maverick_starstrider said:
Yes thank you, I see. So a quantum partition function or statistical field theory partition function includes both thermal and quantum fluctuations provided the Hamiltonian is made of non-commuting operators (i.e. quantum).
Is it then correct to say the thermal partition function of a, say, classical kinetic theory can be described as saddle point + thermal fluctuations and that a quantum path intregral can be described as saddle point + quantum fluctuations but that a quantum partition function integrated from 0 to beta necessarily contains both types of fluctuations.
This sounds right to me.
Furthermore, if one considers high-temperatures (beta goes to zero) and the requirement that all paths start and end at the same point then that terms in the Lagrangian that related to DERIVATIVES in time of the field are quantum fluctuations which are heavily penalized in this high-temperature system leaving only a path integral over the POTENTIAL term with the path integral essentially being the thermal fluctuations?
Yeah, this is right. There's a nice way to think about it if you're familiar with thermal Feynman diagrams: When you develop the perturbation theory of your finite temperature path integral, you develop Feynman diagrams in direct relation to standard QFT, except your periodic imaginary time boundary conditions forces you to develop a Fourier series in time instead of a Fourier transform. Your Fourier frequencies must take discrete values, \omega_n = 2\pi n/\beta, where n is an integer. Now, when T \rightarrow \infty, \beta \rightarrow 0, only the n = 0 frequency will contribute. This is basically what you're saying, but in frequency space.
In essence can we not separate the two types of fluctuations in that quantum fluctuations are "time" fluctuations and thermal fluctuations are in the path integral sum-over-histories at constant time?
EDIT: After writing up this whole post, I realized that I don't really need the next two paragraphs. atyy's point that a finite temperature density matrix is equivalent to a zero temperature reduced density matrix (actually, is this always true?) gets rid of any need to establish a quantum-classical correspondence, which may be ruined by a badly defined Wick rotation. I'll keep them here in case it's new/interesting info to anyone.
Maybe you could elaborate? The problem is that we can find equivalent quantum statistical and thermal statistical ensembles with identical path integrals. This can be seen by reinterpreting the imaginary time path integral for a quantum stat mech system in d spatial dimensions as a classical partition function in d+1 spatial dimensions where the last spatial dimension is periodic with length \beta. If you now take the extreme quantum case, \beta \rightarrow \infty, you find that a quantum statistical system in d spatial dimensions at zero temperature is equivalent to some infinite classical system at finite temperature in d+1 spatial dimensions. (There are some caveats here - for some quantum actions with topological/Berry phase terms the corresponding classical Hamiltonian contains imaginary terms and can't really be properly interpreted).
For example: the zero temperature quantum Ising model in one (spatial) dimension in a transverse field is completely equivalent to Onsager's famous 2D classical Ising model at finite temperature (with some correspondence between the 1D Ising coupling/transverse field and the 2D Ising coupling/temperature). This is discussed in both Sachdev and Fradkin's textbooks: the quantum phase transition in the former case even has identical critical exponents as the classical finite temperature transition in the latter case. In one system we can only have quantum fluctuations, in the other system we can only have thermal fluctuations, but the same universal phase transition.
Ok, now having said all that, you might ask why we care about quantum phase transitions at all, since they're (usually) just equivalent to some classical phase transition, maybe with some funky periodic dimension. This has to do with the major technical problem in the imaginary time path integral: you need to eventually analytically continue your results to real time if you want dynamic correlation functions or really any real-time transport properties. These are dominated by some new time scale related to the phase coherence time, or equivalently, some energy scale (\tau \sim \hbar/E). This relates to what I mentioned in the last sentence of my last post. The "fan structure" that you mentioned earlier (which I suspect you're hinting at in your questions) is determined by different regimes, where either some relevant quantum energy scale/energy gap or the temperature is the scale determining the phase coherence time and the transport properties. The fan occurs because the quantum energy scale must go to zero at zero temperature for a second order phase transition, so you can define different regimes depending on whether temperature dominates over the energy scale. So really, you can talk about the relative thermal and quantum fluctuation effects on your model, but this heavily depends on the specific model you're considering.
atyy is totally correct that there exist reduced density matrices equivalent to thermal density matrices, so I suspect you could come up with some equivalent model to the above which could generate the "temperature" fluctuations via entanglement at zero temperature. I just wanted to explain the "fan" since you mentioned it earlier, since its physical realizations do involve competing quantum and thermal fluctuations.
