Photon

Spatial part depends on L. Parity=(-1)^L.

Spin=1, vector is even. Parity =+1.

Intrinsic is defined to match the classical conventions for E,B,A. Parity =-1.

The classical convention for the photon field A^μ is that the vector piece changes sign under a parity transformation. Then the E field is a vector and the B field is an axial vector. This is merely a convention that could be changed. The parity of the electric field, magnetic field and vector potential could be redefined. The current convention requires the photon to have a negative overall parity. You can look at the photon field in a field theory as

[spatial part x,y,z] [creation op, a a^† ] [spin, S=1]

(See CP Violation pg 39 for an example of the photon field expanded in these terms).

The Fourier sum of plane waves of specific spin shows that a general field can always be written in terms of photons with a simple spin state and a general spatial wave function. Thus the fundamental entity, the photon can be considered quite generally to be a plane wave with a circularly polarized spin piece (Any field can be built from this basic ingredient). For simplicity consider a photon traveling in the z direction or consider the direction of the photon as choosing the coordinate axis so that z point along the photons momentum.

To get the correct parity you need to assign an overall minus sign to the photon creation operator. Spin as stated above doesn’t change sign. The coordinate dependence (spatial wave function) is already cleanly defined. So in order for A to be a vector and view most of the vector character as a result of the spin of the photon you must assign and additional –1 for the intrinsic parity. This is because the spin itself transfers with no change. Therefore you must add this intrinsic part to obtain the normal convention.

The spin part of the photon

Where the relationship between normal spin representation as column vectors and the vector representation as A_x, A_y, A_z is

Which defines the two states of circular polarization as the Ms=+1, –1 as the transverse states where Az=0. The Ms=0 state is identified with polarization along the z-axis. This state is will be linked with the A_o through the gauge condition. Thus the general A^μ will have three independent parts. The free field (no sources) is pure transverse. The longitudinal part will have a scalar J=0 part in addition to the vector spin 1 part.

The E&M fields are often treated as Multipole expansions (see for example DeBennedeti, r Scheck.) These are combinations of space and spin.

A typical wave function will couple these states to get a total angular momentum J M_j .

Considering Spin 1, the states of total J can be built from

J = L+1

J = L-1

J = L

Since we are including a longitudinal piece the associated E&M field will be a virtual field.

You take all of the possible ways to reach a particular total angular momentum state and combine them to obtain a set of Electric, Magnetic and Longitudinal Multipole states. These states are parity eigenstates, angular momentum eigenstates and split the transverse and longitudinal pieces.

The parity is (-1)^L for all of the above states.

There is no J=0 solution for the transverse multipoles. There is a J=0 solution for the longitudinal multipole. This of course constitutes a scalar piece for the E&M field. The longitudinal part of the virtual photon field is related to the Coulomb potential. Correct treatment of the Coulomb piece has some mathematical technicalities.

A plane wave with circular polarization can be written as

You can also define a longitudinal plane wave (Virtual field only).

Helicity is the projection of angular momentum along the momentum direction. The photon doesn’t have a spatial contribution to the helicity because the spatial part is related to the classical interpretation of the angular momentum as the cross product, r x p. Therefore all spatial angular momentum is perpendicular to p. Photon helicity is therefore another representation for the photon spin.

The typical expression for the photon field with multipole moments is consistent with the simple statement that the spatial part provide a parity that depends on L, the spin produces a + parity and an overall intrinsic parity has to added to maintain the standard conventions.