I thought about that, but the argument of the cosh becomes a sum, which really doesn't simplify things at all. Thanks though. I've thought about using Isserlis's theorem, (also known as Wick's theorem) but I can't find a good summary/statement of the theorem that's more precise than Wikipedia's.
I'm dealing with multivariate normal distributions, and I've run up against an integral I really don't know how to do.
Given a random vector \vec x, and a covariance matrix \Sigma, how would you go about evaluating an expectation value of the form
G=\int d^{n} x \left(\prod_{i=1}^{n}...
Now though, there is a pretty clear distinction from philosophy and science as far as I can tell. That's what I'm curious about. Questions about nature are delegated to scientists, although philosophers still consider them as well, but from different perspectives that don't seem to lead to a...
Also, another quick question. Why are philosophical 'theories' rarely required to meet any standards of evidence whatsoever? It seems that if they're talking about our world and how it interacts and such, it should be falsifiable, even if in a modest sense.
I get the feeling that much of the time, philosophers discuss things that make claims about our reality and how it works. I'm of the opinion that claims made about the natural world that can't be tested are a bit irrelevant, so I typically ignore them. It seems though, throughout the history of...
It seems like it's easier to just write Z as an integral over 6N dimensional phase space, do the integrals over p and q. All that's left is a function of z1 and z2, then try using the thermodynamic derivatives to give you the pressure.
Thank you, I didn't think too much about what the Hamiltonian really is. Including the interaction terms gives you a non-trivial calculation, and I think I overlooked the addition of that to the total Hamiltonian.
What's a mass sheet? (Or did you mean mass shell?) Also, I read elsewhere that determining the spectrum corresponds to finding the spectrum of m^{2} or something, but I don't quite understand what that means or why it's important.
The 'partition function' in QFT is written as Z=\langle 0 | e^{-i\hat H T} |0\rangle, but I'm having a difficult time really understanding this. I'm assuming that |0\rangle represents the vacuum state with no particles present. If that's the case, and the Hamiltonian acting on such a state would...
I know in ordinary QM, the spectrum of the Hamiltonian \{ E_{n}\} gives you just about everything you need for the system in question (roughly speaking). So what happens to this spectrum in QFT where |\psi\rangle is now a multiparticle wavefunction in some Fock space? I've been trying to...
I could be wrong, but I think (2) as you've written it doesn't describe a charged scalar field. I think you may have meant |\partial \phi|^{2}=(\partial^{\mu}\phi^{*})(\partial_{\mu}\phi). Anyways, (1) and (3) are equivalent as long as you remember that these quantities appear in an integral, so...
In ordinary QFT, everything is formulated in terms of a Fock basis so when we write |\psi\rangle we mean that this is a product of single particle states covering every momentum mode. This leads to a Hamiltonian that's typically of the form \hat H=\int \frac{d^{3}k}{(2\pi)^{3}} [\omega_{k}(\hat...
I feel a bit silly asking this, but I've been working through some QM lately and there's one aspect of Schrodinger's equation that's puzzling me. I've typically understood the equation as i\hbar \frac{d|\psi\rangle}{dt}=\hat H |\psi\rangle, but I've also seen it written as i\hbar \frac{\partial...