What Is psi_PRIME in the Lorentz Transform of a Wavepacket?

[bjnartowt]

Homework Statement

see attached .pdf. all parts of problem statement are italicized.

Homework Equations

see attached .pdf

The Attempt at a Solution

see attached .pdf


Actually: my question is pretty qualitative. You can look at everything I've done with this problem so far. However, the problem is asking for psi_PRIME, that is, the function in the new frame of reference. As you can see, I've proved the wave equation is invariant under Lorentz transform. Also, I don't think it's a mean feat to transform the x and t arguments of psi to make psi(x_PRIME, t_PRIME). However, what exactly *is* psi_PRIME, beyond the wave in the new frame of reference? I'm not sure how to "get" psi_PRIME. Do I "get" psi_PRIME when I transform its arguments? Then I think psi = psi_PRIME, because I'd be plugging transformed (primed) variables into the same ol' function...

 

[grey_earl]

\psi'(x', t') = A(\Lambda) \psi(x(x',t'), t(x',t')) A^{-1}(\Lambda) where A(Λ) gives you the group theoretical factor which depends on the type of field you have (more exactly, its spin). If ψ is a scalar, then A = 1.

 

[bjnartowt]

Ok, thank you. So you're saying that if psi was a vector, the explicit functional form of psi would transform, and not just the variables x --> xprime and t --> tprime? Because my psi is a scalar (it's a one-dimensional wavepacket).

 

[grey_earl]

Exactly. If your ψ is a scalar, you only transform its arguments (x, t). If it would be a spinor, or a vector, or ..., you would have to multiply by some matrix A which depends on your Lorentz transformation Λ. This is very well explained in Peskin and Schroeder's book "An Introduction to QFT", if you have a library near you.

(BTW: Peskin & Schroeder is one of the most readable books on QFT I have found so far, if you plan to dig into QFT, get this one.)