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arXiv:2603.29201

A Floer Theoretic Approach to Energy Eigenstates on one Dimensional Configuration Spaces

Kevin Ruck

math.SGmath-phquant-phSubmitted Mar 31, 2026• Announced Apr 1, 2026
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Abstract

In this article we consider two classical problems in Quantum Mechanics, namely the 'particle on a ring' and the 'particle in a box' from the viewpoint of symplectic topology. Interpreting the solutions of the corresponding time independent Schrödinger equation as orbits in a suitably chosen time dependent Hamiltonian system allows us to investigate them using Floer theory. More precisely we extend the definition of Rabinowitz Floer homology to non-autonomous Hamiltonians on $\mathbb{R}^{2n}$ with its standard symplectic structure and show that compactness of the moduli space of J-holomorphic curves still holds. With this homology we are then able to prove existence results for energy $E$ eigenstates on the 'ring' or in the 'box' for a big range of exterior potentials.

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The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system with a complex projective Hilbert space as its phase space, thus equipped with a Riemannian metric in addition to a symplectic structure. This paper extends the geometric quantum theory to include aspects of the symplectic topology of the state space by identifying the Robertson-Schrödinger uncertainty relation for pure quantum states as the differential version of the energy identity in the theory of $J$-holomorphic curves. We consider a family of maps from a Riemann surface into a finite-dimensional quantum state space by using the vector fields generated by two quantum observables,and show that the Fubini-Study metric tensor pulls back by such a map to the covariance tensor for the two observables. By calculating the map energy density in the pull-back metric, the uncertainty relation is represented as an equality that compares the map energy differential to the sum of the pull-back of the symplectic form and a positive definite term that vanishes when the map is holomorphic. Saturation of the Robertson-Schrödinger inequality occurs when the map is conformal and the off-diagonal covariance terms vanish. For compact Riemann surfaces where such a map can be globally defined, if the map is holomorphic, it is harmonic and its image is a minimal surface. In this case, the uncertainty product integral is a topological invariant that depends only on the homology class of the curve modulo its boundary.

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SYMPLECTIC STRUCTURES AND QUANTUM MECHANICS

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Constrained Symplectic Quantization: Disclosing the Deterministic Framework Behind Quantum Mechanics

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International Symposium on Lattice Field Theory (LATTICE)•Jul 15, 2026•0 citations

Constrained Symplectic Quantization: Disclosing the Deterministic Framework Behind Quantum Mechanics

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Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a generalized Hamiltonian dynamics in an extra time variable $τ$ which, at large times, samples a microcanonical ensemble. In a previous work we showed that, for an interacting scalar theory in 1+1 dimensions, this framework captures genuine real time features that are inaccessible to Euclidean simulations. That original formulation suffers from two structural limitations, an ill defined non interacting limit and the lack of a direct correspondence between its correlation functions and those generated by the Feynman path integral. To solve these problems we introduced constrained symplectic quantization, a holomorphic reformulation in which fields and action are analytically continued and constraints are imposed on the intrinsic time Hamiltonian flow. The constraints select stable deterministic trajectories and they define convergent holomorphic integration cycles for the corresponding microcanonical measure. In the continuum limit we establish exact equivalence with the Feynman path integral at the level of the generating functional, thus providing a direct link between intrinsic time correlators and real time Green functions. In this contribution, we apply the method to the quantum harmonic oscillator on a real-time 1-dimensional lattice. Testing various observables, we find agreement between numerical and exact results for one- and two-point functions, and we reconstruct characteristic real-time features such as an oscillatory propagator, the discrete energy-gap spectrum, and the evolution of eigenstate probability densities. These tests provide numerical evidence that constrained symplectic quantization can sample real-time quantum observables and offers a practical route beyond Euclidean-time importance sampling.

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Rethinking the Boundary Conditions in the Particle-in-a-Box Mind Experiment

Jixin Chen

In this manuscript, I speculated that the energy density distributions along space and time in a quantum system are uniform. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with a Hamiltonian slightly different than the classical Schrödinger equation. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density function.

•Apr 8, 2021•0 citations
In this article we consider two classical problems in Quantum Mechanics, namely the 'particle on a ring' and the 'particle in a box' from the viewpoint of symplectic topology. Interpreting the solutions of the corresponding time independent Schrödinger equation as orbits in a suitably chosen time dependent Hamiltonian system allows us to investigate them using Floer theory. More precisely we extend the definition of Rabinowitz Floer homology to non-autonomous Hamiltonians on R2n\mathbb{R}^{2n}R2n with its standard symplectic structure and show that compactness of the moduli space of J-holomorphic curves still holds. With this homology we are then able to prove existence results for energy EEE eigenstates on the 'ring' or in the 'box' for a big range of exterior potentials.
The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system with a complex projective Hilbert space as its phase space, thus equipped with a Riemannian metric in addition to a symplectic structure. This paper extends the geometric quantum theory to include aspects of the symplectic topology of the state space by identifying the Robertson-Schrödinger uncertainty relation for pure quantum states as the differential version of the energy identity in the theory of JJJ-holomorphic curves. We consider a family of maps from a Riemann surface into a finite-dimensional quantum state space by using the vector fields generated by two quantum observables,and show that the Fubini-Study metric tensor pulls back by such a map to the covariance tensor for the two observables. By calculating the map energy density in the pull-back metric, the uncertainty relation is represented as an equality that compares the map energy differential to the sum of the pull-back of the symplectic form and a positive definite term that vanishes when the map is holomorphic. Saturation of the Robertson-Schrödinger inequality occurs when the map is conformal and the off-diagonal covariance terms vanish. For compact Riemann surfaces where such a map can be globally defined, if the map is holomorphic, it is harmonic and its image is a minimal surface. In this case, the uncertainty product integral is a topological invariant that depends only on the homology class of the curve modulo its boundary.
Canonical coordinates for the Schrödinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schrödinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a recursion operator which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clear interpretation of Quantum Mechanics in the general setting, provided by Stone-von Neumann theorem, of Symplectic Mechanics. It may give new tools to solve in the general case the inverse problem of quantum mechanics whose solution is given up to now only for one-dimensional systems by the Gel’fand-Levitan-Marchenko formula. 1 Introduction. In the past few years there has been a renewed interest in completely integrable Hamiltonian systems, specially in connection with the study of integrable quantum
Canonical coordinates for the Schrödinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schrödinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a recursion operator which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clear interpretation of Quantum Mechanics in the general setting, provided by Stone–von Neumann theorem, of Symplectic Mechanics. It may give new tools to solve in the general case the inverse problem of quantum mechanics whose solution is given up to now only for one-dimensional systems by the Gel’fand-Levitan-Marchenko formula.
Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a generalized Hamiltonian dynamics in an extra time variable τ\tauτ which, at large times, samples a microcanonical ensemble. In a previous work we showed that, for an interacting scalar theory in 1+11+11+1 dimensions, this framework captures genuine real time features that are inaccessible to Euclidean simulations. That original formulation, however, suffers from two structural limitations, an ill defined non interacting limit and the lack of a direct correspondence between its correlation functions and those generated by the Feynman path integral. To solve these problems we introduced constrained symplectic quantization, a holomorphic reformulation in which fields and action are analytically continued from R\mathbb{R}R to C\mathbb{C}C and constraints are imposed on the intrinsic time Hamiltonian flow. The constraints select stable deterministic trajectories already for the free theory and they define convergent holomorphic integration cycles for the corresponding microcanonical measure. In the continuum limit we establish exact equivalence with the Feynman path integral at the level of the generating functional, thus providing a direct link between intrinsic time correlators and real time Green functions. In this contribution, we apply the method to the quantum harmonic oscillator on a real-time 111-dimensional lattice. Testing various observables, we find agreement between numerical and exact results for one- and two-point functions, and we reconstruct characteristic real-time features such as an oscillatory propagator, the discrete energy-gap spectrum, and the evolution of eigenstate probability densities. These tests provide numerical evidence that constrained symplectic quantization can sample real-time quantum observables and offers a practical route beyond Euclidean-time importance sampling.
Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a generalized Hamiltonian dynamics in an extra time variable τττ which, at large times, samples a microcanonical ensemble. In a previous work we showed that, for an interacting scalar theory in 1+1 dimensions, this framework captures genuine real time features that are inaccessible to Euclidean simulations. That original formulation suffers from two structural limitations, an ill defined non interacting limit and the lack of a direct correspondence between its correlation functions and those generated by the Feynman path integral. To solve these problems we introduced constrained symplectic quantization, a holomorphic reformulation in which fields and action are analytically continued and constraints are imposed on the intrinsic time Hamiltonian flow. The constraints select stable deterministic trajectories and they define convergent holomorphic integration cycles for the corresponding microcanonical measure. In the continuum limit we establish exact equivalence with the Feynman path integral at the level of the generating functional, thus providing a direct link between intrinsic time correlators and real time Green functions. In this contribution, we apply the method to the quantum harmonic oscillator on a real-time 1-dimensional lattice. Testing various observables, we find agreement between numerical and exact results for one- and two-point functions, and we reconstruct characteristic real-time features such as an oscillatory propagator, the discrete energy-gap spectrum, and the evolution of eigenstate probability densities. These tests provide numerical evidence that constrained symplectic quantization can sample real-time quantum observables and offers a practical route beyond Euclidean-time importance sampling.
In this manuscript, I speculated that the energy density distributions along space and time in a quantum system are uniform. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with a Hamiltonian slightly different than the classical Schrödinger equation. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density function.