He announces 4 main Sections: Evolving constants; Toy models; Event POVMs; Extended quantum mechanics
The overall claim will be that :
The two strategies—Eliminate time (Gambini Oporto Pullin) and Promote time (Brunetti & Fredenhagen and HOge)—are less enemies than they seem: they lead to a quantum mechanics using an extended phase space.
SEction 1 Evolving constants:
For classical mechanics of eg 3 point particles, he expounds the basic ideas of partial and complete observables with complete observables commuting with the constraints.
He then describes Dirac constrained quantization of this, invoking the algebraic quantization scheme of Ashtekar et al 1995.
Then he discusses how to treat the basic question
What is the value of the partial observables q or p, given that the time is t = tau?
The proposal of the first strategy (Gambini Oporto Pullin) is to integrate out the parameter tau, in effect by introducing an explcit partial observable, say Qb, for it.
Section 2 Toy models
This is illustrated by a toy model, providing a time-independent conditional probability.
He then pursues the question: what exactly are we conditioning on? This discussion leads in to the idea of Event POVMs, and so to
Section 3: Event POVMs
This is developed in terms of a specific example, and then the general approach of the second strategy is adumbrated. In particular: the operator normalization idea of Brunetti & Fredenhagen and Hoge.
Pashby describes how this construction is committed (through its use of operator normalization) to the event in question happening precisely once; and mentions how this leads to the further question how to model the times of an event that could happen more than once.
Section 4: extended phase space
Any operator in a conventional Hilbert space say L^2(Q) \equiv L^2(R^3) defines and operator in the extended Hilbert space L^2(Q \times R); where of course, R represents time.
He briefly relates the ideas of Brunetti & Fredenhagen and Hoge (2010) to:
a) the Dirac- constrained quantization condition that the Hamiltonian constraint, as an operator, should kill the physical state;
b) Dirac’s own ideas in 1927 for a quantum mechanics on an extended phase space.
He ends by expounding the idea of weights, and quantum partial observables.