The Conditional Probability Interpretation of the Hamiltonian Constraint

TL;DR

This paper refines the Conditional Probability Interpretation (CPI) of the Hamiltonian constraint, addressing past criticisms and extending its applicability to cases with multiple clocks and continuous spectra, thereby strengthening its theoretical foundation.

Contribution

The paper provides a refined formulation of CPI capable of handling multiple clocks and continuous spectra, resolving previous criticisms and enhancing its theoretical robustness.

Findings

CPI can answer questions involving multiple clock times.

Conventional quantum mechanics is recovered in the ideal clock limit.

The approach is rigorously extended to continuous-spectrum cases.

Abstract

The Conditional Probability Interpretation (CPI), first introduced by Page and Wootters, is reviewed and refined. It is argued that in it's refined form the CPI is capable of answering various past criticisms. In particular, questions involving more than one clock time are described in detail, resolving the problems raised in Kuchar's ``reduction ad absurdum''. In the case of Parametrized Particle Dynamics, conventional quantum mechanics is recovered in the ideal clock limit. When E=0 is among the continuous spectrum of the Hamiltonian, the induced inner product is used to construct the physical Hilbert space $\clh_{\rm ph}$ from the generalized eigenvectors in (the topological dual of) $\clh_{\rm aux}$. This allows the CPI to be applied to these `continuous-spectrum' cases in a more rigorous fashion than that described previously. The discrete spectrum case is also treated.