Problem with Path Integral Expressions in Peskin And Schroeder Section 9.1

[maverick280857]

Hi again everyone,

I have some doubts about the path integral expressions given in Section 9.1 of Peskin and Schroeder (pg 281 and 282).

For a Weyl ordered Hamiltonian H, the propagator has the form given by equation 9.11, which reads

$$U(q_{0},q_{N};T) = \left(\prod_{i,k}\int dq_{k}^{i}\int \frac{dp_{k}^{i}}{2\pi}\right)\exp{\left[i\sum_{k}\left(\sum_{i}p_{k}^{i}(q_{k+1}^{i}-q_{k}^{i})-\epsilon H\left(\frac{q_{k+1}+q_{k}}{2},p_{k}\right)\right)\right]}$$

Now, as the authors point out, for the case when [itex]H = \frac{p^2}{2m} + V(q)[/tex], we can do the momentum integral, which is (taking the potential term out)<br /> <br /> $$\int\frac{dp_{k}}{2\pi}\exp{\left(i\left[p_k(q_{k+1}-q_{k})-\epsilon\frac{p_{k}^2}{2m}\right]\right) = \frac{1}{C(\epsilon)}\exp{\left[\frac{im}{2\epsilon}(q_{k+1}-q_{k})^2\right]}$$<br /> <br /> where<br /> <br /> $$C(\epsilon) = \sqrt{\frac{2\pi\epsilon}{-im}}$$<br /> <br /> Now, I do not understand how the distribution of factors $C(\epsilon)$ equation 9.13 (given below) comes up. To quote the authors:<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Notice that we have one such factor for each time slice. Thus we recover expression (9.3), in discretized form, including the <i>proper</i> factors of $C$:<br /> <br /> $$U(q_{a},q_{b};T) = \left(\frac{1}{C(\epsilon)}\prod_{k}\int\frac{dq_{k}}{C(\epsilon)}\right)\exp\left[i\sum_{k}\left(\frac{m}{2}\frac{(q_{k+1}-q_{k})^2}{\epsilon}-\epsilon V\left(\frac{q_{k+1}+q_{k}}{2}\right)\right)\right]$$ </div> </div> </blockquote><br /> So, my question is: how did we get this term:<br /> <br /> $$\left(\frac{1}{C(\epsilon)}\prod_{k}\int\frac{dq_{k}}{C(\epsilon)}\right)$$<br /> <br /> PS -- Is it because the momentum index goes from 0 to N-1 and the coordinate index goes from 1 to N-1? The momentum integral product produces N terms, so to write the coordinate integral with the same indexing as before (in the final expression), i.e. k = 1 to N-1, we factor a $C(\epsilon)$ out?<br /> <br /> Also, in equation 9.12,<br /> <br /> $$U(q_{a},q_{b};T) = \left(\prod_{i}\int \mathcal{D}q(t)\mathcal{D}p(t)\right)\exp{\left[i\int_{0}^{T}dt\left(\sum_{i}p^{i}\dot{q}^{i}-H(q,p)\right)\right]}$$<br /> <br /> shouldn't we just write<br /> <br /> $$\left(\int \mathcal{D}q(t)\mathcal{D}p(t)\right)$$<br /> <br /> instead of<br /> <br /> $$\left(\prod_{i}\int \mathcal{D}q(t)\mathcal{D}p(t)\right)$$<br /> <br /> since the $\mathcal{D}$ itself stands for $\prod$?[/itex]

 

[maverick280857]

Another query I have concerns the derivation of equation 9.14, which is

$$\langle \phi_{b}({\bf{x}})|e^{-iHT}|\phi_{a}({\bf{x}})\rangle = \int \mathcal{D}\phi\exp\left[i\int_{0}^{T}d^{4}x \mathcal{L}\right]$$

from

$$\langle \phi_{b}({\bf{x}})|e^{-iHT}|\phi_{a}({\bf{x}})\rangle = \int\mathcal{D}\phi\int\mathcal{D}\pi \exp{\left[i\int_{0}^{T}d^{4}x\left(\pi\dot{\phi}-\frac{1}{2}\pi^2-\frac{1}{2}(\nabla\phi)^2-V(\phi)\right)\right]}$$

To quote the authors,

The integration measure $\mathcal{D}\phi$ in (9.14) again involves an awkward constant, which we do not write explicitly.

Now, I am a bit confused about the meaning of $\mathcal{D}\phi$ in these two equations. In going from the second equation to the first, we evaluate a Gaussian integral over $\pi$. What happens to the constant factor that originates from this step?

Specifically,

$$\int \exp{(-ax^2 + bx + c)} = \sqrt{\frac{\pi}{a}}\exp{\left(\frac{b^2}{4a} +c\right)}$$

and so

$$\int \exp{\int d^{4}x(-a\phi^2 + b\phi + c)} = \sqrt{\frac{\pi}{a}}\exp{\int d^{4}x\exp{\left(\frac{b^2}{4a} +c\right)}$$

 

[maverick280857]

Anyone?

 

[Avodyne]

What happens to the constant factor that originates from this step?

It gets absorbed into the definition of ${\cal D}\phi$, which changes (by that constant factor) from one equation to the next.

 

[maverick280857]

It gets absorbed into the definition of ${\cal D}\phi$, which changes (by that constant factor) from one equation to the next.

I see, thanks Avodyne...I just wanted to confirm that.