Schwinger’s picture of quantum mechanics

Abstract

In this paper we will present tha main features of what can be called Schwinger's foundational approach to Quantum Mechanics. The basic ingredients of this formulation are the selective measurements, whose algebraic composition rules define a mathematical structure called groupoid, which is associated with any physical system. After the introduction of the basic axioms of a groupoid, the concepts of observables and states, statistical interpretation and evolution are derived. An example is finally introduced to support the theoretical description of this approach.

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

“…This framework allows us to deal with classical and quantum states with the same formalism because the states of a finite-dimensional, Abelian -algebra are in one-to-one correspondence with the probability distributions on a finite-outcome space. Furthermore, it nicely fits into the recently developed groupoidal approach to quantum theories developed in [ 32 , 33 , 34 , 35 , 36 , 37 ].…”

Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

“…Starting from the observation that selective measurements are appropriately described by groupoids, in a series of recent papers [ 2 , 6 , 7 , 8 , 9 , 10 ], a new picture of quantum mechanics was proposed where Schwinger’s algebra of selective measurements was taken one step forward. In this framework, a physical system is described by means of a groupoid , where the set is referred to as the space of “outcomes”, and the elements of the groupoid , with , x being the source and y the target of , are referred to as “transitions”.…”

“…Conceptually speaking, the family of transitions generalizes both Schwinger’s notion of selective measurement previously discussed, the actual transitions used in the statement of the Ritz–Rydberg combination principle [ 23 ], and the experimental notion of “quantum jumps” introduced in the old quantum mechanics. Furthermore, from a modern perspective, we may also say that the transitions represent the abstract notion of amplitudes as “square roots” of probability densities, as argued in [ 6 ], that is specific representations of the groupoid will assign rank-one operators to the transitions , which will represent the “square roots” of standard probabilities.…”

“…We will analyze this question in the groupoid formalism developed recently in [ 2 , 6 , 7 , 8 , 9 , 10 ]. The result presented here will be that, in accordance with Raggio’s theorem, it is actually impossible to build an entangled state out of a separable state of a classical and a quantum system.…”

Differential Geometric Aspects of Parametric Estimation Theory for States on Finite-Dimensional C∗-Algebras

“…While this restriction may seem not particularly relevant for most practical purposes, it is certainly so from the theoretical point of view. Indeed, some recent developments [ 57 , 58 , 59 , 60 , 61 , 62 ] point out the possibility of describing quantum systems whose associated -algebras are groupoid algebras, and thus are in principle more general than the algebra of bounded linear operators. Consequently, a reformulation of the well-known results for an arbitrary -algebra appears to be useful.…”