I read several articles saying that most mathematicians have the peak of their career before 30 and after that they don't do much significant work. Although this is a simplification of the reality, the truth is that in math many people do major breakthroughs before 30, a lot more than in other...
Homework Statement
Calculate the Casimir force in 1D using a Gaussian regulator.
Homework Equations
The Attempt at a Solution
I reached a point where I need to evaluate a sum of the form $$\sum_n n e^{-\epsilon^2n^2}$$ Can someone help me? I didn't really find anything useful online. I...
Hello! In Schwarz's QFT he introduces in chapter 27 the Spin-Helicity formalism as a way of calculating gluon-gluon interactions much easier than going through all the Feynman calculus from the beginning to the end. It seems so amazing, but I am not sure I understand what is the fundamental...
But I am not sure how does he get the first and last terms. He gives a formula for ##\Phi_i##, but ##|0>## is not an eigenstate of ##\hat{\Phi}(x,t)##. What confuses me the most is why don't we have boundary terms for the integral. In QM you have boundaries the for the beginning and end points...
I am not sure about anything right now... But as I said in the post I can't see where he is using the fact that the initial and final sates are the vacuum so technically you would get the same answer no matter what your initial and final states are, which doesn't really make sense to me.
Hello! I am reading from Schwarz book on QFT the Path Integral chapter and I am confused about something. I attached a SS of that part. So we have $$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N exp(i\delta t \int d^3x L[\Phi_j,\partial_t \Phi_j])$$ What happens when we have the left and right...
So, if helicity is not a good quantum number for massive particles and in the massless case helicity is the same as chirality, why do we need helicity in the first place. Why don't we just use chirality?
Yes, that is the book I am using. So in 5.3 as far as I can tell is mainly using helicity...
Hi! I am kinda confused about what gets conserved in QED and what not. So the chirality is always conserved, I got that. So in the massless limit, helicity is too. Now in the massive limit. Are spin and helicity conserved? And if they are, are they at each interaction vertex, or just overall...
Hello! We derived the electron proton scattering differential cross section using QED and I noticed that the equation doesn't depend on the impact parameter. Using classical EM one can calculate the deflection of an incoming electron as a function of the impact parameter, so I was wondering how...
Hello! I am reading some QFT and it is a part about how causality implies spin-statistic theorem. In general, one needs 2 observables to commute outside the light-cone. For scalars, we have $$[\phi(x),\phi(y)]=0$$ outside the light-cone, and by using the operator form of the field you get that...
I tried, but I am confused at certain steps. I would normally post here what I did so far and ask for help, but I am sure they are derived in some book already (also typing down everything would be quite time consuming as there are many indices).
Thank you! That is the book I am using actually and I am at that chapter, but in that book the rules are just listed, I would like a full derivation, the same way it is done in previous chapters for the ##\phi^3## interaction.
Hello! I am reading some QFT and at a point I read that any vector field (here we are working with massive spin 1 particles) can be written as: $$A_\mu(x)=A^T_\mu(x)+\partial_\mu\pi(x)$$ with $$\partial_\mu A^T_\mu(x)=0$$ They don't talk about notation, but from the context I understand that...
Thank you for this! I am sorry I honestly didn't have time to go further yet. However, the creation/annihilation operators, don't they evolve with the full interaction hamiltonian? Shouldn't we have some terms of the form ##e^{iHt}## in between them, to evolve them between the 2 times? Why do we...
Hello! I attached a SS of the part of my book that I am confused about. So there they write the initial and final states in term of creation and annihilation operators, acting on the (not free) vacuum i.e. ##|\Omega>##. So first thing, the value of the creation (annihilation) operators at...
So from the point of view of the S matrix, an interacting particle with momentum p and a non-interacting particle with momentum p are identical at ##\pm \infty##? But the operators used to create a particle with momentum p i.e. ##a_p^\dagger## is different (they evolve differently in time) in...
Hello! I am reading about the S matrix, and I see that one of the assumption that the derivations are based on is the fact that interacting particles are free at ##t=\pm \infty## and I am not sure I understand why. One of the given examples is the ##\phi^4## theory which contains an interaction...
Thank you for your reply. However, what do you mean by tangible? I mean most of the quantities encountered in particle physics are not tangible. Like isospin, or color of quarks, even spin itself, which is not an actual arrow pointing in space, but an inner property of the particle, just like...
So I heard on different occasions that chirality it's a very confusing concept and it is often mixed with helicity. I read some definitions and examples from a book and as far as I can tell (at least for QED), helicity it's an operator that gives the component of the spin along the direction of...
Hello! I found this problem where we are asked what happens to the energy of the outgoing photon in a Compton interaction, if the mass of the electron goes to zero and what is the physical intuition of it. So the formula is this: $$\lambda - \lambda_0 = \frac{h}{m_0 c}(1-cos \theta)$$ So when...
Thank you so much for this. So I did the calculations and I got about ##7*10^{-4}## eV. This is close to what I found in another place online (using some other method), but do you know what's the actual value (just to make sure I did it right)? One more thing, despite getting a value similar to...
I remember in my Stat mech class we defined $$\lambda=e^{\mu \beta}$$ where ##\mu## is the chemical potential. Then, to get the average number of particles, you do $$N=\lambda \frac{\partial log(Z)}{\partial \lambda}$$ which in the case of the photon must be taken at ##\lambda = 1##, as...
Oh right, I need to divide by N, but I will still have V/N on the right. What should I do with it? That would be the number density of the CMB photons. I have to calculate that separately?
Oh I see, thank you! However, what is wrong with my approach. Should we be able, in principle, to obtain the average energy using the partition function (i.e. shouldn't the V cancel, too, in this approach)?
Homework Statement
What is the average energy of the CMB photons, in electronvolts, for ##T=2.73K##?
Homework Equations
The Attempt at a Solution
I used the grand canonical ensemble for photons and after several calculations I get $$<E>=\frac{8\pi V}{c^3}\int_0^\infty \frac{h\nu^3}{e^{\beta...
Hello! In the calculation of the QED matrix element, it says in the book I read that we have to sum over the polarization states of the photon: $$\sum_\lambda \epsilon_\mu^\lambda\epsilon_\nu^{\lambda *}=-g_{\mu\nu}$$ I am a bit confused why do we do a summation over the orthonormal basis...