Are there any ideas in physics which support the argument for eternal return?
I am looking for any mathematical structure supporting observations which could be in support of a cyclical Universe.
I know that Nietzsche's arguments on this were based on classical physics which is now obsolete.
Some things which I could think of which would go in favor of eternal return are:
One which I could think of is that the Universe might not be entirely governed by mathematical rules, or math might not be complete in a sense.( I know this is not physics, and I am looking for physics answers).
There could be a different kind of logic at the base of it all, making time circular.
I cannot however think of an argument through physics for it currently.
https://en.wikipedia.org/wiki/Eternal_return
Eternal return (also known as "eternal recurrence") is a concept that the universe and all existence and energy has been recurring, and will continue to recur, in a self-similar form an infinite number of times across infinite time or space.
Answer
Nietzsche's "argument", for those who don't know it, was that there must be only finitely many possible physical configurations, and that the notion of a beginning of time is unscientific, and so therefore, the universe is eternal, and every physical situation must recur eternally in it.
Of course, he was writing before general relativity and Hubble's discovery of universal expansion. The only time I have seen a modern physicist mention this argument from Nietzsche, was in Frank Tipler's The Physics of Immortality, where it is presented as the alternative to Tipler's own cosmology, in which a computer-god at the end of time resurrects (via simulation) everyone who ever lived.
But as for whether there are any cosmological models taken seriously now, that imply eternal recurrence, whether deterministic or probabilistic... The big bounce of the ekpyrotic universe gets some media, but not so much attention from the community. Anti de Sitter space does imply quantum Poincare recurrence, but isn't taken seriously as a model of reality (unless you could somehow have a de Sitter fluctuation within it).
Anti de Sitter space does illustrate another way you could get a de facto recurrence: you could simply make the time coordinate periodic, so that, rather than the future recapitulating the past, time just "goes in a circle" and the far future is literally also the distant past. Periodic time is a common construct in mathematical physics, but I can't think of any cosmology that is taken seriously and which applies periodic time to the real world. The Goedel universe has this feature, but isn't a serious cosmological model. Maybe you could do this in Penrose's conformal cyclic cosmology, but again, that's a model which exists more in media coverage than as an object of research.
HOWEVER! If we look slightly afield, and ask whether there are scenarios under consideration in which our lives are repeated infinitely throughout space and time, though perhaps not serially, then we come much closer to the mainstream of research.
I believe the core area where this issue arises is in the study of what is called "the measure problem" in cosmology. The measure problem concerns the difficulty of extracting observational predictions in a cosmology that contains infinitely many observers. A probability is a kind of expected frequency of occurrence, so you need to be able to count observers and then say what fraction of them have a particular observation; but when space and time are infinite, these numbers may both be just "infinity", so you are dealing with infinity divided by infinity.
There are ways around this; for example, you may start within a finite volume of space-time, in which numbers are finite and you can define finite ratios, and then you can look at the asymptotic behavior of those ratios as you widen the scope to infinity. But meanwhile, it is incidentally the case, in an infinite universe (or "multiverse") of the kind implied by eternal inflation, that the physical state of our observable region (our Hubble volume) will be repeated infinitely often.
A rather less mainstream scenario, in which infinite duplication may also occur, is the Boltzmann brain problem.
The Boltzmann brain problem is that in an eternally expanding de Sitter universe, even after the expansion has completely attenuated the matter of the universe, the quantum fields will continue to fluctuate, and over infinite time will produce (with appropriately minute, but still greater-than-zero, probability) every physically possible configuration of matter. Only some people are bothered by this, but papers do get written about it.
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