# What is the correspondance between imaginary time and heat?

I apologize for my crude line of questioning, as I am not well versed in physics but there are concepts that interest me. I'm trying to understand the concept of imaginary-time, and I've read in multiple articles of questionable reliability that there is some special relationship between imaginary-time and heat? Or Entropy I believe? Any help is appreciated, but please try to make it as extremely dumbed down as possible, thank you.

• Wikipedia has a brief discussion of the relationship between imaginary time and temperature. I’m not sure that it can be dumbed down, as it is typically a graduate-physics-level idea. – G. Smith Jul 10 '20 at 0:46
• Well I suppose what I mean is, what does it imply? What kind of events can we imagine as a result of this relationship between imaginary-time and statistical mechanics? – Samuel Curry Jul 10 '20 at 1:26
• No events that I know of. The relationship is a formal mathematical one. Time is not actually imaginary. – G. Smith Jul 10 '20 at 1:46
• Well so is the relationship between them that one of them can be described in terms of the other? I'm still unclear after reading the wikipedia article, I'm still trying to grasp what a formal mathematical relationship really means. Again I'm not versed what so ever in this kind of stuff so please forgive my ignorance. – Samuel Curry Jul 10 '20 at 2:08

Consider the Green’s function for a system at a nonzero temperature. It depends on time. One can look at its analytic continuation to imaginary time, even though time isn’t actually imaginary. In imaginary time, the function turns out to be periodic with a period inversely proportional to the temperature.

This analytic behavior of finite-temperature Green’s functions in the complex time plane is merely a formal mathematical relationship between imaginary time and temperature. It doesn’t mean that imaginary time is hot, or heat makes time be imaginary, or anything like that.

This Wikipedia article has some other formal connections between imaginary time and statistical mechanics. They all arise from the formal relationship between the time-evolution operator $$e^{-iHt}$$ (where $$H$$ is the Hamiltonian) and the similar real-instead-of-imaginary exponential $$e^{-H/kT}$$ in the partition function. The replacement $$it\to 1/kT$$ turns the former into the latter.

As far as I know, most physicists today consider this an interesting and beautiful mathematical connection (i.e., both involve an exponentiated Hamiltonian), but one without obvious physical significance, given the fact that we experience real time, not complex or imaginary time.

• "It doesn’t mean that imaginary time is hot, or heat makes time be imaginary, or anything like that." That's basically what I was trying to figure out, lol. I think I understand a bit better now, thank you very much for your answer!! – Samuel Curry Jul 10 '20 at 6:12

Ultimately, the idea boils down to the fact that in quantum mechanics, you evolve states forward in time by applying the time evolution operator

$$U(t) =\exp\left[-it\hat H\right]$$

where $$\hat H$$ is the Hamiltonian (energy) operator. At the same time, the state of a quantum system in contact with a heat bath at temperature $$T$$ is encoded in the thermal density operator

$$\rho = \exp\left[-\frac{1}{T} \hat H\right]$$

Note that $$t$$ and $$T$$ are simply ordinary numbers. It follows that if I can manage to calculate $$U(t)=\big(\text{some formula}\big)$$, then plugging in $$t = -i/T$$ will immediately give me $$\rho$$.

This correspondence is purely formal, in the sense that it doesn't really make any statements about the nature of either time or temperature; however, if you pull on this thread a bit harder, you find that certain problems in $$D$$-dimensional quantum field theory can be re-expressed as quantum statistical mechanics problems in $$D-1$$ dimensions.

In a sense, this provides a deep(?) connection between statistical physics and quantum field theory - not in the sense one that should interpret one in terms of the other, but rather in that the theoretical and computational machinery used in the two quite different subjects are related very intimately.

• Thank you for your answer! I am very math illiterate however, so please forgive me, but what does D denote in D-dimensions? Does it just mean any number like n? And is the − 1 in D − 1 referring to -1 where the square root is the imaginary number i? – Samuel Curry Jul 10 '20 at 6:21
• @SamuelCurry Yes, $D$ is a positive integer like 1,2,3,4. $D-1$ is one less than $D$. – J. Murray Jul 10 '20 at 6:22
• @SamuelCurry I should clarify - $D$ refers to the number of dimensions the problem is cast in. – J. Murray Jul 10 '20 at 6:25
• @SamuelCurry No, it means that e.g. a quantum field theory problem in 3 dimensions can be reduced to a quantum statistical mechanics problem in 2 dimensions. – J. Murray Jul 10 '20 at 6:27
• @SamuelCurry No reason to apologize, it's an interesting question about an interesting subject! – J. Murray Jul 10 '20 at 6:30