# A Question about Path-Integral Measure

+ 2 like - 0 dislike
226 views

I want to do the following path integral.

$$\mathcal{Z}=\int\mathcal{D}x e^{iS[\dot{x}]}$$

The action only denpends on $\dot{x}$. For some reason, I want to replace the integral measure $\mathcal{D}x$ by $\mathcal{D}\dot{x}$.

So I have

$$\mathcal{Z}=\int\mathcal{D}\dot{x}\mathrm{Det}\left(\frac{\delta x}{\delta\dot{x}}\right)e^{iS[\dot{x}]}.$$

The variable $x$ is related with $\dot{x}$ via the linear transformation

$$x(t)=\int_{0}^{t}\dot{x}(s)ds,$$

which implies

$$\mathrm{Det}\left(\frac{\delta x}{\delta\dot{x}}\right)\equiv 1.$$

Am I correct in the above derivation?

No.

@arnold neumaier Can you elaborate?

+ 1 like - 0 dislike

''which implies'' does not follow. Rewrite the derivative as an operator applied to $x$, and perform the functional differentiation by working out the virtual displacement to get the correct answer.

answered Feb 13 by (14,019 points)

Is $\det(\frac{\delta x}{\delta\dot{x}})=\frac{1}{\det(\frac{d}{dt})}$ the correct answer?

Formally yes, but the result is not well-defined without regularization since the determinant of an operator with continuous spectrum makes no sense. On the other hand, the regularized determinant is a constant, hence can be absorbed in the normalization of the functional integral.

Thank you very much. Would you please look at another question I posted here?

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysics$\varnothing$verflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.