When people write down a list of axioms for quantum mechanics, they typically neglect to include a crucial one: positivity (or more generally, boundedness below) of the energy. This is equivalent to saying that something very different happens when you Fourier transform with respect to time versus with respect to space. If $\\psi(t,x)$ is a wavefunction depending on time and space, and you Fourier transform with respect to both time and space
\n$$\\widetilde{\\psi}(E,p)=\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\psi(t,x)e^{iEt}e^{-ipx}dtdx$$
\n(the difference in sign for $E$ and $p$ is just a convention) a basic axiom of the theory is that, while $\\widetilde{\\psi}(E,p)$ can be non-zero for all values of $p$, it must be zero for negative values of $E$.<\/p>\n
This fundamental asymmetry in the theory also becomes very apparent if you want to “Wick rotate” the theory. This involves formulating the theory for complex time and exploiting holomorphicity in the time variable. One way to do this is to inverse Fourier transform $\\widetilde{\\psi}(E,p)$ in $E$, using a complex variable $z=t+i\\tau$:
\n$$\\widehat{\\psi}(z,p)=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty \\widetilde{\\psi}(E,p)e^{-iEz} dE$$
\nThe exponential term in the integral will be
\n$$e^{-iE(t+i\\tau)}=e^{-iEt}e^{E\\tau}$$
\nwhich (since $E$ is non-negative) will only have good behavior for $\\tau <0$, i.e. in the lower-half $z$-plane. Thinking of Wick rotation as involving analytic continuation of wave-functions from $z=t$ to $z=t+i\\tau$, this will only work for $\\tau <0$: there is a fundamental asymmetry in the theory for (imaginary) time.<\/p>\n
If you decide to define a quantum theory starting with imaginary time and Wick rotating (analytically continuing) back to real, physical time at the end of a calculation, then you need to build in $\\tau$ asymmetry from the beginning. One way this shows up in any formalism for doing this is in the necessity of introducing a $\\tau$-reflection operation into the definition of physical states, with the Osterwalder-Schrader positivity condition then needed in order to ensure unitarity of the theory.<\/p>\n
Why does one want to formulate the theory in imaginary time anyway? A standard answer to this question is that path integrals don’t actually make any sense in real time, but in imaginary time often become perfectly well-defined objects that can be thought of as expectation values in a statistical mechanical system. For a somewhat different answer, note that even for the simplest free particle theory, when you start calculating things like propagators you immediately run into integrals that involve integrating a function with a pole, for instance integrating over $E$ integrals with a term
\n$$\\frac{1}{E-\\frac{p^2}{2m}}$$
\nEvery quantum mechanics and quantum field theory textbook has a discussion of what to do to make sense of such calculations, by defining the integral involved as a specific limit. The imaginary time formalism has the advantage of being based on integrals that are well-defined, with the ambiguities showing up only when one analytically continues to real time. Whether or not you use imaginary time methods, the real time objects getting computed are inherently not functions, but boundary values of holomorphic functions, defined of necessity as limits as one approaches the real axis.<\/p>\n
A mathematical formalism for handling such objects is the theory of hyperfunctions. I’ve started writing up some notes about this, see here<\/a>. As I find time, these should get significantly expanded. <\/p>\n One reason I’ve been interested in this is that I’ve never found a convincing explanation of how to deal with Euclidean spinor fields. Stay tuned, soon I’ll write something here about some ideas that come from thinking about that problem.<\/p>\n","protected":false},"excerpt":{"rendered":" When people write down a list of axioms for quantum mechanics, they typically neglect to include a crucial one: positivity (or more generally, boundedness below) of the energy. This is equivalent to saying that something very different happens when you … Continue reading