All Quantum Probability viewed in Complex Projective Geometry

Abstract: In a paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.

• The paper reformulates quantum probabilities solely in terms of the intrinsic geometry of complex projective spaces using the Fubini-Study metric.
• It introduces a novel projection theorem that defines geometric distances and sequential event probabilities without relying on the traditional Hilbert space structure.
• The approach applies to both finite and infinite-dimensional quantum systems, integrating with general von Neumann algebraic frameworks.

Quantum Probability and Complex Projective Geometry: An Intrinsic Geometric Reformulation

Introduction

This work presents a direct reformulation of quantum probability theory solely in terms of the intrinsic geometry of complex projective spaces, detaching quantum probabilistic statements from explicit Hilbert space formulations. The approach rigorously constructs quantum probabilities as geometric functions on projective Kähler manifolds, specifically leveraging the Fubini-Study Riemannian metric, and eschews dependence on the underlying linear Hilbert space structure. This formalism encompasses both finite and infinite-dimensional cases and is compatible with quantum theory based on general von Neumann algebras.

Geometric Structures Underlying Quantum States

Quantum states, both pure and mixed, are traditionally modeled as rays or density operators in a Hilbert space $\mathcal{H}$. The associated complex projective space $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$ inherits a Kähler manifold structure and is equipped with the Fubini-Study metric, whose geodesics and metric properties play a central role. Pure quantum states correspond to points in $\mathbb{CP}(\mathcal{H})$, and quantum events correspond to closed projective subspaces, which in finite dimensions are precisely the connected, totally geodesic, complex submanifolds.

A key construct is the absolute inner product $|\langle x, y \rangle|$ between points in $\mathbb{CP}(\mathcal{H})$, which descends from the modulus of the Hilbert space inner product but is interpreted entirely in geometric terms. Orthogonality, projective angles, and the projection of points onto subspaces all receive intrinsic geometric definitions directly within $\mathbb{CP}(\mathcal{H})$.

Geometric Derivation of Quantum Probabilities

The Born rule, central to quantum probability, is given in the Hilbert space formalism by $|\langle \psi, \phi \rangle|^2$. The geometric reformulation replaces this with $\cos^2 d(x, y)$, where $d(x, y)$ is the Fubini-Study distance between points $x, y \in \mathbb{CP}(\mathcal{H})$.

For more general quantum events, represented by orthogonal projections $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$0 (or equivalently, closed complex projective subspaces $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$1), the probability of observing $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$2 in state $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$3 is $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$4. Geometrically, this is recast as $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$5, where $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$6 and $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$7 is the image of the subspace under the appropriate projective mapping. The result is a closed formula for arbitrary events and states that depends exclusively on the geometry of $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$8.

Projection Theorem for Complex Projective Spaces

A novel projection theorem for complex projective spaces is established, analogous to the familiar Hilbert space projection theorem. Given any projective subspace $$\mathbb{CP}(\mathcal{H}) = \mathcal{H}^{\rm x} / \mathbb{C}^{\rm x}$$9 and any point $\mathbb{CP}(\mathcal{H})$0, there exists a unique or maximally non-unique closest point $\mathbb{CP}(\mathcal{H})$1 (except in degenerate cases where the point is orthogonal to $\mathbb{CP}(\mathcal{H})$2). This geometric projection underpins the purely geometric expressions for conditional and sequential event probabilities in quantum theory.

Consecutive Event Probabilities and Sequential Measurements

The work extends the geometric formalism to sequences of quantum events. The consecutive (Wigner) probability of two time-ordered events $\mathbb{CP}(\mathcal{H})$3 and $\mathbb{CP}(\mathcal{H})$4, given a state $\mathbb{CP}(\mathcal{H})$5, is expressed as

$\mathbb{CP}(\mathcal{H})$6

where $\mathbb{CP}(\mathcal{H})$7 denotes the geometric projection of $\mathbb{CP}(\mathcal{H})$8 onto $\mathbb{CP}(\mathcal{H})$9 using the projective projection theorem. This formula makes explicit the dependence solely on geometric aspects: distances in the Fubini-Study metric and projections in the Kähler manifold. Generalization to finite sequences of events follows naturally within this framework.

Compatibility with von Neumann Algebraic Quantum Theory

The geometric formalism is shown to accommodate quantum systems modeled via arbitrary von Neumann algebras, not restricted to type I factors. Events and states need only be those relevant for the chosen algebraic context, and the geometric definitions proceed unchanged by considering the relevant sublattice of projective subspaces and subset of pure states in $|\langle x, y \rangle|$0. Mixed states and density matrices can also be geometrically represented within this structure. This provides a coherent foundation for the application of the geometric approach in settings such as quantum statistical mechanics and quantum field theory, where Hilbert spaces may be infinite dimensional or have more complicated structures.

Theoretical and Practical Implications

Expressing all quantum probabilities exclusively in terms of the geometry of complex projective spaces provides both a conceptual and technical reformation of quantum theory. This geometric perspective blurs the significance of the underlying linear Hilbert space, instead foregrounding the Riemannian and Kählerian properties of the manifold of pure states. This construction supports the broader programmatic aim of geometrizing the foundations of physics, parallel to symplectic geometry in classical mechanics, Minkowski geometry in relativity, and contact geometry in thermodynamics.

Practically, this geometric approach may motivate new quantum generalizations by considering alternative Kähler or even more general manifolds as the arena for quantum phenomena. It also offers clarity when treating settings with infinite degrees of freedom, as in QFT or statistical mechanics, where conventional Hilbert space technicalities can obscure the underlying structure.

Conclusion

The paper establishes a complete geometric formalism for quantum probabilities, recasting all probabilistic rules in quantum mechanics as functions intrinsically defined on complex projective spaces equipped with the Fubini-Study metric. This work demonstrates the feasibility and power of fully geometrizing quantum theory, potentially opening avenues for further abstraction and generalization by exploring alternative geometric frameworks for quantum probability. The unification of measurement and transition probabilities under this structure provides a mathematically elegant and physically transparent language for foundational research in quantum theory.