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. 12, 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
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