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Field Theory and Statistical Mechanics are closely connected. A Wick rotation t ➝ i /(kT) will take you from one to the other.
This procedure (Wick rotation), which relates classical statistical mechanics and quantum field theory, is, however, somewhat subtle
In flat space-time, the situation is well-understood: if your Hamiltonian has good positivity properties you can analytically continue to imaginary values of time, and when you do this you end up with “Euclidean” path integrals, which actually make sense, unlike QFT path integrals expressed on Minkowski space, which don’t. You can see the problem even in free field theory: the propagator is given by an integral that goes through two poles, so is ill-defined. The correct way to define it to get causal propagation for a theory with positive energies is to go above one pole, below the other, which is equivalent to “Wick rotating” the integration contour 90 degrees to lie on the imaginary time axis.
This is widely used to convert quantum mechanics problems into statistical mechanics problems by means of Wick rotation, which essentially means studying the unitary group exp(−itH/~) by studying the semigroup exp(−βH) and then analytically continuing β to imaginary values.
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.
Unfortunately, relatively little is known about Yang-Mills fields on Minkowski spacetime and, worse yet, the basic objects of interest in quantum field theory (Feynman path integrals) are extraordinarily difficult to make any sense of in this indefinite context. The minus sign in the Minkowski inner product is rather troublesome. Not to be deterred by such a minor inconvenience, the physicists do the only reasonable thing under the circumstances—they change the sign! To lend an air of respectability to this subterfuge, however, they give it a name. Introducing an imaginary time coordinate τ = it is designated a Wick ro- tation and has the laudable effect of transforming Minkowski spacetime into R 4 (x 1 x 2 + y 1 y 2 + z 1 z 2 − t 1 t 2 = x 1 x 2 + y 1 y 2 + z 1 z 2 + τ 1 τ 2 ). What more could you ask? Well, of course, a pedant might ask whether or not any physics survives this transformation. This is a delicate issue and not one that we are prepared to address. The answer would seem to be in the affirmative, but the reader will have to consult the physics literature to learn why (see Section 13.7 of [Guid]). Whether or not there is any physics in this positive definite context is quite beside the point for mathematics, of course. It is only in the positive definite case that (anti-) self-dual connections exist and it is an understanding of the moduli space of these that pays such handsome topological dividends.page 377 in Topology, Geometry and Gauge fields by Naber
[Guid] is Guidry, M., Gauge Field Theories, John Wiley & Sons, Inc., New York, 1991
Another peculiarity of chiral theories arises when one tries to understand how they behave under Wick rotation. Non-perturbative QFT calculations are well-defined not in Minkowski space, but in Euclidean space, with physical observables recovered by analytic continuation. But the behavior of spinors in Minkowski and Euclidean space is quite different, leading to a very confusing situation. Despite several attempts over the years to sort this out for myself, I remain confused, and can’t help suspecting that there is more to this than a purely technical problem. One natural mathematical setting for trying to think about this is the twistor formalism, where complexified, compactified Minkowski space is the Grassmanian of complex 2-planes in complex 4-space. The problem though is that thinking this way requires taking as basic variables holomorphic quantities, and how this fits into the standard QFT formalism is unclear. Perhaps the current vogue for twistor methods to study gauge-theory amplitudes will shed some light on this. On the general problem of Wick rotation, about the deepest thinking that I’ve seen has been that of Graeme Segal, who deals with the issue in the 2d context in his famous manuscript “The Definition of Conformal Field Theory”.
I’ve always thought this whole confusion is an important clue that there is something about the relation of QFT and geometry that we don’t understand. Things are even more confusing than just worrying about Minkowski vs. Euclidean metrics. To define spinors, we need not just a metric, but a spin connection. In Minkowski space this is a connection on a Spin(3,1)=SL(2,C) bundle, in Euclidean space on a Spin(4)=SU(2)xSU(2) bundle, and these are quite different things, with associated spinor fields with quite different properties. So the whole “Wick Rotation” question is very confusing even in flat space-time when one is dealing with spinors.