# Clifford algebra formulation of the Nambu-goto action

Using the wedge product one can pair the generators the Clifford algebra $$Cl_{1,3}(\mathbb{R})$$ to produce 2-vectors (area elements).

The Nambu-Goto action is a statement on the evolution of invariant area elements of the space-time metric.

Is it appropriate to use these 2-vectors to formulate the Nambu-Goto action.

The possible 2-vectors of $$Cl_{1,3}(\mathbb{R})$$ are:

$$\gamma_0\gamma_1,\; \gamma_0\gamma_2,\; \gamma_0\gamma_3,\; \gamma_1\gamma_2,\; \gamma_1\gamma_3,\; \gamma_2\gamma_3$$

The first three bi-vectors are wedge products involving time $$t$$ wedge a position $$x,y$$ or $$z$$. Intuitively they appear relevant to the Nambu-Goto action.

Can we obtain the Nambu-Goto action using the Lagrangian of an invariant sum of area elements:

$$S=\int \mathrm dt\, \mathrm dx\, \mathrm dy\,\mathrm dz \sqrt{\left( A(t,x) \gamma_0 \wedge \gamma_1 \right)^2 + \left( B(t,y) \gamma_0 \wedge \gamma_2 \right)^2 + \left( C(t,z) \gamma_0 \wedge \gamma_3 \right)^2}$$

Or, perhaps one should use another combination of 2-vectors?

• What is the square root of a Clifford matrix, what is the motivation for randomly sticking in Clifford elements into a square root of a well-defined quantity, what would happen if you did this in a QFT context for a point particle. Aug 25 '19 at 19:27

OP considers a field theory in 3+1D, while the Nambu-Goto string action is a field theory in 1+1D. OP probably has a $$p$$-brane action $$S_p~=~-T_0\int d^{p+1}\sigma ~\sqrt{-\det\left(\partial_{\alpha} X\cdot \partial_{\beta} X\right)_{\alpha\beta}}$$ in mind. FWIW, as long as spinor variables doesn't enter the theory, it seems likely that Clifford calculus and gamma matrices are not needed.