John Harding

New Mexico State University
Events as Decompositions

The standard view of quantum mechanics is that a superposition of states corresponding to different outcomes of an event is given by the sum s = s1 + s2 of states of the alternatives. Since vectors are the natural setting to add physical quantities, Dirac viewed this as rational for his formulation of quantum mechanics with states as vectors in a Hilbert space.
We take the view that events correspond to ways to decompose a system as a product of others A = A1 × A2. This is like choosing a coordinate axes to express points in the plane. The superposition of states s1 and s2 for different outcomes is given not by a sum, but by an ordered pair s = (s1, s2).
The collection of ways to decompose a system as a product carries a structure similar to that carried by the projections that are used to model events in the standard Hilbert space setting. This allows development of alternative versions of quantum mechanics based in structures other than Hilbert spaces and motivated entirely by simple physical assumptions.

In the standard Hilbert space formulation of quantum mechanics, events are treated via projection operators of the Hilbert space H associated to the system. These projections form an orthomodular lattice Q(H). Much of quantum theory can be formulated in terms of this structure Q(H) of events. Gleason’s theorem, the spectral theorem, Wigner’s theorem, and Stone’s theorem, tie states, observables, unitaries, dynamics and representations to Q(H).
Projections of H correspond to sums H = H1 ⊕ H2, but also to products H = H1 × H2. Surprisingly, the direct product decompositions X = X1 × X2 of most types of structure form an orthomodular poset Q(X). Portions of a generalized version of quantum theory have been developed using this notion of events as decompositions. This theory is operationally motivated, and is ultimately based on the view that a product provides a superposition s = (s1, s2) of alternatives.
The proposed work involves the continued development of such versions of quantum mechanics. This includes observables, dynamics, group representations, and tensor products. Specific instances of quantum mechanics based on X being a normed group or vector bundle will be studied in detail. The program will also be placed within the setting of a categorical quantum mechanics.

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