iangttymn said:
Oh - one more thing.
As I said, the states in this setup have one + index and - index, so they are
v_{Ab}
Srednicki says to convert these spinor indices to a spacetime index using
\sigma^{\mu}_{Ab}
where \sigma^{\mu}_{Ab} = (I, \sigma_1, \sigma_2, \sigma_3),
so that
v_{Ab} = \sigma^{\mu}_{Ab} v_{\mu}
But it's not particularly clear to me why.
I read section 33, 34 of Srednicki these days too.
He mentioned about something related to representation theory, which I'm not familiar to too.
I also have a few questions:
(1) start from (34.1),
U(\Lambda)^{-1}\psi_a(x)U(\Lambda) = L_a{}^b(\Lambda)\psi_b(\Lambda^{-1}x)
, we have (34.6),
[\psi_a(x),M^{\mu\nu}]<br />
= \mathcal{L}^{\mu\nu}\psi_a(x) + (S_\text{L}^{\mu\nu})_a{}^b \psi_b(x), where \mathcal{L}^{\mu\nu} = -i(x^\mu\partial^\nu - x^\nu\partial^\mu),
by evaluating at x = 0, we arrive at (34.7)
\epsilon^{ijk}[\psi_a(x) , J_k] = (S_\text{L}^{ij})_a{}^b \psi_b(0)
Then, I don't understand why he could deduce that (S_\text{L}^{ij})_a{}^b = \frac{1}{2}\epsilon^{ijk}\sigma_k by just recalling the (2,1) representation of Lorentz group includes angular momentum 1/2 only?
Another question about this is, he started from (34.1), where the operator U(\Lambda) should be the representation of Lorentz group in some Hilbert space, but we haven't quantize the theory, how can we get the Hilber space representation?
(2) He did explain the
invariant symbol a little bit. I read from somewhere else that sometimes the metric which can pull up or lower down the index is an antisymmetric symbol, i.e. Levi Civita symbol. Srednicki gave some explanation here. He said, "whenever the product of a set of representations includes the singlet, there is a corresponding invariant symbol." He gave two examples, however, he didn't explain how he got those. Could anybody shed some light on this? Why he could know
(2,2)\otimes(2,2) = (1,1)_s \oplus (1,3)_A \oplus (3,1)_A \oplus (3,3)_S
implies the existence of g_{\mu\nu} = g_{\nu\mu}.
And, how do we determine the subscripts of summand of direct sums? (The symmetric or antisymmetric)
(3) He showed how to map a 4D spacetime Lorentz representation to a (2,2) representation by the formula given by the OP: A_{a\dot{a}}(x) = \sigma^{\mu}_{a\dot{a}}A_\mu(x) He also gave another example, which is the mapping from (3,1) representation to the antisymmetric rank two, self-dual tensor:
G^{\mu\nu}(x) \equiv (S_\text{L}^{\mu\nu})^{ab}G_{ab}(x)
How did he find the mapping operator \sigma^\mu_{a\dot{a}} and (S_\text{L}^{\mu\nu})^{ab} ?
Thanks very much!