It is not true that a representation (K,π) of U must be a Fock representation in order for states in the Hilbert space K to have an interpretation as particle states. Indeed, one of the central tasks of “scattering theory,” is to provide criteria – in the absence of full Fock space structure – for defining particle states. These criteria are needed in order to describe scattering experiments which cannot be described in a Fock representation, but which need particle states to describe the input and output states.
Haag and Swieca propose to pick out the n-particle states by means of localized detectors; we call this the detector criterion: A state with at least n-particles is a state that would trigger n detectors that are far separated in space. Philosophers might worry that the detector criterion is too operationalist. Indeed, some might claim that detectors themselves are made out of particles, and so defining a particle in terms of a detector would be viciously circular.
If we were trying to give an analysis of the concept of a particle, then we would need to address such worries. However, scattering theory does not end with the detector criterion. Indeed, the goal is to tie the detector criterion back to some other more intrinsic definition of particle states. The traditional intrinsic definition of particle states is in terms of Wigner’s symmetry criterion:
A state of n particles (of spins si and masses mi) is a state in the tensor product of the corresponding representations of the Poincaré group.
Thus, scattering theory – as originally conceived – needs to show that the states satisfying the detector criterion correspond to an appropriate representation of the Poincaré group. In particular, the goal is to show that there are isometries Ωin, Ωout that embed Fock space F(H) into K, and that intertwine the given representations of the Poincaré group on F(H) and K.
Based on these ideas, detailed models have been worked out for the case where there is a mass gap. Unfortunately, as of yet, there is no model in which Hin = Hout, which is a necessary condition for the theory to have an S-matrix, and to define transition probabilities between incoming and outgoing states. (Here Hin is the image of Fock space in K under the isometry Ωin, and similarly for Hout.)
Buchholz and collaborators have claimed that Wigner’s symmetry criterion is too stringent – i.e. there is a more general definition of particle states. They claim that it is only by means of this more general criterion that we can solve the “infraparticles” problem, where massive particles carry a cloud of photons.
The “measurement problem” of nonrelativistic QM shows that the standard approach to the theory is impaled on the horns of a dilemma: either
(i) one must make ad hoc adjustments to the dynamics (“collapse”) when needed to explain the results of measurements, or
(ii) measurements do not, contrary to appearances, have outcomes.
There are two main responses to the dilemma: On the one hand, some suggest that we abandon the unitary dynamics of QM in favor of stochastic dynamics that accurately predicts our experience of measurement outcomes. On the other hand, some suggest that we maintain the unitary dynamics of the quantum state, but that certain quantities (e.g. position of particles) have values even though these values are not specified by the quantum state.
Both approaches – the approach that alters the dynamics, and the approach with additional values – are completely successful as responses to the measurement problem in nonrelativistic QM. But both approaches run into obstacles when it comes to synthesizing quantum mechanics with relativity. In particular, the additional values approach (e.g. the de Broglie–Bohm pilot-wave theory) appears to require a preferred frame of reference to define the dynamics of the additional values, and in this case it would fail the test of Lorentz invariance.
The “modal” interpretation of quantum mechanics is similar in spirit to the de Broglie–Bohm theory, but begins from a more abstract perspective on the question of assigning definite values to some observables. Rather than making an intuitively physically motivated choice of the determinate values (e.g. particle positions), the modal interpretation makes the mathematically motivated choice of the spectral decomposition of the quantum state (i.e. the density operator) as determinate.
Unlike the de Broglie–Bohm theory, it is not obvious that the modal interpretation must violate the spirit or letter of relativistic constraints, e.g. Lorentz invariance. So, it seems that there should be some hope of developing a modal interpretation within the framework of Algebraic Quantum Field Theory…..