# On equivalence of QFT and Quantum Statistical Physics

1. May 17, 2009

### tomkeus

Does fact that QFT in imaginary time is equivalent to QSP represents the proof that many-particle quantum physics is equivalent to quantum theory of fields?

To elaborate a little, I had some discussion with some engineers, and when I was explaining them Standard Model I had to invoke concepts of quantum fields and they immediately turned their noses in despise to "overly abstract" concept.

Since they didn't have problems with quantum particles and statistical physics I've thought of taking the route starting from many-particle quantum physics but I'm not sure that I can do that because I'm not certain how to treat equivalence of imaginary time with temperature. I mean, parameter t in QFT enters from space-time structure but parameter $$\beta$$ is inserted in partition function only to be shown after calculation, what is it's connection with kinetic energy.

In Minkowski space time-component of energy-momentum is energy but I cannot find any formal transformation which would transform it into median kinetic energy at corresponding temperature.

2. May 17, 2009

### malawi_glenn

Why cast pearls before the swine ? ;-)

Why is the quantum field more abstract than the classical field of let's say electromagnetism?

3. May 17, 2009

### tomkeus

Well, beats me, but I kinda came to think of it as an interesting question. One of the first thing you are being taught about QFT is that you cannot do right single-particle relativistic QM because necessarily pair creation-annihilation comes into play. Now, what if we try doing quantum statistical mechanics of relativistic particles? Would we get QFT as a result of formal correspondence of temperature and imaginary time?

4. May 17, 2009

### malawi_glenn

But the fields in Statistical Field Theory is i) not particles and ii) one often use non-relativistic field theory.

5. May 18, 2009

### RedX

I just wanted to point out that there is another way to derive thermal quantum field theory from quantum field theory other than the imaginary-time formalism. There is also the real-time formalism which is more intuitive - an operator based approach rather than path-integral. Of course path-integrals make everything easier, but sometimes you lose the physics if your view is to compare imaginary time in the path integral to inverse temperature and the partition function.