Recent content by copernicus1

If I write the basic scalar field as $$\phi(x)=\int\frac{d^3k}{(2\pi)^3}\frac1{\sqrt{2E}}\left(ae^{-ik\cdot x}+a^\dagger e^{ik\cdot x}\right),$$ this would seem to imply that the creation and annihilation operators carry mass dimension -3/2. That's the only way I can get the total field...

What is the evidence for dark matter being new particles as opposed to some modified form of gravity?

Ah, interesting! Thanks for pointing that out.

I start with $$i\hbar\frac{}d{dt}A_H(t)=[A_H(t),H_H(t)]$$ (subscript means Heisenberg picture) and plug in ##A_H(t)=e^{iH_St/\hbar}A_Se^{-iH_St/\hbar}## and ##H_H(t)=e^{iH_St/\hbar}H_Se^{-iH_St/\hbar}##. (I then replace ##H_S=H_{0,S}+V_S## everywhere and transform both sides of the original...

The standard Heisenberg picture equation of motion is $$i\hbar\frac d{dt}A_H=[A_H,H],$$ assuming no explicit ##t##-dependence on the Heisenberg-picture operator ##A_H##. I've been trying to go directly from this equation to the corresponding interaction-picture equation, $$i\hbar\frac...

Ah ok. Thanks for your help. One last question. When you say that you can construct ##p^2## from the available quantities (i.e. from ##\delta_{\mu\nu}## and ##p_\mu##), I see that you can construct ##p^2=p^\mu p^\nu\delta_{\mu\nu}##, but the magnitude of ##p^\mu## is also invariant under...

Wow thanks for your thorough reply. There are one or two things I am still fuzzy on: 1) When you write that we can construct ##p^4## but then the two-point function is a function of ##p^2##, how did you get from ##p^4## to ##p^2##? 2) How do you know (or how did Peskin and Schroeder know)...

I'm studying renormalization and I have a question about part of a textbook. In P&S at the top of p.324 they show the divergent amplitudes of phi^4 theory, and they say that the two-point vertex (which has superficial degree of divergence D=2 according to the formula they derive) will have a...

Thanks all for your input. vanhees71, thanks a lot, this is making more sense now. I've seen PV regularization done with multiple regulator fields but I'm not quite used to it yet. It seems you are doing something slightly different though, with a derivative \Pi'(\mu^2). Is there a source...

I'm trying to work through the one-loop, one-vertex diagram in \phi^4 theory using Pauli-Villars regularization, and I'm having trouble. Specifically, I can't get the momentum dependence to fall out after integrating, which I think it should. In computing the "seagull" diagram (two external...

For internal photon states, is it necessary to sum over the longitudinal polarization state in addition to the transverse states? And if so, does the ordinary Feynman-gauge propagator take care of this? Thanks!

I think you may have misread my question. My question was referring to the dependence of q and q-dot in the Lagrangian formalism versus the independence of q and p in the Hamiltonian. In Lagrangian dynamics, q and q-dot are not independent. I'm wondering about the difference between the two...

In the lagrangian formalism, we treat the position ##q## and the velocity ##\dot q## as dependent variables and talk about configuration space, which is just the space of positions. In the hamiltonian formalism we talk about canonical positions and momenta, and we consider them independent. Is...

Great thanks. Is there a relationship though between the divergence in the single-vertex interaction and the interactions with higher numbers of vertices? Like, if a single-vertex diagram has a quadratic divergence, would a two-vertex diagram have a quartic divergence?

Can you look at an interaction term in your lagrangian or hamiltonian, like L_{\rm int} or H_{\rm int}, and say immediately how its diagrams will diverge (as in quartic, quadratic, linear, log, etc.)?