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  Two Questions about Path Integral from “Gauge Fields and Strings” by Polyakov

+ 1 like - 0 dislike
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My questions are about worldline path integrals from the book Gauge Fields and Strings of Polyakov.

On page 153, chapter 9, he says

>Let us begin with the following path integral
\begin{align}
&\mathscr{H}(x,y)[h(\tau)]=\int_{x}^{y}\mathscr{D}x(\tau)\delta(\overset{\,\centerdot}{x}{}^{2}(\tau)\boldsymbol{-}h(\tau)) 
\nonumber\\
&=\int\mathscr{D}\lambda(\tau)\exp\left(i\int_{0}^{1}d\tau\lambda(\tau)h(\tau)\right)\int_{x}^{y}\mathscr{D}x(\tau)\exp\left(-i\int_{0}^{1}d\tau\lambda(\tau)\dot{x}^{2}(\tau)\right) \tag{9.8}\label{9.8} 
\end{align}
where $h(\tau)$ is the worldline metric tensor. 

>The action in (9.8) is invariant under reparametrizations, if we transform:
\begin{align}
x(\tau)&\rightarrow x(f(\tau)) \nonumber\\
h(\tau)&\rightarrow\left(\frac{df}{d\tau}\right)^{2}h(f(\tau)) \tag{9.9}\label{9.9}\\
\lambda(\tau)&\rightarrow\left(\frac{df}{d\tau}\right)^{-1}\lambda(f(\tau))
\end{align}

Polyakov continued with the following statement.

>It is convenient to introduce instead of the worldline vector $\lambda(\tau)$, the worldline scalar Lagrange multiplier $\alpha(\tau)$:
\begin{align}
\lambda(\tau)&\equiv\alpha(\tau)h(\tau)^{-1/2} \nonumber\\
\alpha(\tau)&\rightarrow\alpha(f(\tau)) \tag{9.11}
\end{align}
So that:
\begin{align}
&\mathscr{H}(x,y)[h(\tau)] \nonumber\\
&=\int\mathscr{D}\alpha(\tau)e^{i\int_{0}^{1}d\tau\alpha(\tau)\sqrt{h(\tau)}}\int_{x}^{y}\mathscr{D}x(\tau)\exp\left(-i\int_{0}^{1}d\tau\frac{\alpha(\tau)\dot{x}^{2}(\tau)}{\sqrt{h(\tau)}}\right) \tag{9.12}
\end{align}

My first question is about equation (9.12). What Polyakov did there is boldly replace the integral measure $\mathscr{D}\lambda$ by $\mathscr{D}\alpha$. Didn't he miss the Jacobian factor?

$$\mathscr{D}\lambda=\mathscr{D}\alpha\det\left(\frac{\delta\lambda}{\delta\alpha}\right)$$

My second question is the following.

He introduced another parameter $t$, called proper time, defined as

>\begin{align}
t\equiv\int_{0}^{\tau}\sqrt{h(s)}ds;\quad T\equiv t(1) \tag{9.13}
\end{align}
and so
\begin{align}
&\mathscr{H}(x,y)[h(\tau)]\equiv\mathscr{H}(x,y;T) \nonumber\\
&=\int\mathscr{D}\alpha\exp i\int_{0}^{T}\alpha(t)dt\int_{x}^{y}\mathscr{D}x\exp-i\int_{0}^{T}\alpha(t)\dot{x}^{2}(t)dt \tag{9.14}
\end{align}

Can anybody tell me how he derived the equation (9.14) via using the "proper time" parameter $t$?

I also posted my question here here.

asked Feb 14 in Theoretical Physics by New Student (235 points) [ revision history ]
edited Feb 14 by New Student

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