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

Question

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.