On Sunday, May 10, 2020 at 9:09:51 AM UTC-5, Philip Thrift wrote: > > > > On Sunday, May 10, 2020 at 7:20:16 AM UTC-5, Bruno Marchal wrote: >> >> >> On 10 May 2020, at 13:45, Philip Thrift <[email protected]> wrote: >> >> >> >> >> cf. >> https://www.researchgate.net/publication/325907723_A_gentle_introduction_to_Schwinger's_formulation_of_quantum_mechanics_The_groupoid_picture >> >> >> >> >> *Schwinger’s picture of Quantum Mechanics* >> https://arxiv.org/pdf/2002.09326.pdf >> >> >> >> >> Those are good papers. >> >> (I am not entirely sure why you refer to them, but Swinger is very good >> on this). When I teach quantum mechanics, I use an informal version of >> this, based on a textbook by Thomson (but I failed to find it right now). >> This helps to understand than both complex numbers and three dimension >> marry well together to make the quantum weirdness hard to avoid. It >> provides short path to the violation of Bell’s inequality. >> >> Bruno >> >> >> >> *In this paper we will present the 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.* >> >> Finally, we will introduce *a quantum measure* associated with the state >> ρ >> >> First, we realize that the state ρ on C∗(G) defines a decoherence >> functional D on the σ-algebra Σ of events of the groupoid G >> >> We define a quantum measure µ on Σ >> >> ... >> >> >> >> @philipthrift >> Click >> >> >> >> @philipthrift >> >> >> > A measure theory on *the appropriate measure space* underlies both > probability theory and what has been called quantum-probability theory. > > @philipthrift >
*Quantum measures and the coevent interpretation* https://arxiv.org/abs/1005.2242 A quantum measure µ describes the dynamics of a quantum system in the sense that µ(A) gives the propensity that the event A occurs. Denoting the set of events by A, a coevent is a potential reality for the system given by a truth function φ: A → Z₂ where Z₂ is the two element Boolean algebra {0, 1}. Due to quantum interference, a q-measure need not satisfy the usual additivity condition of an ordinary measure but satisfies the more general grade-2 additivity condition (2.1) instead. Let (Ω, A) be a measurable space, where Ω is a set of outcomes and A is a σ-algebra of subsets of Ω called events for a physical system. If A, B ∈ A are disjoint, we denote their union by A ⩁ B. A nonnegative set function µ: A → R+ is grade-2 additive if µ (A ⩁ B ⩁ C) = µ (A ⩁ B) + µ (A ⩁ C) + µ (B ⩁ C) − µ(A) − µ(B)- µ(C) (2.1) for all mutually disjoint A, B, C ∈ A [examples] @philipthrift -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/487195d6-f60c-4b13-9af9-8b9df12ed8f7%40googlegroups.com.
