Jacobi's Principle and the Disappearance of Time

Abstract

Jacobi's action principle is known to lead to a problem of time. For example, the timelessness of the Wheeler-DeWitt equation can be seen as resulting from using Jacobi's principle to define the dynamics of 3-geometries through superspace. In addition, using Jacobi's principle for non-relativistic particles is equivalent classically to Newton's theory but leads to a time-independent Schrodinger equation upon Dirac quantization. In this paper, we study the mechanism for the disappearance of time as a result of using Jacobi's principle in these simple particle models. We find that the path integral quantization very clearly elucidates the physical mechanism for the timeless of the quantum theory as well as the emergence of duration at the classical level. Physically, this is the result of a superposition of clocks which occurs in the quantum theory due to a sum over all histories. Mathematically, the timelessness is related to how the gauge fixing functions impose the boundary conditions in the path integral. Comment: Published version. Significant amendments to presentation. 27 pages

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arXiv:0804.2900v3 [gr-qc] 23 Feb 2010
Jacobi’s Principle and the Disappearance of Time
Sean Gryb
Perimeter Institute for Theoretical Physics
Waterloo, Ontario N2L 2Y5, Canada and
Department of Physics and Astronomy, University of Waterloo
Waterloo, Ontario N2L 3G1, Canada∗
(Dated: February 23, 2010)
Abstract
Jacobi’s action principle is known to lead to a problem of time. For example, the timelessness
of the Wheeler-DeWitt equation can be seen as resulting from using Jacobi’s principle to define
the dynamics of 3-geometries through superspace. In addition, using Jacobi’s principle for non-
relativistic particles is equivalent classically to Newton’s theory but leads to a time-independent
Schr¨ odinger equation upon Dirac quantization. In this paper, we study the mechanism for the
disappearance of time as a result of using Jacobi’s principle in these simple particle models. We find
that the path integral quantization very clearly elucidates the physical mechanism for the timeless
of the quantum theory as well as the emergence of duration at the classical level. Physically, this
is the result of a superposition of clocks which occurs in the quantum theory due to a sum over all
histories. Mathematically, the timelessness is related to how the gauge fixing functions impose the
boundary conditions in the path integral.
PACS numbers: 04.20.Cv
Keywords: Mach’s principle; Jacobi’s principle, the problem of time, Barbour-Bertotti Theory
∗Electronic address: sgryb@perimeterinstitute.ca
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Contents
I. Introduction
3
A. Background3
B. Summary of Results5
C. Earlier Work6
D. Outline6
II. Hamiltonian Formulation
7
A. Jacobi’s Principle and the Timeless Mechanics of Barbour and Bertotti7
1. Action and Equations of Motion7
2. Hamiltonian Formulation9
B. Parameterized Newtonian Mechanics (PNM)10
1. Action and Equations of Motion 10
2. Hamiltonian Formulation12
III. On Gauge Invariance and the Phase Space Path Integral
12
IV. Path Integral Quantization
14
A. PNM14
1. Boundary Conditions 16
2. Connection to Standard Quantum Mechanics16
3. kPNMand Energy Eigenstates 17
B. JBB Theory18
1. Boundary Conditions 19
C. Connection Between kPNMand kJBB
19
V. Time and the Stationary Phase Approximation
20
VI. Outlook
23
Acknowledgments
24
References
25
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I. INTRODUCTION
A.Background
“It is utterly beyond our power to measure the changes of things by time. Quite the
contrary, time is an abstraction, at which we arrive by means of the changes of things...”
In this beautiful quote by Mach from The Mechanics [1] dated 1883, he lays out what
has been called his second principle [2]: that time should be a measure of the changes of
things. Though the argument seems simple, elegant, and epistemologically sound, it took
nearly 100 years before Barbour and Bertotti where able to formulate this principle into
a mathematically rigorous theory of time in 1982 [3]1. The reason for this delay can not
be attributed to technical complications, since the mathematics have been well understood
since Jacobi, but rather to conceptual confusion surrounding how Mach’s principles are
implemented in General Relativity. Though it was clear that Einstein was heavily influenced
by Mach’s ideas, general covariance proved to be misleading as a way of implementing
relationalism. However, through the papers of Barbour and collaborators [3, 5–7], we now
have a clear picture of how Mach’s ideas can be used to derive General Relativity (GR). From
this work, we know that, in regards to time, Mach’s second principle can be implemented
in GR by using Jacobi’s principle to determine the classical dynamics of the system. The
consequences of using such a definition of time in the quantum theory lead to a problem
of time and are the main concern of this paper. Specifically, we find that, in the path
integral description, each path represents a different relational clock. Thus, a sum over all
paths leads to a kind of superposition of clocks (the precise definition of this will be given
in Sec. (V)) resulting in a time independent theory.
Though there are many different problems of time and many different ways of stating
each [8, 9], a simple way of stating the most obvious aspect of the problem of time is to note
that the Wheeler-DeWitt (WDW) equation [10]
?Gabcd(ˆ g3)ˆ πabˆ πcd+?R3(ˆ g3) − Λ??Ψ[g3] = 0,
(1)
in a configuration basis, depends only on the 3-metric g3and variations with respect to it.
1For a crystal clear account of how Jacobi’s principle can be thought as a way of implementing Mach’s
ideas about time, see [4].
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Thus, the formal wavefunctional Ψ[g3] depends only on the configuration space variables
g3, and is completely independent of any variable one could interpret as time. As a result,
solutions to the WDW equation are stationary states. This fact leads to difficulties, for
example, in forming an inner product under which Ψ evolves unitarily and with which one
can define a clear notion of probability.
First encountered in the context of GR, this old but surprising result is characteristic of
any theory that uses Jacobi’s principle for determining its classical dynamics. For example, if
one uses Jacobi’s principle in non-relativistic particle mechanics, one finds (see Sec. (IIA2))
that the analogue of the WDW equation is the time-independent Schr¨ odinger equation
(Eq. (15)). As we will see, the result that one gets a quadratic scalar constraint on the
wavefunction whose solutions are stationary state comes from using Jacobi’s principle to
implement Mach’s second principle in order to define time in a relational way. As a result,
one can study, as we will do here, this aspect of the problem of time in GR by considering the
much simpler case of non-relativistic particles moving through a space dependent potential
using Jacobi’s principle. We will call this theory Jacobi-Barbour-Bertotti (JBB) theory since
it was first written down by Jacobi but was given an interpretation in terms of Mach’s second
principle by Barbour and Bertotti (BB)2. Because the configuration space variables are just
simple functions, and not tensor fields on a 3-manifold as is the case in GR, we will not be
able to reproduce certain aspects of GR such as the infinity of scalar constraints which lead
to Wheeler’s many-fingered time [19]. However, these toy models provide a simple means for
studying the global aspects of the disappearance of time and can be applied, for example,
directly to symmetry reduced models of GR, like mini-superspace, which are of primary
importance in cosmological applications, or any model of gravity with a fixed lapse, such as
Hoˇ rava’s theory [20].
2It should be noted that this is different from what is usually referred to as BB theory [3, 11–17] since
we are not considering the spatial symmetries which produce linear constraints analogous to the 3-
diffeomorphisms of GR. It can be checked [18] that the spacial symmetries add nothing to the discussions
regarding time.
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B. Summary of Results
In this paper we will concern ourselves with two main tasks: 1) the physical mechanism,
from the point of view of the path integral, for the disappearance of time in the quantum
theory, and 2) the conditions underwhich one can recover a notion a time.
Our motivations stem from the following puzzle: as we will see, the canonical quanti-
zation of JBB theory leads to the time-independent Schr¨ odinger equation whose solutions
are stationary states. However, the classical equations of motion give Newton’s laws with
a specific definition of the Newtonian time in terms of the classical trajectories of the par-
ticles in the system (Eq. (6)). Aside from the obvious difficulties associated with defining
an inner product for the Hilbert space, it is even less clear why one should expect to obtain
a classical limit with a specific notion of time, completely equivalent to Newton’s, from a
quantum theory which is completely time independent. How could Newton’s time be hidden
in the time independent Schr¨ odinger equation? Alternatively, why is it that, in Newton’s
theory, time is perfectly well described off-shell while in Jacobi’s theory, which is classically
equivalent, time is only meaningful on-shell? What has happened to time?
In contrast, the path integral for JBB theory is easy to write down, at least at the
formal level. We will find that the analogue of the Newtonian time can be written down in a
straightforward way for this path integral. However, in the full quantum theory, this quantity
contributes to each term in the sum over all histories leading to a superposition of all possible
clocks: the end result being a timeless theory. In the stationary phase approximation, the
path integral is dominated by the classical trajectory picking out a unique clock. We will
find that this clock does indeed read out a time that is equivalent to that used in Newtonian
mechanics. This gives a consistent picture for how a time dependent classical theory could
be the limiting form of a time independent quantum theory. Furthermore, this picture also
tells us how to obtain a unique notion of duration in the quantum theory. For quadratic
potentials, the stationary phase approximation is exact and the Newtonian time along the
classical trajectory servers as a unique notion of duration even in the full quantum theory.
Taking a close look at the path integral thus provides us with a unique notion of time off-shell
(for certain potentials) and intuition as to why this notion of time breaks down in general.
This intuition may be invaluable in suggesting new ways inwhich duration may be defined
in a time independent quantum theory.
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