量子力学的协变几何方法：迈向纯粹几何与拓扑量子理论

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本文主要探讨了"接触几何与量子力学"这一主题，作者Gabriel Herczega 和 Andrew Waldron在《物理评论快报》(Physics Letters B)的781期发表了一篇研究论文。他们提出了一个全新的量子力学一般协变方法，将广义位置、动量和时间变量看作是构成"相时空"的基本坐标。这个概念性的转变使得量子力学的原理不再局限于传统的算符形式，而是与几何学紧密关联起来。在这个新的框架下，作者指出，将量子化过程视为纯粹的几何平坦性条件，意味着量子系统的性质可以通过几何结构来理解和描述。这种视角强调了量子现象与空间-时间结构之间的深刻联系，甚至可能扩展到拓扑学的范畴。这不仅为理解复杂的时间依赖问题提供了新途径，还特别适用于那些对时钟选择或观测者依赖性存在模糊性的系统。例如，在相对论背景下，不同惯性参考系下的量子行为可以通过几何语言进行统一处理。文章的创新之处在于，它不仅仅是一个理论框架，还包含了一个副产品——一个关于如何从几何角度导出量子力学特定方程的推导过程。这可能为解决传统量子力学中的困难，如测不准原理和量子纠缠等问题提供新的思考角度，也可能推动量子信息科学和量子引力理论的发展。这篇论文代表了一种将几何学思维融入量子力学的新尝试，挑战了经典观念，并为未来的研究者探索量子世界的新维度打开了一扇窗户。通过这一一般协变方法，量子力学的许多难题可能会得到重新审视和解答，从而深化我们对微观世界的理解。

Physics Letters B 781 (2018) 312–315

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Contact geometry and quantum mechanics

Gabriel Herczeg

, Andrew Waldron

b,∗

Department of Physics, University of California, Davis, CA 95616, USA

Center for Quantum Mathematics and Physics (QMAP), Department of Mathematics, University of California, Davis, CA 95616, USA

a r t i c l e i n f o a b s t r a c t

Article history:

Received

21 March 2018

Received

in revised form 3 April 2018

Accepted

5 April 2018

Available

online 6 April 2018

Editor:

M. Cveti

We present a generally covariant approach to quantum mechanics in which generalized positions,

momenta and time variables are treated as coordinates on a fundamental “phase-spacetime”. We show

that this covariant starting point makes quantization into a purely geometric ﬂatness condition. This

makes quantum mechanics purely geometric, and possibly even topological. Our approach is especially

useful for time-dependent problems and systems subject to ambiguities in choices of clock or observer.

As a byproduct, we give a derivation and generalization of the Wigner functions of standard quantum

mechanics.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Contact geometry

Mechanics is usually formulated in terms of an even 2n-dimen-

sional

phase-space (or symplectic manifold) with time treated as

an external parameter and dynamics determined by a choice of

Hamiltonian. Yet classical physics ought not depend on choices

of clocks. However, Einstein’s principle of general covariance can

be applied to this situation by introducing an odd (2n + 1)-

dimensional

phase-spacetime manifold Z . Dynamics is now en-

coded

by giving Z a (strict) contact structure—i.e., a one-form α

subject to a non-degeneracy condition on the (phase-spacetime)

volume form:

Vol

:= α ∧ (dα)

∧n

= 0 . (1.1)

Physical phase-spacetime trajectories γ are determined by extrem-

izing

the action

S =



α . (1.2)

Since the integral of a one-form along a path γ is a coordinate

invariant quantity, general covariance (both worldline and target

space) is built in from the beginning [1]. The equations of motion

are

Corresponding author.

E-mail

addresses: Herczeg @ms .physics .ucdavis .edu (G. Herczeg),

wally @math .ucdavis .edu (A. Waldron).

ϕ(

γ ,

) = 0 , (1.3)

where the two-form ϕ := dα is maximal rank by virtue of Eq. (1.1)

and

γ is a tangent vector to the path γ in Z .

The

structure (Z , α) is called a (strict) contact geometry and

Eq. (1.3)determines its Reeb dynamics [2]. In addition to general

covariance, this formulation of mechanics enjoys a Darboux theo-

rem,

which implies the existence of local coordinates (ψ, π

, χ

)

such that α = π

dχ

− dψ (where A = 1, ..., n) that trivialize the

dynamics. Hence one might hope to treat classical and quantum

mechanics as contact topology problems.

2. Goal

We aim to develop a generally phase-spacetime covariant for-

mulation

of quantum mechanics. We ﬁnd a formulation of quan-

tum

mechanics in terms of intrinsic geometric structures on a con-

tact

manifold. Our approach is similar to Fedosov’s quantization of

symplectic manifolds [3], and indeed we were partly inspired by

that work and subsequent applications of Fedosov quantization to

models of higher spins [4]. Quantization based on contact geom-

etry

has been studied before: For example, Rajeev [5] considers

quantization beginning with (classical) Lagrange brackets (the con-

tact

analog of Poisson brackets). Fitzpatrick [6] has extended this

work to a rigorous geometric quantization setting. There is also

earlier work by Kashiwara [7] that studies sheaves of pseudodiffer-

ential

operators over contact manifolds. Investigations motivated

by quantum cosmology of the so-called “clock ambiguity” in the

quantum dynamics of time reparameterization invariant theories

may be found in [8]. Contact geometry has also been employed in

studies of choices of quantum clocks in [9].

https://doi.org/10.1016/j.physletb.2018.04.008

0370-2693/

SCOAP

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