\documentstyle[12pt,doublespace]{article}
\newcommand{\f}{\begin{equation}}
\newcommand{\ff}{\end{equation}}
\newcommand{\blankline}{\vskip .3cm}
\begin{document}
\section{Introduction}
{\bf 1.1 A timeless problem.}
\blankline
A major conceptual problem in quantum gravity [1] is the issue of
what time is, and how it has to be treated in the formalism. The
importance of this issue was recognized at the beginning of the
history of quantum gravity [2], but the problem is still open, and has
recently received increasing attention [3]. The controversy about
time in quantum gravity does not refer to a uniquely defined
problem; the quarrel has been over the questions even more than over
the answers [4-17]. In this paper we propose a point of view on the
puzzle and a physical hypothesis for its solution.
The physical hypothesis that we put forward is the absence of a well
defined concept of time at the fundamental level.
We shall provide a precise mathematical form of this
hypothesis. For the moment, we may illustrate it as
follows. We suggest that at the Plank scale dynamical systems
cannot be described as evolving in a universal time quantity $t$.
More precisely, they cannot be described as hamiltonian systems in
the strict sense. Instead, evolution may only be defined with
respect to physical clock variables.
As will be discussed, this revised concept of time is, in a sense,
implicit in classical general relativity. In this paper, we show that
quantum mechanics can be naturally extended in order to incorporate
it. The idea is not new\footnote{Actually, it is quite old: " Tempus
item per se non est, sed rebus ab ipsis / consequitur sensus,
transactum quid sit in
aevo, / tum quae res instet, quid porro deinde sequantur. / Nec per se
quemquam tempus sentire fatendumst / semotum ab rerum motu
placidaque
quiete. " [18] }; it is implicit in several works (see for instance
refs.[2,5,7,9,17]). But to our knowledge it has never been defined and
studied in detail.
The main assertion of this paper is that there is a natural extension
of canonical Heisen\-berg-picture quantum mechanics, which
remains well-defined in the
absence of a well-defined Schr\"odinger equation, and in the absence
of a fundamental time. This extension is well-defined both in terms
of the coherence of the formalism, and from the point of view of the
viability of the standard probabilistic interpretation.
The key step that allows us to define this extension is a technical
result on the observables of theory. The result is that, even in the
absence of a fundamental time and of an exact Schr\"odinger
equation, there are gauge invariant observables (commuting with the
hamiltonian constraint) which describe evolution with respect to
physical clocks. These observables are self-adjoint operators on the
space of the solutions of the Wheeler-DeWitt equation.
Thus, the solution we propose to the time issue in quantum gravity
is the following. At the fundamental level, there is no absolute time
in terms of which a Schr\"odinger equation could be defined. The
fundamental theory is described by the extended Heisenberg-picture
canonical quantum mechanics, equipped with the standard
probabilistic interpretation. Evolution with respect to physical
clocks is described by self-adjoint operators corresponding to the
observables we mentioned [12].
In the course of the paper, this picture will be motivated, detailed
and shown to be consistent. Of course, only a future complete
quantum gravity theory can establish if it is also realistic.
The problem of time arises in the canonical formulation of the
theory as follows. In quantum general relativity (as in any
diffeomorphism invariant quantum field theory), the Schr\"odinger
equation is replaced by a Wheeler-DeWitt equation, in which the
time-coordinate has disappeared from the formalism. An accepted
interpretation of this fact is that {\it physical\ } time has to be
identified with one of the internal degrees of freedom of the theory
itself ({\it internal time})[4]. Evolution `in time' is identified with
evolution with respect to this internal time. We follow this
interpretation. In this philosophy, it has been shown [5,6,13,14] that
a Schr\"odinger equation may emerge from the Wheeler-DeWitt
equation.
However, it is very likely that, for any choice of the internal
time, only an {\it approximate} Schr\"odinger equation emerges.
In other words, the evolution in the internal time is described by a
Schr\"odinger equation only within some approximation. This
situation is satisfactory as far as
the connection between the theory and the world that we see is
concerned. In fact, whatever experiment we may perform, we are
always well inside this Schr\"odinger approximation. But a theory
that makes sense only within an approximation is not a satisfactory
theory. Thus, the following question is relevant. Does the theory
make sense {\it beyond} the Schr\"odinger approximation~?
If one attempts to take the theory seriously beyond the
Schr\"odinger approximation, several difficulties arise. Just to
mention one of them: if the Schr\"odinger equation is valid only to
first approximation, then the norm of the state is only
approximately conserved. Can a probabilistic interpretation be
maintained, if the norm is not exactly conserved ?
Different attitudes towards the physics of the Wheeler-DeWitt
equation outside the Schr\"odinger approximation can be found in the
literature. An illuminating discussion on the disagreements on the
issue of time is given in ref.[19]. Here, we quote some of these
attitudes; the list and the references are exemplary only and are by
no means exhaustive.
a) The theory makes sense only if an "exact internal time" is
found such that an exact (rather that approximate) Schr\"odinger
equation holds [20]. In this case,
there wouldn't be any non-Schr\"odinger regime.
b) The theory outside the Schr\"odinger regime requires
modifications of the basic structure of quantum mechanics; for
instance we should use infinite norm states, or give up canonical
(i.e. Hilbert space) quantum mechanics [7].
c) Because of these difficulties, a quantum mechanical theory of the
gravitational field does not make sense, and a radical revision of the
basic ideas of quantum mechanics is needed for quantum gravity
[21].
d) Standard quantum mechanics, suitably interpreted, can be used
also for the non-Schr\"odinger regime [2].
The solution we propose in this paper is more or less distinct from
the ones listed. We suggest that the basic structure of canonical
quantum mechanics, namely Hilbert space of states, self-adjoint
operators representing observables, probabilistic interpretation and
wave function collapse, may still accommodate quantum
gravitational physics. An exact internal time
is not required, nor particularly relevant for the quantization.
Rather, it is the concept of time itself that needs to be revised. The
formalism of classical mechanics (as we will elucidate) is already
capable of accommodating this revised concept of time. Canonical
quantum mechanics, in turn, can be very naturally extended in order
to incorporate this revised concept of time.\footnote{Everything we
say is not only valid for general
relativity, but also for any generally covariant field theory. The
same problems appear in the topological quantum field theories [22].
What we say is also relevant for any formulation of string
theory that does not assume a fixed background metric on the target
space.}
For clarity, let us say that here we do not address the
problem of the existence of an exact internal time in general
relativity. Instead, we assume: first, that a way to obtain an
approximate description of the world as we see it (with time) can be
extracted from the theory; second, that this description is valid
only within the approximation. As far as the problem of the choice
of the internal time is concerned, we refer to the literature [5, 8],
and in particular to the recent work of A. Ashtekar on the definition
of an internal time in the weak field limit [6]. See also the works on
the Machian cosmological time [14] and on the observables in general
relativity [16].
The paper is organized as follows. In section 1.2 we introduce the
basic physical hypothesis. In section 1.3 we motivate this
hypothesis by discussing the concept of time in classical general
relativity. In section 2.1, we show that there is a formulation of
classical mechanics which allows us to treat dynamical systems
without making reference to the universal time. In section 2.2,
we discuss the observables that describe the evolution with respect
to the clock time. In section 3.1, the quantum mechanics of the
systems without time is defined. In section 3.2, two technical
issues are investigated: the quantization procedure and the problem
of choosing the scalar product. Section 3.3 extends the results on
the observables that describe evolution in clock time to the quantum
domain. In section 4.1, the proposed solution to the time issue in
quantum gravity is summarized. Section 4.2 contains a discussion
of the difficulties of this solution and some speculations. Section
4.3 contains the conclusions.
There are three papers which are strictly related to the present one
and complementary to it. In the first one [15], a model with
approximate Schr\"odinger equation and no absolute time is
introduced. Its quantization is a concrete example of the ideas
exhibited in the present paper. In the other two papers [16], the
problem of the observables of general relativity is studied,
respectively in the classical and quantum context. Gauge-invariant
observables of the kind introduced in this paper are constructed.
\blankline \blankline \blankline
{\bf 1.2 Clocks and absolute time.}
\blankline
Perception of the flow of time is probably an elementary experience.
In Newtonian physics, as well as in standard quantum mechanics, it
is assumed that this experience corresponds to the existence of an
absolute quantity, the time. This quantity, namely the time of
Newtonian, hamiltonain or quantum mechanics, will be denoted $t$ .
To measure $t$ we use clocks. A clock is a system with
a variables, for instance the position of a hand, which has a simple
behavior in $t$. In this paper, we shall denote a clock variable (the
position of the hand) as $T$ ; we shall denote variables of different
clocks as $T, T', T", ... $ . Good clocks may have, for instance, a
linear behavior in $t$
\f
T(t)= \alpha t.
\ff
It is an elementary-physics-course observation the fact that we
never really measure $t$; rather, we always measure $T$'s. The
value of a physical quantity $Q$, measured at a time $t$, is denoted
$Q(t)$. Since time is determined by measuring a clock variable $T$,
what is actually measured is not $Q(t)$ and $T(t)$, but only the
combined quantity $Q(T)$. Thus, $t$ doesn't ever appear in laboratory
measurements.
The observation that we never reach $t$ in the experiments, but we
only reach $T, T', T", ... $, is not a trivial observation. Since $t$
cannot be observed, eq.(1) can never be verified. We check clocks
one against the other; namely, we measure $T(T'), \ T'(T"), ...$ and so
on. Correspondingly, the problem of constructing clocks has
historically been, and still is, a delicate problem. Galileo used his
pulse to measure the oscillations period of a pendulum and to
discover that they were isochronous. Few years later
doctors were using a pendulum to measure the period of people's
pulse and to check whether they were isochronous.
Indeed, what we have is a large collection of clocks.
The clocks agree one with the other within certain unavoidable
experimental errors. Up to a certain approximation, they provide a
reasonable standard, against which dynamical theories and new
clocks can be checked. But anytime there is need of measuring time
at smaller scales, experimentalists find themselves in the same
situation as Galileo: the pulse for measuring the pendulum and the
pendulum for measuring the pulse. From the experimental point of
view, $t$ can be defined only as the idealized extrapolation of the
(concurring) value of a large ensemble of clock variables.
If $t$ can never be reached experimentally, still it plays a major
role in the conceptual framework of Newtonian and quantum
mechanics. Indeed, Newton or Hamilton equations, as well as the
Schr\"odinger equation, are grounded on the underlying assumption
that there exists a $t$, in which the dynamics is defined.
There are many basic differences between the absolute time
variable $t$ and the clock variables $T, T', T", ... $. Any
realistic physical clock variable satisfies eq.(1) only within some
approximation. $t$ is assumed to run from minus
infinity to plus infinity; clock variables may vary within a bounded
interval. In general, the agreement between the clock variable $T$
and the assumed absolute $t$ is given for granted only down to
certain scale. Below that scale, higher order physical effects,
systematic or statistical errors and quantum
fluctuations\footnote{Jim Hartle has made a detailed study of the
dynamics of clocks under various circumstances, concluding that
ideal clocks capable of surviving quantum gravitational fluctuations
do not exist [7].}, jeopardize the performance of any clock: If I look
carefully at the hand of my watch, I see that it proceeds
in little jumps. These differences between $t$ and $T$ imply
that, given a variable $Q(t)$ and a clock variable $T(t)$, in general it
is not possible to describe the evolution in clock time $Q(T)$ in
hamiltonian form.
Given these observations, we may now state the basic physical idea
of this paper. We put forward the hypothesis that the idealized
absolute hamiltonian, or Schr\"odinger, time $t$ cannot be defined
down to the Planck scale. At the Planck scale it is still possible to
talk of the clock variables $T, T', T", ... $, but it does not make sense
to talk of the absolute time $t$.
More precisely, we suggest that the theoretical framework needed
for understanding quantum gravity requires that one abandons the
idea of the existence of the universal quantity $t$, to which the
specific clock variables are approximations. Only the quantities
$Q(T), Q(T'),.... $ are defined at the fundamental level. Since the
evolution in the clock times does not admit a hamiltonian
description, similarly, we do not expect that a
Schr\"odinger-equation description could be possible.
In the next section we motivate this hypothesis. In the
following ones, we show that the theoretical instruments for
handling the absence of $t$ already exist in classical physics (part
2), and can be easily constructed in quantum physics (part 3).
\blankline\blankline
{\bf 1.3 Time in general relativity.}
\blankline
The first adjustment of the idea of a universal time $t$
follows from special relativity. In special relativity $t$ is replaced
by a class of related times: the Lorentz-times of all the different
Lorentz observers. Equivalently, the hypothesis of the existence
of $t$ is replaced by the hypothesis of the existence of the
Minkowski manifold with its peculiar metric structure.
A much more radical and subtle modification of the concept of time
is implicit in general relativity. In view of the quantization, and in
particular in view of the fact the Schr\"odinger equation requires
$t$, the concept of time in classical general relativity has to be
accurately considered.
As a preliminary step, let us consider the motion in an arbitrary
assigned gravitational field, namely in a given solution
$g_{\mu\nu}$ of Einstein equations. \footnote{In this paper we
assume a compact topology for the spacelike slices of space time.}
Every object travelling along a world line $l$ in $g_{\mu\nu}$
measures a time flow which is given by the proper time along $l$.
Thus, there is a definition of a time quantity for every given
solution $g_{\mu\nu}$ of Einstein equations and every given
trajectory $l$ in this solution. By itself, of course, the independent
time-coordinate $x^o$ (argument of $g_{\mu\nu}(\vec x,x^o)$) is not
a physical time: physics, indeed, can be reformulated in terms of any
reparametrization of $x^o$.
In quantum gravity, we are not concerned with the motion
in given gravitational fields, but with the dynamical evolution of the
gravitational field itself. Einstein equations provide the evolution
of the gravitational field in $x^o$; but $x^o$ is not a physical time. In
which physical time is the evolution of the gravitational field given
? As well known, this question is far from being trivial. We wish
we were able to formulate (compact space) general relativity as a
hamiltonian system evolving in a physical time parameter $t$, but
such formulation has never been constructed.
Let us begin to study this question in physical terms.
In non-gravitational physics, the experimentalist has a clock and
describes the evolution with respect to it. The clock is represented
in the theory by the independent variable $t$. Now let us consider
gravity. Assume the experimentalist has a clock and measures the
evolution of the gravitational field with respect to this clock. To
which variable of the theory does the clock correspond? The clock
cannot be identified with the time-coordinate $x^o$ for the
following reason. The evolution of the gravitational field in the
clock time is uniquely determined by the initial conditions, while
the evolution of $g_{\mu\nu}$ in $x^o$, as given by Einstein
equations, is under-determined.
The solution, of course, is that the clock is a physical objects; its
motion and its rhythm are determined by its equations of motion. If
we consider the equations of motion of the gravitational field {\it
and} the clock, then the problem is not under-determined. But the
gravitational field enters the equations of motion of the clock
(without gravitational field the equations of motion of physical
objects cannot even be written). The dynamics of the clock cannot
be disentangled from the dynamics of gravity itself.\footnote{This
is very different from non-gravitational physics.
In non-gravitational physics, we can first solve the dynamics of the
clock, and forget about it, and then study the dynamics of, say, the
Maxwell field. One can in fact always assume that the interaction
between the field and the clock (how the clock is affected by the
field) can be made arbitrarily small.}
In order to calculate the evolution measured by the experimentalist,
we have to evolve the gravitational
field {\it and} the clock variable together, then solve away
$\vec x, x^o$, and obtain gravitational quantities as functions of the
clock variable.\footnote{For a more detailed version of this
discussion, see ref. [16].} The conclusion is that, in order to predict
the evolution in the physical clock time of the gravitational field,
we have to consider the coupled gravity+clock dynamical system.
The same conclusion can be reached in a formal way as follows. In
any theory in which there is gauge invariance, we must
assume that only gauge-invariant quantities are observable [26].
Because of the general covariance of general relativity,
gauge -invariant quantities must be independent of the coordinates
$\vec x, x^o$. Let us focus on $x^o$. No quantity that depends upon
$x^o$ can be gauge-invariant. Indeed, it is possible to formulate
general relativity without even referring to $x^o$, as in the
Hamilton-Jacoby formulation.
It is not easy to construct (local) gauge-invariant quantities in
general relativity. In principle, nothing forbids that observables in
pure general relativity could be constructed by expressing certain
gravitational degrees of freedom as functions of certain others. In
practice, this has never been completely achieved theoretically, and
seems hopeless experimentally. To our knowledge, the only way to
construct gauge-invariant observables in a gravitational theory is to
consider general relativity coupled with matter and to express the
gravitational degrees of freedom as functions of the matter degrees
of freedom. Gauge invariant quantities obtained in this way are
constructed in ref.[16]. In any case, we have to solve away $x^o$ and
express certain degrees of freedom as functions of others. Among
these others, we identify the physical-time degree of freedom.
The conclusion of both the physical and the formal discussions is
that in general relativity, physical time has to be identified with
one of the degrees of freedom of the theory itself (the "clock"). Such
definition of time is often referred to as internal time.
Internal times differ from a hamiltonian time in many respect.
First of all, the theory does not single out one or the other of these
internal times. Second, none of the (proposed) internal times has all
the features that characterize the $t$ variable of hamiltonian and
quantum mechanics. For instance, reasonable internal-time
variables may grow (in $x^o$) up to a maximum value and then
decrease. More precisely, there is no proposed internal time
such that the theory can be expressed as a well defined hamiltonian
system evolving in this internal time. Third, by definition an
internal time refers to a specific physical variables, unlike the $t$
quantity, which is supposed to be universal. Thus, general relativity
treats time in a peculiar way, as compared to pre-relativistic
physics. The absolute quantity $t$ has disappeared. In its place,
there are different possible internal times, related to specific
physical variables.
Now, the internal times can be identified with the clock variables
$T, T', T", ...$ discussed in the previous section. Thus, maybe quite
surprisingly, general re\-la\-tivity does not provide the evolution
in an absolute time $Q(t)$ and $T(t)$, but only the observable
evolution $Q(T)$. More precisely, there exists a time quantity in the
theory, which is $x^o$, but the evolutions $Q(x^o)$ and $T(x^o)$ are
non-gauge-invariant, and therefore non-observable: the absolute
quantity $t$ has been replaced by an arbitrary and unobservable
gauge parameter $x^o$. The observation made in section 1.2, that
only $Q(T)$ can be observed, is incorporated in the formalism of
general relativity.
As far as the classical theory is concerned, these these fine
distinctions are a bit superfluous. After all, once the metric has
been calculated, a pseudo-riemanian manifold, does not seem to be
conceptually very different from a Minkowski space.\footnote{It is
[16].} However, the consequences of the above discussion are far
reaching at the quantum level. As Wheeler first emphasized, as in
the quantum theory the concept of trajectory disappears, in
quantum gravity there is no pseudo-riemanian manifold at all. More
precisely, quantum observables are attached only to gauge invariant
quantities. Thus, there is no room in the quantum theory for $Q(x^o)$
and $T(x^o)$. Operators correspond only to gauge-invariant
quantities. In the quantum domain, the absence of the absolute time
$t$ is not intuitively remedied by a picture of the pseudo-riemanian
manifold.
The only way out that we see, is to completely abandon the idea of
absolute time. Only the evolution with respect to clocks makes
sense. In certain physical situation, or for particular solutions of
Einstein equations \footnote{in particular, of course, for a flat
solution}, we may idealize these clocks in terms of $t$. At the
Planck scale, we may not.
In standard quantum mechanics, the Schr\"odinger equation {\it
requires} the existence of a $t$, which corresponds to the classical
hamiltonian time. To do quantum gravity, an alternative formulation
of quantum mechanics is needed. This formulation should not require
the idealized quantity $t$ as part of the basic formalism; istead, it
should be able to directly deal with $Q(T)$ quantities.
But, before going to quantum mechanics, if we abandon the idea that
time is one of the conceptual bedrocks of the theory, does it still
make sense to do physics, calculate measurable quantities, and
develop a consistent and satisfactory picture of an evolving
universe~?
\blankline
\blankline
\blankline
\section{Classical dynamical systems without time.}
{\bf 2.1 Mechanics without time: pre-symplectic mechanics.}
\blankline
The possibility of describing dynamical systems without
hamiltonian time has to be first explored in the context of classical
mechanics.
Mechanics may be defined as the general theory of the evolution of
physical systems in time. From this point of view, time is required
for the very definition of the elementary mechanical concepts. For
instance, the state of the system is defined at a given time. In such
a conceptual framework, $t$ is required.
However, there exists an alternative starting point for mechanics.
This is provided by presymplectic mechanics. This formulation does
not require the absolute time for defining the basic concepts of the
theory.
We shall illustrate presymplectic mechanics by first showing
that hamiltonian mechanics admits a reformulation in terms of a
presymplectic space, and then noticing that this reformulation does
not require the variable that represents time to be specified, or even
defined. Readers familiar with presymplectic mechanics may skip
this presentation.
In presymplectic mechanics, which is an elegant generalization of
standard hamiltonian mechanics, a dynamical system is just
defined by a presymplectic manifold $(C,\omega)$. Let
$(S,\omega_S,H)$ be a hamiltonian system: $S$ is the phase space,
$\omega_S$ is the symplectic form, and $H$ is the hamiltonian. Let
$q_i,p^i$ be canonical coordinates on $S$ ($\omega_S=dp^i\wedge
dq_i$). The dynamical system is completely described on the space
$C=S\times R$, with coordinates $q_i,p^i, t$, by the presymplectic
form
\f
\omega=\omega_S-dH(p,q,t)\wedge dt.
\ff
The motions of the system are the integral lines of the null
vector field of $\omega$ (orbits, or trajectories, of $\omega$). We
denote this vector field $Y$
\f
i_Y \omega \equiv Y^a \omega_{ab} = 0 .
\ff
In the coordinates on $C$ that we are considering, the variable $t$
has a preferred role, as it is clear from eq.(2). This preferred role
identifies $t$ as the time variable. The presymplectic space,
however, has a geometric, coordinate-independent meaning, like the
phase space. In a different coordinate system on $C$ (say
$p',q',t'$),\ \ $\omega$ may have the same form as in eq.(2), but with
$t$ substituted by a different variable, say $t'$.
\f
\omega=\omega_S-dH'(p',q',t')\wedge dt'.
\ff
Thus, the presymplectic formulation may accommodate different
time variables. For instance, it may accommodate the different
Lorentz times of special relativity. The presymplectic formalism,
ideed, provides us with the only way to write a relativistic
dynamical system in canonical form without destroying manifest
Lorentz covariance.
Time evolution is described in the presymplectic formulation in a
peculiar way. Each orbit of $\omega$ represents a possible motion
of the system. An orbit defines a correlation between two different
variables of the system. For instance, every orbit defines a
function $q_i(t)$. If $t$ is our time variable, then this function
describes the evolution of $q_i$ in $t$. But the same orbit also
defines the function $q_i(t')$. Thus, had we chosen $t'$ as our time
variable, the presymplectic formulation would equally well provide
the evolution in $t'$.
A {\it state} of the system is defined as an orbit. Note that this
definition of state does not refer to a particular choice of the
time variable, nor to a particular moment of time. Rather, it
represents, in a sense, the entire history of that particular state.
Looking ahead at the quantum context, it is meaningful to refer to
this kind of definition of state as {\it Heisenberg state}.
The {\it observables} of the system are defined as the scalar
functions $Q$ on $C$ that are constants along the trajectories (the
orbits)
\f
Y(Q) \equiv Y^a\partial_a Q = 0.
\ff
The functions like $q_i(t)$, which describe time evolution, are also
observables in the sense of eq.(5). This statement may seem
strange, but it will be carefully clarified in the next section.
Note that there is no observable corresponding to a generic variable
$q_i$ (unless $q_i$ is a constant of the motion). The observable is
the {\it function} $q_i(t)$. More precisely, there is one different
observable for every real value of $t$.
As a simple example, consider the presymplectic description of a
free relativistic particle. ($x^o,x^a,p_a$) are coordinates on $C$ and
\f
\omega=\sum_{a=1}^3 dp_a\wedge dx^a-d\sqrt{\vec p^2+m^2}\wedge
d x^o
\ff
($\mu = 0,1,2,3;\ a= 1,2,3$ from now on). The well known constants
of motion $P_a=p_a$, $P_o=\sqrt{\vec p^2+m^2}$ and $M^{\mu\nu}
=x^\mu P^\nu-x^\nu P^\mu$ are constant along the trajectories
generated by $\omega$. They are physical observables. The
observables that describe the evolution in $x^o$ will be constructed
in next section.
As the example suggests, presymplectic systems often arise in
theoretical physics in the form of constrained hamiltonian systems
with weakly vanishing canonical hamiltonian.\footnote{In the
Lagrangian formalism, these systems can be described by a
reparametrization invariant action, in term of a fictitious non-
physicalparameter.} Indeed, the
constraint surface of these systems, equipped with the (degenerate)
two-form induced
by the unconstrained-phase-space symplectic two-form, is the
presymplectic manifold. This presymplectic structure incorporates
all the relevant information on the system. $\omega$ in eq.(6), for
instance, is the two-form on the constraint surface
$K(x^\mu,p_\mu)=p^2-m^2=0$ induced by the unconstrained phase
space symplectic form $dx^\mu\wedge dp_\mu$. In these systems,
the constraint is often denoted hamiltonian constraint.
In the next section we will show that the observables $q(t)$ satisfy
\f
i_X \omega = -dH , \ \ {d\over dt} q(t) = X(q) = \{ q(t) , H \}
\ff
(where we have used also the more familiar Poisson brackets
notation). Note that time evolution can be described as the
existence of a particular structure on the set of the observables:
there exists a class of observables of the form $q(t)$ such that the
$t$ evolution is generated by an hamiltonian function $H$ (in the
sense of eq.(7)). We call this structure on the set of the
observables a {\it time structure}. Equivalently, a time structure is
a splitting of $\omega$ as in eq.(2).
In conclusion, in presymplectic mechanics the states of the
system are represented by the orbits of $\omega$, and do not evolve.
Observables are represented by scalar functions on $C$, constant
along the orbits. Note that the definition of state and of
observable does not make any reference to the concept of time.
This is why presymplectic dynamics may describe different times
for the same system.
What interests us here is a possibility offered by the presymplectic
formalism which is more radical than the possibility of
accommodating different times. Since presymplectic mechanics does
not require the existence of a time $t$ for the definition of the
basic mechanical concepts, it can also describe systems in which
there is no hamiltonian time $t$ at all.
More precisely, there are presymplectic dynamical systems
$(C,\omega)$ which are {\it not} hamiltonian systems\footnote{In
this paper we use "hamiltonian system" in the strict sense, which
does not include the cases (which are in fact presymplectic
systems) in which the hamiltonian
is weakly vanishing.}. These system do not admit a time
structure\footnote{If a presymplectic system does not admit a time
structure, there cannot be a corresponding hamiltonian system,
because the hamiltonian of the hamiltonian system defines a time
structure on the corresponding presymplectic system.}. For instance,
the trajectories of $\omega$ may be closed. We define a
presymplectic dynamical system that has no corresponding
hamiltonian formulation {\it dynamical system without time}.
A simple example of a presymplectic dynamical system without
time is given by the constraint surface $C$ of the constraint
\f
K= q_1^2+q_2^2+p_1^2+p_2^2-M \sim 0.
\ff
Since $C$ is compact, it does not admit a time structure, because a
time structure implies the decomposition $C=\Sigma\times R$,
where $R$ is the real line. Thus, there is no hamiltonian system
corresponding to this presymplectic system.
In the dynamical systems without time, one can still talk of states
(the orbits of $\omega$), of observables and also of evolution.
In fact, the orbits still determine a functional relation between
different variables. One variable, say $q_1$ in the model (8), can be
interpreted as a clock variable. Every orbit defines the evolution
$Q(q_1)$ for any other variable $Q$ as a function of $q_1$. But
$q_1$ does not have the properties that characterize the
hamiltonian time $t$, so that the evolution in $q_1$ cannot be
described as a hamiltonian evolution.
For instance, there are values of $q_1$ which are not reached by
certain trajectories. This cannot happen in a hamiltonian system.
The physical interest of the systems with no time is that they can
be interpreted as systems describing the evolution with respect to
physical clocks, as opposed to the evolution with respect to the
absolute time $t$. Dynamical systems of this kind arise in
theoretical physics. Examples are given by certain cosmological
models [2], by any topological field theory [22], by the Barbour-
Bertotti model [13], and many others.
The example {\it par excellence}, however, is of course general
relativity (on a compact space). The ADM constraint surface,
equipped with the two-form induced by the symplectic two-form of
the ADM phase space is the presymplectic manifold.
General relativity can do without an absolute hamiltonian time
because as a dynamical system general relativity is not a
hamiltonian system, but a presymplectic system.
We do not regard the lack of a (natural) hamiltonian description of
general relativity as a failing of the formalism. Instead, we take it
as a profound indication that the absolute time $t$ is not a physical
quantity, and that only evolution with respect to clocks is
observables.
The suggestion of this paper is that this indication has to be taken
seriously. If so, it has to be extended also to quantum mechanics.
\blankline \blankline \blankline
{\bf 2.2 The description of evolution: constants of the motion that
evolve.}
\blankline
In the previous section, a problem was left pending. The general
definition of observable, namely eq.(5), seems to be in contradiction
with the statement that the evolution of a variable $q_i$ as
a function of another variable $t$ is observable. In this section, we
show that this contradiction does not exist. The analysis will be
rather technical; but it is on a technical point concerning the
existence of these observables that the hypothesis we are proposing
relies. \footnote{We believe that many difficulties in the canonical
quantization of parametrized systems follow from the confusion on
this point.}
We assume for simplicity that the presymplectic system is defined
by a (hamiltonian) constraint $K$ on a phase space $q_n, p_n,\
n=1,...,N$. We single out a variable on the phase space, say $q_1$,
which we use a clock variable. For the moment, we assume that
$q_1$ is a hamiltonian time.
The definition (5) of observable, is equivalent to the requirement
that an observable $Q$ is given by a function $Q(q_n, p_n)$, which
has vanishing Poisson brackets with the constraint $K$. The question
we address is: how can such quantities, constant along the
trajectories generated by $K$, describe the evolution ? The answer
is that there do exist observables that satisfy eq.(5) {\it and}
represent the evolution in $q_1$. We now define these observables.
Let us focus on another variable, say $q_i,\ i=2, ...,N$. We want the
observable that represents the evolution of $q_i$ as a function of
$q_1$. As we mentioned, this will not be a single observable, but
rather a one parameter family of observables, each one representing
the value of $q_i$ at a different value, say $t$, of the clock variable
$q_1$. Let us call $Q_i(t)$ these observables. There is one
observable $Q(t)$ for every real number $t$; $Q(t)$ is an
observables, namely a function on $C$ (or the restriction to $C$ of a
function on the phase space)
\f
Q_i(t) = Q_i(t; q_n,p_n).
\ff
The $t$ dependence of $Q_i(t)$ should not be confused with its
dependence on $C$. $Q_i(t)$ must be constant on the orbits {\it for
every $t$} in order to be observable. And it should be equal to the
function $q_i(q_1)$ {\it in any point of $C$} in order to describe
what we want it to describe. The key point is that the two
requirement are not contradictory. In fact, we define the observable
$Q_i(t)$ for every real number $t$, as follows.
{\it $Q_i(t)$ is constant along each trajectory, and on each
trajectory it has the numerical value equal to the value of the
variable $q_1$ in the point P where that trajectory intersects
$q_1=t$.}
Equivalently, $Q_i(t)$ is defined by the two equations
\f
\{Q_i(t;q_n,p_n), K(q_n, p_n)\} =0, \label{prima}
\ff \f
Q_i(t;t,q_2, q_3, ... ,p_n)= q_i. \label{seconda}
\ff
The first equation implies that $Q_i(t)$ is observable, namely
constant along the trajectories. The second determines the value of
$Q_i(t)$ on any trajectory. This value is the numerical value
obtained by looking for the point $q_1=t$ along that trajectory and
reading out $q_i$ in that point. In any point P that lies in a
trajectory $l$, the function $Q_i(t)$ is equal to the function
$q_i(q_1)$ determined by $l$. We denote the observables defined by
equations like eqs.(10-11) {\it evolving constants of the motion} or
{\it evolving constants}.\footnote{The fact that observables of this
kind can describe evolution is discussed for instance in [17,23]. }
Consider the presymplectic system that represents the relativistic
particle, defined in the previous section. The observables $X^a(t)$
that describe the evolution of $x^a$ as a function of $x^o$ are
\f
X^a(t;x^\mu,p_\mu) = x^a - {p^a \over \sqrt{\vec p^2 +m^2}}(x^o-
t)
\ff
It is easy to check that eqs.(10-11) are satisfied. The second one is
immediate. The first,
\f
\{X^a(t; x^\mu, p_\mu), p^2-m^2\}=0,
\ff
follows directly from the fact that $X^a(t)$ can be expressed as a
function of the well known constants of the motion $P_{\mu}$,
$M^{\mu,\nu}$:
\f
X^a(t; x^\mu, p_\mu) = {P_a\over P_o} t + {M^{ao}\over P_o}.
\ff
The last equation shows that in any point of any trajectory, $X^a(t)$,
seen as a function of $t$, is equal to the function $x^a(x^o)$
determined by that trajectory. Thus, $X^a(t)$ is a function on the
phase space which is constant along the trajectories generated by
$K$, but describes the evolution of $x^a$ in $x^o$.
Evolving constants can be constructed for any function $q=
q(q_n,p_n)$, and for different choices of the time variable $q_T=
q_T(q_n,p_n)$. The evolving observable $Q(T)$ which gives the
evolution of $q$ in the clock time $q_T$ is defined by the two
equations
\f
\{Q(T), K\} =0,
\ff \f
Q(q_T) = q,
\ff
where the dependence upon the coordinates is not indicated. The
explicit form of these observables is obtained by solving the
dynamics generated by the constraint (geometrically, this
amounts to construct the orbits) and inverting the solutions
of the equations of motion. From the defining equations, we get
\f
{\partial Q(T)\over\partial T} \{q_T,K\}=\{q,K\},
\ff
which can be used to propagate $Q(T)$ in $T$.
In a generic dynamical system, one is not able to solve the dynamics
exactly and to construct the evolving constant in explicit form.
However, the evolving constant are always defined by eqs.(15-16).
In the presymplectic theory, in a sense dynamics has been reduced to
kinematics. If we know the expression for any observable $Q(t)$ for
every $t$, then we know everything about the system, and the
dynamics is solved. This is analogous to what happens in Heisenberg
mechanics. If we know every Heisenberg observable $\hat Q(t)$ for
every $t$, the dynamics is solved. Kinematics becomes non-trivial:
it is non-trivial to construct the observables.
If the dynamical system admits a time structure and $q_t$ is a good
hamiltonian time, then $K=p_t+H$, where the hamiltonian $H$ does
not depend on the momentum $p_t$, conjugate to $q_t$. In this case,
eq.(17) becomes
\f
{\partial Q(t)\over\partial t} =\{q,H\}.
\ff
Since $H$ does not depend on $p_t$, the commutator of $q$ with $H$
is the same as the commutator of $Q(t)$. Therefore
\f
{\partial Q(t)\over\partial t} =\{Q(t),H\}.
\ff
This is the Hamilton equation of motion. Thus, in a hamiltonian
system the evolving constants are nothing more than the usual
observables, seen as functions of $t$.
In a system without time, it is still possible to define evolving
constants analogous to the ones just defined. These are the
observables that describe the evolution in the clock time $q_T$. The
evolving constants of the system without time eq.(8) have been
constructed in ref.[15]. In the general case, the Hamilton equation
of motion (19) does not hold. Hamilton equation is replaced by
equation (15), which is more general .
To summarize, the definition of state and observable in the
Hamiltonian formalism requires the existence of a time $t$, which
is absolute and fixed once and for all. By contrast, the definition of
the formal structure of the dynamical theory in the presymplectic
formalism does not require $t$. Evolution with respect to a
dynamical variable $q_T$, chosen as a clock, is described by a
particular class of observables $Q(T)$. The basic equation that
$Q(T)$ satisfies is eq.(15). If the system admits a hamiltonian
formulation in the time variable $q_t$, this equation reduces to the
Hamilton evolution equation.
\blankline \blankline \blankline
\section{Quantum mechanics without time.}
%718
{\bf 3.1 Extending quantum mechanics.}
\blankline
Can the hypothesis of the absence of the absolute time be
incorporated in quantum mechanics ? Is there an existing
formulation of quantum mechanics which does not require $t$ to
exist~? Is there a form of quantum mechanics that extends
Schr\"odinger-equation quantum mechanics, in the same sense in
which the presymplectic mechanics extends the Hamilton equations
mechanics~?
The answer is: almost. To define the Schr\"odinger picture a time
variable $t$ is needed. A Schr\"odinger quantum state
is defined as the state of the system at time $t$, precisely as a
point of the phase space represents a state of the classical system
at a time $t$. However, in the Heisenberg picture $t$ is not
required to define the basic concepts of the theory. Because of that,
it is possible to define an extremely natural extension of the
Heisenberg picture, which may deal with system in which there is no
hamiltonian time $t$.
The Heisenberg states are often introduced as the
states at $t=0$. But they can also be interpreted, in a
more fundamental way, as a global (time unrelated) characterization
of the state. These states are the quantum analog of the
trajectories in the presymplectic formalism. The
interpretation of the Heisenberg states as states representing the
entire history of the system, has been stressed by Dirac
[24]. If the system admits a Schr\"odinger picture, we
may represent the state space of the Heisenberg picture
in terms of the space of the Schr\"odinger states at $t=0$. This is
the analog of labelling the presymplectic trajectories by means of
their $q_i$ coordinates at $q_1=0$.\footnote{The opportunity of
using Heisenberg states was vigorously advocated by Dirac in the
first edition of his celebrated book on quantum mechanics [24]. In
sec.I.3 Dirac argues that special
relativity {\it forces} us to use Heisenberg states. His physical
definition of the Heisenberg observable "at a given time"(sec.II.9) is
precisely the one we use here. It is interesting to notice that in
later editions Dirac shifted from the Heisenberg definition of state
(which he calls the relativistic one) to the Schr\"odinger one (which
he calls the non-relativistic one). He does that in order to gain in
simplicity (after all, he is doing non-relativistic quantum mechanics
!), but he complains (in the preface) that ``it seems a pity'' to give up
with the relativistic notion. At the end of his life, Dirac returned to
advocate Heisenberg states, and in 1981 he gave a talk in Erice
(Sicily) using a single transparency, on which there was written
only: ``$\imath\hbar {dA\over dt}=[A,H]$: HEISENBERG MECHANICS IS
THE GOOD MECHANICS ''.}
Similarly, Heisenberg observables correspond to the presymplectic
observables. There is no Heisenberg observable corresponding to the
variable $q$; rather, there is a one parameter set of observables
$\hat Q(t)$ corresponding to the values of $q$ at $q_t = t$.
When the system admits a classical hamiltonian formulation, there
is an hamiltonian operator $\hat H$ and the Heisenberg observables
are related by
\f
\hat Q(t+t')
= e^{i\hbar t'\hat H } \ \hat Q(t)\ \ e^{-i\hbar t' \hat H},
\ff
Eq.(20) is the quantum
realization of the time structure. In differential form, it becomes
\f
i\hbar\ \partial_t \hat Q(t)= [\hat Q(t),\hat H].
\ff
It is extremely important to emphasize that this equation is the
quantum version of the Hamilton equation of motion (19), and it is
also the Schr\"odinger equation, as it looks in the Heisenberg
picture.
Now we arrive to our main point. In the classical systems without
time (in the technical sense defined in sec.2.1) the Hamilton
equation (19) does not hold. It is reasonable to expect that in their
quantum physics the corresponding eq.(21) would noy hold either. In
those systems, an evolving constant $Q(T)$ would correspond to a
quantum operator $\hat Q(T)$ which does not satisfy eq.(20). The
key point is that this fact does not disturb the Heisenberg picture at
all. The Heisenberg picture is well defined also if the relation
between $\hat Q(T)$ observables at different $T$'s is not given by
eq.(20).
What may go wrong in these systems, is that the set of all the
observables $\hat Q_i(T)$ at a fixed $T$ may not form a complete
set. If so, a state is not characterized by its projection on the
eigenstates of a family of $\hat Q_i(T)$ for a fixed $T$. This means
that the outcome of the measurements of $\hat Q_i(T)$ for a fixed
$T$ does not uniquely characterize the state. Namely, one cannot
define a Schr\"odinger picture ! (See ref.[15] for a concrete example
in which all that happens.)
Suppose a definition of the Hilbert space $\cal H$ of the Heisenberg
states is given. Suppose the definition of Heisenberg operators $\hat
Q$ as self-adjoint operators on $\cal H$ is also given. And suppose
that among these operators there are also evolving constants $\hat
O_i(T)$. Then, we can run the entire standard machinery of the
probabilistic interpretation of quantum mechanics: the outcome of
the measurement of a quantity $Q$ on a state $\psi$ is an eigenvalue
$q$ of the $\hat Q$ operator; the probability of getting $q$ is the
modulus square of the projection of $\psi$ on the $q$-eigenvector;
and so on. All this makes sense also if the operators $\hat O_i(T)$
do not satisfy equation (21). (The equation they satisfy will be
studied in sec.3.3.)
The Heisenberg states are the quantum version of the presymplectic
states: they represent "hystories" of the system. The Heisenberg
operators $\hat O$ correspond directly to the presymplectic
observables $O$. Among these, there are the quantum evolving
constants $\hat O_i(T)$, corresponding to the classical evolving
constants $O_i(T)$. If the classical system admits also an
hamiltonian formulation, then we have eq.(20), and we may define a
Schr\"odinger picture. If it doesn't (it is a system without time),
everything still makes sense. But the Schr\"odinger picture cannot
be defined.
Quantum Mechanics may be synthesized in axiomatic form (see, for
instance, [25]). A set of axioms refers to the definition of state and
observable, to the identification of the expectation value of the
measurement with the mean value of the operator and to the
collapse of the wave function. One of the axioms (Postulate P3 in
[25]) refers to time evolution; let us call it Time Axiom. In the
Heisenberg picture, the Time Axiom requires that all the observables
depend on the a time variable $t$, and that an operator $\hat H$
exists such that that eq.(21) holds.
In the Heisenberg picture, the Time Axiom can be dropped without
compromising the other axioms and the probabilistic interpretation
of the theory. Thus, we may formulate our basic proposal on the
quantization of the classical systems without time.
\vskip .5cm
{\it
1. We define the structure given by the axioms of Heisenberg
picture quantum mechanics, excluding the Time Axiom, {\rm quantum
mechanics without time}.
2. We suggest that the quantum physics of the presymplectic
systems that don't have a hamiltonian version, is governed by {\rm
quantum mechanics without time}, as defined in 1.}
\vskip .5cm
General relativity is one of these systems; thus, we suggest,
non-perturbative quantum gravity is to be constructed in the
framework of quantum mechanics without time.
A quantum system without time is constructed in ref.[15]. It
quantizes the system (8). The Hilbert space is defined, and the
operators that represent the evolution of $q_2$ as a function of the
clock time $q_1$ are constructed. (More precisely, the corresponding
self-adjoint projection operators are defined.)\ We urge the reader
to refer to that paper for a concrete implementation of the general
theory discussed here.
In a quantum system without time, there may be an approximation
within which the Schr\"odinger equation (21) holds. This is the way
we expect standard quantum mechanics may be recovered.
If there is an approximate Schr\"odinger equation, the fact that the
Schr\"odinger norm is only approximately conserved is just a
consequence of the approximation and does not disturb the full
theory. The quantum states of the model defined by eq.(8) admit [15]
a representation of the form
\f
\psi(q_1,q_2).
\ff
They satisfy an {\it approximate} Schr\"odinger equation
\f
-i\hbar{\partial\over\partial q_1}\psi(q_1,q_2)= \hat H
\psi(q_1,q_2)+ {\rm small\ terms}.
\ff
The norm
\f
||\psi||(q_1)= \int dq_2 |\psi(q_1,q_2)|^2,
\ff
obtained by fixing the internal time $q_1$ and integrating on
the remaining variables, is not conserved in $q_1$. But this norm is
{\it not} the one defined by the correct scalar product of the
theory. The fact that this norm is not conserved in $q_1$ does not
contradict the probabilistic interpretation. It is as harmless as the
fact that the integral in $q_2$ of the modulus square of the wave
function of a two-dimensional harmonic oscillator depends on $q_1$.
Finally, let us discuss the wave function collapse. The measurement
of the quantity $Q$ at the clock time $T$ is accompanied by the
projection of the state on an eigenstate of the operator $\hat Q(T)$.
The Heisenberg states get projected at any measurement. The
information that the measurement is performed at the clock time
$T$ is contained in the fact that the eigenstates of $\hat Q(T)$, on
which the state gets projected, depend on $T$. The question "when"
the projection occurs is meaningless, since the state does not
evolve.
But there seems to be a problem here. Projectors do not commute.
Even if it is meaningless to say when the projections occur,
nevertheless, the order in which they occur is not meaningless. But,
unlike the hamiltonian time $t$, a clock time $T$ may (classically)
increase and then decrease along a trajectory. Thus, in general $T$
does not define an ordering relation. How do we know the {\it order}
in which to perform the wave function projections ? If we replace
the well-behaved $t$ by the ill-behaved $T$, how do we know how to
order the collapses ?
In order to answer this question, we should notice that $t$ and the
ordering of the collapses are not necessarily related. This fact was
emphasized by Dirac in ref.[24] and is clearly discussed by Jim
Hartle in ref.[7]. The following example shows that time and
collapse-ordering may be unrelated. The formalism of quantum
mechanics allows a sequence of measurements not ordered in the
time in which the system evolves. We can measure $B(t)$ and {\it
then\ } $A(t')$ with $t'$, where $\hat Q$ commutes with the
Wheeler-DeWitt constraint, have physical meaning.
\blankline \blankline \blankline
{\bf 4.2. Problems and comments.}
\blankline
We do not think that the proposed solution of the time issue is clear
and complete. Both at the technical and at the conceptual level,
there are points that remain open or unclear.
Physically, not any variable can be used as a clock. We relaxed the
requirement that a clock variable $T$ must be a
hamiltonian time. However, we did not provide any alternative
definition of a "time" variable. What does characterize the physical
variables $T$ that can be used as clocks ? To our view, this is an
open question.
A related technical question is the compatibility of the two
equations (\ref{prima}) and (\ref{seconda}) that define the evolving
observables. It is clear from the geometrical picture that if
$\{q_T,K\}=0$, equation (17) cannot be integrated. In the
general case, the functional relation that an orbit defines between
two variables is only implicit. $Q(T)$ may be multivalued.
Physically, there is nothing wrong with that, but the definitions
should be adjusted. If an orbit intersects $q_T=T$ in M points
$P_m$, we may introduce M observables $Q^{(m)}$ such that
\f
O^{(m)}(q_T) = q(P_m),
\ff
and so on. In other words, eq.(16) should be replaced by a weaker
equation. The details should be worked out.
$Q(T)$ may defined only on a bounded interval. This interval may
depend on the orbit. Thus, there are values of $T$ for which $Q(T)$
is defined only in certain regions of $C$. Outside these regions,
$Q(T)$ may becomes complex. In the quantum domain, this implies
that the operator $\hat Q_i(T)$ may have complex eigenvalues, and
therefore is not self-adjoint.
A way out is provided in ref.[15]. The idea is to use the projection
operators on the eigenstates of $\hat Q(T)$ ($\hat Q(T)$ is still
symmetric) corresponding to real eigenvalues instead of
$\hat Q(T)$ itself. The projectors are self-adjoint and correspond
to the basic yes/no experimental observations. The present paper is
more conceptual than technical and, on the purpose of clarity, we did
not use the projectors formalism. But we think the projector
formalism is very likely to be the correct formalism in the general
case.
Finally, assuming that the hypothesis we are presenting is realistic,
developing a physical intuition of the systems with no time is a non-
trivial problem. Simple models [15,17,32] may
help. In the model in ref.[15], the clock time fails to be a good time
because of global properties of the orbits. Locally, the system
behaves as a hamiltonian system; but on the entire orbit one has to
patch different times. The topology of the orbit is closed and
there is no way to map $R$ smoothly onto an orbit.
An interesting relation between global properties of the orbits and
small scale measurements emerges in this model. If one measures a
variable in a quasi-classical state with a high precision, the state
is severely affected by the collapse. The collapse excites
components of the wave function, which correspond to classical
trajectories in which the clock variable reverses its direction right
away.
It is tempting to speculate that this behavior could be
general. Suppose that in gravity we take the growing radius
$R$ of the universe as a clock variable. Since the universe may
recollapse, there are orbits for which $R$ "goes back". One can
speculate that a measurement which implies a Plank scale precision
may project the state of the gravitational field on components of
the wave function which correspond to a universe that would
immediately start to recontract. Thus, Plank scale measurements
may destroy the unitarity of the evolution in $R$.
Apart from these (wild) speculations, the example suggests two
ways in which a realistic gravitational system without time may be
concretely thought. One is to recall that it is difficult to imagine
that in general relativity there could be a good clock that may run
forever on {\it any} solution of Einstein equations (recall that most
solutions develop singularities). The second (maybe related) one is
to think that non-time behavior may appear at very
short time intervals. More precisely, that there may be physical
reasons for which there are no good clocks that resolve time below
the Planck time.
\blankline \blankline
{\bf 4.3 Conclusions.}
\blankline
In this paper, we propose a solution to the problem of time in
quantum gravity. We make the hypothesis that the concept of
absolute time $t$, as used in hamiltonian mechanics as well as in
Schr\"odinger quantum mechanics, is not relevant in a fundamental
description of quantum gravity.
This time has to be replaced by arbitrary clock times $T$ in terms
of which the dynamics may not be of the Schr\"odinger form. The
motivation for this hypothesis is that in general relativity there is
no observable absolute time. A hamiltonian formulation of gravity
in strict sense (choice of a clock time $T$ and the identification of
$T$ with the hamiltonian time) is contrary to the basic physical
ideas of general relativity and {\it irrelevant} for the quantization.
An extension of quantum mechanics, which does not need $t$, is
required, in order to incorporate in quantum mechanics the physical
ideas of general relativity. This extension (quantum mechanics
without time) is defined in a very natural way, by just dropping the
Time Axiom from the Heisenberg picture.
In a quantum mechanical system without time, the Schr\"odinger
equation (which in the Heisenberg picture is $\dot{\hat Q} = i\hbar
[\hat Q,\hat H]$) is replaced by the equation $[\hat Q, \hat K]= 0$.
There are observables $\hat Q(T)$ (evolving constants) that describe
the evolution with respect to a physical clock variable. In spite of
the fact that evolution in $T$ may be non-unitary, the probabilistic
interpretation is viable. Unitary evolution and Schr\"odinger
equation may be recovered within an approximation.
As far as the problem of time is concerned, in a quantum theory of
gravity there is no need to give up the probabilistic interpretation of
the wave function, Hilbert space, finite norm states and self-
adjoint operators corresponding to observables. The notion of
absolute time is not necessary, and we think that the difficulties in
dealing with a theory without time are only psychological. We
suggest that, in looking for a quantum gravity theory, "time" should
simply be forgotten.
Our proposal is in a sense conservative and in a sense radical.
It is conservative, since we keep as much of general relativity, and
as much of standard quantum mechanics as possible. Our
philosophy is that general relativity and quantum mechanics
summarize our basic knowledge of the world, and we shouldn't
change them, unless forced by experiments or by a requirement of
internal consistency. It is radical, because we assume that at the
fundamental level time is not defined. Thus, a radical revision of a
familiar concept is required. However, this modification of the
concept of time (with respect to the time of Hamilton mechanics), is
forced by general relativity itself, and implicit in its formalism.
The proposed extension of quantum mechanics is nothing but the
insertion of this revised concept of time in the basic structure of
the quantum formalism. The fact that this can be done so naturally
is, for us, a good sign.
Of course, the proposed solution of the time issue is only an
hypothesis. In order to verify this hypothesis, a non-perturbative
quantum gravitational theory has to be constructed. In spite of
recent progress in this direction [33], it is well
known that there are major technical difficulties in the actual
construction of a canonical quantum theory of gravity. In this paper
we have tried to resolve an {\it a priori} difficulty, which could have
undermined canonical quantization. We have shown that a conceptual
framework in which time in quantum gravity is not a problem for the
canonical theory, does exist. Whether nature chooses this
conceptual framework or not, is an open question.
\blankline
\blankline
\blankline
\blankline
\blankline
\blankline
\vfil
I am particularly grateful to Abhay Ashtekar, Chris Isham, Karel
Kuchar and Lee Smolin for many interesting discussions. I also thank
Daniele Amati, Bruno Bertotti, Bryce DeWitt, Al Janis, Jim Hartle,
Carlos Kozameh, Ted Newman, Raphael Sorkin and Marco Toller for
useful suggestions and interesting exchanges of ideas. This work
was supported by the NSF grant PHY-9012099.
\blankline
\blankline
\blankline
\blankline
\blankline
\blankline
{\bf References}
\blankline
[1] For a review of the present status of perturbative and
nonperturbative methods in quantum gravity see
C.J. Isham, {\it Quantum gravity}, in {\it General Relativity and
Gravitation, proceedings of the 11th International conference on
General Relativity and Gravitation (GR11), Stockholm 1986},
Cambridge University Press, Cambridge 1987.
A. Ashtekar, {\it Quantum Gravity, self duality, Wilson loops and
all that}, invited talk at the GR12, to appear on the proceedings,
Boulder, Colorado 1989. E. Alvarez, {\it Quantum gravity: an
introduction to some recent results}, Rev. Mod. Phys. {\bf 61}, 561
(1989).
[2] B.S. DeWitt, Phys. Rev. {\bf 160}, 1113 (1967). J.A. Wheeler, in
{\it Battelle Rencontres 1987}, ed. C. DeWitt, J.A. Wheeler, Benjamin
N.Y. 1968.
[3] L. Smolin, concluding remarks at the conference 'Approaches to
Quantum Gravity', Santa Barbara 1986. D.G. Boulware, concluding
remarks at the Quantum Gravity section of the GR11, Stockholm
1986. To the issue of time was dedicated also the discussion of the
Quantum Gravity section at the ``International Conference of
Gravitation and
Cosmology'', Goa 1987, and a section of
the Osgood Hill conference
{\it Conceptual problems in quantum gravity} Boston 1988,
proceedings
edited by A.Ashtekar and J.Stachel. (Birkhauser, Boston, in press).
[4] K. Kuchar in `Quantum Gravity 2' eds. C.J. Isham, R. Penrose,
D.W. Sciama. Oxford University Press, Oxford 1982.
[5] T. Banks, Nucl. Phys. {\bf B249}, 332 (1985). H.D. @eh, Phys. Lett.
{\bf
A116}, 9 (1986). A.Peres, Am. J. Phys. {\bf 48}, 552 (1980). D.N.
Page, W.K.
Wootters, Phys. Rev. {\bf D27}, 2885 (1982).
[6] A. Ashtekar, ``Old Problems in the light of New variables,
proceedings
of the Osgood Hill conference {\it Conceptual problems in quantum
gravity}, op. cit.
[7] J.B. Hartle, `Quantum Kinematics of spacetime', Santa Barbara
University pre\-print UCSBTH-87-79; Phys.Rev. {\bf D37}, 2818
(1988).
{\it Quantum Cosmology}, talk at the GR12 conference, Colorado
1989.
[8] W.G. Unruh, ``Time and Quantum gravity'', unpublished notes
(1988).
Sorkin, ``On the Role Of Time In The Sum-Over-Histories Framework
For Quantum
Gravity'', Syracuse university report (1988). R.M. Wald, Phys. Rev.
{\bf D21}, 2742 (1980).
J.J. Halliwell, Phys. Rev. {\bf D36}, 3626 (1987). T.Padmanabhan, Int.
Journ. Mod. Phys. {\bf A4}, 4735 (1989). T.P. Singh, T.Padmanabhan,
{\it Notes on semiclassical gravity}, Ann. of Phys. (1989).
[9] J.B. Hartle, S.W. Hawking, Phys. Rev. {\bf D28}, 2960 (1983).
S.W. Hawking, Comm. Math. Phys. {\bf 87}, 395 (1982).
Don N. Page, Phys. Rev. {\bf D34}, 2267 (1986).
[10] J.B. Barbour "Leibnizian time, Machian dynamics and quantum
gravity" in "Quantum concepts in space and time", edited by C.J.
Isham and R. Penrose, Cambridge University press 1986. J.B. Barbour
and L. Smolin, ``Can quantum mechanics be sensibly applied to the
universe as a whole ?'' Yale preprint, April 1988. R. Sorkin ``On
the role of time in the sum over histories framework for gravity'',
Syracuse preprint, 1988.
[11] P. Hajicek, Phys. Rev. {\bf D34}, 1040 (1986).
[12] The viewpoint of this paper is discussed also in
C. Rovelli, {\it Is there incompatibility between
the way time is treated in General Relativity and in standard
quantum mechanics ?}, proceedings of the Osgood Hill conference
{\it Conceptual problems in quantum gravity} Boston 1988, op. cit. .
The reports C. Rovelli, {\it What is time ?} Roma University preprint
(1988), and {\it Time in quantum gravity: physics beyond the
Schr\"odinger regime}, University of Pittsburgh preprint PITT-90-02
(1989), are earlier incomplete versions of the present work.
[13] C. Rovelli, {\it Quantum mechanics without time: group
quantization of the Barbour-Bertotti model}, University of Roma
preprint, 1988; and in the proceedings of the Osgood Hill conference
{\it Conceptual problems in quantum gravity} Boston 1988, op. cit.
[14] B. Bertotti, C.Rovelli, {\it Time in Quantum Machian Cosmology},
in preparation, 1989.
[15] C. Rovelli, {\it Quantum mechanics without time: a model},
University of Pittsburgh preprint PITT-90-03, in print on Phys. Rev.
D. (1990).
[16] C. Rovelli, {\it What is observable in classical and quantum
gravity ?}, University of Pittsburgh preprint Pitt-90-10 (1990); {\it
Quantum reference systems}, University of Pittsburgh preprint Pitt-
90-11 (1990).
[17] S. Carlip, {\it Observables, Gauge Invariance and Time in 2+1
dimensional quantum gravity}, Princeton preprint, IASSNS-HEP-
90/41 (1990); {\it Measuring the metric in 2+1 dimensional
quantum gravity}, Princeton preprint, IASSNS-HEP-90/46 (1990).
[18] Tito Lucretius Caro, {\it De rerum natura}, I-459. "Time does
not exist by itself. Time gets meaning from the objects: from the
fact
that events are in the past, or that they are here now, or they will
follow
in the future. It is not possible that anybody may measure time by
itself; it may only measured by looking at the motion of the objects,
or at their peaceful quiet."
[19] D.Ruelle, Comm. Math. Phys. {\bf 85}, 3 (1982).
[20] K. Kuchar, discussion at the "International Conference of
Gravitation and Cosmology", Goa 1987; discussion at the Osgood
Hill meeting on Conceptual Problems of Quantum Gravity, op. cit.
1988; and private discussions, 1990.
[21] R. Penrose, talk at the conference `Approaches to Quantum
Gravity'
Santa Barbara 1986. L. Smolin, Class. Quantum Grav. {\bf 3}, 347
(1986). S.W. Hawking, Comm. Math. Phys. {\bf 55}, 133 (1977).
R. Penrose in {\it General Relativity: an Einstein Centenary
Survey}, edited by S. Hawking and W. Israel, Cambridge University
Press, Cambridge 1979.
[22] E. Witten, Comm.Math.Phys. {\bf 117}, 353 (1989).
[23] D.N. Page, W.K.Wootters, Phys. Rev. {\bf D27}, 2885 (1982). W.K.
Wootters Int. Journ. Theor. Phys. {\bf 23}, 701 (1984).
[24] P.M. Dirac, {\it The principles of quantum mechanics}, first
edition, Oxford at the Clarendon Press, Oxford 1930.
[25] J. Glimm, A. Jaffe, {\it Quantum Physics}, Springer Verlag, New
York
1981.
[26] P.M. Dirac, {\it Lectures on Quantum Mechanics} Belfer
Graduate School of Science, Yeshiva University, New York 1964.
[27] A. Ashtekar, private discussion.
[28] C. Rovelli, Il Nuovo Cimento {\bf B100}, 343 (1987).
[29] C. J. Isham in `Relativity Groups and Topology II', eds. B.S.
DeWitt and R. Stora, North-Holland, Amsterdam 1984.
[30] J.S. Bell, {\it First class and second class difficulties in
quantum gravity}, talk at the Conference on ``Frontiers in Physics,
High Technology and Mathematics'', Trieste 1989.
[31] C. Rovelli, Phys. Rev. {\bf D35}, 2987 (1987).
[32] S. Deser, R. Jackiw, G. t'Hooft, Ann. Phys. NY {\bf 152}, 221
(1984). S.
Deser, R. Jackiw, Ann. Phys. NY {\bf 153}, 405 (1984). E. Witten, Nucl.
Phys. B
{\bf 311}, 46 (1989). A.Ashtekar, V. Husain, C. Rovelli, J. Samuel,
L.Smolin,
Class. Quantum Grav., {\bf 6}, L185 (1989).
[33] A. Ashtekar, `Recent developments in canonical gravity' Part
IV and V. Bibliopolis, Naples, 1989. T. Jacobson, L. Smolin, Nucl.
Phys. {\bf B266}, 295 (1988). C. Rovelli, L. Smolin, Phys. Rev. Lett.
{\bf 61}, 1155 (1988); Nucl. Phys. B331, 80 (1989).
\vfil\eject
\thispagestyle{empty}
\vfil
\bf
\centerline{\Large \bf TIME IN QUANTUM GRAVITY:}
\blankline
\centerline{\Large \bf AN HYPOTHESIS.}
\rm
\vskip1.5cm
\centerline{\bf Carlo Rovelli}
\vskip.6cm
\centerline{\it Department of Physics, University of Pittsburgh,
Pittsburgh, 15260 U.S.A.}
\centerline{\it and}
\centerline{\it Dipartimento di Fisica, Universita' di Trento; INFN,
Sezione di Padova; Italy.}
\vfil
\centerline{{\bf Abstract}}
\blankline
A solution to the issue of time in quantum gravity is proposed. The
hypothesis that time is not defined at the fundamental level (at the
Plack scale) is considered.
A natural extension of canonical Heisenberg-picture
quantum mechanics is defined. It is shown that this extension is
well defined and can be used to describe the ``non-Schr\"odinger
regime'', in which a fundamental time variable is not defined.
This conclusion rests on a detailed analysis of which quantities are
the
physical observables of the theory; a main technical result of the
paper is the identification of a class of gauge-invariant observables
that can describe the (observable) evolution in the absence of a
fundamental definition of time.
The choice of the scalar product and the interpretation of the wave
function are carefully discussed. The physical interpretation of the
extreme ``no time'' quantum gravitational physics is considered.
\vfil
\rightline{{\today}.}
\eject
\end{document}