0

I wonder if anyone could shed some light on the representation theory of the Lorentz group. In particular, I would like to understand unitary and spinorial representations of boosts better. To my understanding the action of boost on a 2-spinor is given by $ \exp(-a \sigma_i)$, where $a \in \bf{R}$ and $\sigma_i$ is a Pauli matrix (see e.g. page 8 in http://www.weylmann.com/spinor.pdf). This action is clearly not unitary. I am aware that similar questions have been posted before, but I am still confused. In the answers given it is stated that boosts do indeed act unitarily on spinors, but what does such a representation look like explicitly? I have read about every source I can find on the representation theory of the Lorentz group but haven’t come across a representation of boosts that act unitarily on 2-spinors. So my question is simply, given that such a representation exists, what does it look like explicitly?

Any answer would be very much appreciated!

Qmechanic
  • 201,751
egj
  • 9
  • Link to answers? – Qmechanic Jan 03 '22 at 09:06
  • 1
    Related: https://physics.stackexchange.com/q/632964/2451 and links therein. – Qmechanic Jan 03 '22 at 09:10
  • I don't know much on the subject, but I can suggest that the unitary representation is for the four-dimensional representation of the Lorentz group. By the fact that it is the $\text{O}(1,3)$ group, it should leave the norm of a four-vector invariant in the Minkowski space ($(1,3)$ metric). So the spinor you are talking about should be four-dimensional. For a two dimension spinor I don't know how Lorentz group could be represented. – Rob Tan Jan 03 '22 at 11:08
  • I also thought about the fact that the four dimensional representation of the Lorentz group is the fundamental one, that's why it called $\text{O}(1,3)$ even if you represent it in an arbitrary number of dimensions. For this, you should not be able to define it in a number of dimensions inferior than $4$. I also considered that to define properly the Lorentz group you should use all 6 generators of SO$(1,3)$, that means $6$ independent matrices, but in the two dimensional representations you would have four-elements matrices, and you cannot define $6$ independent matrices of $4$ element each. – Rob Tan Jan 03 '22 at 11:17
  • 1
    States transform in unitary representations of the Lorentz group which are necessarily infinite dimensional. Fields transform in highest weight representations and these have finite dimension... 2 in the case of the Weyl spinor you're talking about. – Connor Behan Jan 03 '22 at 14:04

0 Answers0