Schwinger's Picture of Quantum Mechanics III: The statistical interpretation
arXiv:1909.07265 · doi:10.1142/S0219887819501652
Abstract
Schwinger's algebra of selective measurements has a natural interpretation in the formalism of groupoids. Its kinematical foundations, as well as the structure of the algebra of observables of the theory, was presented in two previous papers (arXiv:1905.12274 and arXiv:1907.03883). In this paper, a closer look to the statistical interpretation of the theory is taken and it is found that an interpretation in terms of Sorkin's quantum measure emerges naturally. It is proven that a suitable class of states of the algebra of virtual transitions of the theory allows to define quantum measures by means of the corresponding decoherence functionals. Quantum measures satisfying a reproducing property are described and a class of states, called factorizable states, possessing the Dirac-Feynman `exponential of the action' form are characterized. Finally, Schwinger's transformation functions are interpreted similarly as transition amplitudes defined by suitable states. The simple examples of the qubit and the double slit experiment are described in detail, illustrating the main aspects of the theory.
39 pages. Comments are welcome!
References in corpus (5)
Cited by in corpus (20)
- From the Jordan product to Riemannian geometries on classical and quantum states
- Schwinger's picture of Quantum Mechanics
- Schwinger's Picture of Quantum Mechanics IV: Composition and independence
- Quantum States, Groups and Monotone Metric Tensors
- Differential geometric aspects of parametric estimation theory for states on finite-dimensional C*-algebras
- Schrödinger's problem with cats: measurements and states in the Groupoid Picture of Quantum Mechanics
- Feynman's Propagator in Schwinger's picture of Quantum Mechanics
- Information geometry on groupoids: the case of singular metrics
- Symmetries and Reduction -- part I -- Poisson and symplectic picture
- Covariant Variational Evolution and Jacobi Brackets: Fields
- Groupoid and algebra of the infinite quantum spin chain
- Schwinger's picture of quantum mechanics: 2-groupoids and symmetries
- Covariant Variational Evolution and Jacobi Brackets: Particles
- Covariant reduction of classical Hamiltonian Field Theories: From D'Alembert to Klein-Gordon and Schrödinger
- An abstract theory of physical measurements
- Lagrangian description of Heisenberg and Landau-von Neumann equations of motion
- On the categorical foundations of quantum information theory: Categories and the Cramer-Rao inequality
- Geometry from divergence functions and complex structures
- Multidimensional Contours à la Fröhlich-Spencer and Boundary Conditions for Quantum Spin Systems
- Some remarks on the notion of transitions