Recent content by tomdodd4598

I realize now that my EOMs were only valid for when the potential was a function of only ##x_\mu##, so that was another mistake on my part... regardless, I think things make sense now :)

Thanks for the help :)

I think I realize where I've been unclear: when I wrote "##V##", I didn't necessarily mean something that was just a function of position, i.e. it could be something like your scalar field term. However, I don't see why we're necessarily restricted to such a scalar field term, or indeed the...

I understand that this is a legitimate Lagrangian, but I don't understand why we've introduced a scalar field, or indeed a Lagrangian that can't describe massless particles if, for example, ##\Phi(x)=0##. The reason I chose ##L = \dot{x}_\mu \dot{x}^\mu / \left(2e\right) - em^2 / 2 - V## is that...

I was confused by what you wrote/shared, because as you say, those Lagrangians don't really work for a massless particle, but I'm trying to use one that does :P Regardless, I think understand what your point is now: my choice of ##V=z## is not Lorentz invariant. It seems to me, then, that the...

Could you explain what you mean by "invariant" in this context? I'm not sure I follow. In the massive case, I can use the Lagrangian $$L = \frac{\dot{x}_\mu \dot{x}^\mu}{2e} - \frac{em^2}{2} - V,$$ and the EOM for ##e## is $$m^2 e^2 = - \dot{x}_\mu \dot{x}^\mu,$$ which when substituted into the...

Unfortunately I can't edit the OP now, so to clarify, the material in the spoiler was part of the original post which Orodruin pointed out was erroneous. The example of an inconsistency (using the ##V=z## potential) is the new content which I think explains my original doubt a bit more clearly.

Ah, good spot! I have updated the original post to more clearly reflect my concern about inconsistency in light of this, but it does make me wonder... is this equation telling me that there is a constraint on what the potential can be, or at least what the initial conditions can be?

The Lagrangian for a massless particle in a potential, using the ##(-,+,+,+)## metric signature, is $$L = \frac{\dot{x}_\mu \dot{x}^\mu}{2e} - V,$$ where ##\dot{x}^\mu := \frac{dx^\mu}{d\lambda}## is the velocity, ##\lambda## is some worldline parameter, ##e## is the auxiliary einbein and...

This thread might be evidence that time loops into the past are indeed possible.

I'm familiar with the self-energy calculation yielding the ##\beta##-function, and the argument for why that happens (Peskin & Schroeder), but what I'm looking for is getting the ##\beta##-function from the vertex, as we would with something such as ##\phi^4## theory where we don't have the...

Sorry if bumps are not a respectable move, but I still feel a little bit lost on these issues.

It's been a fair few months now, and I am still struggling to find an example of how this calculation of the QED vertex can reveal the positive beta function of QED. Does anyone know what I'm doing wrong? I'm assuming P&S's result is correct, and I have reproduced it.

In that case, I must be making some sort of mistake. I have successfully found the Dirac form factor as given on page 196 of Peskin and Schroeder: ##m## is the electron mass and ##\mu## is a fictitious photon mass. However, focussing on the logarithmic piece, it is easy to see that the...

Hey there, I am a little confused about the way most textbooks and notes I've read find the beta function for QED. They find it by looking at how the photon propagator varies with momentum ##q##, in particular in the context of a ##2\rightarrow2## scattering process which is proportional to...