The Direction of Time
In August 2007 I began to blog about the book of H.D.Zeh " The
Physical Basis of The Direction of Time".
It is in parts available online here.
Introduction
As for any good mystery novel, the author introduces the enigma of
time,
as it presents itself to a physicist, already on the first page.
"The asymmetry of Nature under a 'reversal of time' appears only too
obvious,
as it deeply affects our own form of existence. If physics is to
justify the hypothesis that its laws control everything that happens in
Nature, it should
be able to explain this fundamental asymmetry [..]
Surprisingly, the very laws of Nature are in pronounced contrast to
this fundamental
asymmetry; they are essentialy symmetric under 'time reversal'. It is
this discrepancy that defines the enigma of the direction of time [..]"
Also on the first page we already read how physicists commonly approach
this enigma.
"It has indeed proven appropriate to divide the formal description of
Nature into
laws and initial conditions [..]
Initial conditions are usually understood as conditions [..] which
select particular
solutions of the equations of motion. They could just as well be
formulated as
final conditions, even though this would not represent the usual operational
(hence asymmetric) application of the theory." [*]
It is only a small step now to the consensus opinion that the enigma of
time simply
reduces to an empirical fact about the initial condition and all that
is left to do
is to find a convincing explanation for the observed initial
conditions. [x]
However, the author goes on to explain that there are different views
about this
issue and a group of physicists (e.g. C.F. Weizsaecker) and
philosophers are of the opinion that the asymmetry
of time is a much more fundamental property and mentions
"The argument that the historical nature of the world be a
pre-requisite (in the Kantian
sense) for the fact that we can make experience [..]"
If one prefers Popper over Kant, one can point out that the concept of
time cannot be
falsified by an observation or experiment, since the very notion of an
experiment requires a state before (when we are uncertain of its
outcome) and after (when we know the result of
the experiment). [o]
This issue plays an important role in the debate about the
interpretation of quantum mechanics,
since
"The extra-physical time arrow appears in all operational
formulations of quantum theory, such as those describing probabilistic
relations connecting preparations and
subsequent measurements - thus restricting quantum theory to laboratory
physics performed
by humans."
But it is important to understand that the goal of Zeh's book is not to
'explain' or 'derive' a concept of time, rather
"The prime intention of this book is to discuss the relations between
various arrows of time,
and to search for a universal master arrow. To this end,
certain open problems which have often been pragmatically put aside in
the traditional theories will have to be clearly worked out. They may
indeed become essential in more general theories [..]"
[*] As far as I am aware, only Frank Tipler
has seriously considered the use of a final condition, in his book 'The
Physics of Immortality'. It is no surprise that his conclusions
are rather unusual.
[x] These
blog posts
of Sean Carroll are typical and recent examples.
[o] If I am not mistaken, this
essay by Lubos Motl also emphasizes this point of view, that the
'arrow of time' is a fundamental (logical) pre-requist and not just
empirical fact. Lubos
wrote "A related arrow of time is the logical arrow of time. You should
always assume that you know the initial conditions in the past - or the
present - and use the physical laws to
predict the future. [..] you should never do it in the opposite way."
chapter 1, The Physical Concept of Time
The first chapter is the shortest of the book; In the original 1984
version it was
just 1 1/2 pages, now it is six pages, but still shorter than the
introduction.
I suspect one reason is that physics does not have a good concept of
time (yet).
At first, the author introduces the mechanistic concept of time
which "is also based on this representation of time by the real
numbers, but it
avoids any subjective foundation; it is defined in terms of objective
motion [..]
all motions qi(t) in the Universe can be replaced by
'timeless' trajectories
qi(q0) in a global configuration space, where the
hand of an appropriate 'clock' may be used as q0. [..]
These timeless trajectories may also be described by means of a physically
meaningless parameter x in the form qi(x) for all i,
where equal values
of x characterize the simultaneity of different q's. [..]
If Jacobi's principle is applied to Newton's theory, absolute time can
be recovered
as a specific parameter x that simplifies the equations of
motion (Poincare 1902)." [*]
Notice how the term 'simultaneity' sneaks in without further
explanation. Also the
notion of a 'clock' is used without further discussion, which I find
very unfortunate.
A (mechanical) clock usually consists of an oscillator (e.g. a
pendulum) and a counting mechanism, registering the 'ticks' of this
oscillator.
The symmetry of the oscillator (returning repeatedly to its initial
state) ensures that each
tick measures equal amounts of time, but the crucial part for our topic
is the counting mechanism.
In general, a non-reversible mechanism (e.g. hands of a clock plus
calendar) is used and it would
have been interesting to see a discussion if one can introduce 'clocks'
without assuming already the 'direction of time', even within Newtonian
physics only.
The remainder of chap. 1 discusses the generalization of the mechanical
time concept
and the theory of relativity. In between there is a brief discussion
about 'the present',
which will later reappear in the Epilog.
"Newton's mechanistic time [..] specifies
neither a direction in time nor a specific present. [..]
The concept of a present thus seems to have as little to do with the
concept of time itself as color has to do with light [..]
Both the present and color characterize our subjective perception
of time and light,
respectively."
Obviously, this is an old issue, long debated by philosophers and
perhaps St.
Augustine said it already best, several hundred years ago...
The 'subjective perceptions' are obviously taking place in (our) brains
and are
thus part of the physical universe; But the question how 'the present'
(or 'colors') emerge as
features of brains is largely without an answer. We simply do not know
and it does not
really help to (try to) get rid of those issues by calling them
'subjective'.
In Fig. 1.1 a lightcone is depicted, with the usual explanations
"space-time past and future are defined relative to every event P [a
space-time point]", "What we observe as [..] the subjective here-and-now
P."
This leads to a naive question: If the here-and-now is a subjective
perception generated by a brain, how does that brain and its activity
fit into a single space-time point?
[*] In the book the symbol 'lambda' is used not an 'x'.
chapter 2, The Time Arrow of Radiation
Usually, if one studies physics, electromagnetism and thermodynamics
are two different lectures.
Therefore, the 2nd chapter was an important reason for me to buy the
book in 1984,
to answer some questions lost in between those lectures.
"after an electric current has been switched on, one finds a
retarded
electromagnetic field that is coherently propagating away from
its source.[..]
However, the reversed phenomena are never observed in Nature."
"one may write [the four potential] Au as a functional of
the sources ju.
[..] one obtains the retarded and advanced
potentials [..] related to one another by a reversal of retardation
time [..]"
"many textbooks argue somewhat mysteriously that 'for reasons of
causality' [..] only the
retarded fields [..] occur in Nature."
The author calls this the intuitive notion of causality:
"correlated effects (that is,
nonlocal regularities, such as coherent waves) must always
possess a local common cause in their past"
and further "this asymmetric notion of causality is a major explanandum
of the physics of time asymmetry".
This issue was already discussed by Einstein and Ritz in a famous
controversy. While Ritz
"conjectured that the thermodynamical arrow of time might be explained
by the retardation of
electromagnetic forces [..]", Einstein thought that it was the other
way around.
In section 2.1 the retarded and advanced form of the boundary value
problem are presented in some detail and in section 2.2 the
thermodynamical and cosmological properties of absorbers are discussed.
"A spacetime region is called an '(ideal) absorber' if any radiation
propagating within its boundaries is (immediately) thermalized at the
absorber temperature T( =0)".
At this point the thermodynamic 'arrow of time' is used to explain the
retardation of electromagnetic fields.
With the assumption of absorbing boundaries in a "laboratory situation
the radiation arrow is a
simple consequence of the thermodynamical arrow characterizing
absorbers". [*]
In general and "in particular in astronomy" the "night sky does in fact
appear black" and plays a similar role.
After discussing briefly Olber's paradox, we read that "The cosmic
expansion [..] is thus
also essential for the non-equilibrium formed by the contrast between
cold interstellar space and the hot stars.[..]
The expansion of the Universe has therefore often been propsed as the master
arrow of time." (We will read more
about this in chapter 5.)
The subsequent section 2.3 discusses radiation dampening and the
difficulties to obtain a consistent classical
description of electrodynamics. "[..] Dirac's equation of motion [..]
represents a
Newtonian (second order) equation of motion which depends on a force
that acts ahead of time. How
could this 'acausal' result be derived using retarded fields alone?
Moniz and Sharp (1977) demonstrated
that the pathological behavior of this 'classical' electron is a
consequence of a mass renormalization
that exceeds the physical electron mass".
Section 2.4 elaborates on the absorber theory of Wheeler and Fenman
(1945); The latter described a seminar
about this proposal and Pauli's reaction to it in his autobiography.
Feynman: "I wish I had remembered what Pauli
said, because I discovered years later that the theory was not
satisfactory when it came to making the quantum theory.
It's possible that the great man noticed the difficulty immediately.."
[*] see also my
brief discussion of the 'arrow of time' and the role of absorbing
laboratory walls.
chapter 3 , The Thermodynamical Arrow of Time
This chapter discusses the derivation of classical master equations,
the Stosszahlansatz of
Boltzmann,
then the coarse graining of
Gibbs and finally the general master equation of Zwanzig. The key is
that
one discards information about the microstate, which then leads to time
asymmetric results. In the case
of Boltzmann's H-theorem, information about the correlation of
particles is lost and replaced with an
assumption of molecular chaos.
The Gibbs distribution is derived by averaging over small (but fixed)
volume elements in 6N dimensional configuartion space. (By the way,
some 'clever' textbooks at this
point refer to the semi-classical Bohr-Sommerfeld equation to justify
this, which is misleading at best.)
On p.56 the physics which explains why this works is discussed
[*]:
"Even very small uncertainties in
the Hamiltonian may be sufficient to completely destroy fine-grained
information within a short time interval.
Borel (1924) estimated the effect of a gravitational force that would
arise here on earth by the displacement
of a mass of the order of a few grams by a few centimeters at the
distance of Sirius. He thereby pointed
out that this would lead to a completely different microscopic state
for the molecules forming a gas in a
vessel under normal conditions within seconds. Although distortions of
the individual molecular trajectories
are extremely small, they would be amplified in each subsequent
collision by a factor of the order of l/R,
the ratio of the mean free path over the molecular radius."
Unfortunatley, most physicists make (implictly) the assumption that
there really is a microstate of the world
and we just do not know it. I have explained earlier why I think this
is very misleading and that it would be
appropriate to acknowledge that physics requires multiple
descriptions
of reality. This would in my opinion
make further progress on various issues much easier, in particular a
better understanding of the 'arrow of time'.
My 2nd remark concerns section 3.5 which discusses the observation of
Weizsaecker and others that the
probability pR that our memories and
documents were actually formed in the historical process they describe,
is very small compared to the probability pA
that they were formed randomly. In other words, pR
stands
somehow for the probability that the reality we experience is actually
'real', while pA denotes the probability
that everything is just absurd. I discussed a similar conclusion here ("there is
(more or less) only one way how
reality can match
with our memories. But there are many ways how our memories could be
false and our
experiences fake, either by
accident or conspiracies") and would
like to add a bit to it.
The author notes that "David Hume's fundamental insight that we can
never predict anything with certainty
applies to the past as well", but fails to acknowledge that the claim
of Weizsaecker's argument is much stronger
than that: not only is pA larger than zero,
but it is much larger than pR.
However, notice the strange logical structure of the argument: If
{P}shall denote our knowledge of physics,
then we can write Weizsaecker's argument as: {P} => pA
>> pR or "our knowledge of
physics leads us to
conclude that the absurd is much more probable than reality as we
experience it."
However, {P} relies on R, since if we cannot trust our memories, then
we cannot trust our physical laws and
the conclusions we reach using them: R <=> {P} => pA
>> pR and thus p( pA
>> pR ) = pR
<< pA.
I leave it as an exercise for the reader to resolve this issue [x].
[*] Unfortunately, quite often statisticians use nowadays the concept
of (maximizing) entropy without always
examining why it actually works. In section 3.3 the author
discusses 'thermodynamics and information' and
notices that "a star cluster posesses meaningful
temperature and entropy from the point
of view that the
motion of individual stars is regarded as 'microscopic'." But notice
that in this case one can use (at least in
principle) photons to determine the 'microstate', i.e. positions and
velocities of
all stars.
[x] The recent discussion of simulated worlds leads to a similar
puzzle. Extrapolating the laws of physics, one
reaches the conclusion that it is likely that we live in a simulated
world. Thus it is unlikely that physics as we
know it describes the real world.
chapter 4 , The Quantum Mechanical Arrow of Time
In order to follow chap. 4 one needs to be familiar with the concept of
decoherence.
The chapter begins with "the formal
transition from classical to quantum statistical mechanics". The author
proceeds to derive the Pauli equation,
i.e. a quantum version of the master equations discussed in the
previous chapter. On page 91 we find a core argument: "Erich Joos
(1984) was able to show that the off-
diagonal elements pmn between states from such
macroscopically different subspaces disappear by interaction
with the environment ('decoherence')."
In section 4.3 the role of 'decoherence' is discussed: "It is this
universality and unavoidability of entanglement
with the natural environment that seems to have been overlooked for the
first 50 years of quantum theory.
All attempts to describe macroscopic objects quantum mechanically as
being isolated, and therefore by means
of a Schroedinger equation, were thus doomed to failure - even when
including environment-induced dynamical
terms that might describe a distortion. Decoherence is different, and
extremely efficient, since it does not require
an environment that disturbs
the system. The distortion of the
environment by the system affects the density
matrix of the system, too, because of quantum nonlocality, but on a
much shorter time scale than thermal relaxation
or dissipation." Some examples of decoherence are then discussed in
some detail.
While I fully agree with the statement above, I do not like the
following statement
made earlier on the same page:
"Classical concepts emerge
approximately in the form of apparent ensembles of narrow wave packets
through
unavoidable and practically irreversible interaction with the
environment." [see also this text].
Concepts cannot emerge approximately. In my opinion it is an important
fact (already recognized by Niels Bohr)
that our description of the environment, which includes the observer, is necessarily
different from the description
of a quantum system. Thus classical concepts do
not emerge from the quantum description, rather the two are independent
but related due to correspondence principle(s)
- and decoherence describes the switch from one to
the other(*).
In a very dense section 4.6 the author discusses 'the time arrow in
various
interpretations of quantum theory' and
I would recommend various papers by the author on this and similar
topics [e.g. 1,
2, 3] if one wants to
find more
clarity on this issue.
(*) In the introductory
text about decoherence (which I linked to at the beginnig) Kiefer
and Joos write on p.5:
"... the evolution [of system and environment] could in
principle be reversed. Needless to say that such a reversal
is
experimentally extremely difficult, but the interpretation and
consistency of a
physical theory must not depend on
our present technical abilities."
I disagree completely. Our knowledge of the environment, which includes
us, the observer(s), is necessarily
incomplete and thus cannot be reversed, independent of our technical
abilities.
chapter 5, 6 and Epilogue
I will not write about chapter 5 (time arrow of space-time geometry)
and 6 (time arrow in quantum cosmology) in the same detail as the
previous chapters.
The author outlines the thermodynamics of black holes, but a meaningful
discussion of the time arrow in general relativity would, in my
opinion, have to
include a careful examination of several open problems:
e.g. cosmic censorship and counter examples, existence of closed
timelike loops and the interior of rotating black holes. And of course
the question of how to define the entropy of gravitational radiation
and spacetime in general. (see also)
A discussion of the thermodynamics of acceleration (i.e. the Unruh
effect) follows, but only
the (unphysical) case of uniform acceleration. The much more
interesting case of non-uniform acceleration, the
backreaction of the detector etc. are not examined.
Finally, the discussion of quantum cosmology is affected by the
author's rejection of string theory, which is so far the only
consistent framework known to contain a description of quantum gravity.
"A simple toy model of a quantum universe" is discussed instead, but it
is not clear to me how much one can really learn from this exercise.
In the Epilogue, the topic of the here-and-now
is discussed again.
"According to Carnap, "Einstein said that the problem of the Now
worried him seriously. He explained that the
experience of the Now means something special for man, something
essentially different from the past and the future,
but that this important difference does not and cannot occur within
physics. ..." [..]
Carnap emphasized , however, that Einstein agreed with him (..) that
this situation does not indicate a defect of the physical concept of
time. (..)
The situation should rather be understood as reflecting the undefined
role of the observer [..]."
The
Statistical Mechanic