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Journal of Physics Conference Series

Quantum Potential and the Canonical Group of Generalized Coherent States

Glen Dennis, Maurice A. de Gosson

Published Mar 1, 2026• 0 citations
Publisher

Abstract

Abstract I nspired by our work with Basil Hiley, we continue the exploration of Fermi’s Ansatz. Back in 1930, Fermi showed that a wavefunction could be associated with a hypersurface in phase space. Fermi’s approach therefore leads to a geometric picture of quantum states that is based on Fermi ellipsoids. Our investigations with Basil Hiley revealed that the internal energy provided by Bohm’s quantum potential sustains these Fermi blobs whenever a quantum process is unfolding. Moreover, this internal energy is associated with a noncommutative quantum phase space (coarse-grained by quantum blobs - minimum uncertainty phase-space cells that are in a bijective correspondence with the squeezed coherent states familiar from quantum optics). In this paper, we use Fermi’s method in order to gain further insights into the invariance of quantum blobs under canonical transformations. Starting with a real Gaussian, we multiply it by a complex exponential (“chirp”) and thereby obtain a generalized coherent state. The associated change of gauge enables us to make contact with Fermi’s approach. As in de Broglie-Bohm theory, the phase and magnitude of the wavefunction feature explicitly here and are revealed to play a key role. Subsequently, we find the Fermi Hamiltonian operator and associated eigenvalue equation which the generalized coherent state satisfies. Considering Hamilton’s equations written in compact form, we show that the Hamiltonian flow determined by the Fermi Hamiltonian consists of symplectic matrices that belong to a one-parameter subgroup of Sp( n ) - we call this subgroup the canonical group of the generalized coherent state. The lift of this canonical group to the metaplectic group is found to leave the generalized Gaussian invariant; whereas the canonical group itself leaves the associated phase-space quantum blob invariant. We conclude that Fermi’s Ansatz provides an instructive method for proving these invariance results. It sheds light on subtle aspects of quantum processes that other approaches might miss.

DOI
10.1088/1742-6596/3189/1/012005

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Abstract I nspired by our work with Basil Hiley, we continue the exploration of Fermi’s Ansatz. Back in 1930, Fermi showed that a wavefunction could be associated with a hypersurface in phase space. Fermi’s approach therefore leads to a geometric picture of quantum states that is based on Fermi ellipsoids. Our investigations with Basil Hiley revealed that the internal energy provided by Bohm’s quantum potential sustains these Fermi blobs whenever a quantum process is unfolding. Moreover, this internal energy is associated with a noncommutative quantum phase space (coarse-grained by quantum blobs - minimum uncertainty phase-space cells that are in a bijective correspondence with the squeezed coherent states familiar from quantum optics). In this paper, we use Fermi’s method in order to gain further insights into the invariance of quantum blobs under canonical transformations. Starting with a real Gaussian, we multiply it by a complex exponential (“chirp”) and thereby obtain a generalized coherent state. The associated change of gauge enables us to make contact with Fermi’s approach. As in de Broglie-Bohm theory, the phase and magnitude of the wavefunction feature explicitly here and are revealed to play a key role. Subsequently, we find the Fermi Hamiltonian operator and associated eigenvalue equation which the generalized coherent state satisfies. Considering Hamilton’s equations written in compact form, we show that the Hamiltonian flow determined by the Fermi Hamiltonian consists of symplectic matrices that belong to a one-parameter subgroup of Sp( n ) - we call this subgroup the canonical group of the generalized coherent state. The lift of this canonical group to the metaplectic group is found to leave the generalized Gaussian invariant; whereas the canonical group itself leaves the associated phase-space quantum blob invariant. We conclude that Fermi’s Ansatz provides an instructive method for proving these invariance results. It sheds light on subtle aspects of quantum processes that other approaches might miss.
We extend the notion of quantum blob studied in previous work to excited states of the generalized harmonic oscillator in n dimensions. This extension is made possible by Fermi's observation in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction Ψ or by a certain function g_{F} on phase space canonically associated with Ψ. We study Fermi's function when Ψ is a Gaussian (generalized coherent state). A striking result is that we can use the Ekeland--Hofer symplectic capacities to characterize the Fermi functions of the excited states of the generalized harmonic oscillator, leading to new insight on the relationship between symplectic topology and quantum mechanics.
Fermi observed in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction Ψ\PsiΨ or by a certain function gFg_FgF​ on phase space canonically associated with Ψ\PsiΨ. In this Note we study we study Fermi's function when Ψ\PsiΨ is a squeezed coherent state. We relate it with the Wigner transform of Ψ\PsiΨ, thus generalising a previous observation of Benenti and Strini. We show that the symplectic capacity of the phase space ellipsoid gF(x,p)≤0g_{F}(x,p)\leq0gF​(x,p)≤0 is bounded by h/2h/2h/2 and nh/2nh/2nh/2 (n the number of degrees of freedom).
Fermi observed in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction ΨΨΨ or by a certain function gFg_FgF​ on phase space canonically associated with ΨΨΨ. In this Note we study we study Fermi's function when ΨΨΨ is a squeezed coherent state. We relate it with the Wigner transform of ΨΨΨ, thus generalising a previous observation of Benenti and Strini. We show that the symplectic capacity of the phase space ellipsoid gF(x,p)≤0g_{F}(x,p)\leq0gF​(x,p)≤0 is bounded by h/2h/2h/2 and nh/2nh/2nh/2 (n the number of degrees of freedom).
In earlier work, we introduced quantum blobs as minimum-uncertainty symplectic ellipsoids in phase space. These objects may be viewed as geometric monads in the Leibnizian sense, representing the elementary units of phase-space structure consistent with the uncertainty principle. We establish a one-to-one correspondence between such monads and generalized coherent states, represented by arbitrary non-degenerate Gaussian wave functions in configuration space. To each of these states, we associate a classs of Toeplitz operators that extends the standard anti-Wick quantization scheme. The mathematical and physical properties of these operators are analyzed, allowing for a generalized definition of density matrices within the phase-space formulation of quantum mechanics.
Abstract In earlier work, we introduced quantum blobs as minimum-uncertainty symplectic ellipsoids in phase space. These objects may be viewed as geometric monads in the Leibnizian sense, representing the elementary units of phase-space structure consistent with the uncertainty principle. We establish a one-to-one correspondence between such monads and generalized coherent states, represented by arbitrary non-degenerate Gaussian wave functions in con…guration space. To each of these states, we associate a class of Toeplitz operators that extends the standard anti-Wick quantization scheme. The mathematical and physical properties of these operators are analyzed, allowing for a generalized definition of density matrices within the phase-space formulation of quantum mechanics.
The manifold of one-particle operators has a rich group structure. It is the Lie algebra of the complex Lie group GL(2s,), (2s = dimension of one-particle spin orbital basis) realized as second-quantized operators acting in fermion fock space. This representation is reduced by the subspaces referring to N particles (N = 0, 1, 2,…). One can associate families of coherent states within each of these subspaces with real forms, Gr, of GL(2s,). These families have isotropy groups, Hr, which are associated with subgroups of these real forms (isotropy groups are the groups of transformations that leave a state invariant). The cosets GL(2s,)/Hr can be given an exponential parameterization that labels the coherent states. This label space is a complex vector space which is in correspondence with the nonlinear manifold of coherent states in the hilbert space ℋ︁N. These complex variables can be made time dependent, and by utilizing the time-dependent variation principle one can obtain a nonlinear approximation to the time-dependent Schrödinger equation which has the form of classical equations of motion in a generalized phase space 𝒫 = Grc/Hrc. This classical dynamics can be given a Hilbert space description, where the Hilbert space is a subspace of square integrable holomorphic functions f: ↦ 𝒫 → and is isomorphic with ℋ︁N. Different pairs {Gr,Hr} are associated with different families of coherent states. Some examples of these constructions are given, and the structure of the approximate time evolution is discussed and related to time-dependent approximation schemes and generalized random phase approximations (RPA).