Spin tensor and Lorentz group operator in bispinor case

Question

For infinisesimal bispinor transformations we have

$$\delta \Psi = \frac{1}{2}\omega^{\mu \nu}\eta_{\mu \nu}\Psi , \quad \delta \bar {\Psi} = -\frac{1}{2}\omega^{\mu \nu}\bar {\Psi}\eta_{\mu \nu}, \quad \eta_{\mu \nu} = -\frac{1}{4}(\gamma_{\mu}\gamma_{\nu} - \gamma_{\nu}\gamma_{\mu}). \qquad (.1)$$ Then, by compairing $(.1)$ with transformation by the generators of the Lorentz group, $$\delta \Psi = \frac{i}{2}\omega^{\mu \nu}J_{\mu \nu}\Psi ,$$ we can make the conclusion that in bispinor representation $$J_{\mu \nu} = -i\eta_{\mu \nu}. \qquad (.2)$$ By the other way, from Noether theorem we can get spin tensor, $$S^{\mu, \alpha \beta} = \frac{\partial L}{\partial (\partial_{\mu}\Psi)}Y^{\alpha \beta} + \bar {Y}^{\alpha \beta}\frac{\partial L}{\partial (\partial_{\mu}\bar {\Psi})}.$$ Then, by having $(.1)$ and Lagrangian $$L = \bar {\Psi}(i \gamma^{\mu}\partial_{\mu} - m)\Psi ,$$ it's easy to show that $$S^{\mu, \alpha \beta} = i\bar {\Psi}\gamma^{\mu}\eta^{\alpha \beta}\Psi .$$ It's clearly that I can get $(.2)$ by $$S^{\alpha \beta} = \int S^{\mu, \alpha \beta}dx_{\mu},$$ but for me it's not obvious how to compute it. Can you help me?

This post imported from StackExchange Physics at 2014-12-14 11:30 (UTC), posted by SE-user PhysiXxx

Answer 1

By Noether’s theorem, the generators of the Lorentz group are the zero components of the currents, i.e., the Lorentz charges:

$$S^{\alpha\beta} = S^{0,\alpha\beta} = i\bar {\Psi}\gamma^{0}\eta^{\alpha \beta}\Psi = \Psi^{\dagger}\eta^{\alpha \beta}\Psi$$

These charges generate the Lorentz transformations on the spinors by the canonical Poisson brackets:

$$\left \{ \Psi, \Psi^{\dagger} \right \}_{P.B.} = -i \mathbb{I}$$

(With all other Poisson combinations vanishing). The Poisson brackets can be obtained from the time derivative term in the Dirac Lagrangian:

$$i \Psi^{\dagger}\partial_0\Psi$$

Which implies that $i \Psi^{\dagger}$ is the canonical momentum of $\Psi$, thus satisfies the canonical Poisson brackets.

The action of the Lorentz charges correctly generates the Lorentz transformation:

$$\delta \Psi = \left \{ \frac{1}{2} \omega_{\alpha\beta }S^{\alpha\beta}, \Psi^{\dagger} \right \}_{P.B.} = \frac{1}{2}\omega^{\alpha \beta}\eta_{\alpha \beta}\Psi$$

This post imported from StackExchange Physics at 2014-12-14 11:30 (UTC), posted by SE-user David Bar Moshe

Comments

"...Which implies that $i\Psi^{+}$ is the canonical momentum of $\Psi$, thus satisfies the canonical Poisson brackets...", - did you make this conclusion by the connection $$L_{d} = \pi \partial_{0}\Psi - H_{d},$$ where $H$ is given from Dirac equation, $$i\partial_{0}\Psi = H\Psi , \quad H = (\gamma \cdot \hat {\mathbf p}) + m, \quad H_{d} = \bar {\Psi} H \Psi ?$$

This post imported from StackExchange Physics at 2014-12-14 11:30 (UTC), posted by SE-user PhysiXxx

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And what physical sense has bracket $$\delta \Psi = \left[\frac{1}{2}\omega_{\alpha \beta}S^{\alpha \beta}, \Psi^{+}\right]_{P.B.}?$$

This post imported from StackExchange Physics at 2014-12-14 11:30 (UTC), posted by SE-user PhysiXxx

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Oh yes, I understand. It is generator of infinitesimal transformations through Poisson's brackets.

This post imported from StackExchange Physics at 2014-12-14 11:30 (UTC), posted by SE-user PhysiXxx

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Unfortunately, it generates false infinitesimal transformations for $i\Psi^{+}$.

This post imported from StackExchange Physics at 2014-12-14 11:30 (UTC), posted by SE-user PhysiXxx

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If the problem is a sign problem, please notice that the $\Psi$s are Grassmann variables and they acquire minus signs when they are commuted.

This post imported from StackExchange Physics at 2014-12-14 11:30 (UTC), posted by SE-user David Bar Moshe

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Maybe, it's such worse. The transformations for $i\Psi^{+}$ can be represented as $$(\delta \Psi )^{+} = \frac{\omega^{\mu \nu}}{2}\Psi^{+}\eta_{\mu \nu}^{+} = \frac{\omega^{\mu \nu}}{2}\Psi^{+}\gamma_{0}\eta_{\mu \nu}\gamma_{0} \qquad (.1)$$ (analogical ways of transforamation's derivation give the same). But then $$[\gamma_{0}, \eta_{\mu \nu}] = \frac{1}{2}(g_{0\nu}\gamma_{\mu} - g_{0\mu}\gamma_{\nu}),$$ and it's convolution with $\omega^{\mu \nu}$ gives $\frac{1}{2}\omega^{\mu 0}\gamma_{\mu}$. So I can't transform $(.1)$ to form which is given by Poisson's brackets.

This post imported from StackExchange Physics at 2014-12-14 11:30 (UTC), posted by SE-user PhysiXxx