I address this question of ontology in my book, and I strongly suggest you take a look at that. (I know the book is a bit pricey, but you can always get it from a library! ;)
But here’s a reply in a nutshell.
First, the whole point of PTI is the idea that QM describes REAL possibilitites that do not live in spacetime—i.e., that spacetime is not ‘all there is’. So the QM objects DO exist, in my interpretation. That’s the basic ontology. The mathematical object that describes these real possibilitites is Hilbert space. Again: ‘what exists’ is not the same as ‘what is in spacetime’. Not being in spacetime does not disqualify an entity from existing. This is where I think ‘mainstream’ attempts to interpret QM stumble, because they automatically assume that because the quantum state (or ‘wavefunction’) does not live in spacetime, it therefore necessarily describes something that ‘doesn’t physically exist’, i.e., it only describes knowledge. I think that’s a false choice. Remember that it was Heisenberg who first suggested that QM states describe a previously unsuspected kind of physical reality. That’s the idea I’m pursuing.
There are no ‘particles’ in TI or PTI. So at a basic level, it is interacting field currents that are fundamental. These are the physical possibilitites.
As for the actual events, these comprise a discretized spacetime corresponding to the transactions that have been actualized. This is a definite history of energy exchanges between emitters and absorbers, and is the emergent ‘classical’ world. I invite you to Chapter 8 of my book for further details. A specific example of the emergence of a ‘classical’ trajectory is given at the end of Chapter 4.
Again, the main point: ‘physically real’ is not equivalent to ‘existing in spacetime’. Quantum states describe physically real possibilitites that do not live in spacetime, but have their existence in a realm mathematically described by Hilbert (actually Fock) space. Spacetime is just the set of actualized events—i.e. emitters and absorbers that have exchanged energy via actualized transactions.Each of these defines an invariant spacetime interval. But note that this is a relational view of spacetime—the latter is not a substantive, independently existing structure. It’s just a map we use to describe the set of actualized events.
To address your final question directly: the things that can be actualized are described by the weighted projection operators in the von Neumann mixed state ([process 1′) occurring on measurement—the weights are just the Born Rule. (TI is the only interpretation that can physically explain this ‘measurement’ transformation.) The thing that is actualized is described by the projection operator ‘left standing’ while the other ones have disappeared. These are ‘just’ properties, if you like, but they are supported (as a substratum) by the emitter and absorber involved in their actualization. So in PTI, the spacetime arena of phenomena, i.e., observed properties, is rooted in a pre-spacetime substratum of physical possibilitites.
Thanks Mitchell—it’s only at the nonrelativistic limit that there is a timelike partial ordering in this sense, and that emerges stochastically from the relativistic level. I.e., there is no temporal causal relationship in the basic field propagation. So my picture isn’t quite captured by the formulation in this paper (which also doesn’t appear to address wf collapse and the possible relation of collapse to an emergent spacetime). But in any case, thanks again for your interest and I hope you will take a look at the book. The main dividend you get from the TI picture is a robust solution to the measurement problem, in contrast to the ‘FAPP’ quasi-solution obtainable from decoherence approaches. In particular, decoherence never gives true irreversibility, since you never get real collapse with decoherence alone. In PTI you get true collapse, which also sheds light on macroscopic irreversibilty. I discuss this in my book as well.