Trying to understand constraint equation from paper by Brink

[chenkr3]

I am trying to fully understand the spectrum structure in the paper "A LAGRANGIAN FORMULATION OF THE CLASSICAL AND QUANTUM DYNAMICS OF SPINNING PARTICLES " by Brink, Vecchia and Howe. (I attached the file)
I am having a problem with equation (4.13). It writes [itex]p \cdot b|\psi_{phys}\rangle=0[/itex] but was supposed to be [itex]p \cdot \psi |\psi_{phys}\rangle= p \cdot \frac{1}{2}(b + b^{\dagger})|\psi_{phys}\rangle=0[/itex] by constraint (4.3). Why is the [itex]b^{\dagger}[/itex] omitted from equation (4.13)? Am I missing something in understating how [itex]b^{\dagger}[/itex] acts on the spectrum? Am I missing something with the identification of [itex]b[/itex] and [itex]b^{\dagger}[/itex] as creation and destroying operators?

Thanks.

 

[AndyW]

Thank you for bringing up your question about equation (4.13) in the paper "A LAGRANGIAN FORMULATION OF THE CLASSICAL AND QUANTUM DYNAMICS OF SPINNING PARTICLES" by Brink, Vecchia and Howe. I understand that you are trying to fully understand the spectrum structure in this paper and are having trouble with equation (4.13). I have reviewed the paper and can provide some insights to help clarify this equation.

Firstly, I want to point out that the authors have used a particular notation in their paper where they have defined b as a shorthand notation for \frac{1}{2}(b + b^{\dagger}). This is mentioned in the introduction of the paper where they state that "the operator b is shorthand for the pair (b,b^{\dagger})". Therefore, equation (4.13) can be rewritten as p \cdot b |\psi_{phys}\rangle=0, which is consistent with the constraint (4.3).

Furthermore, in the same section where equation (4.13) appears, the authors have also stated that "b and b^{\dagger} are creation and annihilation operators, respectively, for the physical states". This means that b^{\dagger} acts as the creation operator on the physical states |\psi_{phys}\rangle. Therefore, the inclusion of b^{\dagger} in equation (4.13) is redundant, as it is already taken into account in the definition of b.

In conclusion, you are not missing anything in your understanding of how b and b^{\dagger} act on the spectrum. The authors have used a shorthand notation and the inclusion of b^{\dagger} in equation (4.13) is not necessary. I hope this clarifies your understanding of the spectrum structure in this paper. If you have any further questions, please do not hesitate to ask.