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I am trying to do this functional integral:

$$F_{2D}[\alpha,\phi] = \int\limits_{ \Phi(\cos(\theta),\sin(\theta))=\phi(\theta)} \exp \left( i\alpha\int\limits_{x^2+y^2<1}\Phi(x,y)(\partial_x ^2 + \partial_y ^2)\Phi(x,y) dx dy \right) D\Phi$$

In other words it is a functional integral where the boundary is a circle which and the field has values $\phi(\theta)$ on the boundary. Inside the circle the fields can take on any values.

I would imagine this would correpond to some kind of "quantum drum". The answer would depend only on the constant $\alpha$ and the height of the drum on the boundary $\phi(\theta)$. I feel like this should be solvable interms of elementary(ish) functions.

This is a generalisation of the simpler example of a the 1D case:

$$F_{1D}[\alpha,A,B] = \int\limits_{\Phi(0)=A}^{\Phi(1)=B} \exp \left( i\alpha\int\limits_{0}^{1}\Phi(x)\partial_x ^2 \Phi(x) dx \right) D\Phi = \sqrt{\alpha/2\pi i}\exp(-\alpha(A-B)^2/2i)$$

But I can't find if the 2D case this been solved anywhere or even how to begin? If not a circular drum, maybe a square drum would be easier?

Actually I do have one idea and that is to expand the fields $\Phi(x,y)=\sum a_{nm}(\phi) H_{nm}(x,y)$ into a set of orthogonal functions that satisfy the boundary conditions. But that is as far as I got. In other words it would be summing over all vibration modes of a circular membrane.

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    Have you solved the classical equations of motion with your given boundary conditions? If yes, you could then proceed as follows. Expand $\Phi=\Phi_{\rm cl}+\tilde{\Phi}$ in the path integral, identify $\Phi_{\rm cl}$ with the classical solution, and $\tilde{\Phi}$ with quantum fluctuations that are in turn required to vanish at the boundary (so that $\Phi$ satisfies the specified BC's). Pick convenient coordinates (e.g., polar), mode expand $\tilde{\Phi}$ in a complete set, Wick rotate, integrate, and evaluate resulting infinite products with (e.g.) zeta function regularisation. – Wakabaloola Feb 22 '21 at 10:41
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    see, e.g., the DETAILS section in https://physics.stackexchange.com/a/384919/83405 for an elaboration on each of these steps in a similar context. incidentally, i'm not sure i see any dynamics; where is time? so i don't see the sense in which this is really a "quantum drum". I would expect you need at least 3 dimensions for that problem. – Wakabaloola Feb 22 '21 at 10:46
  • @Wakabaloola You are right there are no dynamics since this is 2D Euclidean space-time. I am mainly doing it as an exercise in ways to solve path integrals with boundaries. One could Wick rotate one dimension I guess. I think this would be classified as topological field theory perhaps. One could call this a topological quantum drum. –  Feb 22 '21 at 16:29
  • I was reading Rovellis Quantum Gravity book about functional integrals with arbitrary boundaries but he doesn't offer any examples of solvable cases except for the infinite strip. I will check out your link –  Feb 22 '21 at 16:37
  • I tried something similar in the 1D case but wasn't sure if the whole function $\Phi$ had to be expanded in a complete set or just the "fluctuations" $\tilde \Phi$. In the 1D case I get $\Phi(x) = A + x(B-A) + \tilde \Phi$ which I then tried to expand in terms of sines. Does this seem right? –  Feb 22 '21 at 16:40
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    Yes, exactly, that sounds right in the 1D case – Wakabaloola Feb 22 '21 at 16:49
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    I think you can also think of it as a 0-pt disc amplitude in $c=1$ string theory. Although strictly speaking you would really need to add a few more ingredients to get a well-defined theory on arbitrary surfaces (such as a Liouville scalar of appropriate central charge so that the classical Weyl symmetry that is evidently present is not anomalous, as well as Fadeev-Popov ghosts). Because if you can have discs, then why not annuli, or why not other surfaces; ideally you want the theory to make sense on all of them since there is nothing special about the disc. – Wakabaloola Feb 22 '21 at 16:49

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After a lot of investigating I think the answer is likely to be:

$$F_{2D}[\phi] \propto \exp\left(i \int\limits_0^{2\pi}\int\limits_0^{2\pi} \frac{\phi(\theta)\phi(\theta')}{1-\cos(\theta-\theta')} d\theta d\theta \right)$$

but I can't prove it yet. If not this then something very similar with $\cos$ replaced with a similar function.