Past exam of the particle physics course of the University of Oxford / 2015 / 6

Use $g^{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ and restore $\hslash$ temporarily. The free Dirac equation is $(i\hslash {\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$, with $c=1$. Its Hermitian conjugate is $-i\hslash ({\partial }_{\mu }{\psi }^{†})({\gamma }^{\mu }{)}^{†}-m{\psi }^{†}=0$. Multiplying on the right by ${\gamma }^0$ and using $({\gamma }^{\mu }{)}^{†}{\gamma }^0={\gamma }^0{\gamma }^{\mu }$ gives the adjoint equationThe Dirac adjoint and original equations give $({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }\psi =im\bar{\psi }\psi /\hslash$ and $\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi =-im\bar{\psi }\psi /\hslash$. Adding gives the continuity equationThe Dirac current transforms as a four-vector; the positive density is the time component, not a Lorentz scalar. With ordinary units the number current is $c\bar{\psi }{\gamma }^{\mu }\psi$ and the electron charge current is $-e$ times it.

For either normalized two-spinor ${\chi }^s$, $(\mathbf{\sigma }\cdot \mathbf{p}{)}^2=p^{2}I$ givesThe Dirac equation alone does not fix $N$: a normalization convention is needed. In the usual covariant external-spinor convention ${\bar{u}}_r{u}_s=2m{\delta }_{rs}$, choose real positive $N=\sqrt{E+m}$. Then $u_r^{†}{u}_s=2E{\delta }_{rs}$ and ${\bar{u}}_s{\gamma }^{\mu }{u}_s=2p^{\mu }$. A second common convention is unit rest density, $\bar{u}u=1$, giving $N=\sqrt{(E+m)/(2m)}$ and density $E/m$. Both explain why the boosted density is not one particle per unit coordinate volume: the density is a four-current component and changes under a Lorentz transformation.

One may instead choose unit laboratory density, $N=\sqrt{(E+m)/(2E)}$, or box normalization $N=\sqrt{(E+m)/(2EV)}$ for one particle in volume $V$. An infinite plane wave is not normalizable to one particle over all space; a localized wave packet is required for that. Thus a unique $N$ cannot be inferred from the prompt without specifying which of these conventions is intended.

Let ${\chi }_{\pm }=(1,\pm 1{)}^T/\sqrt{2}$. The eigenvalues of ${\Sigma }_1$ are $\pm 1/2$, each with multiplicity two, and orthonormal bases for their eigenspaces areA general linear combination of the two positive-energy solutions has upper spinor $\xi =a{\chi }^1+b{\chi }^2$ and lower spinor $(\mathbf{\sigma }\cdot \mathbf{p})\xi /(E+m)$. An x-spin eigenvector needs $\xi \propto {\chi }_{\pm }$ and the lower spinor to have the same ${\sigma }_1$ eigenvalue. The Fixed-axis spin obstruction for positive-energy Dirac spinors isIts norm is $2\sqrt{p_y^2+p_z^2}$, so it vanishes exactly when $p_y=p_z=0$. For example with $\mathbf{p}=p\hat{z}$, the upper ${\chi }_{+}$ produces lower ${\chi }_{-}$, excluding a common eigenvector for every $p\ne 0$. In the nonrelativistic limit, the lower components are $O(p/m)$, and $({\psi }_1\pm {\psi }_2)/\sqrt{2}$ approach x-spin eigenvectors. At rest they are exact.

The prompt's unrestricted claim of impossibility at relativistic energies needs qualification: for momentum parallel to x these combinations remain exact eigenvectors at any energy. For a beam transverse to x, such as the conventional z-directed beam, the intended impossibility holds, and the lower components are no longer negligible. A transported relativistic spin observable differs from this fixed matrix; helicity is the conserved projection along momentum for a free particle.

An electron gun releases electrons by thermionic emission, field emission or photoelectric effect, accelerates them through an electric potential, and uses electrostatic or magnetic lenses to form a beam in vacuum. A hot unpolarized cathode does not by itself select spin.

For a polarized source, illuminate a suitable semiconductor photocathode, commonly strained GaAs, with circularly polarized light near its band-gap threshold. Optical angular-momentum selection rules preferentially populate one conduction-electron spin; emission from a negative-electron-affinity surface extracts the selected population. Reversing the light's circular polarization reverses the beam polarization. Strain and a suitable excitation wavelength suppress transitions that dilute the selection. Solenoids and spin rotators orient the spin after acceleration, and a scattering polarimeter measures the achieved polarization; it need not be $100\%$.