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I'm trying to understand induced representation / little group stuff in Weinberg QFT vol. 1 chapter 2 (around page 64, 65, 66). So is this the correct way of thinking about it:

We have the Poincaré group of symmetries; we wish to know how to represent operators (derived from these symmetries) that act on physical states, [itex] \Psi [/itex], in the Hilbert space.

Since 4-momenta - p - commute as shown in the Lie Algebra of the group, classify the state-vectors according to p and someother quantities [itex]\sigma[/itex]. Then use a Lorentz transformation to re-write a general momentum state [itex]\Psi_{p,\sigma}[/itex] in terms of a finite number of distinct "standard momenta" [itex]k_{\mu}[/itex], as in equation (2.5.5). This standard momentum [itex]k_{\mu}[/itex] is invariant under a certain group symmetry [itex] W^{\mu}_{\nu}[/itex] by construction.

The set of W that satisfy this [itex]Wk=k[/itex] are called the little group.

Correct so far? I guess my question is then what to make of this? To take an example off of Table 2.1 page 66 if my state vector is dependent on k-momentum (already in standard form), [itex](0,0,0,M)[/itex] (case (a)) then are the only transformations that leave it invariant (i.e. can produce an eigenvalue-eigenvector) those of SO(3), the rotation group?

As you can see I have lots of the pieces but just not quite sure what the whole point is, but really want to get it properly. Thanks

We have the Poincaré group of symmetries; we wish to know how to represent operators (derived from these symmetries) that act on physical states, [itex] \Psi [/itex], in the Hilbert space.

Since 4-momenta - p - commute as shown in the Lie Algebra of the group, classify the state-vectors according to p and someother quantities [itex]\sigma[/itex]. Then use a Lorentz transformation to re-write a general momentum state [itex]\Psi_{p,\sigma}[/itex] in terms of a finite number of distinct "standard momenta" [itex]k_{\mu}[/itex], as in equation (2.5.5). This standard momentum [itex]k_{\mu}[/itex] is invariant under a certain group symmetry [itex] W^{\mu}_{\nu}[/itex] by construction.

The set of W that satisfy this [itex]Wk=k[/itex] are called the little group.

Correct so far? I guess my question is then what to make of this? To take an example off of Table 2.1 page 66 if my state vector is dependent on k-momentum (already in standard form), [itex](0,0,0,M)[/itex] (case (a)) then are the only transformations that leave it invariant (i.e. can produce an eigenvalue-eigenvector) those of SO(3), the rotation group?

As you can see I have lots of the pieces but just not quite sure what the whole point is, but really want to get it properly. Thanks

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