Quantum-Theoretic Schrödinger Equation in Abstract Hilbert Space

If quantum theory could ‘see’ the intricate structure of the invariant set, it would ‘know’ whether a particular putative measurement orientation θ was counterfactual or not. It must be noted that counterfactuals in quantum mechanics appear in discussions of (a) non-locality, (b) pre- and post-selected systems, and (c) interaction-free measurement; Quantum interrogation. Only the first two issues are related to counterfactuals as they are considered in the general philosophical literature:

If it were that A, then it would be that B.

The truth value of a counterfactual is decided by the analysis of similarities between the actual and possible counterfactual worlds. The difference between a counterfactual (or counterfactual conditional) and a simple conditional: If A, then B, is that in the actual world A is not true and we need some “miracle” in the counterfactual world to make it true. In the analysis of counterfactuals out of the scope of physics, this miracle is crucial for deciding whether B is true. In physics, however, miracles are not involved. Typically:

A : A measurement M is performed
B : The outcome of M has property P

Physical theory does not deal with the questions of which measurement and whether a particular measurement is performed? Physics yields conditionals: “If Ai, then Bi“. The reason why in some cases these conditionals are considered to be counterfactual is that several conditionals with incompatible premises Ai are considered with regard to a single system. The most celebrated example is the Einstein–Podolsky–Rosen (EPR problem) argument in which incompatible measurements of the position or, instead, the momentum of a particle are considered. Stapp has applied a formal calculus of coun- terfactuals to various EPR-type proofs and in spite of extensive criticism, continues to claim that the nonlocality of quantum mechanics can be proved without the assumption “reality”.

However, since, by hypothesis, quantum theory is blind to the intricate structure of the invariant set I, it is unable to discriminate between factual and counterfactual measurement preparations and therefore admits them all as theoretically valid. Hence the quantum-theoretic notion of state is defined on a quantum sub-system in preparation for any measurement that could conceivably be performed on it, irrespective of whether this measurement turns out to be real or counterfactual. This raises a fundamental question. If we interpret the quantum-theoretic notion of state in terms of a sample space defined by a h ̄ neighbourhood on the invariant set, how are we to interpret the quantum-theoretic notion of state associated with counterfactual world states of unreality, not on the invariant set, where no corresponding sample space exists?

Hence, when p ∈ I (the real axis for the Gaussian integers), then α|A⟩ + β|B⟩ can be interpreted as a probability defined by some underlying sample space. However, when p ∉ I (the rest of the complex plane for the Gaussian integers), we define a probability-like state α|A⟩ + β|B⟩ from the algebraic properties of probability, i.e. in terms of the algebraic rules of vector spaces. Under such circumstances, α|A⟩+β|B⟩ can no more be associated with any underlying sample space. This ‘continuation off the invariant set’ does not contradict Hardy’s definition of state, since if p ∉ I, then its points are not elements of physical reality, and hence cannot be subject to actual measurement. A classical dynamical system is one defined by a set of deterministic differential equations. As such, there is no requirement in classical physics for states to lie on an invariant set, even if the differential equations support such a set.  As a result, for a classical system, every point in phase space is a point of ‘physical reality’, and the counterfactual states discussed above are as much states of ‘physical reality’ as are the real world states. Hence, the world of classical physics is perfectly non-contextual, and is not consistent with the invariant set postulate.

The following interpretation of the two-dimensional Hilbert Space spanned by the vectors |A⟩ and |B⟩, emerges from the Invariant Set Postulate. At any time t there corresponds a point in the Hilbert Space, where α|A⟩ + β|B⟩ can be interpreted straightforwardly as a frequentist probability based on an underlying sample space of trajectory √h ̄ neighbourhood on the invariant set. However, since the invariant set and hence its underlying deterministic dynamics are themselves non-computable, it is algorithmically undecidable as to whether any given point in the Hilbert Space can be associated with such a sample space or not; as such, each point of the Hilbert Space is as likely to support an underlying sample space as any other. For points in the Hilbert Space which have no correspondence with a sample space on the invariant set, α|A⟩ + β|B⟩ must be considered an abstract mathematical quantity defined purely in terms of the algebraic rules governing a vector space.

Consistent with the rather straightforward probabilistic interpretation of the quantum-theoretic notion of state on the invariant set, it is reasonable to suppose that, on the invariant set, the Schrödinger equation is itself a Liouville equation for conservation of probability in regions where dynamical evolution is Hamiltonian. Since quantum theory is blind to the intricate structure of the invariant set, the quantum-theoretic Schrödinger equation must be formulated in abstract Hilbert Space form, i.e. in terms of unitary evolution, using algebraic properties of probability without reference to an underlying sample space.

One algebraic property inherited from the Schrödinger equation’s interpretation as a Liouville equation on the invariant set is linearity: as an equation for conservation of probability, the Liouville equation, is always linear, even when the underlying dynamics X ̇ = f(X) are strongly nonlinear. This suggests that attempts to add nonlinear terms (deterministic or stochastic) to the Schrödinger equation, e.g. during measurement, are misguided.

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