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Modeling Position and Momentum in Finite-Dimensional Hilbert Spaces via Generalized Clifford Algebra

Ashmeet Singh, Sean M. Carroll

Published Jun 25, 2018• 6 citations
OpenAlex

Abstract

The finite entropy of black holes suggests that local regions of spacetime \nare described by finite-dimensional factors of Hilbert space, in contrast with \nthe infinite-dimensional Hilbert spaces of quantum field theory. With this in \nmind, we explore how to cast finite-dimensional quantum mechanics in a form \nthat matches naturally onto the smooth case, especially the recovery of \nconjugate position/momentum variables, in the limit of large Hilbert-space \ndimension. A natural tool for this task is the generalized Clifford algebra \n(GCA). Based on an exponential form of Heisenberg's canonical commutation \nrelation, the GCA offers a finite-dimensional generalization of conjugate \nvariables without relying on any a priori structure on Hilbert space. We \nhighlight some features of the GCA, its importance in studying concepts such as \nlocality of operators, and point out departures from infinite-dimensional \nresults (possibly with a cutoff) that might play a crucial role in our \nunderstanding of quantum gravity. We introduce the concept of "Schwinger \nlocality," which characterizes how the action of an operator spreads a quantum \nstate along conjugate directions. We illustrate these concepts with a worked \nexample of a finite-dimensional harmonic oscillator, demonstrating how the \nenergy spectrum deviates from the familiar infinite-dimensional case.

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The finite entropy of black holes suggests that local regions of spacetime \nare described by finite-dimensional factors of Hilbert space, in contrast with \nthe infinite-dimensional Hilbert spaces of quantum field theory. With this in \nmind, we explore how to cast finite-dimensional quantum mechanics in a form \nthat matches naturally onto the smooth case, especially the recovery of \nconjugate position/momentum variables, in the limit of large Hilbert-space \ndimension. A natural tool for this task is the generalized Clifford algebra \n(GCA). Based on an exponential form of Heisenberg's canonical commutation \nrelation, the GCA offers a finite-dimensional generalization of conjugate \nvariables without relying on any a priori structure on Hilbert space. We \nhighlight some features of the GCA, its importance in studying concepts such as \nlocality of operators, and point out departures from infinite-dimensional \nresults (possibly with a cutoff) that might play a crucial role in our \nunderstanding of quantum gravity. We introduce the concept of "Schwinger \nlocality," which characterizes how the action of an operator spreads a quantum \nstate along conjugate directions. We illustrate these concepts with a worked \nexample of a finite-dimensional harmonic oscillator, demonstrating how the \nenergy spectrum deviates from the familiar infinite-dimensional case.
The finite entropy of black holes suggests that local regions of spacetime are described by finite-dimensional factors of Hilbert space, in contrast with the infinite-dimensional Hilbert spaces of quantum field theory. With this in mind, we explore how to cast finite-dimensional quantum mechanics in a form that matches naturally onto the smooth case, especially the recovery of conjugate position/momentum variables, in the limit of large Hilbert-space dimension. A natural tool for this task are the Generalized Pauli operators (GPO). Based on an exponential form of Heisenberg's canonical commutation relation, the GPO offers a finite-dimensional generalization of conjugate variables without relying on any a priori structure on Hilbert space. We highlight some features of the GPO, its importance in studying concepts such as spread induced by operators, and point out departures from infinite-dimensional results (possibly with a cutoff) that might play a crucial role in our understanding of quantum gravity. We introduce the concept of "Operator Collimation," which characterizes how the action of an operator spreads a quantum state along conjugate directions. We illustrate these concepts with a worked example of a finite-dimensional harmonic oscillator, demonstrating how the energy spectrum deviates from the familiar infinite-dimensional case.
In this thesis, we lay out the goal, and a broad outline, for a program that takes quantum mechanics in its minimal form to be the fundamental ontology of the universe. Everything else, including features like space-time, matter and gravity associated with classical reality, are emergent from these minimal quantum elements. We argue that the Hilbert space of quantum gravity is locally finite-dimensional, in sharp contrast to that of conventional field theory, which could have observable consequences for gravity. We also treat time and space on an equal footing in Hilbert space in a reparametrization invariant setting and show how symmetry transformations, both global and local, can be treated as unitary basis changes. Motivated by the finite-dimensional context, we use Generalized Pauli Operators as finite-dimensional conjugate variables and define a purely Hilbert space notion of locality based on the spread induced by conjugate operators which we call "Operator Collimation." We study deviations in the spectrum of physical theories, particularly the quantum harmonic oscillator, induced by finite-dimensional effects, and show that by including a black hole-based bound in a lattice field theory, the quantum contribution to the vacuum energy can be suppressed by multiple orders of magnitude. We then show how one can recover subsystem structure in Hilbert space which exhibits emergent quasi-classical dynamics. We explicitly connect classical features (such as pointer states of the system being relatively robust to entanglement production under environmental monitoring and the existence of approximately classical trajectories) with features of the Hamiltonian. We develop an in-principle algorithm based on extremization of an entropic quantity that can sift through different factorizations of Hilbert space to pick out the one with manifest classical dynamics. This discussion is then extended to include direct sum decompositions and their compatibility with Hamiltonian evolution. Following this, we study quantum coarse-graining and state-reduction maps in a broad context. In addition to developing a first-principle quantum coarse-graining algorithm based on principle component analysis, we construct more general state-reduction maps specified by a restricted set of observables which do not span the full algebra (as could be the case of limited access in a laboratory or in various situations in quantum gravity). We also present a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. Throughout the thesis, we discuss implications of our work in the broader goal of understanding quantum gravity from minimal elements in quantum mechanics.
In this work we take a closer look at the algebraic-operator correspondence between the momentum space and the position space which defines the form of the canonical momentum operator in position space in Quantum Mechanics (QM). Starting from the Fourier transform (FT) relationship, we present a Hermitian generalization of the canonical momentum operator in position space. The action of the generalized operator is found to generate a local flow accompanied by position-dependent rescaling, rather than a global translation. Explicit eigenfunctions are obtained for representative cases and are shown to possess a well-defined limit to the plane-wave solution in QM. As an illustration, the infinite square well problem is solved using the generalized operator, yielding a deformed spectrum that has a smooth limit to the standard QM result.
Quantum mechanics of a linear harmonic oscillator in a finite-dimensional Hilbert space satisfying the correct equations of motion is studied. The connections to Weyl's formulation of the algebra of bounded unitary operators in finite space as well as to a truncated quantized linear harmonic oscillator are discussed. It is pointed out that the discrete Fourier transformation (DFT) plays a central role in determining the actual form of the position, the momentum, the number and the Hamiltonian operators. The explicit form of these operators in different bases is exhibited for some low values of the dimension of the Hilbert space. In this formulation, it is shown that the Hamiltonian is indeed the logarithm of the DFT and that by modifying Weyl's framework to include position and momentum operators with non-uniformly spaced spectra the equations of motion are satisfied.
We make a detailed study of the infinite dimensional Galilean Conformal Algebra (GCA) in the case of two spacetime dimensions. Classically, this algebra is precisely obtained from a contraction of the generators of the relativistic conformal symmetry in 2 d . Here we find quantum mechanical realisations of the (centrally extended) GCA by considering scaling limits of certain 2d CFTs. These parent CFTs are non-unitary and have their left and right central charges become large in magnitude and opposite in sign. We therefore develop, in parallel to the usual machinery for 2 d CFT, many of the tools for the analysis of the quantum mechanical GCA. These include the representation theory based on GCA primaries, Ward identities for their correlation functions and a nonrelativistic Kac table. In particular, the null vectors of the GCA lead to differential equations for the four point function. The solution to these equations in the simplest case is explicitly obtained and checked to be consistent with various requirements.
Schwinger's finite ( D ) dimensional periodic Hilbert space representations are studied on the toroidal lattice with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-preserving canonical transformations are examined. The generalized representations of the Wigner function are examined in the finite-dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair in is studied and, based on the admissibility of the underlying q -oscillator subalgebra, an algebraic approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the Susskind-Glogower-Carruthers-Nieto phase operator formalism as well as standard action-angle Wigner function formalisms are examined in the infinite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.