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!
O brave new quantum world!
This confused me also, when I saw it the first time…
I am not sure that substititutin changing the argument of exponentials from imaginary to real is really switching to the classical.
One does “similar” [but not exactly] contour rotations elsewhere in math. physics and basically where a Green function is under consideration, but wait a minute, now I remember that I had asked my QFT teacher long time ago, if the Wick rotation was the same as the contour rotation for Green functions, and he said that there is a profound difference, but I couldn’t firgure out what he said afterwards 🙁
I discuss the Wick rotation in detail here:
http://motls.blogspot.com/2005/02/wick-rotation.html
I am not a Quantum theorist, nor a string theorist, but you may find interest in
http://www22.pair.com/csdc/pdf/spie2ext.pdf.
in which experimental evidence is given for the existence of a 3 space topological defect that corresponds to a zero mean curvature surface in a 3D geometrical space of signature plus,plus,minus.
The defect occures in a fluid, and does not depend upon scales. The defect is quantized in that its closed integrals are rational.
I wonder if relative to x,y,z,t, the Minkowski plus, plus, plus minus signature can be “rotated”
to the same signature but of the type plus, plus,minus, plus.
In open flat Mintkowski Space one can substitute -ict for the time dimension.Oscillating or harmonic functions with a variable of N oscillations which have an infinite product of a smaooth harmonic function which alternate between + and -can be made to converge rather than diverge such as F(x)=sin x times N.
Quote:
A Spring in Imaginary Time
Math 241 Homework
John Bxxxx
One of the stranger aspects of Lagrangian dynamics is how it turns into statics when we replace the time coordinate t by it — or in the jargon of physicists, when we ‘Wick rotate’ to ‘imaginary time’ !
People usually take advantage of this to do interesting things in the context of quantum mechanics, but the basic ideas are already visible in classical mechanics. So, let’s look at them!
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Einstein had established that “TIME” IS a dimension, interchangeable in some fashion with the 3 “metric-like” dimensions.
However, this two different types of “dimension”, the foundation of physic, form an awkward relationship.
Up to now, most if not all question this “TIME-like” axis, “Metric axis” are taken for granted .
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The 3D 1T is just a model.
What is Metric-Like stand for ? Is it making any physics-sense ?
Ds’ … “meter-like” are historical believes based on “rigid ruler” … which doesn’t exist.
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21st century, “time” is practically used for all measurement.
1) Every physical ruler, in it basic form, is a stick of clocks.
2) A “Meter” IS, ~3.3x 10^-9 sec.
3) Triangulation by using TIME and Time-Angles for position by GPS.
4) Astrophysicist using, understanding, subconsciously maybe …. “distance” “out” is moving INTO another time position.
5) Even a speeding ticket using laser, is a quantity of time divide by time … velocity … in fact should be a ratio !!
Think about it, your “front” and “side” is making 90 degree of TWO time axis…
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All Dimensions …. are TIME-LIKE dimensions…..
In other word, today’s physics and it’s mathematic , are an