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I want to define the following two object with spinor-type indices:

$${\hat{e}}^{\alpha\beta}(x)\equiv e_n^\mu(x) \gamma_n^{\alpha\beta}\partial_\mu$$

$$\omega^{\alpha\beta}(x) \equiv e^\mu_p(x)W_\mu^{nm}(x) (\gamma_p\sigma_{nm})^{\alpha\beta}$$

e.g. the electron action in curved space can be written using these as:

$$\int e\overline{\psi}^{\alpha}(x) (\hat{e}^{\alpha\beta}(x)+\frac{i}{4} \omega^{\alpha\beta}(x))\psi^\beta(x)dx^4$$

Where $W$ is the spin connection and $e$ is the vierbein. I want to construct the scalar curvature tensor from these objects.

My instinct would be to try something like:

$$f_{\alpha \beta \sigma \tau}[\hat{e}^{\alpha\beta}(x) + \frac{i}{4}\omega^{\alpha\beta}(x),\hat{e}^{\sigma\tau}(x) + \frac{i}{4}\omega^{\sigma\tau}(x)]$$

for some set of constants $f$.

Do you think this is possible? Is there an article somewhere with the formula?

(I think you may need to allow also the a version of $\hat{e}$ with the derivative operator applying backwards.)

Edit: I just realised this is somewhat equivalent to defining the covariant derivative:

$\mathbf{\nabla}_\mu = \partial_\mu + e^n_\mu \mathbf{P}_n + W_\mu^{nm}\mathbf{M}_{nm}$

where $P$ and $M$ are the operators for translation and rotation. Hence the answer should be similar to: $[\nabla_\mu, \nabla_\nu]^2$ if I remember rightly. Except my operator is $e^\mu_n \gamma^n \nabla_\mu$. But of course $\nabla e =0$. So it should work. Maybe that helps.

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  • $\begingroup$ 1. What sort of object are $\hat{e}$ and $\omega$ supposed to be? What space do they live in? What motivates their definition? 2. What is $\sigma$? 3. What's going on with the indices and their summation convention there? Some indices are contracted in the canonical upper-lower convention, but there are also indices which seem to be both lower or both upper and yet are summed? 4. Why would your instinct be that instead of taking the expression of the curvature in terms of the vielbein and the spin connection and expressing it in terms of your objects? $\endgroup$ – ACuriousMind Jan 31 at 20:34
  • $\begingroup$ They are operators that live in Clifford space. My motivation is because I'm working on a lattice theory which requires no-vector quantities except derivatives. I need to express things to do with curvature of the lattice in terms of objects within the clifford algebra of dirac matrices. I might have a read of Penrose book on spinor descriptions of General Relativity. $\endgroup$ – zooby Jan 31 at 21:17
  • $\begingroup$ What's "Clifford space"? Do you mean the Clifford algebra? How is $\partial_\mu$ supposed to be an element of that space? $\endgroup$ – ACuriousMind Jan 31 at 21:38
  • $\begingroup$ You're being very pedantic. It says that in the title. $\endgroup$ – zooby Jan 31 at 21:50
  • $\begingroup$ You are asking about relations between mathematical objects. If you do not provide a proper rigorous definition of these objects - or at least a physicist's approximation to one - it is impossible to answer the question as you intend it to be answered. One cannot do mathematics on objects whose definition one does not know. This is not pedantry, but diligence. Again, $\partial_\mu$ is not an element of the Clifford algebra nor a scalar for which multiplication with an element of the algebra would be defined, so it is unclear how $\hat{e}$ is supposed to be one. $\endgroup$ – ACuriousMind Jan 31 at 21:54
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There are already issues in your first two equations, some of which are mentioned in @ACuriousMind comments.

  1. Your definition of $\omega$ has both vierbein and connection. This is especially wrong in Palatini formalism where vierbein and connection are supposed to be independent variables. Also as a result $\omega$ is a (Clifford algebra-valued) scalar, while $\hat e$ is a vector, which would invalidate some of your following equations.

The rest of the issues are relatively minor but they are not just about being pedantic, but about (in)consistency of the notation.

  1. In your definition of $\hat e$ you sum over repeated lower tetrad indices, in your definition of $\omega$ you sum over upper and lower indices.

  2. If you use “hat” to indicate Clifford-algebra valued objects, then $\omega$ also should have it.

  3. $\sigma_{mn}$ is not an element of Clifford algebra (it has extra imaginary unit), so better use $\gamma_{mn}=\gamma_{[m}\gamma_{n]}$, it would avoid propagation of $i$ in equations.

  4. Also for consistency, better use the forms language rather than dealing both with contravariant and covariant components of tensors. This way, the forms would “hide” Lorentzian indices, while the graded vector space structure of Clifford algebra would “hide” the tetrad indices, resulting in a much simpler notation.

Implementing these notation suggestions we would arrive at the language of clifforms, or Clifford algebra valued forms. A good starting point would be a paper:

  • Dimakis, A., & Muller-Hoissen, F. (1991). Clifform calculus with applications to classical field theories. Classical and Quantum Gravity, 8(11), 2093, doi:10.1088/0264-9381/8/11/018.

The paper contains everything that OP was trying to achieve. Here are some equations to illustrate this notation.

In a fairly standard approach we start with coframe $\theta^a$ and connection $\omega^{ab}=-\omega^{ba}$ (ordinary) 1-forms, so that the metric is given by $g\equiv\eta_{ab}\theta^a\otimes\theta^a$. Then we define the clifforms: $$\theta:=\gamma_a\theta^a,\qquad \omega:=\frac14 \gamma_{ab}\omega^{ab} ,$$ ($\gamma_a$ are Clifford algebra generators).

Torsion and curvature 2-forms which in vierbein notation are defined as $$\Theta^a:=d\theta^a+\omega^a{}_b\wedge\theta^b,\qquad \Omega^a{}_b:=d\omega^a{}_b+\omega^a{}_c\wedge\omega^c{}_b,$$ while in the language of clifforms they are written as: $$ \Theta:=D\theta:=d\theta+\omega\wedge\theta+\theta\wedge\omega,\qquad \Omega:=d\omega+\omega\wedge\omega,$$ where $$ \Theta:=\gamma_a\Theta^a,\qquad\Omega:=\frac{1}{4}\gamma_{ab}\Omega^{ab}.$$

Einstein-Hilbert Lagrangian in clifform notation becomes $$ \mathcal{L}_1= \kappa_1 \Omega \cap \theta \wedge \theta, $$ where $\cap$ is an exterior product of Clifford algebra and $\kappa_1$ is a constant. Varying it by regarding $\theta$ and $\omega$ as independent variables would give us standard (vacuum) Einstein field equations (with vanishing torsion).

Note, that EH action is not the only one possible to construct using this language (that would produce equivalent of the standard vacuum EFE). For example, it is possible to construct quadratic spinor Lagrangian:

This Lagrangian differs from EH by a total derivative, but has some nice properties, such as finiteness of the total action for an isolated gravitating body.

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