Dear Ken,

Very enjoyable essay, well explained and insightful. You did a great job dispelling some superstitions even physicists have about the origin of the time arrow. I fully agree that no matter what one tries, the least problematic remains the past hypothesis. What amazes me is that I see once in a while people trying to explain the arrow of time by introducing some time asymmetry (with or without care to not be compensated by P and T symmetries) in the evolution equations themselves! In fact, even Penrose's Weyl Curvature Hypothesis is in my opinion not an explanation of the second law, but it may be a prediction of it, in the sense that at the Big Bang the sources of gravity exist, but the field didn't spread yet, somehow like the retarded solution of Maxwell's equations. Imposing WCH is equivalent to imposing law entropy for gravity, and it is just a partial restatement of the problem, not the explanation.

And I think you are right to propose that the boundary conditions may have the same status as the dynamical laws themselves. Now maybe this will lead, in the case of a universe which big-crunches itself (excluded by measurements of the cosmological constant) that the time arrow reverts when the universe reaches the mid-life crisis. But since it expands forever, maybe the initial and final boundary conditions manifest differently, even if they may be subject to the same principle. Maybe this can be connected to Penrose's Conformal Cyclic Cosmology (CCC), although there may be some complications here.

Your essay motivated me to organize some random thoughts I have about this subject.

So here is an idea I have about boundary conditions. I can call it "boundary law without boundary law", but is not as one may expect just another way to obtain boundary conditions out of the dynamics, as in fluctuations or Janus point. It is based on conformal symmetry. You know Maxwell's equations on Minkowski spacetime (backreaction ignored) are invariant to conformal transformations, which extend the Poincaré group. The conformal boundary of the Minkowski spacetime is mapped by conformal inversion to the lightcone at the origin, and vice-versa. So on the conformal boundary the (conformal transformations of the) solutions have the same kind of regularity as on the lightcone at origin. What's beautiful is that this conformal invariance applies to the entire Standard Model as long as all masses are 0, so his is equivalent to the absence/vanishing of the Higgs field (I don't know yet if this holds for the neutrino too). Now turn on gravity, the global properties of spacetime change. Maybe it is asymptotically flat, in which case the boundary remains similar. The conformal invariance is broken. But we know that there is a richer conformal invariance, the local one, consisting in rescaling the metric tensor independently at each point of spacetime. So I suspect that there is also a generalization of translations and in fact of the full conformal group. The reason I suspect this is that we still have conservation of momentum, because the stress-energy tensor is locally conserved, but it is local now. So I am in particular interested to see how I can make local the full conformal group. Now going back to the boundary, I suspect that some regularity persists even after we turn on gravity and the Higgs field. Even if maybe the conformal boundary changes because of the cosmological constant. So I think it worth seeing what are the effects on this regularity, what remains of it, after the breaking of the conformal symmetry. Note that maybe this can be connected to Penrose's CCC, but not necessarily, because maybe a scale inversion happens, maybe not, when crossing the final boundary to go back to the initial boundary. This needs to be investigated, maybe the asymmetry between the initial and final boundaries allow it, maybe not. If the asymmetry is too strong to admit scale inversions as in Penrose's CCC, I think this will be instead more like Penrose's WCH. Anyway, it seems to me that conformal symmetry may hold a key to obtain some boundary law from the dynamical law, in a fundamental way, as opposed to assuming fluctuations and anthropic principles.

About Boltzmann's brains, they indeed seem to be much more probable than any sort of stable brain which is the product of evolution. But there are two factors which I don't think were taken into account when making such calculations, at least not to my knowledge. Brains belonging to evolving species crowd together, both in time and space. In our world at least, life is finite, brains die and new brains are born, and they are born from one another. This implies a huge number of brains crowded together. One can speculate that it is possible to have species which are immortal, but it seems plausible that they are much rarer than species whose individuals have a reasonably short life span (but not too short), because this allows selection and evolution. So if we take this into account, could it be possible that the Boltzmann brains are in fact much less likely to exist than brains enrolled in a species? Moreover, a brain which is part of a surviving and evolving crowd, being subject to natural selection, it is much better adapted to observe the environment. So a crowd of n brains has a much larger probability than the probability of n independent Boltzmann brains. In the index counting of such brains we should also take into account not only the number of brains, but also their life span, so an ephemeral Boltzmann brain will be a very brief fluctuation very little connected to the environment, while for an observer it may be more likely to be part of a crowd of brains having longer lives. (of course the life span affects in two opposite ways this calculation). I think this argument relies on more parameters, and they are very difficult, if not impossible to estimate, and in addition we encounter similar problems as in the doomsday argument. Similarly for single planet orbiting a single star in a high-entropy background, we should take stability into account, Boltzmann brains may seem more likely, but it may be more likely for a mind to be in a stabler brain, and also stars tend to crowd too, although life on one planet seems not to be significantly correlated with life on other planets in the same galaxy. So although I think a boundary law or special initial conditions are a better explanation, it is not as easy to reject the explanation based on fluctuations based on Boltzmann brains.

After this long comment which turned into a mini-essay, how can I now invite you to also read my actual essay? :)))

Again, excellent essay! Good luck in the contest!

Best wishes,

Cristi Stoica, Indra's net

Dear Ken,

We agree pretty well that the current situation in which our more fundamental physical theories involve far too many adjustable parameters. For one, I do not believe, as the essay implies a standard theorist must, that these parameters were somehow (chosen at) random.

You seem to agree, but react in an altogether different way. I (as in my essay) seek to explain the respective systems of equations as the result of a more fundamental formula, EVALUATED AT "|" the respective space-time. In the traditional sense, that expands the meaning of |H> to include other scales, besides the traditional weak scale. *It is clear from your consideration of BC that you understand the fundamentality of Hamiltonian, H and in QM |H>, and Classical mechanics.

I emphasize that this is a re-interpretation of Dirac's notatation, but it aligns somewhat with your thinking on the subject. After all, a space-time average is simply taken over designated space-time 'boundaries'. So I derive GR and QC/ED from a generic basis via choosing the 'boundary conditions' used in "|", i.e. the traditional |weak, and in particular |astrophysical in which GR (at the time=present) is derived.

The resulting |H> at strong scale is quite revealing, and can avoid the "appearance of 'randomly selected' adjustable parameters".

However, I must note that this work is based on and constructed in 100% agreement with Hartle, Hertog and Hawking's work on the cosmological "no-boundary condition" basis. I would suggest that you consider their work quite highly, as it also helps me eliminate both that pesky "initial state of cosmology" AND "cosmological coincidence" problems.

J.B. Hartle, S.W. Hawking and T. Hertog, "The Classical Universes of the No-Boundary Quantum State" hep-th/0803.1663v1 March 2008.

I'd be willing to come out to San Jose to explain better via a seminar... but please feel free to read my work here and online arxiv.

Wayne

https://fqxi.org/community/forum/topic/3092

Hi Ken,

Great, provocative read. You say an appeal to randomness should not appear in fundamental explanations. But what if the universe really has some fundamental randomness built into it? Isn't this one way the world could be? And if so, then shouldn't this be part of our fundamental picture?

I also was unsure about your conductor analogy. As you concede, in that case, we do think a more fundamental dynamical explanation is possible. So I am not sure that this gives precedent for boundary conditions being explanatory *on their own* as you want them to be. It gives precedent for boundary conditions being explanatory in contexts where we know there are more fundamental explanations that are being black-boxed. But that is not where we are in the cosmological case.

Best,

Alyssa

Dear Ken,

I highly appreciate your well-written essay in an effort to understand.

Your essay allowed to consider us like-minded people.

I hope that my modest achievements can be information for reflection for you.

Vladimir Fedorov

https://fqxi.org/community/forum/topic/3080

I just increased your rating from 6.6 to 6.8, but someone two again put you down.

The classical second law cannot be fundamental, but we have microscopic analogs of the second law: for instance the Austin-Brussels condition

[math]\Im(\Theta) \leq 0[/math]

Everywhere around me I see time-asymmetric phenomena. Time-symmetric laws were created in early physics, because physics was born from the observation of simple planetary motions; however biology and chemistry were born from the observation of complex processes and time-symmetry was never an option for the theorist. This is why the mass action law in chemical kinetics was irreversible since its very beginning.

The ordinary derivation of the second law in kinetic theory of diluted gases doesn't assume the equal a priori probabilities postulate; at contrary the probabilities are all different and given by the Maxwell & Boltzmann distribution.

The equal a priori probabilities postulate is *truly* valid on equilibrium. The argument that "at any given time the actual system is in 1 microstate, with 100% certainty, and all others with 0%" doesn't invalidate the postulate because it is about the ensemble, not about individual systems in the ensemble. Each individual system has properties that differ from the ensemble as Qi = Qensemble + dQi.

There are several proposals for time-asymmetric dynamical laws; for instance modifications of the Schrödinger equation to make it irreversible.

Time symmetric dynamics cannot explain irreversible phenomena. Some people run computer simulations in such a way that it reproduces the expected result, such as a positive diffusion coefficient. But inverting the simulation gives results invalid with real-life experiment. Computer simulations with time-reversible dynamics can show entropy increasing or decreasing independently of the initial state. That people has been on safe situation up to now because they have run simulations in cases where previous knowledge did exist. So they could just get the expected numerical answer. The problem starts when one want to make a simulation in regimes outside experience. Time-symmetric simulations cannot differentiate real phenomena from invalid phenomena. That is the reason for the recent emphasis on introducing directly time-asymmetry on the equations of motion that will be solved by the computer. For instance in DPD the simulation uses the next decomposition for the total force

F = FC + FD + FR

FD are dissipative forces which break time-reversibility and ensure the systems evolve in agreement with the second law, independently of the initial state (low or high entropy).

The problem with fixing the final boundary is that evolution is not deterministic, and so it is not defined at early times. Why would fix the initial entropy to be lower and the final entropy to be higher? This would work for certain kind of processes such as evolution from an initial equilibrium state to a final equilibrium state. But this two-times boundary will fail in other regimes, for instance in non-Markovian regimes where production of entropy can be *negative* without violating the classical thermodynamics laws.

The correct approach consists on developing time-asymmetric equations and left the system evolve from an initial state t=0; not say to the system which has to be the state at some t>0, based in prejudices like that the final entropy at t>0 always has to be higher than the initial entropy at t=0.

My comment about the Liouville equation (or its von Neumann quantum analog) did mean the next. Integrating the mechanical equation of motion we obtain

[math]\rho(t) = exp(-iLt)\rho(0)[/math]

by Liouville theorem this mechanical equation of motion predicts entropy S is constant. This is obvious because the equation is time reversible. Then some people defines a coarse-grained entropy Scg, which is then allowed to vary. The problem is that the equation continues being-time reversible and for each dScg>0 there exists a dScg

...is negative.

What people do in practice is to select the evolution compatible with the classical second law and ignore the other. The problem is not only that the basic equation gives a set of incorrect solutions. The problem is when we want study dynamical regimes whre the classical second law doesn't apply. In those regimes the second law cannot be used as a consistency check to select the valid solutions; the solutions compatible with Nature, and discard the others.

"There are no irreversible events at a micro scale (Or so most physicists believe; maybe we're all wrong.)". Research made on last 50 years show irreversibility has microphysical roots

http://onlinelibrary.wiley.com/doi/10.1002/0471619574.ch17/summary

https://books.google.es/books?id=Px7Wnx0K_EQC&pg=PA301&lpg=PA301&dq=MICROPHYSICAL+IRREVERSIBILITYAND+TIME+ASYMMETRIC+QUANTUM+MECHANICS

Boltzman was wrong and when pressed by critics he vacilated and gave inconsistent answers. Unfortunately a part of physicists continue repeating his mistakes forever. A macro description of time-reversible microscopic physics doesn't introduce irreversibility aparent or otherwise. Rigorous tracing or coarse-graining procedures generate macroscopic equations incompatible with experience, what Boltzmanians do is to produce mathematically invalid derivations where time-symmetry is explicitly broken by introducing some extra-dynamical ad hoc concept that selects the correct description compatible with the second law. The resulting macro equations no longer are compatible with the initial micro equations. Irreversibilty hasn't been derived fmr reversibility. Irrversibility has been imposed by breaking microscopic reversibility. Moreover those ad hoc procedures are oly valid for simple systems such as diluted gases, markovian dynamics in heat baths, and so. That is why Boltzmanians have never produced an equation of motion valid for arbitrary regimes.

15 days later

Dear Ken; further to my earlier comments, please: Since we cannot both be right, would you mind commenting on my half-page refutation of Bell's theorem?

See ¶13 in More realistic fundamentals: quantum theory from one premiss.

NB: I clarify Bell's 1964-(1) functions by allowing that, pairwise, the HV (λ) heading toward Alice need no be the same as that (μ) heading toward Bob; ie, it is sufficient that they are highly correlated via the pairwise conservation of total angular momentum. Thus, consistent with Bell's 1964-(12) normalization condition:

[math]\int\!d\boldsymbol{\lambda}\:\rho(\boldsymbol{\lambda})=\int\!d\boldsymbol{\mu}\:\rho(\boldsymbol{\mu})=1.\;\;\;(1)[/math]

Further, in my analysis: after leaving the source, each pristine particle remains pristine until its interaction with a polarizer. Then, in that I allow for perturbative interactions, my use of delta-functions represents the perturbative impact of each such interaction.

My equation (26) then represents the distribution of perturbed particles proceeding to Alice's analyzer. Thus (with b and μ similarly for Bob):

[math]\int\!d\lambda\;\rho(\lambda)_{Alice}=\tfrac{1}{2}\int\!d\,\lambda[\delta\;(\lambda\sim a^{+})+\delta\,(\lambda\sim a^{-})]=1.\;\;\;(2)[/math]

PS: Bridging the continuous and the discrete -- and thus Bell's related indifference -- integrals are used here by me for generality. Then, since the arguments of Bell's 1964-(1) functions include a continuous variable λ, ρ(λ) in Bell 1964-(2) must include delta-functions. Thus, under Bell's terms, my refutation is both mathematically and physically significant.

PLEASE: When you reply -- or if you will not -- please drop a note on my essay-thread so that I receive an alert. Many thanks; Gordon

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