If the theory of everything indicates a singularity-free and finite universe, will Poincare recurrence be relevant to the universe? If so, is there any interesting physical consequence, e.g. in superstring or quantum gravity?

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Yes, our Universe is approaching the empty de Sitter space – we are already pretty close to it, actually, because the cosmological constant dominates the vacuum energy (68% of it). It has a cosmic horizon (the boundary behind which we can't see) and the degrees of freedom are formally living on that surface.

Via the holographic principle (in a somewhat less tested context), one may claim that this means that de Sitter space has a finite entropy so a finite-dimensional Hilbert space is enough to describe everything that happens in it (including the matter in not-yet-empty mostly de Sitter space, like the present Universe). If that's so, the Universe we inhabit behaves much like any system with finitely many degrees of freedom, and it has Poincaré recurrences.

The Poincaré recurrence time is extremely long, something like $\exp(10^{120})$ billion years – it is because the entropy of the de Sitter horizon is $10^{120} k_B$ (the cosmological constant is $10^{-120}$ in Planck units or so). After this superlong time, approximately, events start to repeat themselves. At least in some sense, it is fair to say that the time is literally periodic.

This very long timescale is the ultimate "maximum duration" that may be discussed in physics. For example, if a calculation in field theory or string theory implies that the lifetime of a vacuum is longer than this Poincaré recurrence time, the decay is considered unphysical because "it cannot happen in time", anyway.

The timescale is vastly longer than anything we have a chance to experimentally test. Each 10 billion years or so, the linear distances between galaxies double and the density of normal matter decreases by an order of magnitude. In hundreds of billions of years, a vast majority of the currently active stars will be inactive and even "new generations" of the stars will already be gone or dying. In trillions or certainly quadrillion years, there will be nothing left to energize star-powered life as we know it, and similar "local, more modest" sources of useful energy will be diminishing in similar ways.

It is hard to imagine that there will be any intelligent beings in a quadrillion years. This is still vastly smaller than the Poincaré recurrence time. And this result of the comparison is no coincidence. Of course that things must have a chance to "destroy any pattern" before the chaos has a chance to reassemble itself into the patterns again.

  • The cosmic horizon depends on the observer, so does the finite-dimensional Hilbert space depend on the observer ? If yes, how are the Hilbert spaces related ? – jjcale Jan 17 '14 at 21:29
  • As the second question correctly assumes, it's really the same Hilbert space for all observers. It's just being interpreted - the observables are being chosen - in a way that depends on the observer. The key fact is about observables, not the Hilbert space itself. And the fact here is that the field operators that are behind the horizon of each other don't quite commute with one another. The fields within a single horizon are building blocks for a maximum commuting set of observables, kind of. Locality breaks further. – Luboš Motl Jan 18 '14 at 6:47
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    Are you THAT Lubos Motl who discovered Matrix String theory ? – Physics Guy Sep 24 '16 at 22:19
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    Thanks for formulating the question in this flattering way. The correct original name is Screwing String Theory, however. – Luboš Motl Sep 27 '16 at 10:27

The Poincaré recurrence theorem will hold for the universe only if the following assumptions are true:

  1. All the particles in the universe are bound to a finite volume.
  2. The universe has a finite number of possible states.

If any of these assumptions is false, the Poincaré recurrence theorem will break down.

Yes, the recurrence time for this universe is about $10^{{10}^{{10}^{{10}^{2.08}}}}$. The unit (seconds, or years) isn't important.

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