Sign of the time derivative of the Majorana Lagrangian

[JustMeDK]

Let $\gamma^{\rho} \in M_{4}(\mathbb{R})$ be the Majorana representation of the Dirac algebra (in spacetime signature $\eta_{00} = -1$), and consider the Majorana Lagrangian $$\mathcal{L} = \mathrm{i} \theta^{\mathrm{T}} \gamma^{0} (\gamma^{\rho} \partial_{\rho} - m) \theta,$$ where $\theta$ is a Grassmann-valued four-spinor. The associated gravitational energy-density, the 00-component of the Belinfante energy-momentum tensor, I calculate to be $$\Theta^{00} = \frac{\mathrm{i}}{2} [ \theta^{\mathrm{T}} (\partial^{0} \theta) - (\partial^{0} \theta)^{\mathrm{T}} \theta].$$ Inserting into it the stationary plane, wave solution $\theta = \mathrm{exp}(\gamma^{0}Et)\eta$, where $\eta$ is some spacetime-independent, Grassmann-valued four-spinor, yields $\Theta^{00} = \mathrm{i} E \eta^{\mathrm{T}} \gamma^{0} \eta$. Due to $(xy)^{*} \equiv y^{*}x^{*}$ for Grassmann-valued quantities, this expression for $\Theta^{00}$ is complex self-conjugate (and nonvanishing), as it should be, but it is not real-valued.

In comparison, for the Dirac Lagrangian, $$\mathcal{L}_{D} = -\mathrm{i} \psi^{\dagger} \gamma^{0} (\gamma^{\rho} \partial_{\rho} - m) \psi,$$ also in spacetime signature $\eta_{00} = -1$, a similar calculation of the gravitational energy-density yields for a plane wave solution $\psi = \mathrm{exp}(-\mathrm{i}Et)\psi_{0}$ the real-valued expression $\Theta^{00} = E \psi_{0}^{\dagger} \psi_{0}$. The exact sign of $-\mathrm{i} \psi^{\dagger} \gamma^{0} \gamma^{0} \partial_{0} \psi = +\mathrm{i} \psi^{\dagger} \partial_{0} \psi$ in $\mathcal{L}_{D}$ is essential for this energy-density to be positive-definite.

And now to my question: Is it nonsensical to analogously contemplate what the sign of the time derivative should be in the case of the Majorana Lagrangian? And if not, what is it?

 

[JustMeDK]

Extracting observables

Perhaps more concretely the following general question is what I am asking: How are any observables - ordinary, real numbers - to be extracted from a classical (i.e., non-quantum) theory that uses Grassmann numbers?

Using complex self-conjugate quantities like $\mathrm{i}\theta_{1}\theta_{2}$, say, where $\theta_{1},\theta_{2}$ are Grassmann numbers, is not a solution, for even though such a product does commute with everything, it is not an ordinary, real number, because it squares to zero.