms of the phase and time of oscillation at the Fermi energy. The density so obtained has the correct quantum oscillations and nonclassical tails and is a distinct improvement over the Thomas-Fermi density. Two simple examples are presented and implications for electronic structure calculations are discussed.A new approach has been used to determine directly the reduced single particle density matrix for noninteracting fermions in a one-dimensional potential. A time evolution operator closely related to the classical limit of the Feynman path integral propagator is derived which has superior behavior in turning point and nonclassical regions. This is used to determine the reduced density matrix in terms of the phase and time of oscillation at the Fermi energy. The density so obtained has the correct quantum oscillations and nonclassical tails and is a distinct improvement over the Thomas-Fermi density. Two simple examples are presented and implications for electronic structure calculations are discussed.

The Journal of Chemical PhysicsJan 15, 197332 citations

O(\\epsilon)

, where

\\epsilon

pq

- and

qp

-ordeeing. As an explicit example the fermionic oscillator is considered in detail.

{\mathbb{Z}_d} \times {\mathbb{Z}_d}

(

d

an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (

d=3

), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.