Category: Physics

Talking about different physical theories in different dimensions is one of physicists favorite pastimes. Thus, for instance, we often move the theory to the infinite dimensional case or to the case of dimension slightly greater or less than some integer dimension. Often we work with low dimensional theories because of constraints–for example we do this when we talk about an electron gas trapped in a two surface or a one dimensional line or even a zero dimensional dot. But sometimes we work in different dimensions to see if we can gain insight into the theory in a dimension where we can’t seem to make much progress. Such is the case for a quantum theory of gravity. Our success in understanding quantum gravity in two spatial dimensions plus one time dimension (2+1) has been far better than our success in understanding quantum gravity in three spatial dimensions plus one time dimension (3+1). Why?
Well there is an easy way to see why quantum gravity in 2+1 dimensions is quite a different beast than quantum gravity in 3+1 dimensions. Let’s look at the 3+1 dimensional case first. At a fixed time, the spatial metric has six degrees of freedom (it’s a real tensor and it’s a symmetric tensor and it’s in dimension three.) But the laws of general relativity are invariant under general coordinate changes. This means that there are four gauge degrees of freedom which correspond to the choice of four spacetime coordinates. Thus there are two physical degrees of freedom in 3+1 dimensional gravity.
But what happens in 2+1 dimensional gravity? At a fixed time, the spatial metric now has three degrees of freedom (it’s a real tensor and it’s a symmetric tensor and it’s in dimension two.) But now there are three gauge degrees of freedom. Thus in 2+1 dimensional gravity there are no physical degrees of freedom!
In fact what happens in 2+1 is that the curvature tensor vanishes! Now recall that if we parallel transport a vector around an area where the curvature tensor vanishes, then the vector doesn’t change. Thus if we work in a spacetime where there all loops encose areas, then parallel transport will be path independent and there will be a global notion of parallelism. Quite a boring theory right? Well yes, if you only consider simply connected spacetimes (i.e. spacetimes where all loops are continuously contractible to a point.) But if you consider spacetimes which have noncontractible loops (think of a torus and the circles which form circumferences of this torus) then the parallel transport around one of these noncontractible loops doesn’t enclose an area. Now the geometry of this flat spacetime is characterized by the results of parallel transport around noncontratible loops (holonomies.) So right away we see that gravity in 2+1 dimensions will be an interesting theory when we allow topologically nontrivial spacetimes. In fact, when we construct the solutions and quantize gravity in 2+1 dimensions we are led to a topological quantum field theory! Actually things get quite interesting in 2+1 dimensions when we try to quantize the theory. In fact there are many different approaches to this quantizations, and, strangely, not all of these are consistent (this is why you have to pay attention when all these mathematical physicists go on and on about all these different methods to quantize classical theories!)
Well enough quantum gravity for today. Just remember, theories of physics are never as complicated as most theoretical physicists would like you to believe.

In cryptography there has been a long tradition of calling two parties involved in a protocol Alice and Bob. This tradition has been proudly maintained in quantum information science, no doubt in large part because quantum cryptography was one of the first ideas in quantum information science. One of the nicest things about the Alice/Bob labeling scheme is that it allows one to use gender to distinguish parties. I suspect that since gender is dear to our animal hearts, this concise way of refering to parties has significant syntactic advantage. David Mermin once said something along the lines of “if quantum information contributes nothing else to physics it least it will have given us Alice and Bob.” Actually, where I’ve found the Alice/Bob labeling scheme most efficient is in special relativity where it allows one to give gender to different reference frames (note also that since all frames are equal…) It is much clearer to most students when you refer to “his reference frame” or “her reference frame.”
Along these lines, when you have to introduce a new party, it is traditional to call this party Eve. This is usually done because in cryptography, Eve is the eavesdroping malicious third party. But this screws up the whole gender roles efficient labeling. I therefore propose that instead of calling the third party Eve, we should call the third party E.T., or Elephant, or Eagle such that we can use “it” to refer to this party. Now what do we do for four parties?

In quantum theory, we are interested in calculating the amplitude for starting in some initial state |i> and ending in some final state |f>. For a Hamiltonian H evolving for a time t, this amplitude is given by <f |exp(-iHt)|i>. In the path integral formulation of quantum theory, we rewrite this as the path integral &#092int dq exp(i S(q)) where this integration is performed over all paths q and S(q) is the action (&#092int_0^t L(q,dot{q}) dt ). Often what we’re really interested in is the long time propogators, so our action is really integrated from minus infinity to plus infinity. What has always astounded me is that often times we can calculate this path integral by performing a Wick rotation: we substitute -it for t in the path integral and thus we obtain a path integral with terms which don’t oscillate wildly. This often results in a situation where we can then either explicitly calculate the integral, or where we can numerically integrate the path integral by standard Monte Carlo methods. In fact, you will recall, what we’ve done is transformed the path integral into a partition function from classical statistical mechanics.
So here is my question. Is this anything more than a trick or is there something profound going on here? In particular I’m thinking about hidden variables. Since we have taken a quantum system and transformed it into a classical system, we’ve effectively made the transition to a hidden variable theory. Sampling from the classical statistical mechanical system described after the Wick rotation is now sampling from some hidden variable theory. Why doesn’t this immediately work? Well the first problem is that we have transformed the amplitude into a partition function. The probability of going from the state |i> to the state |f> is the magnitude squared. But does this really mess us up? We now have something which looks like int dq exp( S[q] ) int dq’ exp (S'[q’]) for the probability. The S’ comes about because the action is now the action going from plus infinite to minus infinity. But this still looks like a partition function: however now we aren’t sampling over all paths q but instead all paths which start with |i> go to |f> and then return back to |i>. So our hidden variables are not paths from minus infinity to plus infinity, but instead are now spacetime “loops” which go from minus infinity to plus infinity and then back to minus infinity. What does this mean? Now that is an interesting question!