A Transfer Operator Approach to Relativistic Quantum Wavefunction Igor Mezi c

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A Transfer Operator Approach to Relativistic

Quantum Wavefunction

Igor Mezi´c

University of California, Santa Barbara, CA 93106-5070, USA

mezic@ucsb.edu

October 25, 2022

Abstract

The original intent of the Koopman-von Neumann formalism was

to put classical and quantum mechanics on the same footing by in-

troducing an operator formalism into classical mechanics. Here we

pursue their path the opposite way and examine what transfer oper-

ators can say about quantum mechanical evolution. To that end, we

introduce a physically motivated scalar wavefunction formalism for a

velocity ﬁeld on a 4-dimensional pseudo-Riemannian manifold, and

obtain an evolution equation for the associated wavefunction, a gen-

erator for an associated weighted transfer operator. The generator of

the scalar evolution is of ﬁrst order in space and time. The probability

interpretation of the formalism leads to recovery of the Schr¨odinger

equation in the non-relativistic limit. In the special relativity limit,

we show that the scalar wavefunction of Dirac spinors satisﬁes the new

equation. A connection with string theoretic considerations for mass

is provided.

Contents

1 Introduction 2

2 Preliminaries 3

arXiv:2210.12891v1 [quant-ph] 24 Oct 2022

3 Wavefunction Evolution 4

3.1 Postulates ............................. 5

3.2 Relationship with the Schr¨odinger Equation . . . . . . . . . . 7

3.3 Relationship to Weighted Composition Operators . . . . . . . 8

4 Special Relativity Case: Dirac Equation 8

5 Examples 10

5.1 The non-relativistic case of ﬂat 1-dimensional conﬁguration

space................................ 10

5.2 TheLagrangian.......................... 13

5.3 Dispersion relationship . . . . . . . . . . . . . . . . . . . . . . 14

5.4 deBroglie relationships . . . . . . . . . . . . . . . . . . . . . . 14

5.5 The relativistic wavepacket . . . . . . . . . . . . . . . . . . . 15

5.6 Non-relativistic dispersion relationship . . . . . . . . . . . . . 17

6 Discussion and Conclusions 21

A Mass and the Wavefunction 21

B Relationship to String Theory 22

1 Introduction

Dynamical systems theory can be pursued in the phase-space (Poincar´e) for-

malism [1], or alternatively in the Koopman formalism [2, 3, 4]. The Koop-

man formalism applied in phase space leads to the probability interpretation

of the associated phase-space wavefunction consistent with the Born inter-

pretation in quantum mechanics [5]. Born’s proposal on interpretation of

the square of the wavefunction as probability led to successful application of

quantum mechanics to a broad swath of problems. The dichotomy between

the phase-space domain of the classical wavefunction and the physical space-

time nature of the quantum wavefunction recently led to a number of eﬀorts

to reconcile the two (see e.g. [6, 7, 5, 8, 9, 10, 11, 12, 13] and the rest of

the articles in this volume). These works are pursued in the nonrelativistic

context. A diﬀerent approach was pursued in [14], where the spectrum of

the quantum harmonic oscillator was related to the Koopman operator spec-

trum of the classical harmonic oscillator by a construction involving a pair

of harmonic oscillators with Hamiltonians of opposite sign.

In this paper we pursue the operator-theoretic approach to derive an

equation of motion - the relativistic quantum transfer equation (RQTE) - for

the resulting quantum-theoretical wavefunction starting from a relativistic

dynamical system on a 4-dimensional space-time. Namely, we start from the

spacetime manifold, and not the phase space, and utilize Fock’s proper-time

formalism [15]. The key idea is that the RQTE arises from the projection of

a 4-dimensional conserved ﬁeld through a complex scalar ﬁeld. The resulting

equation - when presented in the probabilistic interpretation - has solutions

that reduce to the Schr¨odinger equation in the nonrelativistic limit, and the

euation for the Dirac scalar in the special relativity limit.

The paper is organized as follows: in section 2 we introduce the relativis-

tic setting and the notation. In section 3 we derive RQTE under several

postulates. We describe the class of operators - the weighted composition

operators - that are generated by RQTE. In section 4 we discuss the rela-

tionship between RQTE and the Dirac equation. In section 5 we consider

several examples treated within the RQTE formalism: harmonic oscillator,

particle in a box and Gaussian wavepacket. We discuss the relationship of the

RQTE wavefunction with mass in Appendix A and relationship with notion

of mass in string theory in Appendix B.

2 Preliminaries

Let Mdenote a 4-dimensional space-time pseudo-Riemannian manifold en-

dowed with a metric tensor g. Consider the section of its tangent bundle

T M, the proper velocity ﬁeld (the four-velocity ﬁeld) V=dX/dτ [16] where

X(τ) : R→Mis the time-like world line parametrized by the proper time τ.

We deﬁne the level sets of proper time τon Mto be able to use it for evolu-

tion of the ﬂow of V. Any vector ﬁeld on Mcan be rectiﬁed near a point X

with V(X)6= 0 [17]. Since the four-velocity ﬁeld Vis nonzero everywhere,

there exists a neighborhood NXof any point Xin which it can be rectiﬁed

by a local choice of coordinates (x0(X), ..., x3(X)) on M. In the coordinates

(x0, ..., x3), the four velocity ﬁeld has components V= (c, 0,0,0). Note that

(x1, x2, x3) label points on the x0= 0 intersection of an individual world line.

Let σ=τ(0, x1, x2, x3) be the proper time ﬁeld over the section x0= 0. We

can deﬁne a new parameter s=τ−σin a small neighborhood of X. In this

way, the 0 proper time is synchronized for all trajectories in a neighborhood.

Absent topological obstructions, this can be extended to the whole of Mto

deﬁne a space slice Mτ

S. With topological obstructions, the construction is

still valid on subsets of M. In this case, we redeﬁne Mto be such a subset.

We keep the notation τfor the reparametrized proper time. The norm of

Vdeﬁned using the metric tensor gon Mis constant, ||V||2=−c2,where

cis the speed of light in vacuum [15] (we are using the (−1,1,1,1) metric

convention). We denote by Gτ:M→Mthe ﬂow of Von M. We denote by

Dτfthe proper time derivative (i.e. the Lie derivative [17]) of f, representing

the change of a scalar physical quantity in the direction of V. The manifold

is equipped with the volume form with density p|det g|.

The ﬂow Gτcan be used to deﬁne the family of Koopman composition

operators [2] parametrized by τacting on (in general, complex) functions

f:M→Cby

Uτf(X) = f◦Gτ(X).(1)

Note that, in contrast with Koopman’s original formulation on the phase

space, Uτacts on functions deﬁned on the spacetime M. The operator Dτ

is the generator of the evolution Uτ. The functions in the eigenspace at 0 of

Dτare conserved quantities, since

Dτf= 0 (2)

implies fis conserved on the world line X(τ). In terms of the Koopman

operator evolution, for such fwe get

Uτf(X) = f(X).(3)

In line with the terminology used in Koopman operator theory [18, 3] we

call functions g:M→Cobservables. By identiﬁcation with the associ-

ated, position-dependent operators, the terminology is consistent with that

of quantum mechanics.

3 Wavefunction Evolution

Consider a ﬁeld ρconserved under trajectories of Von M. Its restriction

onto level sets of the complex ﬁeld eiY of modulus 1,with phase Yreads

DeiY =ρ

i|DY |eiY .(4)

We assume that the density ρis not observed directly, but is projected via

a complex scalar ﬁeld eiY , as indicated by equation (4) and shown in ﬁgure

1. The geometry can be described as that of a ﬁber bundle over Mand ρeiY

is a horizontal lift of the spacetime trajectory. This construction renders the

appearence of complex numbers in quantum mechanics a natural consequence

of geometry.

Figure 1: The geometry of the ﬁber bundle over M.

Given this geometric formulation, we use the following postulates:

3.1 Postulates

1. There is a function ρ:M→Rthat is constant on trajectories of V

satisfying

Dτρ= 0.(5)

We argue in the Appendix A that ρis physically the oscillation wavenum-

ber and is related to mass (and thus energy).

2. The observable wavefunction ψis the pushforward of ρby an observable

eiY given by

ψ=ρ

i|DY |eiY =ρ

iKeiY =ρ

iK e−iY .(6)

where Yis a phase and K=|DY |. This, in turn, implies

ρ=iKeiY ψ. (7)

3. ρ/|DY |is an invariant density for V.

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ATransferOperatorApproachtoRelativisticQuantumWavefunctionIgorMezicUniversityofCalifornia,SantaBarbara,CA93106-5070,USAmezic@ucsb.eduOctober25,2022AbstractTheoriginalintentoftheKoopman-vonNeumannformalismwastoputclassicalandquantummechanicsonthesamefootingbyin-troducinganoperatorformalismintoclassi...

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