by H. D. Zeh

April 6, 1993

This appeared in the *Mathematical Intelligencer.*

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In this book Zeh brings some clarity to a very murky problem:
why are the future and the past so different? One need only to read the
physics journals to see that this is a multi-faceted and very real
issue that vexes the experts even now.
To understand its seriousness,
it is first necessary to see how *similar* the future and the past are.
They don't seem so in everyday life: we remember the past but not
the future, our actions affect the future but not the past, and so on.
From this standpoint it is really quite surprising that the dynamical
laws of physics — with one small exception — seem to be symmetrical
under time reversal.

Before we go any further,
it's important to get a clear idea of what time reversal symmetry really means.
At the simplest level, we may think of the laws of physics as equations
involving a time variable , and say that they are symmetric
under time reversal if given any solution, and making the substitution , we obtain another solution. To take an easy example, consider
a point particle of mass in three-dimensional space with no forces
acting on it. If we write its position as a
function of time, , Newton's second law says that

If satisfies this equation then so does .

Typically the laws are more complicated, and one may have to be more
careful in defining time reversal symmetry.
Different laws of physics involve very different mathematical
structures, but they are almost always separated into two components, the
"kinematics'' and the "dynamics''. The kinematics consists of the
description of the set of states that the system can be in at any
given time. For example, in classical mechanics, we can specify the
state of a point particle in three-dimensional space by giving its
position and its velocity ,
so
.
The dynamics tells how states change with time. In a theory where
we can predict both the future and past from the present,
and where there are no time-dependent external influences, we usually describe
dynamics with a family of maps
, where .
If the state of the system is at some time , the state is
at time .
The maps should form a "one-parameter group,'' that is,

We say that the physical system given by has "time-reversal symmetry'' if there is a map , called time reversal, such that

For example, our point particle with no forces on it moves with constant velocity, so

It's easy to check that is a one-parameter group, and that the system has time-reversal symmetry, where

Note, by the way, that time-reversal symmetry in the sense described
above is different from requiring that a given state be invariant
under time-reversal:

Our world is evidently in a state that is not even approximately invariant under time reversal; there are many processes going on whose time-reversed versions never seem to happen. But this is logically independent from the question of whether the dynamical

At this point, we could go through all theories of physics and
check to see whether they have time reversal symmetry.
But let us simply turn to the most up-to-date and complete laws of
physics we know: the standard model and general relativity. The
"standard model'' is a complicated theory of quantum fields that
describes the most fundamental particles we know
(mainly leptons and quarks) and the forces — electromagnetism and the
weak and strong nuclear forces — by which they interact. In other
words, it treats everything except gravity. The standard model
has time-reversal
symmetry *except* for effects involving
the weak force. This is the force that
permits a proton and electron to turn into a neutron and a neutrino, as
happens in some radioactive atoms, or vice-versa, as in some others.

In fact, in quantum field theory, time reversal, or is
one of a trio of possible symmetries, the others being charge conjugation or
which amounts to interchanging particles with their
antiparticles, and parity, or which is related to spatial
inversion

in somewhat the same way that time reversal relates to the map

In a quantum field theory states are given by unit vectors in a Hilbert space . The symmetries and are given by unitary operators on — if the theory in question admits these symmetries — while is given by an

On the other hand, general relativity treats gravity, which is a great puzzle in its own right, since it seems very difficult to unify with the rest of the forces. Unlike all the other forces, it is not at all natural to formulate its dynamics in terms of a one-parameter time evolution group. Essentially, this is because it treats of the geometry of spacetime itself, and how it wiggles around. While the dynamics of general relativity is by now moderately well understood, the modifications required for a quantum theory of gravity are still very poorly understood, and seem to require a radical rethinking of the very notion of time. In his last chapter, "The Quantization of Time,'' Zeh tours this fascinating subject. While a quantum theory of gravity would be likely to have profound implications for the study of time reversal, one can fairly say that so far the dynamics of gravity seems to admit time reversal symmetry.

It's worth noting that there are some cases where at first glance it
looks as if the laws of physics are asymmetric under time reversal, but on
closer inspection it turns out to be the fault of the particular *state* of the universe we are in. The two most famous examples are the
"time arrow of radiation'' and the "time arrow of thermodynamics.''
Here an "arrow of time'' is used loosely to denote something that is
not symmetric under time reversal.

The time arrow of radiation refers simply to the fact that when we
shake an electrically charged object, it emits waves of radiation that
ripple outwards as time progresses into the *future,* rather than
the past. This is expressed mathematically in terms of what are called
Green's functions. To understand these, it's
easier to consider the scalar wave equation rather than Maxwell's equations of
electromagnetism in their full glory. Thus we have a "field''
being produced by a "source''
, and we assume both are smooth functions and that

where

The source does not uniquely determine the field, but it is possible to write down formulas that give us for any source a field with . In particular, we say that is a Green's function (actually a distribution) for the scalar wave equation if

where is short for , implies that . Two Green's functions are the "advanced'' one,

and the "retarded'' one,

where and is the Dirac delta distribution. In electromagnetism one typically uses the retarded Green's function, so that if is nonzero only for times , then is typically nonzero after , but is zero before .

It may seem odd that while the equation
is
preserved by the transformation
, we are
solving it in a way that doesn't respect this symmetry. But there are
two things that help resolve this puzzle.
First, it is worth noting that working with the retarded rather than
the advanced Green's function is, at least for vanishing outside a
bounded set, equivalent to an assumption about the nature of the field
, namely that it vanish as . In short, we are
making a time-asymmetric assumption about the *state* of the system
when we are choosing the retarded Green's function. Why do we make
this assumption? For a quite interesting reason: because it's dark at
night. In a sense, light radiates out from the sun and from our flashlights,
rather than coming into them from the distance, because the universe is
a rather dark and cold place. The very fact that space is mostly dark and
empty, with a speckling of hot bright stars that radiate *outwards*,
is blatantly time-asymmetric, so the time arrow of radiation appears to be
cosmological in origin. This fact about the universe is crucial to life
as we know it, since all life on earth is powered by the outgoing
radiation of the sun, and the earth in turn dumps its waste heat
into the blackness of space.

A second, subtler point is that the equation
does not fit into the general framework of
one-parameter groups, because the field is subject to an
arbitrary time-dependent external influence, the source .
Here one wants to
imagine oneself, the experimenter, as being able to do whatever one
wants with the source , and see what it does to the field .
This is related
to the notion of free will: we like to think that the laws of physics
govern the behavior of everything *else*, but that we are free to do
whatever we want. However, in the most fundamental laws of physics we know -
the standard model and general relativity — no "arbitrary external
influences'' appear. In these laws, there is no need to choose
between a retarded and advanced Green's function (or some other Green's
function, for that matter). There is only the need to choose the
state that best matches what we observe.

The time arrow of thermodynamics is perhaps the most famous aspect of
time reversal symmetry — so I will treat it very briefly here.
Why is it so much more likely that
a porcelain cup will fall to the floor and smash to smithereens, than it
is for a pile of porcelain smithereens to form into a cup and jump into
ones hand? Disorder seems to be always on the increase. In fact, in
thermodynamics there is a quantity called entropy, , which is a
a measure of disorder — although one must be very careful not to fall for
the negative connotations of "disorder,'' which here is interpreted in
a very precise and sometimes counterintuitive sense. The second law of
thermodynamics is that

This law appears utterly time-asymmetric, and reconciling it with the (almost) time-symmetric fundamental laws of physics has exercised the minds of many physicists for many years. But the final resolution seems to be a simple one: this law is not true

As with the time arrow of radiation, in the last analysis it appears to
be nothing but
a raw experimental fact that the entropy of our universe
is increasing.
In a sense this is not surprising, because pondering chemistry and
biology a bit it becomes apparent that life as we know it
requires the entropy to be changing monotonically, rather than staying
about the same.
One might ask why
rather than
, but this is essentially a matter of convention.
Processes like remembering and planning, which
define the psychological notions of future and past, are only able to
occur in the direction of increasing entropy. That is, a memory at time
can only be of an event at time for which , while
a plan at time can only be for an action at time for which
. Since we have settled on using calendars for which the
number of the years increase in the direction of plans, rather than memories, we
have *chosen* a time coordinate
for which implies .

The main remaining mystery, then, is why the state of the universe is
grossly asymmetric under time reversal, even though the dynamical laws
of physics are almost — but not quite! — symmetric. If
the reader wishes to puzzle over this some more, or wants supporting
evidence for some of the (perhaps upsetting) claims I've made above,
he or she could not do better than to read Zeh's book.

© 1993 John Baez

baez@math.removethis.ucr.andthis.edu