In the world about us, the past is distinctly different from the future. More precisely, we say that the processes going on in the world about us are asymmetric in time, or display an arrow of time. Yet, this manifest fact of our experience is particularly difficult to explain in terms of the fundamental laws of physics. Newton’s laws, quantum mechanics, electromagnetism, Einstein’s theory of gravity, etc., make no distinction between the past and future - they are time-symmetric.
Einstein’s theory of general relativity goes further and says that time has no objective meaning. The world does not, in fact, change in time; it is a gigantic stopped clock. This freaky revelation is known as the problem of frozen time or simply the problem of time.
The problem of time is not simply that we never have enough of it, but that we do not quite know what it is we do not have enough of.
The debate about the nature of time has been going on for at least two and a half millenia, since Heraclitus, Parmenides, and other pre-Socrates ( [Kirk-1957], [Barbour-2000]).
If time is a river, some see it from the point of view of a white-water rafter, caught up in the moment; others from the perspective of a river surveyor, mapping the river as a whole.
With 2500 years to work up a running start, literature is enormous. Among the discussions found helpful: [Reichenbach-1956], [Gold-1958], [Penrose-1962], [Davies-1974], [Cramer-1983], [Penrose-1986], [Horwich-1987], [Earman-1989], [Hawking-1993], [Halliwell-1994], [Macey-1994], [Oaklander-1994], [Omnes-1994b], [Thorne-1994], [Davies-1995], [Savitt-1995], [Price-1996], [Hawking-1996], [Hawking-1996b], [Atmanspacher-1997], [Schulman-1997], [Krasnikov-1998b], [Nahin-1999], [Zeh-1999], [Barbour-2000], [Stenger-2000], [Zeh-2001], [Muga-2001b], [Galison-2003], [Zeh-2003c], [Greenberger-2005], [Kiefer-2005b], [Gross-2007], [Toomey-2007], [Barbour-2008], [Halpern-2008], [Kaku-2008].
The debate has sharpened considerably in the last century, since our two strongest theories of physics – relativity and quantum mechanics – take almost opposite views.
Time and space are treated symmetrically in relativity: they formally indistinguishable, except for entering the metric with opposite signs. Even this breaks down going through the Schwarzschild radius of a black hole: consider the line element:
Here time and the radius shift the sign with which they enter the metric when ( [Adler-1965]). Also [Novikov-1998].
Note problem resolved by Georges LeMaitre in 1932 (per [Kaku-2005], p116).
In relativity, it takes (significant) work to recover the traditional forward-facing time. We have to construct the initial spacelike hypersurface, it does not appear naturally, see for instance [Barbour-2008].
In quantum mechanics we have the mantra: time is a parameter, not an operator. Time functions like a butler, escorting wave functions from one room to another, but not itself interacting with them.
This is alien to the spirit of quantum mechanics: why should time, alone among variables, escape being quantized?
In quantum mechanics, defining the spacelike foliations across which time marches is problematic.
No evidence these foliations are well-defined, given that uncertainty in time precludes exact knowledge of which hypersurface you are on at any one time.
They are difficult to reconcile with relativity. If Alice and Bob have different surfaces, the quantum fluctuations purely in space for one, will be partly in time for the other.
There is a nice analysis of the difficulties in a series of papers by Suarez: [Suarez-1997] [Suarez-1997b] [Suarez-1998] [Suarez-1998b] [Suarez-1998c] [Suarez-1998d] [Suarez-1998e] [Suarez-2000] [Suarez-2003]. Per the abstract for [Suarez-2000]:
It is argued that: 1) Quantum Mechanics implies the preferred frame also because of the collapse delayed at detection, 2) forthcoming experiments with moving beam-splitters will allow us to decide between Preferred Frame and Multisimultaneity, and 3) if the Preferred Frame prevails, superluminal communication is in principle possible. |
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-- Antoine Suarez |
Suarez's specific response, Multisimultaneity, was not confirmed experimentally ( [Stefanov-2001] [Stefanov-2002]) but his objections remain.
Uncertainty in time/energy assumed by Heisenberg [Heisenberg-1927]. This is a much explored subject, see: [Hilgevoord-1996], [Hilgevoord-1996b], [Oppenheim-1997], [Oppenheim-1998b], [Oppenheim-1998c], [Oppenheim-1999], [Oppenheim-2000], [Busch-2001]. To over-summarize: there is an uncertainty relationship between time and energy, but it does not stand on quite the same basis as the uncertainty relation between space and momentum. Great precision in the definition of terms is essential.
As Feynman has noted, if any experiment can break down the uncertainty principle, the whole structure of quantum mechanics will fail. The lack of an exact parallel between the time/energy and space/momentum uncertainty relations is therefore troubling.
Although the path-integral formalism provides us with manifestly Lorentz-invariant rules, it does not make clear why the S-matrix calculated in this way is unitary. As far as I know, the only way to show that the path-integral formalism yields a unitary S-matrix is to use it to reconstruct the canonical formalism, in which unitarity is obvious. |
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-- Steven Weinberg [Weinberg-1995] |
One can argue that one does not expect this symmetry in non-relativistic quantum mechanics. But, the problem does not go away in quantum electrodynamics.
In canonical quantization we still have the problem of a special role for time; in Feynman path integrals, apparently not so, however unitarity no longer obvious.
If no single perspective shows both unitarity and time/space symmetry, than it is possible that the underlying theory is incomplete.
We are looking at a situation a bit like Enron bookkeeping, where the presence of off-shore accounts and un-tracked money makes suspect the overall presentation.