Relativistic particle phase-space action ($I=\int dt[p_m{\dot{x}}^m-\frac{e}{2}(p^2+M^2)]$)

The worldline einbein imposes the relativistic mass-shell condition. With the Minkowski metric of signature $(-,+,\dots ,+)$, its first-class constraint is $\phi =(p^2+M^2)/2$.

A pseudoclassical spinning particle supplements ordinary position and momentum by odd Grassmann variables. An even multiplier imposes $p^2=0$ and an odd multiplier imposes $p\cdot \zeta =0$. The odd symplectic term gives $\{{\zeta }^m,{\zeta }^n{\}}_D=-i{\eta }^{mn}$. Quantization represents these variables by a Clifford algebra, producing a spinor wavefunction.

Quantizing the odd Dirac brackets yields $\{{\hat{\zeta }}^m,{\hat{\zeta }}^n{\}}_{+}={\eta }^{mn}$. The constraint $p\cdot \zeta =0$ is then the massless Dirac equation on a spinor. Its square enforces $p^2=0$, matching the even constraint. The spin bilinear is represented by $-i[{\gamma }^m,{\gamma }^n]/4$ in this action convention.

The odd particle variables form an even antisymmetric tensor through their bilinear. Adding it to orbital angular momentum gives a conserved Lorentz Noether charge. The particle and odd-variable equations make the two time derivatives cancel. After spinning-particle Dirac quantization, the bilinear becomes the spinor Lorentz generator rather than an ordinary commuting classical spin vector.

With $d{s}^2=-2d{x}^{+}d{x}^{-}+d{x}_i^2$, $x^{+}=t$, and $p_{-}\ne 0$, the mass-shell condition gives $H=-p_{+}=-(p_i^2+{\mu }^2)/(2p_{-})$. In the future-directed sector $p^{+}=-p_{-}>0$, this is $(p_i^2+{\mu }^2)/(2p^{+})$. Its Schrödinger equation is equivalent to the Klein-Gordon equation after multiplication by $2p_{-}$.

An einbein specifies a one-dimensional metric tensor along a worldline. In the relativistic particle phase-space action it is the Lagrange multiplier for the mass-shell condition and transforms as $\delta e=\dot{ϵ}$ under the corresponding canonical gauge transformation. After eliminating the momentum, its geometric interpretation depends on the normalization of the particle action.

For the relativistic particle action $S=(m/2)\int [g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }/e-e]d\lambda$, varying the worldline einbein gives $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-e^2$. On the positive-einbein branch the proper time satisfies $ds=ed\lambda$. A reparametrization with ${\lambda }^{\prime }=s$ sets the transformed einbein to one, making the geodesic equation affinely parametrized. This gauge choice is made after retaining the einbein mass shell constraint; fixing the action first would discard that equation. If the parameter interval is held fixed at both ends, its integrated proper length is a modulus and cannot be arbitrarily changed by endpoint-preserving transformations.

The constant mode of the worldline einbein is not removed by gauge transformations whose parameters vanish at the ends of a fixed interval. This Schwinger proper-time parameter remains integrated over after gauge fixing. For $\phi =(p^2+M^2)/2$, the geometric on-shell proper length is $M\int edt$.

On a unit interval with gauge parameters vanishing at its ends, $e=s+\dot{ϵ}$ separates the worldline einbein into its invariant average $s$ and a canonical gauge transformation. The Jacobian for nonconstant modes is $\det {\partial }_t$ between the appropriate boundary-condition spaces. On a circle its prime excludes the constant gauge-parameter zero mode. This Faddeev-Popov determinant can be represented by anticommuting $b,c$ fields with an action proportional to $\int b\dot{c}dt$.