18 July 2007

The problem of time

One of my campers asked me about the problem of Time: why is it different from space, why can't there be hidden time dimensions.

Here's my response; please comment with corrections/additions:

The problem of Time is a very difficult one, and you've asked some deep questions, many of which do not yet have good answers. I'll answer bits of them, but my answers will be necessarily incomplete.

First of all, why aren't there hidden time dimensions? In fact, lots of physicists have asked about this, and a priori there could be hidden time dimensions. In special/general relativity, the only difference between space and time dimensions is a minus sign somewhere; you could make a manifold with two time dimensions and d spatial dimensions if you want. Then you can still do general (and special) relativity just fine --- the laws are perfectly consistent with any (nonnegative) numbers of time and space dimensions.

But problems arise in quantum field theory. Many QFTs have what are called "anomalies": while the original classical laws are symmetric in a certain way, it becomes impossible to quantize and keep the symmetry. So, for instance, without a Higgs field, it's impossible to quantize a chiral gauge theory with massive fermions (most of the universe is a non-chiral gauge theory, which can be quantized with massive fermions, but the Weak Force is chiral, meaning that if you look at it in a mirror, you get different effects; this is why physicists believe that there are Higgs bosons). There's a very important example in string theory: the _only_ anomaly-free bosonic string theories exist in 26 dimensions (25 space, one time), and the _only_ anomaly-free superstring theories are in 10 (9+1) dimensions.

In fact, for an anomaly-free QFT --- i.e. a QFT that respects special relativity --- you cannot have both at least two space and at least two time dimensions. You could give up special relativity, but then you'd be giving up the entire ballgame: the whole idea of QFT is to respect a symmetry that definitely appears in nature.

It seems that abstractifying (not a word) time is different from abstractifying space. Geometry takes properties of real space, and idealizes them into abstract spaces; of course, the whole point of special/general relativity is to take properties of spacetime and idealize them into abstract spacetimes.

What about phase space? Don't we normally abstractify space into phase space, and import time in the most stupid "\times\R" way? This is where the meat of the matter is. In some sense, I'd argue that time is inherently _more_ abstract than space. "Time" is a measurement of how things _change_ --- a function should be thought of as a one-dimensions object (with a "front" or "in" end, and "back" or "out" end; functions can be glued together so long as you respect this orientation) pointing in the "time" direction. What's going on with things like "phase space" is that phase space exactly describes the _state_ of a particle, and you need a time dimension for the state to change. Thus Hamiltonian formalism in classical or quantum mechanics.

Incidentally, Lagrangian formalism is manifestly special-relativistic invariant --- in Lagrange (equals "least action principle"), the law is that particles evolve in such a way as to minimize a certain total path length. This is perfectly easily expressible in a spacetime, and that expression is probably more natural than in just space, because the path the particle takes is through spacetime. Lagrangian formalism is what most modern QFT folks use.

One other point is worth making: Roger Penrose, one of the most brilliant (and crackpottish) physicists alive today, invented a physics called "Twistor Theory". For twistor theory, he has to replace SO(3,1), the symmetry group of (our) three space dimensions and one time dimension, with SO(4,2), which most naturally describes a universe with two time dimensions. Twistor theory, which takes light rays as basic rather than spacetime points, has not been widely adopted; it's not clear to me what the philosophical corollaries of this change are.

Sorry I don't know any references. Hope this gives you some more ideas where to look.

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