A groupoid measure-theoretic formulation of QM

[Philip Thrift]

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

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

[Bruno Marchal]

On 10 May 2020, at 13:45, Philip Thrift <cloud...@gmail.com> 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

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[Philip Thrift]

On Sunday, May 10, 2020 at 7:20:16 AM UTC-5, Bruno Marchal wrote:

On 10 May 2020, at 13:45, Philip Thrift <cloud...@gmail.com> 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

[Lawrence Crowell]

I read a paper on how spacetime on the level of special relativity can be built from "magma." This is not hot molten rock in the Earth, but a groupoid construction. This was a while ago, almost 10 years, but I read it well; I had to for I was a referee on it. I can't find it right now, but I will try to search it out.

LC

[Philip Thrift]

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