Did Boltzmann understand all about time?

Lubos Motl wrote a pedagogical review about the notion of time. The title of the posting is "The arrow of time: understood for 100 years". As a conservative Lubos believes that all interesting things about time were said by Boltzmann already before the birth of quantum theory. Second law would summarize all that is interesting. Lubos is also impatient about the fact that there are still people who feel that the nature of time is not fully understood.

The core of the Boltzmannian view about time is simple to summarize.

1. The time development can be seen as analogous to Markov process. To continue, let us introduce discrete time t_n = n×Δt. One could regard this as a technical simplification but will find that Δt has interpretation as time scale of quantum coherence.

2. Physical events are identified as transitions from state i to state j taking place during time interval Δt and the probabilities P(i,j) characterize them. The probability of state i at time t=t_n is given by p(i,t_n) and once the probabilities p(i,0) for initial state are given by summing the probabilities over all paths j→k→...→i leading from j to i and calculating the average over the initial states

   p(i,t_n)= ∑_j (Pⁿ)(i,j)× p(j,0).

   This formula leads to the second law stating the increase of entropy defined by Shannon formula.

There are several unpleasant questions that Lubos leaves out of consideration.

1. Boltzmann's approach was developed before the advent of quantum mechanics but involves classical probabilistic approach not very natural in classical physics. Quantum theory indeed allows to calculate the transition probabilities p(i,j) from first principles. There are however deep interpretational problems. Can one say when quantum transition -state function reduction - breaking the deterministic Schrödinger evolution really takes place? How long is the period of non-determinism if it occurs? Is the non-determinism really assignable to the geometric time or could it be that the relationship between the geometric time and experienced time is what we do not actually understand? These are basic problems of quantum measurement theory which is a useful bag of calculational recipes but does not really deserve to be called a theory.

2. At the technical level these problems are avoided by assuming that transitions take place during an infinitely long time interval. This idealization means that quantum coherence is present in infinitely long time scale. The resulting probability proportional to a square of energy conserving delta function is transformed to a rate by dividing with the length of this infinitely long time interval dividing away one delta function expressing the conservation of energy. One can feed the resulting rates to kinetic equations and assume that quantum coherence is present only in infinitely short time scales. Do not get frustrated: this is the situation! One can guess that there must be some finite time scale for which quantum coherence and sum over amplitudes makes sense and in longer time scales one must use Boltzmann's discretized approach and sum over probabilities.

There are also problems related to the justification of probabilistic approach.

1. To some degree this picture could be formally justified by using path integral formulation of quantum field theory. What one must do is to replace the modulus squared for the path integral approximated with a sum over discrete paths with a sum in which all interference terms are neglected. This means a de-coherence. This kind of approximation can be defended by saying that due to the small value ofΔt in the example discussed. Do we really understand the origin and mechanisms of de-coherence? Could de-coherence have a more detailed description involving perhaps new physics?

2. De-coherence assumption is rather strong in the many-sheeted space-time of TGD. TGD based view about dark matter as a hierarchy of phases partially labeled by Planck constant predicts macroscopic quantum coherence even in astrophysical time and length scales so that the Markovian view can be used only if one restricts the consideration to processes in definite time scale below the natural time scale characterizing the time intervals during which observations are made.

The basic problem of this approach is that the observer is not part of the Universe. In classical physics observer was a complete outsider and in quantum measurement theory the situation remains the same although the measurement interaction leading to state function reduction affects the measured system.

TGD inspired theory of consciousness can be seen as a generalization of the quantum measurement theory to resolve its basic paradox by making observer part of the Universe via the notion of self as well as to understand the differences and relation between the time of physics (geometric time) and experienced time by identifying the chronon of latter as quantum jump defining moment of consciousness.

1. The outcome is what I call zero energy ontology. Zero energy states are pairs of positive and negative energy states localizable to the upper and lower boundaries of causal diamonds defined as intersections of future and past directed light-cones of Minkowski space (and taking Cartesian product with CP₂). There are CDs within CDs and they form a hierarchy.

2. The hierarchy of CDs is also a correlate for a hierarchy of conscious entities which I refer to as selves. CD represents the perceptive field of self. CD represents correlate for quantum jump identified as the chronon of experienced time and there is an infinite hierarchy of chronons. CD also defines also quantum coherence region inside which the sum over probabilities must be replaced with the sum over amplitudes so that Boltzmann's kinetic description fails.

3. The notion of time measurement resolution reduces to the time scale of CD. If this time scale comes as powers of 2, p-adic length scale hypothesis follows. One would have a hierarchy of physics realized in different p-adic length and time scales characterized by primes near integer powers of 2.

4. Although one has quantum coherence in a given time scale (CD), it is possible to have de-coherence in shorter time scale (sub-CDs). The description of hadronic reactions in terms of quarks and gluons using kinetic distributions defined in relatively short time and length scales and the description of hadrons using wave functions defined in considerably longer scales is a good example about de-coherence within coherence.

5. This gives hopes about improved understanding of the second law in living matter. The essentially new notion is that of scale: when one speaks about second law one must specify the time scale in which it is applied. Only if applier is CD modeling what happens in the ensemble of sub-CDs this description works. If one tries to understand what happens in CDs characterized by time scale longer than the natural time scale of the observation- the approach fails. These CDs are expected to be highly relevant in biology.

I will not continue here further but give instead a link to the article About the Nature of Time and also a link to a video summarizing the recent view about the relation between geometric and subjective time: this includes explanation for the emergence of the arrow of time and for the fact that the contents of sensory experience are about very narrow time interval although one would expect that entire CD determines the contents of also sensory experience. I hope that I do not sound too authoritative and that my badly broken English is not too painful an experience;-).