#### selfAdjoint

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In J.W. van Holten's essay

Anybody know?

*Aspects of BRST Quantization*,hep-th/0201124, he begins (Section 1.1) with the free relativistic particle. In order to derive the dynamics from the action principle he introduces an auxiliary variable:a bit later he notesIn addition to the co-ordinates [itex]x^{mu}[/itex], the action depends on an auxiliary variable e; it reads

[tex] S[x^{\mu};e] = \frac{m}{2} \int_1^2 (\frac{1}{e} \frac{dx_{\mu}}{ed\lambda}\frac{dx^\mu}{d\lambda} - ec^2) d\lambda [/tex]

I thought I was really familiar with the relativistic free particle, but I've never encountered this einbein before. I can't make out from the rest of his discussion just why it's there. It's a constraint on the action to be sure, and that is very apposite to his future development of BRST symmetry. But why have it at all?...the action is invariant under a change of parametrization of the real interval [itex]\lambda \rightarrow \lambda'(\lambda)[/itex], if the variables [itex] (x^{\mu},e)[/itex] are transformed simultaneously to [itex](x'^{\mu},e')[/itex] according to the rule

[tex]x'^{\mu}(\lambda') = x^{\mu}(\lambda), e'(\lambda')d\lambda' = e(\lambda)d\lambda [/tex]

Thus the co-ordinates [itex]x^{\mu}[/itex] transform as scalar functions on the real line [itex]\mathbf{R}^1[/itex], whilst [itex]e(\lambda)[/itex] transforms as the (single) component of a covariant vector (1-form) in one dimension. For this reason it is often called theeinbein.

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