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    Early career in math

    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...
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    Casimir effect with Gaussian regulator

    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...
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    A Spin-helicity formalism for gluon-gluon amplitudes

    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...
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    I Path Integral in QFT

    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...
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    I Spin and helicity conservation in QED

    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...
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    I Long distance QED

    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...
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    I Why are spinors not observables?

    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...
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    I LRZ for scalar QED

    Hello! Can someone direct me towards a reading where the Feynman rules for scalar QED are derived? Thank you!
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    I Transverse mode of a field

    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...
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    I Further S matrix clarifications

    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...
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    I S matrix and vanishing fields

    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...
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    I Helicity vs Chirality

    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...
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    I Compton scattering on massless electron

    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...
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    CMB average energy

    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...
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    I QED sum over polarization states

    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...
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    I Wigner-Eckart theorem

    Hello! I am a bit confused about the dimensionality of the vectors in Wigner-Eckart theorem. Here it is how it gets presented in my book. Given a vector space V and a symmetry group on it G, with the representation U(G) we have the irreducible tensors $${O_i^\mu,i=1,...,n_\mu}$$ (where ##n_\mu##...
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    Other First year PhD in theoretical physics

    Hello! What can a first year PhD student (in an American university) do, research related, in the field of Theoretical (mainly high-energy) physics? I know in the USA you need to start research from the first semester of PhD (even if you formally choose a definite field of research at the end of...
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    I Space and time translations

    Hello! I am a bit confused about the sign in space and time translation operators acting on a state. I found it with both plus and minus sign and I am not sure which one to use when. The equations I am talking about are: $$U(t)=e^{\pm iHt/\hbar}$$ and $$T(x)=e^{\pm ixp/\hbar}$$. Is it a plus or...
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    I Sign of the terms in a circuit

    Hello! I am a bit confused about the sign of the ##L\frac{dI}{dt}## term in the circuits (DC circuits). In my book it is defined with a minus, on wikipedia it is defined with a plus and I am not sure which one should I use. I can pick any sign I want and the result will come out right...
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    I Force in an inertial frame

    Hello! The angular velocity in the non-inertial frame of a rotating body of mass m is ##\Omega## and I need to find the force acting on the body (in the non-inertial frame associated with the body). In the book they say (without any derivation, they just state it) that the force is...
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    I Forces of constraint

    Hello! I have this Lagrangian: $$L=\frac{1}{2}m\dot{r}^2(1+f'(r)^2)+\frac{1}{2}m\dot{\phi}^2r^2-mgf(r)+\lambda(\phi-\omega t)$$ This represents the motion of a point-like object of mass m along a curved wire with shape $$z=f(r)$$ The wire rotates with constant angular velocity around the z axis...
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    I Singlet state

    Hello! I got a bit confused about the fact that the whole the description of spin (and angular momentum) is done in the z direction. So, if we are told that a system of 2 particles is in a singlet state i.e. $$\frac{\uparrow \downarrow -\downarrow \uparrow }{2}$$ does this mean that measuring...
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    Relativistic Rocket

    Homework Statement A rocket moving with speed v passes a stationary observer. The observer waits a time T (according to his clock) after the rocket passes and send a pulse of light in the direction of the rocket. The rocket pilot notes that, according to her clocks, the time elapsed between the...
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    Off center cylinder

    Homework Statement A cylinder of radius a and mass m contains a point mass, also of mass m, located a distance ##a/2## from the symmetry axis. The cylinder is placed on an incline, which is initially horizontal, but is very slowly raised. Assuming the cylinder cannot slide on the incline, at...
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    Not a perfect blackbody

    Homework Statement Two parallel plates plates are maintained at temperatures ##T_L## and ##T_R## respectively and have emissivities ##\epsilon_L## and ##\epsilon_R## respectively. Given the Stephan-Boltzmann constant ##\sigma##, express the net energy transfer rate per area from the left plate...
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    I Different approximations for the same problem

    Hello I am looking at Stat Mech problem 2 from here (page 8) with solution here. I am confused about their approximations. They are all valid, but they are different. For example in part a) they use $$\frac{\partial P}{\partial x}(x+\frac{1}{2}\Delta x,t)=\frac{P(x+\Delta x,t)-P(x,t)}{\Delta...
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    Other MIT Physics General Exams

    Hello! I solved the MIT physics graduate general exams (the Part 2 from here) and they are super useful to consolidate your knowledge in the areas they cover. Does anyone know where (and if) I can find the exams between 2002 and 2012 and after 2012?
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    Frequency of oscillations

    Homework Statement A particle moves in 1D in a potential of the form $$U=Ax^2+Bx^4$$ where A can be either positive or negative. Find the equilibrium points and the frequency of small oscillations. Homework Equations The Attempt at a Solution So the equilibrium points are obtained by setting...
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    I Lagrangian term

    Hello! I have a classical Lagrangian of the form $$L=A\dot{x_1}^2+B\dot{x_2}^2+C\dot{x_1}\dot{x_2}cos(x_1-x_2)- V$$ the potential is irrelevant for the question and A, B and C are constants. When doing $$\frac{d}{dt}\frac{\partial L}{\partial \dot{x_1}}$$ the solution gives this...
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    I Extreme length contraction

    Hello! If we have a 1m stick (as measured in its stationary reference frame, call it S) and we (S') move with a high enough velocity, we can make the length of the stick in our frame as small as we want. So for high enough velocities, the stick will appear so small in our frame, S', that it will...
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