Observables with transcendental eigenvalues

Question

Are there any "natural" physical observables which have non-empty point spectrum which consists of numbers which are not algebraic numbers?

Answer 1

I don't have a complete answer, but maybe the following is useful for your purposes:

Consider the Laplacian $\Delta$ on a circular drum of unit radius. As explained on the wikipedia page, the axially symmetric eigenvectors $\Delta u(r) = -\lambda^2 u(r) $ are Bessel functions $u(r)=J_0(\lambda r)$. Obviously, the boundary condition requires $J_0(\lambda)=0$. In other words, the eigenvalues of this Laplacian correspond to zeroes of Bessel functions, and I would be very surprised if these numbers are not transcendental. In fact, Mathworld mentions that the first zero has been proven to be transcendental by Le Lionnais.

The corresponding quantum mechanical situation would be the Hamilton operator $H=-\hbar^2/2m\cdot \Delta$ of a free 2D electron confined to the unit disk. The boundary conditions are the same, $\psi|_{ \partial\Omega}=0$.

Of course, the electron has the problem that it's not clear whether $\hbar= h/2\pi$ should be counted as transcendental or algebraic, physicists frequently set $\hbar = 1$.

For instance, consider a 1D electron inside a box of length $L$, i.e. $\psi(0)=\psi(L)=0$. The eigenstates are simply standing waves, and the eigenvalues are

$$ E_n = \frac{\hbar^2}{2m_e} \left(\frac{\pi n}{L}\right)^2.$$

If you count $h$ as algebraic, then this is algebraic. But if you count $\hbar$ as algebraic, then this is transcendental. Your way out is probably to turn the question into a relative one: is there a physical observable whose eigenvalues are algebraically independent of the $E_n$? Clearly, you only need to consider the eigenvalues of the Laplace operator now.

Answer 2

Transcendental numbers often pop out in higher-loop Feynman diagram calculations. One example is the spectrum of local operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory, where the anomalous dimensions at higher loops in general contain $\zeta$-functions. As an example, the dimension of the operator $\mathop{Tr} \phi^a\phi^a$ is given perturbatively by $$\gamma = 4 + 12g^2 - 48g^4 + 336g^6 + (-2496+576\zeta(3)-1440\zeta(5))g^8 + \mathcal{O}(g^{10})$$ (I don't think there is a rigorous proof showing that $\zeta(n)$ is non-algebraic for positive and odd values of $n$, but it seems likely to me...)