Effect of Gauge Choice on Correlators


A BRST transformation acting on the coordinate $q$ gives $\delta q = \Lambda c \frac{\partial H}{\partial p}$. So in the case of FRW, we have $\frac{\partial H}{\partial p} = - \frac{p}{24 \pi^{2} a}$ so the BRST transformation gives $\tilde a = a - \Lambda c \frac{p}{24 \pi^{2} a}$, so

\begin{align} \langle a \left( t_{1} \right) a \left( t_{2} \right) \rangle_{ \tilde \chi } =& \int \mathcal D \tilde \mu \ \ \tilde a \left( t_{1} \right) \tilde a \left( t_{2} \right) \exp \left( \frac{i}{\hbar} S_{T} \right) \Bigg|_{ \tilde \chi} \\ =& \int \mathcal D \mu \exp \left( \frac{i}{\hbar} \int dt \{ \bar c \left( \tilde \chi - \chi \right) , \Omega \} \right) \left( a \left( t_{1} \right) - \Lambda c \frac{p \left( t_{1} \right)}{24 \pi^{2} a \left( t_{1} \right)} \right) \left( a \left( t_{2} \right) - \Lambda c \frac{p \left( t_{2} \right)}{24 \pi^{2} a \left( t_{2} \right)} \right) \exp \left( \frac{i}{\hbar} S_{T} \right) \Bigg|_{\tilde \chi} \\ = & \int \mathcal D \mu \left( a \left( t_{1} \right) - \Lambda c \left( t_{1} \right) \frac{p \left( t_{1} \right)}{24 \pi^{2} a \left( t_{1} \right)} \right) \left( a \left( t_{2} \right) - \Lambda c \left( t_{2} \right) \frac{p \left( t_{2} \right)}{24 \pi^{2} a \left( t_{2} \right)} \right) \exp \left( \frac{i}{\hbar} S_{T} \right) \Bigg|_{ \chi} \\ = & \left< a \left( t_{1} \right) a \left( t_{2} \right) \right>_{ \chi } + \left< \left( - \frac{ \Lambda }{24 \pi^{2}} \left( \frac{c \left( t_{2} \right) p \left( t_{2} \right) a \left( t_{1} \right) }{ a \left( t_{2} \right)} + \frac{c \left( t_{1} \right) p \left( t_{1} \right) a \left( t_{2} \right) }{a \left( t_{1} \right) } \right) + \frac{\Lambda^{2} c \left( t_{1} \right) c \left( t_{2} \right) }{24^{2} \pi^{2}} \frac{p \left( t_{1} \right) p \left( t_{2} \right) }{a \left( t_{1} \right) a \left( t_{2} \right)} \right) \right>_{ \chi } \end{align}

where $\Lambda = -\frac{i}{\hbar} \int dt \ \bar c \left( \tilde \chi - \chi \right)$.

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