Why Representation of Clifford algebra are constant for an orthonormal frame?

Question

Let $e_\alpha$ be a basis of the tangent bundle $TM$ and $ \rho: T_x M \rightarrow \operatorname{End}\left( W\right)$ a representation of a Clifford algebra.

In this text Field theory from a bundle point of view by Laurent Claessens he claims at page 122 that if the frame $e_\alpha$ are orthonormal everywhere, then we have the matricial equality $$ \rho\left(\left(e_\alpha\right)_x\right)_{i j}=\rho\left(v_\alpha\right)_{i j} $$ where $v_\alpha$ is a basis on $\mathbb{R}^n$

Why this claim is true?

Answer 1

As explained in my comment, the claim can only hold with respect to a particular local trivialization of $W$. Suppose you picked such a trivialization, then you can define a representation $\rho$, which is constant with respect to it. Any other representation $\rho'$ will then differ from $\rho$ by an automorphism of $\operatorname{End}(W)$. But the fibers of $\operatorname{End}(W)$ are just matrix algebras, and by the Skolem-Nöther theorem any such automorphism is inner, i.e., induced by a transformation of the fiber of $W$. So $\rho$ and $\rho'$ will differ by a change of trivialization of $W$. I invite you consider the exercise of showing that the change of trivialization is continuous/smooth when $\rho'$ is.