Probability Manifolds

Stanley Gudder

Ken‐ichi Sugiyama

The Born rule is a rule that a probability we observe a small particle like an electron is proportional to the square of the absolute value of the wave function. In this paper, we try to derive the Born rule from the many-worlds interpretation. Many researchers have tried to derive the Born rule (also called Born's rule, Born's law, or probability interpretation) from Many-Worlds Interpretation (MWI). However, no one succeeds. Thus, the derivation of Born’s rule had become an important issue for MWI. We try to derive Born’s rule by introducing an elementary event of probability theory to the quantum theory as a new method. We interpret the wave function as a manifold like a three-dimensional sphere, and interpret the absolute value of the wave function as the surface area of the manifold. We suppose that the manifold exists in the discrete space that has lattice points. We interpret a point on the surface of the manifold as a state that we cannot divide any more, an elementary state. We draw an arrow from any point to any point. We interpret an arrow as an event that we cannot divide any more, an elementary event. Probability is proportional to the number of elementary events, and the number of elementary events is the square of the number of elementary states. The number of elementary states is proportional to the surface area of the manifold, and the surface area of the manifold is the absolute value of the wave function. Therefore, the probability is proportional to the square of the absolute value of the wave function.

Stanley Gudder, Nino Zanghı̀

F. M. Ciaglia, A. Ibort, J. Jost +1 more

Abstract The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of $$C^{*}$$ C∗ -algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given $$C^{*}$$ C∗ -algebra $$\mathscr {A}$$ A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $$\mathscr {S}$$ S of a possibly infinite-dimensional, unital $$C^{*}$$ C∗ -algebra $$\mathscr {A}$$ A is partitioned into the disjoint union of the orbits of an action of the group $$\mathscr {G}$$ G of invertible elements of $$\mathscr {A}$$ A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space $$\mathcal {H}$$ H are smooth, homogeneous Banach manifolds of $$\mathscr {G}=\mathcal {GL}(\mathcal {H})$$ G=GL(H) , and, when $$\mathscr {A}$$ A admits a faithful tracial state $$\tau $$ τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through $$\tau $$ τ is a smooth, homogeneous Banach manifold for $$\mathscr {G}$$ G .

Stanley Gudder

The concept of a probability manifold M is introduced. The global properties of M inherited from its local structure are then considered. It is shown that a deterministic spin model due to Pitowski falls within this general framework. Finally, we construct a phase-space model for nonrelativistic quantum mechanics. These two models give the same global description as conventional quantum mechanics. However, they also give a local description which is not possible in conventional quantum mechanics.

Manuel Hohmann