This posting is about the problems with the idea that you can simply formulate quantum mechanical systems by picking a configuration space, an action functional S on paths in this space, and evaluating path integrals of the form
\n$$\\int_{\\text{paths}}e^{iS[\\text{path}]}$$<\/p>\n
Necessity of imaginary time<\/strong><\/p>\n This section has been changed to fix the original mistaken version. <\/em> Trying to do the same thing for Yang-Mills theory, again one gets something ill-defined for real time, with the added disadvantage of no way to actually calculate it. Going to imaginary time and discretizing gives a version of lattice gauge theory, with well-defined integrals for fixed lattice spacing. This is conjectured to have a well-defined limit at the lattice spacing is taken to zero.<\/p>\n Not an integral and not needed for fermions<\/strong><\/p>\n Actual fundamental matter particles are fermions, with an action functional that is quadratic in the fermion fields. For these there’s a “path integral”, but it’s in no sense an actual integral, rather an interesting algebraic gadget. Since the action functional is quadratic, you can explicitly evaluate it and just work with the answer the algebraic gadget gives you. You can formulate this story as an analog of an actual path integral, but it’s unclear what this analogy gets you.<\/p>\n Phase space path integrals don’t make sense in general<\/strong><\/p>\n Another aspect of the fermion action is that it has only one time derivative. For actions of this kind, bosonic or fermionic, the variables are not configuration space variable but phase space variables. For a linear phase space and quadratic action you can figure out what to do, but for non-linear phase spaces or non-quadratic actions, in general it is not clear how to make any sense of the path integral, even in imaginary time.<\/p>\n In general this is a rather complicated story (see some background in the part I post<\/a>). For an interesting recent take on the phase-space path integral, see Witten’s A New Look At The Path Integral Of Quantum Mechanics<\/a>.<\/p>\n Update<\/strong>: A commenter pointed me to this very interesting talk by Neil Turok<\/a>. The main motivation that Turok explains at the beginning of the talk (and also in the Q and A afterwards) is exactly one that I share. He argues that the lesson of the the last 40 years is that one should not try and solve problems by making the Standard Model more complicated. All one needs to do is look more closely at the Standard Model itself and its foundations. If you do that, one thing you find is that there’s a “trouble with path integrals”. In Turok’s words, the problems with the path integral indicate that “the field is without foundations” and “nobody knows what they are doing”.<\/p>\n I do though very much part company with him over the direction he takes to try and get better foundations. He argues that you shouldn’t Wick rotate (analytically continue in time), but should complexify paths, analytically continuing in path space. For some problems doing the latter may be a better idea than doing the former, and in his talk he works out a toy QM calculation of this kind. But the model he studies (anharmonic oscillator) doesn’t at all prove that going to the imaginary time theory is a bad idea, for some calculations that works very well. He’s motivated by defining the path integral for gravity, where Euclidean quantum gravity is a problematic subject, but the gravitational version of the toy model I think will also be problematic. The ideas I’ve been pursuing involving the way the symmetries of spinors behave in Euclidean signature I think give a promising new way to think about this, and you won’t get that from just trying to complexify the conventional variables used to describe geometries.<\/p>\n","protected":false},"excerpt":{"rendered":" This posting is about the problems with the idea that you can simply formulate quantum mechanical systems by picking a configuration space, an action functional S on paths in this space, and evaluating path integrals of the form $$\\int_{\\text{paths}}e^{iS[\\text{path}]}$$ Necessity … Continue reading
\nIf one tries to do this path integral for even the simplest possible quantum field theory case (a non-relativistic free particle in one space dimension), the answer for the propagator in energy-momentum space is
\n$$G(E,p)=\\frac {1}{E-\\frac{p^2}{2m}}$$
\nFourier transforming to real-time is ill-defined (the integration goes through the location of the pole at $E=\\frac{p^2}{2m}$). Taking $t$ complex and in the upper half plane, for imaginary $t$ the Fourier transform is a well-defined integral. One gets the real-time propagator then by analytic continuation as a boundary value. For a relativistic theory one has
\n$$G(E,p)=\\frac{1}{E^2-(p^2+m^2)}$$
\nand two poles (at $E=\\pm \\sqrt{p^2+m^2}$) to deal with. Again Fourier-transforming to real-time is ill-defined, but one can Fourier transform to imaginary time, then use this to get a sensible real-time propagator by analytic continuation.<\/p>\n