Computing harmonic oscillator propagator via path integral

[naele]

Homework Statement

Show that
$$G(q_2,q_1;t)=\mathcal{N}\frac{e^{iS_{lc}}}{\sqrt{\det A}}$$
where $\mathcal{N}$ is a normalization factor independent of q1, q2, t, and w. Using the known case of w=0, write a formula for G such that there is no unknown normalization factor.

Homework Equations

I previously showed that
$$G(q_2,q_1;t)=e^{iS_{cl}/\hbar}\int \mathcal{D}xe^{iS[x]/\hbar}$$

verified that $(x_1,x_2)=\int_0^Tx_1(t)x_2(t)dt$ was a scalar product over the real vector space of trajectories x(t) and showed that $S[x]=-\frac{m}{2}(x,A\cdot x)$ where
$$A=\frac{d^2}{dt^2}+\omega^2$$
and showed that A was self-adjoint. We're also given the following identity
$$\int\! dX\, e^{-\frac{1}{2}(X,A\cdot X)+(J,X)}=\frac{(2\pi)^{n/2}}{\sqrt{\det A}}e^{\frac{1}{2}(J,A^{-1}\cdot J)}$$
where n is the dimension of the of the vector space.

The Attempt at a Solution

If I just blindly apply the identity that's given to us, I get confused about the exponent of the prefactor. I know that it should simply be a square root, but when I apply the identity (with J=0) I get N/2 ie
$$G(q_2,q_1;t)=\left(\frac{2\hbar\pi}{im}\right)^{N/2}\frac{e^{iS_{cl}}}{\sqrt{\det A}}$$
I could collect all that into the normalization factor and get the desired expression anyway but that makes me a little uncomfortable.

As far as the 2nd part I'm not really sure if the problem is asking me to deduce the harmonic oscillator propagator from the free-particle or whatever. I'm confused because in the next question we're asked to find the eigenvectors and eigenvalues of A, and then from there calculate G. At least that part seems really straightforward, especially using the Fourier series method from Feynman & Hibbs.

Thanks for any help.

 

[fzero]

I think that part of your confusion is that you can't just "blindly apply" the identity

$$\int\! dX\, e^{-\frac{1}{2}(X,A\cdot X)+(J,X)}=\frac{(2\pi)^{n/2}}{\sqrt{\det A}}e^{\frac{1}{2}(J,A^{-1}\cdot J)}$$

This is an integral over a vector $$X$$ of scalars, whereas the propagator involves an integral over paths. You can make a connection between the two by defining a lattice, but then the normalization factor is only defined up to the limit in which the lattice spacing goes to zero and $$N\rightarrow\infty$$. I believe the 2nd part of the problem involves either defining this limit or using some previously known result to rewrite it. I'm not sure what "known case" you've covered so far though.

 

[naele]

Yea, I was thinking of doing something like this
$$\int\!\mathcal{D}x\, e^{iS[x]/\hbar}&=\lim_{N\to\infty} B^{(N-1)/2}\int\,\mathcal{D}x\,\exp \left[-\frac{m}{2i\epsilon\hbar}\sum_{i=0}^{N-1}[(x_{i+1}-x_i)^2-\epsilon^2\omega^2x_i^2] \right]$$
where B is an unknown prefactor to be determined. I can then introduce a a vector $X=(x_1,\cdots,x_{N-1})$. The only problem I see is a discretized version of A, but I might just wave my hands over that. Then, if I apply the identity I can collect everything into one factor independent of everything.

As for the second part, the only other known case we've seen is the free particle. So I suppose if I naively introduce the same pre-factor instead of B like I have up there, and apply the same procedure I can get the harmonic oscillator result (but with undetermined det A).

Then the next question after that asks to determine the eigenvalues of A so getting the full expression should come right out, I hope.