A new Quantum Mechanics interpretation

The composability interpretation

We have reached the end of this series of how we should understand the quantum mechanics’ wavefunction: ontological or epistemological in the standard sense. The short answer is that it is neither. The long answer is that a new interpretation is emerging from the project of recovering quantum mechanics from natural axioms http://arxiv.org/abs/1303.3935.

Wikipedia presents the table of comparison between existing interpretations. 

Before showing how the new interpretations measures up in that table I need to state that the answers are not unique, but they depend on the context. For example: Is quantum mechanics deterministic? In quantum mechanics evolution is pure unitary including during the apparent collapse http://arxiv.org/abs/1305.3594 so in the elliptic composability context, the answer is yes. As a reminder, composability means that the laws of nature are invariant under tensor composition, and elliptic composability is one of the three possible composability classes: elliptic (quantum mechanics), parabolic (classical mechanics) and hyperbolic (not present in nature). However, in the usual sense of looking at predicting concrete experimental outcomes, quantum mechanics is not deterministic.

	Elliptic composability	Parabolic composability
Deterministic?	Yes: the evolution is pure unitary	No: the individual experimental outcomes are unpredictable
Wavefunction real?	Yes, the wavefunction is elliptic-ontic: it has a similar reality with for example the electromagnetic field with the core difference that the wavefunction exists in a configuration space.	No in the sense of Heisenberg: the wavefunction represents a probability, but not an objective reality itself in space and time.
Unique History?	Yes	Yes
Hidden Variables?	Does not apply	No
Collapsing wavefunctions?	No	No
Observer role?	No	No
Local?	Locality independent.	No
Counterfactual definiteness?	Yes in a new sense: we can speak about observable algebras before measurement. We still cannot speak about results of measurements that have not been performed.	No
Universal wavefunction exists?	Unclear/Undecided	Unclear/Undecided
Contextual?	No in the sense of Gleason’s theorem	Yes in the sense of K-S theorem

The mathematical framework for quantum mechanics is that of C* algebras/modules. Observables form an algebra and one can define a unique norm of an element T using the spectral radius, which is the maximum norm of a complex number lambda such that “(T-lambda I)” does not have an inverse. If the norm of T: ||T|| satisfies the C* condition:

||T^{dagger} T|| = ||T||^2

then one can show using GNS construction that T is an operator in a Hilbert space H.

It is an elementary exercise to show the reverse implication, but constructing a Hilbert space from the C* condition is a nontrivial and was first shown by Gelfand in 1942.

Now what struck me the most from that paper is that it is signed: “Moscow 1942”. In 1942, Nazi troops were about 20 miles away from Moscow ready to burn down the city (before the winter weather stopped their advance), and instead of packing up and leaving for safety, Gelfand was making fundamental contributions to functional algebra. Remarkable.

There is a big subtlety relating to Hilbert spaces in quantum mechanics. There are actually two Hilbert spaces and they are linked through the Choi-Jamiolkowski isomorphism, or the channel-state duality http://en.wikipedia.org/wiki/Channel-state_duality.

Naively this come from identifying: |psi><phi| with <phi|psi> but they are very different beasts with different norms. T belongs to a Hilbert space of its own with the inner product given by Trace(A^dagger B) where A and B are observables. T is also an operator on the usual Hilbert space H of the wavefunction. In the information theoretical approaches for deriving quantum mechanics one starts from the states and the experimental procedures one can do in the lab in an instrumentalist fashion (see for example Hardy’s famous 5 reasonable axioms paper http://arxiv.org/abs/quant-ph/0101012).

The aim of composability approach and interpretation is much larger however. No concepts should be assumed from the real world, and everything should be fully derived mathematically in a very rigorous fashion.

One subtle point is that the actual representation of the wavefunction’s abstract Hilbert space does not have any ontological interpretation. The same abstract Hilbert space can be represented by very different mathematical constructs which cannot be put into any isomorphism. The case in point is the Dirac delta function which is a distribution and not a proper function. As a function it has contradictory properties and to properly understand it, knowledge of spectral theory is required.

The wavefunction is simply a mathematical non-contextual (i.e universal) tool of computing probabilities. What have ontological value are the operators and they correspond to actual experimental procedures in a lab. Through channel-state duality the wavefunction indirectly acquires (elliptical) ontological value too. This is the sense that the wavefunction can be called elliptic-ontic.

For a one particle wavefunction (when the configuration space becomes the space-time), the fact that the wavefunction is defined up to a phase in a particular case can be understood as a gauge degree of freedom corresponding to U(1). The wavefunction in this case has the same ontological value as the electromagnetic field. Now this may sound too good to be true, but it is natural in the context of C* modules (the natural generalization of C* algebras). In C* modules, the quantum mechanics number system are no longer complex numbers but C* algebras. Also the typical Hilbert C* module corresponds to vector bundles and the gauge group corresponds to the connection on the bundle. Expanding on this, in one case the Born's interpretation can be upgraded from probability density to probability current density and naturally arrive at Dirac's equation and the electroweak gauge symmetry SU(2)xU(1). This justifies the minimal coupling.  The mathematical framework for this generalization is that of a spectral triple in non-commutative geometry. Classifying all possible realizations of the elliptic composability is a work in progress. Looks like nature picked one particular realization which gives rise to the Standard Model. Why this particular realization was picked cannot be answered in the composability framework and additional ideas are needed.

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