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.