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arXiv:2503.01417 · Journal of Physics Conference Series

Beyond Born-Oppenheimer Green’s function theories: absolute and relational

Ville J. Härkönen

Published Jun 1, 2025• 0 citations
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Abstract

Abstract We consider quantum field theoretic many-body Green’s function approach to solve the Coulomb many-body problem. The earlier beyond Born-Oppenheimer Green’s function theories are absolute in nature and are based on the non-reduced Hamiltonian. Motivated by the issues following this approach we have developed a reduced Green’s function theory which is relational in nature. The central differences between these approaches trace back to the relational-absolute debate, which has continued for over two thousand years since the time of Aristotle and still persists today. We highlight that these aspects of the theories are connected to several areas of modern physics, including relational quantum mechanics, quantum reference frames, superselection rules, the separation of different types of motion and spontaneous symmetry breaking. The starting point of any such exact theory must be absolute, given that the global conservation laws hold. However, for the observables to be meaningful, they must be defined with respect to the relative space. Approximations, such as the one introduced by Born and Oppenheimer, can break global symmetries and make certain parts of the absolute description well-defined. We highlight that spontaneous symmetry breaking is not necessary to explain the existence of solids, but entities such as phonons can be naturally explained within the relational theory.

DOI
10.1088/1742-6596/3017/1/012023

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Abstract We consider quantum field theoretic many-body Green’s function approach to solve the Coulomb many-body problem. The earlier beyond Born-Oppenheimer Green’s function theories are absolute in nature and are based on the non-reduced Hamiltonian. Motivated by the issues following this approach we have developed a reduced Green’s function theory which is relational in nature. The central differences between these approaches trace back to the relational-absolute debate, which has continued for over two thousand years since the time of Aristotle and still persists today. We highlight that these aspects of the theories are connected to several areas of modern physics, including relational quantum mechanics, quantum reference frames, superselection rules, the separation of different types of motion and spontaneous symmetry breaking. The starting point of any such exact theory must be absolute, given that the global conservation laws hold. However, for the observables to be meaningful, they must be defined with respect to the relative space. Approximations, such as the one introduced by Born and Oppenheimer, can break global symmetries and make certain parts of the absolute description well-defined. We highlight that spontaneous symmetry breaking is not necessary to explain the existence of solids, but entities such as phonons can be naturally explained within the relational theory.
The method of many-body Green's functions is developed for arbitrary systems of electrons and nuclei starting from the full (beyond Born-Oppenheimer) Hamiltonian of Coulomb interactions and kinetic energies. The theory presented here resolves the problems arising from the translational and rotational invariance of this Hamiltonian that afflict the existing many-body Green's function theories. We derive a coupled set of exact equations for the electronic and nuclear Green's functions and provide a systematic way to approximately compute the properties of arbitrary many-body systems of electrons and nuclei beyond the Born-Oppenheimer approximation. The case of crystalline solids is discussed in detail.
We combine the recently developed many-body Green's function theory for electrons and nuclei with the exact factorization of the wave function. The existing Born-Oppenheimer Green's functions are shown to be special cases of our exact approach. We consider the limitations of the laboratory frame formulation of the Green's function theory and discuss why the body-fixed frame formulation is needed in order to go beyond the Born-Oppenheimer theory. We give exact forms of the electronic and nuclear Green's functions written in terms of the exact factorized states, providing a systematic approach beyond the Born-Oppenheimer approximation. The lowest order approximation to the exact electronic Green's function is found to be an expected value of the Born-Oppenheimer electronic Green's function with respect to the nuclear density.
We derive an exact relational quantum field theory whose structure is dictated by invariance postulates motivated by the symmetries of the Hamiltonian and by the physical equivalence of inertial frames. The postulates require the physically meaningful observables of an isolated system to be invariant under translations, rotations and changes of inertial frame, and therefore replace the conventional absolute description by one formulated in terms of relational observables. For non-relativistic many-body systems, the resulting physical Hamiltonian is the internal Hamiltonian, which we formulate entirely in terms of the original elementary fields. We show that conventional local densities and currents are not physical observables, whereas relational spatial observables generally acquire a higher particle and field rank. Consequently, even the simplest relational densities and currents require higher-order Green's functions. The one-particle Green's function therefore loses its conventional role as the fundamental carrier of physical spatial information, forcing a fundamental change in the structure of the quantum field theory. The theory reproduces the internal formulations known from atomic and molecular physics as special cases while providing a unified relational quantum field theory for arbitrary isolated non-relativistic electron-nuclear systems.
We consider symmetries and reduction in non-relativistic many-body quantum mechanics, with the aim of identifying physically meaningful observables in systems such as molecules and crystalline solids. To this end, we propose a unified framework based on two additional postulates supplementing the standard quantum-mechanical formalism. For stable systems, the physically relevant states are normalizable stationary states, while physically meaningful observables are required to be invariant under a selected subgroup of the symmetry group and under Galilean boosts. In addition, we postulate the existence of a map from the set of all observables allowed by quantum mechanics to the corresponding invariant physically meaningful observables. The originality of the present work does not lie in specific reductions, but in the unified framework that connects symmetry reduction and relational many-body quantum theory. We interpret entities like superselection rules and quantum reference frames as important parts of the postulated process of obtaining the physically meaningful relational description. In particular, the requirement of Galilean-boost invariance added strengthens the criterion for physical observability by excluding quantities that depend on the choice of inertial frame. An important consequence of the postulates is that in the considered cases every physically meaningful observable necessarily depends on more than one non-invariant observable, the latter being typically associated with degrees of freedom assigned to a single particle. The postulates thus lead to theories that are well aligned with the literature on reduction and the description of molecules, while at the same time being consistent with the relational interpretation of quantum mechanics, according to which the complete physical description of a system is defined only relative to other systems.
We consider symmetries and reduction in non-relativistic many-body quantum mechanics, with the aim of identifying physically meaningful observables in systems such as molecules and crystalline solids. To this end, we propose a unified framework based on two additional postulates supplementing the standard quantum-mechanical formalism. For stable systems, the physically relevant states are normalizable stationary states, while physically meaningful observables are required to be invariant under a selected subgroup of the symmetry group and under Galilean boosts. In addition, we postulate the existence of a map from the set of all observables allowed by quantum mechanics to the corresponding invariant physically meaningful observables. The originality of the present work does not lie in specific reductions, but in the unified framework that connects symmetry reduction and relational many-body quantum theory. We interpret entities like superselection rules and quantum reference frames as important parts of the postulated process of obtaining the physically meaningful relational description. In particular, the requirement of Galilean-boost invariance added strengthens the criterion for physical observability by excluding quantities that depend on the choice of inertial frame. An important consequence of the postulates is that in the considered cases every physically meaningful observable necessarily depends on more than one non-invariant observable, the latter being typically associated with degrees of freedom assigned to a single particle. The postulates thus lead to theories that are well aligned with the literature on reduction and the description of molecules, while at the same time being consistent with the relational interpretation of quantum mechanics, according to which the complete physical description of a system is defined only relative to other systems.
We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox “absolute” quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject.