mathematical physics - What's the point of having an einbein in your action?

One often comes across actions written with an extra auxiliary field, with respect to which if you vary the action, you get the equation of motion of the auxiliary field, which when plugged into the original action lets you retrieve a more familiar looking action without the auxiliary field. An example is

$\begin{equation*} S=\int d\tau e^{-1}(\tau)\eta_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau} \end{equation*}$

which when solved for the einbein $e$ gives you the familiar square root form of the action. Why would you want this extra auxiliary field apart from having an action which is not in an annoying square root form? Why, for example, does GSW mention on Page 18 of "Superstring Theory" that "the role of $e(\tau)$ is to ensure the action is invariant under reparametrizations of $\tau$" when you can have reparametrization invariance even in the square root form of the action?

Answer

first, you and Moshe may have somewhat misunderstood the comment that "the purpose of introducing $e$ is to guarantee the reparameterization symmetry of the action". This wasn't meant to express the circular argument that we need to add a redundant field in order to have a redundancy.

Instead, I think that the authors simply wanted to write that "in order for this action - which we already chose to study - to be equivalent to the einbein-frei form, the new field $e$ has to be unphysical, so the reparameterization has to be insured, so the factor of $e$ has to be included." If the $e$ were simply omitted in this form of the action, the action wouldn't be reparameterization-invariant, so it wouldn't be equivalent to the "proper length" of the world line we want to describe: the proper length of a world line is independent of the way how we parameterize the world line to calculate the integral (the proper length), so any equivalent description has to have the same property.

Motivation for gauge redundancies

In this particular case, the main technical reason is that by introducing the new auxiliary variable, the einbein, you get rid of the square root. Note that without this variable, the action would depend on the standard Lorentz gamma factor, $1/\sqrt{1-v^2/c^2}$. From the Lagrangian, it would imprint itself into the Hamiltonian as well. Hamiltonians that depend on square roots of functions of observables are hard - relatively to what you get with $e$. In the world sheet generalization, the Hamiltonian would even include an integral of a square root which is really bad - especially if you want to Fourier-transform things.

On the other hand, the underlying physics actually doesn't need the messiness as can be seen by introducing clever redundancies. There is no square root in the action including the einbein - but it's still equivalent. By introducing this fake degree of freedom, or a redundancy, we actually convert the equations of motion for $x$ to a simple wave equation, $\nabla_\mu\nabla^\mu x^\alpha=0$. Well, it's a wave equation in at least $1+1$ dimensions. In $0+1$ dimensions, it's just an equation for a uniform motion of $x^\alpha$.

Case of other gauge symmetries

The reparametrization of $\tau$ is a redundancy of the system, but so is any gauge symmetry, for example the colorful $SU(3)$ in QCD or the diffeomorphism group in general relativity (which is nothing else a higher-dimensional generalization of this einbein example: $e(\tau)\equiv \sqrt{g_{00}}$ is the redundant field, and gravity has no physical components in $0+1$ dimensions). You could also work without it - and the twistor description of $N=4$ $d=4$ $SU(N)$ gauge theory actually has no locality so it has no local gauge symmetry in spacetime. Other descriptions that don't have any gauge redundancy include the unitary gauge in spontaneously broken gauge theories; AdS/CFT that replaces the redundancy by a totally different one in the dual description; the S-dual description that replaces the gauge group with the magnetic monopoles' dual gauge group; and many others.

The purpose of having local symmetries - or gauge redundancies - is always either to simplify the equations or to make some of their symmetries manifest.

In particular, if you want elementary particles with spin equal to one or higher, and if you want a manifest Lorentz symmetry in spacetime and manifest locality, you need local vector- or tensor-valued fields such as $A_\mu$ and $g_{\mu\nu}$ that also have time-like components. In a quantum theory, the time-like components would produce negative-norm states: the theory could predict negative probabilities if these polarizations would stay in the spectrum.

Of course, gauge invariance is the only consistent way to remove them. Because gauge symmetry is a "symmetry" in the sense that it commutes with the Hamiltonian, the gauge invariance of the initial state guarantees the gauge invariance of the final state, so unphysical, gauge-dependent states are not produced by the evolution.

At any rate, the choice of a redundant equivalent description is always made in order to make some or several of the properties below manifest:

1. Unitarity combined with manifest Lorentz invariance: gauge symmetry is needed to remove the ghosts


2. Locality - one needs local tensor fields


3. Simple, harmonic-oscillator-like underlying character of the dynamics that would be masked by square roots in the einbein-frei formulation (typical in the case of diffeomorphisms of the world volumes)

Cheers, LM