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.
\nSo 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!<\/p>\n","protected":false},"excerpt":{"rendered":"
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 … <\/p>\n