7360/7361 Updates

May 3

Posted lectures are now up to date.


Apr 27

For the current set of problems, I've removed AH 6.9 and replaced
it with AH 6.3.


Apr 18

The posted lecture notes are now up to date, including some corrections 
in the section on particle exchange and potentials.


Apr 6

There's a typo in the statement of problem 2.2 in AH.
The sin(theta/2) at the bottom of pg 49 should be sin^2(theta/2).
(Thanks, Yuriy.)


Apr 4

For the last problem, you may assume that in both cases, the
dominant part of the integral as epsilon vanishes comes from
large momentum, and so the mass m may be neglected.  To get
the dependence on epsilon, I found it easiest to consider the
timelike and spacelike cases separately.  Also, note that
I've reworded the cutoff slightly in the problem statement.


Mar 31

I've updated the posted lectures, and posted a new homework.
If you start on AH 2.6 now, you might enjoy the seminar on 
Monday a bit more.


Mar 24

For the time dependence of the boost generator, recall that
there's an extended version of the Heisenberg equations of
motion.

The next homework due date is postponed from Wed to Fri, by
popular demand.  I should mention that I'll be out of town
on Thursday, but will be here Wed and Fri if you have questions.


Mar 8

I've updated the lecture notes on our site.  Note that I changed some conventions
in my notes since the last time I posted them, to be consistent with what we did
in lecture.  So if you've printed out the third set of lectures on scalar fields,
you should toss those and refer to the newly posted set.


Mar 6

For the string problem, here are the standard 
orthogonality and completeness relations for momentum-space 
(Fourier) expansions satisfying periodic boundary conditions.  
Please ask if you're not sure what these mean or why they're 
useful, or if you find an error.


Mar 2

For the problem about the transformation properties of 
the conjugate momentum, you can figure out how pi should
transform a couple ways.  First, you can think of how
the transformed pi' would be defined in terms of d L/ d phidot', 
compare it to pi = d L/d phidot, and use the chain rule.
You could also assume that H is invariant (it almost always
is for internal symmetries), and recall L = pi phidot - H.
This will tell you immediately how pi changes.  The two results 
should be consistent.  (Roughly, it should be the inverse of the 
trans. for phi.  Incidently, this is intended to be a short problem; 
ask me if you're spending a lot of time on it.)


Mar 2

I've posted solutions for set 2.  Please read through these
and ask questions if anything's unclear.


Feb 28

For the current associated with Lorentz transformations, to repeat
what we did for translations, you'll need a form of the infinitesimal
parameter eps_mu_nu that is only non-zero in one particular plane
(say, alpha-beta).  To do this for translations, we set the vector
eps_mu equal to a const eps times delta_mu^alpha = g_mu^alpha.
For eps_mu_nu, you'll need to do something similar in both indices,
and also explicitly account for its antisymmetry.  If eps_alpha_beta
is non-zero, so is eps_beta_alpha.

I hope this wasn't more confusing than it was worth.


Feb 24

Lecture notes are now up to date.


Feb 11

I've corrected the first term in the vector field lagrangian
to include a leading minus sign.  It doesn't affect much of
anything in this problem, but makes it consistent with everyone
else.


Feb 8

It might also help to know that Det[A B] = Det[A] Det[B] (which no doubt
you already knew).


Feb 7

I've got my lecture notes posted up to about where we'll be tomorrow,
plus a little.


Feb 7

It might be helpful to know that, for an NxN matrix M

 eps_{i1 i2 .... iN} M^i1_j1  M^i2_j2 ... M^iN_jN = (Det[M]) eps_{j1 j2 ... jN}

(Here, ^ and _ are the usual tex notations for raised and lowered indices.  If
you find this unreadable, look here.

Just as we did for the metric, it's useful to turn this around and define a
proper Lorentz transformation as any transformation that leaves both g and epsilon
invariant.  (An improper transformation, such as parity and time reversal, leaves
only g invariant, but changes epsilon by a sign.)

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