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 ($A_{mu}$)?
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
Get help from others!
Recent Questions
Recent Answers
© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP