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.