Correlators in Proper Time Gauge


As in Halliwell 1988, we can gauge-fix to proper time at the level of the action, although as demonstrated above it seems that the correlators will depend on the gauge fixing. Nevertheless we can compute the correlators with this understanding, writing

\begin{array} {l c l} $\left< a \left( t_{1} \right) a \left( t_{2} \right) \right>_{ \chi = 0 }$ & = & $\int \mathcal D \mu \ a \left( t_{1} \right) a \left( t_{2} \right)$ \\ \ \ & \ \ & \ \ $\times \left \exp \left( \frac{i}{\hbar} \int^{ t''}_{t'} dt \left( p \dot a - NH + \Pi \left( \dot N - \chi \right) + \bar \rho \dot c + \bar c \dot \rho - \bar \rho \rho + c \{ \chi , H \} \bar c + \rho \frac{\partial \chi}{\partial N} \bar c \right) \right) \right| _{\chi = 0}$ \\ \ \ & = & $\int \mathcal D \mu \ a \left( t_{1} \right) a \left( t_{2} \right) \exp \left( \frac{i}{\hbar} \int^{t''}_{t'} dt \left( p \dot a - NH + \Pi \dot N + \bar \rho \dot c + \bar c \dot \rho - \bar \rho \rho \right) \right)$ \\ \ \ & = & $\int \mathcal D p \mathcal D a \mathcal D \Pi \mathcal D N \left\{ a \left( t_{1} \right) a \left( t_{2} \right) \exp \left[ \frac{i}{\hbar} \int^{t''}_{t'} dt \left( p \dot a - NH + \Pi \dot N \right) \right] \right\}$ \\ \ \ & \ \ & \ \ $\times \int \mathcal D \rho \mathcal D \bar c \mathcal D \bar \rho \mathcal D c \left\{ \exp \left[ \frac{i}{\hbar} \int^{t''}_{t'} dt \left( \bar \rho \dot c + \bar c \dot \rho - \bar \rho \rho \right) \right] \right\}$ \end{array}

From Halliwell, the ghost integral has value $\left( t'' - t \right)$, and performing the integral over $\Pi$ has the result of reducing to ordinary integration over $N$. Thus we are left with

\begin{align} \int dN \left( t'' - t' \right) \int \mathcal D p \mathcal D a \ a \left( t_{1} \right) a \left( t_{2} \right) \exp \left[ \frac{i}{\hbar} \int^{t''}_{t'} dt \left( p \dot a - NH \right) \right]. \end{align}

Reparametrize the path $a(t)$ with the parameter $\sigma = N \left( t - t' \right)$, and define $T = N \left( t - t' \right)$. Using the fact that $\frac{da}{dt} = \frac{da}{d \sigma} \frac{d \sigma}{dt} = N \frac{da}{d \sigma}$, we can rewrite the above path integral as

\begin{align} \int dT \int \mathcal D p \mathcal D a \ a \left( r_{1} T \right) a \left( r_{2} T \right) \cdot \exp \left[ \frac{i}{\hbar} \int^{T}_{0} d \sigma \left( p \dot a - H \right) \right] \end{align}

where $r_{1} = \frac{t_{1} - t'}{t'' - t'}$ and $r_{2} = \frac{t_{2} - t'}{t'' - t'}$.

Note, then, that we are actually computing the correlator at the two points in proper time determined by the elapsing of fixed ratios of the total (proper) time.

A question about this form of the correlators is whether the usual proof for the path integral expression of correlators goes through when ghost integration is involved. The usual proof for the phase space path integral form of the correlators depends on a resolution-of-identity rewriting of the (canonical) multiplication operators $\hat a \left( t_{1} \right)$ and $\hat a \left( t_{2} \right)$, and on the composition law for phase space path integrals:

\begin{align} \int ^{q(t'') = q''}_{q(t') = q'} \mathcal D [ q(t) ] \mathcal D [ p(t) ] = \int ^{ \infty}_{ - \infty} d q_{1} \int ^{q_{II}(t'') = q''}_{q_{II}(t_{1}) = q_{1}} \mathcal D [ q_{II}(t) ] \mathcal D [ p_{II}(t) ] \int ^{q_{I}(t_{1}) = q_{1}}_{q_{I}(t') = q'} \mathcal D [ q_{I}(t) ] \mathcal D [ p_{I}(t) ] \end{align}

so that we can write

\begin{align} \left< q'', t'' \left| \hat q \left( t_{1} \right) \right| q', t' \right> = \int^{ \infty}_{ - \infty} dq_{1} \left< q'', t'' \left| q_{1}, t_{1} \right> q_{1} \left< q_{1}, t_{1} \left| q', t' \right>. \end{align}

But it is unclear whether this should hold when the propagators involve integration over ghost fields, since in that case the propagator is defined as

\begin{align} G_{ \chi } \left( a'' \left| a' \right \right) = \int \mathcal D \mu \ \exp \left\{ \frac{i}{\hbar} \int_{t'}^{t''} dt \left[ p_{\alpha} \dot q ^{\alpha} - NH + \Pi \left( \dot N - \chi \right) +\bar \rho \dot c + \bar c \dot \rho - \bar \rho \rho + c \{ \chi , H \} \bar c + \rho \frac{\partial \chi}{\partial N} \bar c \right] \right\} \end{align}

with the restrictions $c \left( t' \right) = 0 = c \left( t'' \right)$ and $\bar c \left( t' \right) = 0 = \bar c \left( t'' \right)$, and if we naively break this at a time $t_{1}$ intermediate between $t'$ and $t''$, it doesn't seem that the boundary conditions on $c$ and $\bar c$ should hold at $t_{1}$, so that the composition law does not hold at least in the obvious way.

The upshot of this and the previous section is that constructing correlators in a path integral format taking into account the gauge freedom is problematic, in particular because of the composition law. However there are some possible remedies for this. There are some papers about what happens to the composition law in a reparametrization invariant setting:



and also there is the option to work, as Varadarajan does, not with the lapse-function expression for parametrization, but with embedding variables, which allow one to put the constraint in a form which does not involve the Lagrange multiplier and thus circumvents the necessity of ghost integration.

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