1. Pg 38. In (4.14) is there a reason for the choice of the ordering of terms and their complex conjugates? In the question sheet it seems to be a different order but I thought maybe it is important?

The order is not important. This is all classical physics, so all the fields are just numbers which commute with each other (in quantum mechanics, the fields become operators and do not necessarily commute with each other).

2. Pg 46. In (4.78) do we already know 1/N(w) is 2pi * 2 w ?

Yes, the definition of N(ω) should be taken as given. This was not derived here. As mentioned in lectures, for relativistic systems it is the natural (Lorentz invariant) phase space density.

3. Pg 51. Im confused about the derivation of (5.16). The first term appears to be the result of differentiating the first term of (5.14) w.r.t derivatives of A. However isnt this exactly the same thing as in part of the first term in (4.48) where you get F contravariant. If so is this equal to the first term of (5.16)? Is it obvious?

You need to compute the E-L equation ν = . Indeed, the derivative appears in (4.48) (up to replacing the labels on indices). However, now there is the ν acting on it, which means that the term ends up rather different from those in (4.48). NB you need also to use the Lorenz gauge condition.

4. Pg 72. Is susceptibility defined as the derivative when B=0 or are we holding B at 0 for a specific reason?

Indeed, the susceptibility is defined as the derivative at B = 0, hence the form of (7.38).

5. I was wondering: on page 81 of the notes, we change the heavy step function phita(t) to exp[i epsilon *t] and 1) heavy step is not a delta function 2) even if it can be expressed like this (?), it should be just written delta (t ) rather than exp because we are looking at the time domain , i.e. expressing phita (t) in time domain should be just epsilon

The equation in the middle of page 81 sets the Heaviside function Θ(t) to be 0 for t < 0 (NB limit of integral) and to be the limitϵ 0 of e-ϵt for t > 0 (i.e. this is indistinguishable from 1 for any finite t and sufficiently small ϵ). So it is still the step function.

The identificiation is made in the time domain (it occurs within the integral). It is put into a Fourier transform to generate the FT Θ(ω). [Unhelpfully, the notes do not use a different symbol for the Heaviside function, Θ(t), and its Fourier transform, Θ(ω), other than implicitly by the change in argument. The FT is, of course, a different function, so perhaps should better have been denoted (ω).]

6. on same page where we write epsilon +i w / w^2 + epsilon ^2 = pi delta (w) + P i / w is this a result which we should not know the proof of ?

The first (real) part is a standard representation of the Dirac delta function. This should have been mentioned in the IB NatSci Maths course. It appears in equation (3.5b) of these notes. The second part may, similarly, be taken as a representation of the Cauchy principal part. This may well be new to those taking this course, but should be understood in a similar manner to the Dirac delta function. (i.e. you can view this limiting expression as a definition of what one means by the principal part P of the integral of 1∕ω, just as one can take the other limiting expression to be what you mean by the Dirac delta function inside an integral.)

7. p 78, How can the relation G_tilda=i h_bar G derived? It seems to use partial integration, but I have no idea in detail.

Since the Schrödinger equation is first-order in time, for times t > t the wavefunction ψ(x,t) can be taken to be the solution of the equation i + 2ψ = iδ(t - t)ψ(x,t) with the boundary condition that ψ(t) = 0 for t < t. [To see this: (i) note that for t > tthe right hand side is zero, so ψ(x,t) satisfies the time-dependent Schrödinger equation; (ii) by integrating over an infinitesimal interval of time either side of t = t, one recovers the initial condition that ψ(x,t = t) = ψ(t).]

However, this equation is equivalent to (8.12) with a “source” F(x,t) = ψ(x,t)δ(t - t). Putting this into (8.13), where the Green’s function G is defined, and comparing with the definition of via ψ(x,t) = (x,t;x,t)ψ(x,t) dx, shows that = iG.

8. p 78, I am confused with ψ(x,t) = (x,t;x,t)ψ(x,t) dx, How can this integral be a funtion of x, t only? I think it should be a function of x, t, and t’.

It is not necessarily a function of t. The tdependences of (x,t;x,t) and ψ(x,t) cancel. For example, for a time-independent Hamiltonian, in ket notation, we have in which the dependence on tcancels on the right hand side. Converting to the position representation, ψ(x,t) = x|ψ(t), we have the propagator (x,t;x,t) = x|e-iĤ(t-t)|x′⟩.
9. p 79, Fig 11. During the integration to get G(p,t-t’), why did we conducted on asymmetric path for t<t’ and t>t’(The first diagram)? Does same result comes when we conduct integral with semicircle for both situation? Also, can we apply Cauchy’s integral formula in this situation? I searched the strict statement of the formula and its conditions. It says that the pole should be in the ’interior’ of the integral, and the pole is in the boundary of the t>t’ semicircle in the first diagram. What is the reason that we can apply this formula?

When there is a pole on the real axis, the integral along the real axis is not defined. We need to decide how to define it. The top diagram is a particular choice of how to define this integral: it deforms the integral along the real axis to move into the complex plane and to pass just above the pole. With this definition the propagator becomes causal. (For t < tthe integral can be completed by closing the contour in the upper half plane. The pole is not enclosed so Cauchy’s theorem says the contour integral vanishes. For t < tthe integral can be completed by closing the contour in the upper half plane; Cauchy’s theorem says that the contour integral picks up the residue at the enclosed pole.)

Cauchy’s theorem can be applied because the pole lies either inside the contour (for t > t) or outside the contour (for t < t).

10. Also, I am confused with ’pole-moving trick’, too. : G(p,E)=1/(E-p^2/2m+ie) Are we assuming e>0 just for the causality(Which looks quite arbitrary but makes sense with relativity.)? What does e stands for? Is it just a parameter that we assume lim e->0? Is this related with defining Dirac delta function as a limit of distribution?(so when the parameter e goes to 0, the distribution goes to Dirac delta ftn, and corresponding G(p,E,e)=1/(E-p^2/2m+ie) goes to 1/(E-p^2/2m).)

Yes ϵ > 0 for causality. Yes, it just a parameter for which we assume limϵ 0. Take care: when ϵ goes to zero G(p,E,ϵ) = 1(E -p22m + ) has a real part that is (proportional to) a Dirac delta function and an imaginary part that is (proportional to) the Cauchy principal part of 1(E - p22m). [See middle of page 81.]

11. p 68 How do you get from the first equation to the second when estimating H on pg 68? Are we just ignoring the small (si - s)? If so then why isn’t it (2dJs^2 + B)?

This involves an expansion of the interaction energy to first order in (si- s). The zeroth order term would be -(J∕2) i,δs2, which is a constant -(J∕2)N(2d)s2. The first order term would be -(J∕2) i,δ2s (si - s), which is equal to -(J∕2)(2d)2 is (si - s). (The second order term is dropped.) Taking the zeroth and first order terms together gives: a constant term, +JNds2 (not shown in the handout); a term that is linear in si, given by -2Jd isis, which is one of the terms given (with s = s ). The other term shown, linear in si, is just the coupling to B, as in the initial H.

12. p 68 In estimating the partition function, why is summing over all possible Boltzmann factors equivalent to integrating over all angles and raising to the power N?

For H({si}) = i=1Nh(si), we can write exp(-βH) = i=1N exp(-βh(si)). Then Z = {si}exp(-βH({si})) = i=1N . Since all sites i are equivalent (h the same for them all), we can write this as Z = N. The integrals over θ and ϕ denote the sum over all orientations of the classical vector s representing the spin.

13. p68 Similarly, how do you get to the equation for <si . z>, where one integral is raised to the power N-1?

In computing si.z, the same reasoning as above applies for all N -1 sites other than the site i on which you compute the average spin. For that one spin, you have an extra si.z = cosθ in the integral.

14. TP1 Exam 2017, Q1b: How do you find Q2? The solutions say integrate (d L/d psi_dot) * delta_psi - tL. Where has this -tL term come from?

For a more detailed worked answer, see these notes.