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Let's start with a single photon first. A basis of the one-photon vector space (we'll explain why it's not a Hilbert space below) is given by the eigenstates |k,εμ〉 where k, the 4-momentum is null (k2=0) and the k0 component, the energy, is positive and εμ is the unit polarization vector and the index μ ranges from 0 to 3. So, k is uniquely determined by the spatial momentum . Using the bra–ket notation, we equip this space with a sesquilinear form defined by
where the factor is to implement Lorentz covariance. We are using the +−−− metric signature here. However, this sesquilinear form gives positive norms for spatial polarizations but negative norms for time-like polarizations. Negative probabilities are unphysical, not to mention a physical photon only has two transverse polarizations, not four.
If we include gauge covariance, we realize a photon can have three possible polarizations (two transverse and one longitudinal (i.e. parallel to the 4-momentum)). This is given by the restriction . However, the longitudinal component is merely unphysical gauge. While it would be nice to define a stricter restriction than the one given above which only leaves the two transverse components, it is easy to check that this can't be defined in a Lorentz covariant manner because what is transverse in one frame of reference isn't transverse anymore in another.
To resolve this difficulty, first look at the subspace with three polarizations. The sesquilinear form restricted to it is merely semidefinite, which is better than indefinite. In addition, the subspace with zero norm turns out to be none other than the gauge degrees of freedom. So, define the physical Hilbert space to be the quotient space of the three polarization subspace by its zero norm subspace. This space has a positive definite form, making it a true Hilbert space.
This technique can be similarly extended to the bosonic Fock space of multiparticle photons. Using the standard trick of adjoint creation and annihilation operators, but with this quotient trick, we come up with the free field vector potential operator valued distribution satisfying
with the condition
for physical states |χ〉 and |ψ〉 in the Fock space (it is understood that physical states are really equivalence classes of states which differ by a state of zero norm).
It should be emphasised that this is not the same thing as
Note that if O is any gauge invariant operator,
does not depend upon the choice of the representatives of the equivalence classes, and so, this quantity is well-defined.
This is not true for non-gauge-invariant operators in general because the Lorenz gauge still leaves residual gauge degrees of freedom.
In an interacting theory of quantum electrodynamics, the Lorenz gauge condition still applies, but no longer satisfies the free wave equation.
- S. Gupta, Proc. Phys. Soc. v. A63, nr. 267, p. 681–691, 1950
- K. Bleuler, Helv. Phys. Acta, v. 23, rn. 5, p. 567–586, 1950