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My question is somewhat related to this post. I wish to know how the infinite dimensional representations are induced from Little group (via action of Boost matrices?). From what I understood was that we have two Casimir Operator

  1. $P^\mu P_{\mu}$
  2. $W^\mu W_{\mu}$

These are used to classify the representation based on mass and spin. Little Group is the subgroup which leaves the momentum invariant, which in case of massive particles are rotations but since we want to include spinors and spinors are the fundamental representation of $SU(2)$ which can be used to construct other representations for higher spins. However this is where my doubt begins, we have completely forgotten about the boost. How does the boost actually look like in this unitary representation and is the rotation still in finite dimensional representation here? Also, when we talk about the lorentz transformation of dirac equation we are actually using finite dimensional representation of lorentz group. Schwartz in his chapter 10, section 3.1, he constructed the lagrangian using the invariance under finite dimensional representation for Lorentz algebra which somewhat makes sense due to isomorphism between the representations but I wanna understand why don't we discuss these things mostly in unitary representation since they are so important in Quantum Mechanics.

Qmechanic
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Chandra Prakash
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    You are confused about the role of the two different representations - one finite-dimensional and one infinite-dimensional - here; see e.g. this answer of mine – ACuriousMind Feb 03 '24 at 18:50
  • What about transforming fields inside the correlation functions? Atleast the states have to transform under unitary representation and so should fields, right? – Chandra Prakash Feb 03 '24 at 19:02
  • Schwartz’ book is good for some things, but for this particular subject it is terrible in my opinion. The best one is Weinberg’s Chapter 2. I will try to write an answer later. – Gold Feb 03 '24 at 19:03
  • @ChandraPrakash I don't understand your question - as my answer explains, the fields transform both under a finite-dimensional representation on their classical target space and the infinite-dimensional representation on the quantum space of states. – ACuriousMind Feb 03 '24 at 19:15
  • Yes, thanks alot but I wanna know more about the infinite dimensional representation now. Specifically how'd they look like. – Chandra Prakash Feb 03 '24 at 19:44

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