Schrodinger paper 1931, time operator

Question

Reading General Principles of Quantum Mechanics by Pauli, I came across the Pauli argument about the impossibility of existence of time operator in classical quantum mechanics. On p. 63 he mentions a Schrödinger article about this argument:

• E. Schrödinger, Berl. Ber. (1931) p. 238,

that I think it’s

• E. Schrödinger, Spezielle Relativitätstheorie und Quantenmechanik Sitzungsberichte der Preußischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse, (1931), 238-247.

I searched for it in the internet and I wasn’t able to find it. Could someone indicate me where I can find it if there is in anyplace online?

However from https://arxiv.org/abs/quant-ph/9908033 I found an explicit demonstration of the argument, but I have a doubt. The argument is based on the assumption that the parameter $\beta \in R$. If it’s so then the eigenvalues of $H$ will be continuous and unlimitated. But what if I restrict the domain of $\beta$ to a discrete subset of $R$ such that the eigenvalues of $H$ would be inferiorly limited and eventually discrete in relation to the specific Hamiltonian? In this way I would bypass the Pauli argument and could have a time operator self adjoint with a unitary energy translation operator! Could someone correct me?

Answer 1

This is impossible, it's the subject of the paper:

• W. Unruh & R. Wald, "Time and the interpretation of quantum gravity". Phys. Rev. D40 (1989) 2598–2614 doi:10.1103/PhysRevD.40.2598

The argument is reviewed in Section 2.1 of Isham's review paper on Time in Quantum Gravity starting on page 11.