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Springer eBooks

Functional (Path) Integral Formalism as a Functional Fourier Transform of the Functional Differential One

Edward B. Manoukian

Published Jan 1, 2020• 1 citation
Publisher
DOI
10.1007/978-3-030-51081-7_24

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Quantum Hamilton-Jacobi theory, spectral path integrals, and exact WKB analysis

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We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi (HJ) theory. In classical mechanics, Hamilton-Jacobi theory is a powerful formalism, however, its utility is not explored in quantum theory beyond approximation schemes. The canonical transformation enables one to set the new Hamiltonian to constant or zero, but keeps the information about solution in Hamilton’s characteristic function. To benefit from this in quantum theory, one must work with a formulation in which classical Hamiltonian is used. This uniquely points to phase space path integral. However, the main variable in HJ formalism is energy, not time. Thus, we are led to consider the Fourier transform of the path integral, the spectral path integral Z ˜ ( E ) . The evaluation of path integrals reduces to determining the quantum Hamilton characteristic functions (which can be achieved via an asymptotic analysis) and a discrete sum over the quantum period lattice, generalizing Gutzwiller’s sum.

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The path integral is a method that often used in the quantum problems calculation. For example; the calculation of quantum system energy that has complex potential form. The method gives more easily than perturbation method. The method is also used to derive Green function, which usually used Fourier transformation. The Green function has widely application in quantum physics, since it used to compute solution of inhomogen differential equation as Schrodinger equation. In the particle physics, the Green function used as propagator in Feynman’s diagram. Considering the importance of Green function, and the powerfull of path integral method, in the paper, the method used to derive the formula of Green function for quantum harmonic oscillator system. The system has widely application to give more information of physical phenomena, for example, the atomic vibration in solid state. The result was also compared with Fourier transformation method and both give the same result as hoped.

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Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms

Mathieu Beauvillain, Blagoje Oblak, Marios Petropoulos

Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.

quant-pharXiv:2512.19840•Dec 24, 2025
We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi (HJ) theory. In classical mechanics, Hamilton-Jacobi theory is a powerful formalism, however, its utility is not explored in quantum theory beyond approximation schemes. The canonical transformation enables one to set the new Hamiltonian to constant or zero, but keeps the information about solution in Hamilton’s characteristic function. To benefit from this in quantum theory, one must work with a formulation in which classical Hamiltonian is used. This uniquely points to phase space path integral. However, the main variable in HJ formalism is energy, not time. Thus, we are led to consider the Fourier transform of the path integral, the spectral path integral Z ˜ ( E ) . The evaluation of path integrals reduces to determining the quantum Hamilton characteristic functions (which can be achieved via an asymptotic analysis) and a discrete sum over the quantum period lattice, generalizing Gutzwiller’s sum.
The path integral is a method that often used in the quantum problems calculation. For example; the calculation of quantum system energy that has complex potential form. The method gives more easily than perturbation method. The method is also used to derive Green function, which usually used Fourier transformation. The Green function has widely application in quantum physics, since it used to compute solution of inhomogen differential equation as Schrodinger equation. In the particle physics, the Green function used as propagator in Feynman’s diagram. Considering the importance of Green function, and the powerfull of path integral method, in the paper, the method used to derive the formula of Green function for quantum harmonic oscillator system. The system has widely application to give more information of physical phenomena, for example, the atomic vibration in solid state. The result was also compared with Fourier transformation method and both give the same result as hoped.
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.