Hello,
Thank you for your interesting question about the application of the annihilation/creation operator formalism derived for bosons and fermions in the context of orbits of the Poincaré group, and specifically how this applies to photons.
### Background and Context
As you noted, the standard construction of Fock spaces for bosons and fermions using creation and annihilation operators relies on the representation theory of the Poincaré group, where one considers unitary irreducible representations associated with particular orbits in momentum space. The Hilbert space is constructed from single-particle states carrying such representations, and then symmetrized or antisymmetrized tensor products build the multi-particle Fock space.
Photons, as massless spin-1 particles, are associated with a **different orbit** of the Poincaré group than massive particles. The little group for massless particles is isomorphic to the Euclidean group \(E(2)\), rather than the rotation group \(SO(3)\) that appears for massive particles. This leads to subtleties in the representation theory and the construction of physical states, notably including the gauge degrees of freedom.
### Why the Standard Construction Does Not Apply Directly to Photons
- The derivation in Folland and similar texts often assumes orbits corresponding to massive particles with well-defined helicity or spin representations.
- Photons are described by **massless representations with helicity ±1**, and the representation space includes gauge redundancies which are not captured by the straightforward orbit-based construction.
- The Hilbert space for photons cannot be simply taken as \(L^2(\mathcal{O}, d\mu)\) over the orbit \(\mathcal{O}\) as done for massive particles.
### How the Formalism Can Be Adapted for Photons
Despite these complications, the creation and annihilation operator formalism **can and is applied to photons**, but with important modifications:
1. **Gupta-Bleuler Formalism:**
- This is a method to quantize the electromagnetic field by allowing indefinite metric states to appear in the Hilbert space and then imposing subsidiary conditions to restrict to the physical subspace.
- Here, creation/annihilation operators correspond to the four-potential \(A^\mu\), including unphysical polarizations, with the physical photon states selected afterward.
2. **BRST Quantization:**
- A more sophisticated and mathematically rigorous framework using cohomological methods to handle gauge redundancies, where the physical Hilbert space is defined via BRST cohomology.
- Creation/annihilation operators are extended to include ghost fields, and physical states are identified through BRST invariance.
3. **Direct Construction via Helicity States:**
- Photons can be described by helicity eigenstates with helicities ±1. Creation and annihilation operators act on these helicity states directly, with the appropriate modifications to the representation space to include only physical degrees of freedom.
4. **Use of Rigged Hilbert Spaces and Distributional Representations:**
- Since the usual \(L^2\) space does not fully capture photon states, one often works in more generalized function spaces (rigged Hilbert spaces) or uses test function spaces to accommodate the subtleties.
### Summary
- The derivation of creation/annihilation operators for massive particles using \(L^2\) functions over Poincaré orbits does not directly carry over to photons due to their massless nature and gauge symmetries.
- However, through methods like Gupta-Bleuler, BRST quantization, or helicity state constructions, one can successfully extend the formalism to photons.
- The key is to carefully handle gauge redundancies and the structure of the little group \(E(2)\) associated with massless particles.
### Further Reading
- *Quantum Field Theory* by Steven Weinberg, Vol. 1 (especially the discussion on massless particles and gauge fields).
- *The Quantum Theory of Fields* by Weinberg for a thorough treatment of massless particle representations.
- *Lectures on Quantum Field Theory* by Folland, where some of these constructions are discussed.
- For mathematical rigor, the papers on BRST quantization and gauge theory representation theory.
I hope this clarifies why the straightforward orbit-based construction does not directly apply to photons, and how the formalism is nevertheless extended in practice.
Best regards!
