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Interpretation of the electromagnetic field strength tensor as a spin-1 field

Physics Asked on April 10, 2021

If I understand correctly the electromagnetic field strength tensor $ F_{munu}$ could be considered as a spin-1 field. In that case, what can one say about the total spin and the $z$-component of the spin for this field? Also, how is $F_{munu}$’s spin related to that of the photon field ($A_{mu}$)?

One Answer

The spin of the electromagnetic field tensor $F_{munu}$ is best understood by writing it as a spinor. A spin 1 field is a represented by a symmetric spinor $xi^{AB}$ or by a dotted symmetric spinor $eta_{dot{A}dot{B}}$. In order to get the field transforming correctly under parity, the electromagnetic field has to be a direct sum using the symmetric spinor and it's complex conjugate dotted spinor. begin{equation} F_{munu} sim xi^{AB}oplus [xi^{*}]_{dot{A}dot{B}} end{equation} The symmetric spinor $xi^{AB}$ has three independent complex components $xi^{11},xi^{12}=xi^{21},xi^{22}$. Linear combinations of these components correspond to the three complex components of the electromagnetic field $B^{r}+iE^{r}$. The source free Maxwell equations are obtained by acting on the symmetric spinor with the Hermitian momentum operator $hat{p}^{dot{A}}_{ B}$ begin{equation} hat{p}^{dot{A}}_{ B}xi^{BC}=0 end{equation} This equation is similar to the Dirac equation for a massive spin 1/2 field. The photon is massless, so it has helicity = $pm 1$ instead of spin (essentially spin 1 along or against the direction of flight). The photon has two helicity degrees of freedom, but the spin 1 field $F_{munu} sim xi^{AB}oplus [xi^{*}]_{dot{A}dot{B}}$ has six real components. The Maxwell equations $hat{p}^{dot{A}}_{ B}xi^{BC}=0$ project the spinor onto a two-dimensional subspace.

This is as far as I know how to answer the question at present. The gauge field $A^{mu}$ is a four vector so it ought to be a Hermitian spinor field of type $X^{dot{A}}_{ B}$ which is the tensor product of two spin 1/2 fields. It has four components, so the gauge fixing must come in to reduce four to the two helicity degrees of freedom.

Answered by Stephen Blake on April 10, 2021

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