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 \int dq exp(i S(q)) where this integration is performed over all paths q and S(q) is the action (\int_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!
Who's Your Papa
Google for “pontiff”!
Big Tesuque Run
Saturday I did the 12 mile 10,000-12,000-10,000ft Big Tesuque Run. My time was 1:45:30. I was 25th out of 110 and came in 5th in my age group. My time was almost 15 minutes faster than when I ran the course the previous week! Yeah for addrenaline.
A Philosophical Argument
Bell’s theorem tells us that there is no local hidden variable theory which reproduces the statistics of quantum theory. Fine. One way to think about moving onward given Bell’s theorem is to given to look for nonlocal hidden variable theories which reproduce quantum theory. But now there is something strange that happens. If you have nonlocal hidden variables, i.e. quantities describing the state of the universe but which are jointly accessible by two spacelike seperated observers, what is the difference between this and assuming that your notion of spacelike separation is not correct. Suppose you come up with a nonlocal theory. What prevents anyone from reinterpreting your nonlocal theory as a totally local theory in which spacelike separation is defined different? Well there is, as far as I can tell, exactly one difference: in quantum theory we cannot use entangled particles to communicate between spacelike separated observors. But this difference doesn’t disallow interpretting a nonlocal hidden variable theory as simply spacelike separation being defined differently, it just tells us the spacelike sepearation of entangled particles must force a nonsignaling constraint (and reproduce quantum theory!) So why don’t we spend more time thinking about where the structure of our spacetime manifold comes from?
Quantum Gravity?
Patrick Hayden points me to hep-th/0410036. If I understand this paper correctly and the paper is correct, this seems to me to be a BIG deal. In this paper, the author take the Hilbert-Palatini action for GR and adds two terms, both of a topological nature. These terms don’t change the fact that the classical theory derived from this action is classical GR. However, the author shows that these terms make this action the same as the action for an so(4,1) [or so(3,2) depending on whether the cosmological constant is positive or negative] Donaldson-Witten topological quantum field theory. The Donaldson-Witten TQFT is an “exactly solvable” quantum theory. What does this mean? Can anyone say a theory of quantum gravity? I knew you could. As the author puts it, “this proves that exact, non-perturbative calculations can be preformed in 3+1 dimensional quantum gravity.”
[Update 10/13/04: Well that was the quick fall. As Nathan Lundblad notes, the paper has been withdrawn!]
I'm the Gingerbread Man
This coming weekend, I’ve signed up for
I ran the course last weekend and finished in just under two hours. It’s a very spectacular run this time of year because the aspens have all colored up. Just as long as it doesn’t snow on me it should be fun!
Nobel Closed Timelike Curve
I will have you note, that one of today’s Nobel prize winners in physics, David Politzer, has written articles on closed timelike curves (a.k.a. time travel). Does my paper on closed timelike curves look crazy now? Huh? Oh yes, it does.
Hot Enough To Melt…Err I Mean Unmelt…
From an IOP news article:
Law-breaking liquid defies the rules
Monday 27 September 2004
Physicists in France have discovered a liquid that “freezes” when it is heated. Marie Plazanet and colleagues at the Université Joseph Fourier and the Institut Laue-Langevin, both in Grenoble, found that a simple solution composed of two organic compounds becomes a solid when it is heated to temperatures between 45 and 75°C, and becomes a liquid when cooled again. The team says that hydrogen bonds are responsible for this novel behaviour (M Plazanet et al. 2004 J. Chem. Phys 121 5031).
It's Four!
When physicists say “3+1″they are not talking about the number “4”, but usually they are talking about spacetime: three spatial dimensions and one temporal dimension. One question which plagues theoretical physicists’ poor little minds is the question of why we see a macroscopic world of 3+1 dimensions. Mostly this is because physicists believe that at small enough length or time scales (large enough energies) the geometry of spacetime itself can exist in nontrivial states of connectivity. Thus we think of spacetimes at small enough scale as existing in all sorts of strange configurations (in some poor little region the spacetime may look like a 10 dimensional manifold, for instance.) “Spacetime foam” is what we call this strange state of affairs. How do we get from this spacetime foam up to where our experiments live and we seem to see a four dimensional universe?
Concerning this problem, I just today read the paper “Emergence of a 4D World from Causal Quantum Gravity,” by J. Ambjorn, J. Jurkiewicz, and R. Loll which was published in Physical Review Letters, (Volume 93, page 131301, 2004.) This paper attempts the following. Construct spacetime by glueing together a bunch of little four dimensional simplical spacetimes. Like I said earlier, if we glue a bunch of these four dimensional simplical spacetimes together, we get something which is not necessarily four dimensional. Now when we do this glueing we should insist on maintain causality (i.e. no closed time like curves and such.) So we can construct these crazy spacetimes, but what do they mean. Well now we associate with each of these spacetimes an amplitude. So there is some notion of an action S for the given simpical spacetime we have created and we assign to this an amplitude, Exp[iS]. Now what one would love to do is to sample over all of these crazy spacetimes and hence calculate the propogators for different such spacetimes. But this is hard. This is hard because of the fact that we have to sample over this crazy oscillating Exp[iS]. But sometimes it is not so hard. Sometimes it is possible to perform a “Wick” rotation and change Exp[iS] into Exp[-S]. This means the problem of calculating the total amplitude looks like adding up a bunch of different spacetimes with weights Exp[-S]: this looks just like classical statistical mechanics! What the authors of the above paper do is they insist that it is possible to perform such a rotation. They then perform Monte Carlo simulations of the resulting statistical mechanical system. And what do they find? They argue that what they find is that the resulting spacetime is indeed dominated by a spacetime of dimension “3+1!”
So starting out from something which had only a totally local sense of dimension (the original building blocks are “3+1”) you glue them together in pseudo-arbitrary (preserve causality, able to Wick rotate) ways (this is what is called “background independence”) and yet, you find, at the end of the day, that you have effectively a global “3+1” spacetime! Amazing, no?
Was It Just Me?
Have I been asleep while the following argument has sprung up among Republicans: the war on Iraq is right because the death and destruction is occuring in Iraq and not in America. Bush used this argument at least two times and after the debate I saw Giuliani make exactly the same argument.
I must have been asleep.
On another note, I was able to mimic Bush almost word for word on a few questions before he answered the questions. It’s like he’s become a song. Or maybe just a broken record.