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arXiv:2111.14236 · Quantum Studies Mathematics and Foundations

An interpretation of quantum foundations based on density functional theory and polymer self-consistent field theory

Russell B. Thompson

Published Jul 19, 2022• 8 citations
arXivHTMLPDF

Abstract

Abstract The Feynman quantum-classical isomorphism between classical statistical mechanics in 3+1 dimensions and quantum statistical mechanics in 3 dimensions is used to connect classical polymer self-consistent field theory with quantum time-dependent density functional theory. This allows the theorems of density functional theory to relate non-relativistic quantum mechanics to a classical statistical mechanical derivation of polymer self-consistent field theory for ring polymers in a 4 dimensional thermal-space. One dynamic postulate is added to two static postulates which allows for a description of quantum physics from a 5 dimensional thermal-space-time ensemble perspective. A connection with aspects of classical field theory can be made in the classical limit.

DOI
10.1007/s40509-022-00276-y

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Often considered separated worlds, classical and quantum mechanics share numerous connections with one another. Indeed, as classical mechanics corresponds to a limiting case of quantum mechanics, certain concepts and elements of physical intuition developed in one theory can be and have been used to tackle issues in the other. Nevertheless, these connections do not only cover conceptual issues but also numerous techniques. Indeed, the inherently probabilistic nature of quantum mechanics and its close resemblance to Markovian stochastic processes opens the door to the application of a broad range of powerful methods, initially developed for quantum mechanics, in classical equilibrium and non-equilibrium statistical mechanics. In this thesis we develop progress in three subjects by taking advantage of either a conceptual or a methodological connection between quantum and classical mechanics. 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Lastly, the explicit connection between the auxiliary Hamiltonian and the observables of the system allows for a perturbation scheme in cases in which an exact solution is not easily achievable. This scheme shows that, even in situations in which an exact local auxiliary term cannot be achieved, under special circumstances it is possible to achieve an operator that is local and still approximately stifles non-adiabatic transitions.

•Apr 7, 2017•0 citations
Abstract The Feynman quantum-classical isomorphism between classical statistical mechanics in 3+1 dimensions and quantum statistical mechanics in 3 dimensions is used to connect classical polymer self-consistent field theory with quantum time-dependent density functional theory. This allows the theorems of density functional theory to relate non-relativistic quantum mechanics to a classical statistical mechanical derivation of polymer self-consistent field theory for ring polymers in a 4 dimensional thermal-space. One dynamic postulate is added to two static postulates which allows for a description of quantum physics from a 5 dimensional thermal-space-time ensemble perspective. A connection with aspects of classical field theory can be made in the classical limit.
Abstract Chapter 14 covers the statistical mechanics of driven quantum systems and the non-equilibrium ensembles associated with them. The topic is presented both from the state-vector and density matrix perspectives. The chapter begins with a discussion of driving forces resulting from light-matter interactions and their treatment via time-dependent perturbation theory. The Fermi Golden Rule is derived, and it is shown how to incorporate the Golden Rule into the calculation of frequency spectra within the canonical ensemble, which then connects to canonical quantum time correlation functions. The formalism is derived again from the quantum Liouville equation. The discussion then turns to the challenge of computing quantum time correlation functions and approximate techniques based on the Feynman path integral of Chapter 12. These schemes include centroid and ring-polymer molecular dynamics as well as open-path integral methods
The equations of time-dependent density functional theory are derived, via the expression for a quantum weak value, from ring polymer self-consistent field theory using a mathematical correspondence between time and imaginary time. The imaginary time path integral formalism of Feynman, in which inverse temperature is seen to be a Wick rotation of time, allows one to write the equilibrium partition function of a quantum system in a form mathematically isomorphic with the path integral expression for the dynamics. Therefore the self-consistent field theory equations which are solutions to the equilibrium partition function are Wick rotated back into a set of dynamic equations, which are shown to give an expression for a quantum weak value of the one-particle density. Remarkably, weak values emerge naturally here without being postulated, as an intermediate step before recovering the standard expression for the density. The weak value expression in turn leads to the equations of time-dependent density functional theory. This first-principles derivation does not use the theorems of density functional theory, which are instead applied to guarantee equivalence with standard quantum mechanics. An expression for finite-temperature dynamics is also given, which shows that a ring polymer model for quantum particles holds for time-dependent systems as well as equilibrium situations. Issues arising in time-dependent density functional theory, such as causality, initial state dependence, and vvv-representability, are discussed in the context of the ring polymer derivation.
The advent of the Hohenberg-Kohn theorem in 1964, its extension to finite- T , Kohn-Sham theory, and relativistic extensions provide the well-established formalism of density-functional theory (DFT). This theory enables the calculation of all static properties of quantum systems without the need for an n -body wavefunction ψ. DFT uses the one-body density distribution instead of ψ. The more recent time-dependent formulations of DFT attempt to describe the time evolution of quantum systems without using the time-dependent wavefunction. Although DFT has become the standard tool of condensed-matter computational quantum mechanics, its foundational implications have remained largely unexplored. While all systems require quantum mechanics (QM) at T =0, the pair-distribution functions (PDFs) of such quantum systems have been accurately mapped into classical models at effective finite- T , and using suitable non-local quantum potentials (e.g., to mimic Pauli exclusion effects). These approaches shed light on the quantum → hybrid → classical models, and provide a new way of looking at the existence of non- local correlations without appealing to Bell's theorem. They also provide insights regarding Bohmian mechanics. Furthermore, macroscopic systems even at 1 Kelvin have de Broglie wavelengths in the micro-femtometer range, thereby eliminating macroscopic cat states, and avoiding the need for ad hoc decoherence models.
From a discretization of the path integral formulation of quantum mechanics, it is possible to relate equilibrium quantum many body theory to classical statistical mechanics. In this paper, we significantly extend and analyze this well known isomorphism in terms of the equilibrium theory of classical molecular fluids composed of flexible polyatomic species. We show how quantum influence functionals are isomorphic to classical cavity distribution functions. The former describe the influence of surrounding media on the dynamics of quantal degrees of freedom, and the latter describe environmental effects for classical models of flexible molecules and chemical equilibria. The connection allows the use of classical theories to perform nonperturbative calculations of influence functionals which treat the influence functionals and many body correlation functions in a self-consistent fashion. We illustrate the computational advantages of the method by studying its predictions for a hard sphere model of liquid helium above the l transition. The nature of quantum indistinguishability of identical particles (i.e., quantum exchange) is treated in our theory in terms of an exact isomorphism with chemical equilibria. This connection allows the treatment of exchange in condensed phases in terms of the classical law of mass action, and provides a computational advance over existing methods for interacting systems. By picturing exchange in terms of classical association equilibrium, we arrive at a view (hinted at long ago by Feynman and by Penrose and Onsager) in which the Bose condensation is related to an equilibrium polymeric sol–gel transition. Thus, below the l transition, the correlations in liquid helium are equivalent to those in a classical fluid containing a finite concentration of macroscopic polymers. We stress how the path integral aspect of the isomorphism leads to useful geometrical interpretations of quantum phenomena. For example, tunneling phenomena can be viewed in terms of solitonic (or instantonic) configurations or flexible chain molecules. With this picture, we show how the isomorphism can be employed to understand both adiabatic and nonadiabatic solvent effects on chemical bonding. For concreteness, we provide a detailed analysis of a particular model of the chemical bond for which a partial summation over intermediate quantum paths leads to an Ising model problem in the isomorphism. While applications like this are presented in the form of qualitative illustrations, a variety of methods can be employed to produce quantitative results. We sketch how these calculations can be performed for various problems, making connections with methods like the renormalization group (RG) technique. The classical isomorphism together with the modern theory of classical polyatomic systems provides a powerful framework for quantitative solutions of condensed matter quantum mechanical problems.
Abstract The connection between statistical mechanics on the one side and quantum mechanics and quantum field theory on the other side is based on the analogy between thermal and quantum fluctuations. Formally, the connection is expressed through the mathematical equivalence between the partition function in statistical mechanics and the propagator in quantum field theory. This chapter explores the equivalence between statistical mechanics and quantum mechanics or quantum field theory in general terms using the Feynman path integral and with the example of the equivalence between the classical XY Heisenberg model and the sigma model of quantum field theory. Invoking the concepts of the partition function and the transfer matrix, an example demonstrates the passage from the quantum mechanics of a single degree of freedom, a zero-dimensional system, to the statistical mechanics of a one-dimensional system represented by classical variables.
Often considered separated worlds, classical and quantum mechanics share numerous connections with one another. Indeed, as classical mechanics corresponds to a limiting case of quantum mechanics, certain concepts and elements of physical intuition developed in one theory can be and have been used to tackle issues in the other. Nevertheless, these connections do not only cover conceptual issues but also numerous techniques. Indeed, the inherently probabilistic nature of quantum mechanics and its close resemblance to Markovian stochastic processes opens the door to the application of a broad range of powerful methods, initially developed for quantum mechanics, in classical equilibrium and non-equilibrium statistical mechanics. In this thesis we develop progress in three subjects by taking advantage of either a conceptual or a methodological connection between quantum and classical mechanics. First, we develop the well-known mapping between systems of strongly repelling, two-dimensional directed lines and systems of one-dimensional fermions in order to extend the directed polymer problem to richer geometries. By expressing path integrals in generalized curvilinear coordinates, we successfully generalize the model to settings such as polymers anchored to curved edges, polymers constrained to uneven walls, and polymers constrained to curved surfaces. In each case, we identify the Hamiltonian of an analogous quantum system, which, because of the new geometry of each setting, develops features such as a time- and position-dependent mass and an external electromagnetic vector potential. Along the way, we perform an in-depth analysis of the approximations made and establish regimes of their validity. Finally, in order to obtain analytical results, we employ an extension of the time-dependent perturbation theory scheme of quantum mechanics to imaginary time. Complementing this with the assumption of ground-state dominance, we obtain compact expressions for universal shifts in the free energies of the various systems that allows for isolation of the effects of the distinct geometrical properties of each system. The second piece of work presented into this thesis relates to the non-equilibrium setting of active particles. Strongly interacting, self-propelled particles can form a spontaneously flowing, polar (i.e. motionally aligned), active fluid. The study of the connection between the microscopic dynamics of a single, self-propelled particle and the macroscopic dynamics of a liquid comprising such particles can yield insights into experimentally realizable active flows, but this connection is well understood in only a few select cases. Here, we introduce a model of self-propelled particles that is based on an analogy with the motion of an electron subject to strong spin-orbit coupling. We find that, within our model, self-propelled particles experience an analog of the Heisenberg uncertainly principle that instead relates positional and rotational noise. An extension to many-component (and hence more classical) spinors, under which this uncertainty relation vanishes, contributes to the justification of this interpretation. Furthermore, by coarse-graining the microscopic model, we find expressions for the coefficients of the hydrodynamic Toner-Tu equations, established some time ago to describe an active liquid composed of “active spins.” The connection between self-propelled particles and quantum spins may possibly provide a route for realizing exotic phases of matter using active liquids, via inspiration hailing from systems composed of strongly correlated electrons. The third and final piece of work to be presented on this thesis consists of a semi-classical approach to Transitionless Quantum Driving (TQD). TQD is a method, developed by means of a reverse engineering strategy, under which non-adiabatic transitions in time-dependent quantum systems are, in M. V. Berry's words, “stifled” through the introduction of a specific auxiliary Hamiltonian. This Hamiltonian comes, however, expressed as a formal sum of outer products of the original instantaneous eigenstates and their time-derivatives. Generically, how to actually create such an operator in the laboratory is thus rarely evident. The operator may even be non-local. By following a semi-classical approach, we obtain a recipe that yields the auxiliary Hamiltonian explicitly, in terms of the fundamental operators of the system (e.g., position and momentum). By using this formalism, we are able to ascertain criteria for the locality of the auxiliary Hamiltonian, and also to determine its exact form in certain special cases. Lastly, the explicit connection between the auxiliary Hamiltonian and the observables of the system allows for a perturbation scheme in cases in which an exact solution is not easily achievable. This scheme shows that, even in situations in which an exact local auxiliary term cannot be achieved, under special circumstances it is possible to achieve an operator that is local and still approximately stifles non-adiabatic transitions.