- #1

jdstokes

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In relativistic quantum field theory the Dirac spinors can be chosen to be eigenstates of the helicity operator [itex]\vec{\Sigma}\cdot \vec{p} /|\vec{p}|[/itex].

I want to show that [itex]\vec{\Sigma}\cdot \vec{a}[/itex] commutes with the Dirac Hamiltonian only if [itex]\vec{a}\propto \vec{p}[/itex]. As usual I'm using Einstein summation everywhere.

[itex][\vec{\Sigma}\cdot \vec{a}, H_\mathrm{Dirac}]\psi=[\vec{\Sigma}\cdot \vec{a}, -i\hbar c \gamma^i\partial_i + mc^2]\psi =-i\hbar c[\Sigma^ia_i, \gamma^i \partial_i]\psi[/itex]

[itex][\Sigma^i a_i, \gamma^l \partial_l]\psi=[\frac{i}{2}\epsilon^{ijk}\gamma_j\gamma_k a_i, \gamma^l \partial_l]\psi= \frac{i}{2}\epsilon_{ijk}[\gamma^j\gamma^k a^i, \gamma^l \partial_l]\psi = \frac{i}{2}\epsilon_{ijk}(\gamma^j\gamma^k a^i \gamma^l \partial_l-\gamma^l \partial_l\gamma^j\gamma^k a^i ]\psi[/itex]

since [itex]\epsilon^{ijk} = \epsilon_{ijk}(-1)^3[/itex] and [itex]\gamma_i = -\gamma^i,\, a_i = -a^i[/itex] in flat spacetime with (+,-,-,-) signature.

Now suppose [itex]a^i = \partial^i[/itex]. Then using equality of mixed partials and dividing out any constants gives

[itex]\epsilon_{ijk}(\gamma^j\gamma^k\gamma^l \partial_l\partial^i\psi - \gamma^l\gamma^j\gamma^k \partial_l\partial^i \psi)[/itex]

Using the relation [itex]\{ \gamma^\mu,\gamma^\nu \} = 2\eta^{\mu\nu}[/itex] twice and the fact that [itex]\eta^{ij}\partial_i = \partial^j[/itex] in flat spacetime gives

[itex]\epsilon_{ijk}(\gamma^k \partial^j - \gamma^j\partial^k)\partial^i \psi[/itex]

Now, in the sum over k,j the term in brackets is anti-symmetric but so is [itex]\epsilon_{ijk}[/itex] so I don't see why this should vanish.

Any help would be appreciated.

I want to show that [itex]\vec{\Sigma}\cdot \vec{a}[/itex] commutes with the Dirac Hamiltonian only if [itex]\vec{a}\propto \vec{p}[/itex]. As usual I'm using Einstein summation everywhere.

[itex][\vec{\Sigma}\cdot \vec{a}, H_\mathrm{Dirac}]\psi=[\vec{\Sigma}\cdot \vec{a}, -i\hbar c \gamma^i\partial_i + mc^2]\psi =-i\hbar c[\Sigma^ia_i, \gamma^i \partial_i]\psi[/itex]

[itex][\Sigma^i a_i, \gamma^l \partial_l]\psi=[\frac{i}{2}\epsilon^{ijk}\gamma_j\gamma_k a_i, \gamma^l \partial_l]\psi= \frac{i}{2}\epsilon_{ijk}[\gamma^j\gamma^k a^i, \gamma^l \partial_l]\psi = \frac{i}{2}\epsilon_{ijk}(\gamma^j\gamma^k a^i \gamma^l \partial_l-\gamma^l \partial_l\gamma^j\gamma^k a^i ]\psi[/itex]

since [itex]\epsilon^{ijk} = \epsilon_{ijk}(-1)^3[/itex] and [itex]\gamma_i = -\gamma^i,\, a_i = -a^i[/itex] in flat spacetime with (+,-,-,-) signature.

Now suppose [itex]a^i = \partial^i[/itex]. Then using equality of mixed partials and dividing out any constants gives

[itex]\epsilon_{ijk}(\gamma^j\gamma^k\gamma^l \partial_l\partial^i\psi - \gamma^l\gamma^j\gamma^k \partial_l\partial^i \psi)[/itex]

Using the relation [itex]\{ \gamma^\mu,\gamma^\nu \} = 2\eta^{\mu\nu}[/itex] twice and the fact that [itex]\eta^{ij}\partial_i = \partial^j[/itex] in flat spacetime gives

[itex]\epsilon_{ijk}(\gamma^k \partial^j - \gamma^j\partial^k)\partial^i \psi[/itex]

Now, in the sum over k,j the term in brackets is anti-symmetric but so is [itex]\epsilon_{ijk}[/itex] so I don't see why this should vanish.

Any help would be appreciated.

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