# Causality in Schwinger’s Picture of Quantum Mechanics

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^{4}

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## Abstract

**:**

## 1. Introduction: Causal Structures vs. Quantum Mechanics

## 2. The Geometric Theory of Causality

## 3. Algebraic Causality: A Categorical Approach

#### 3.1. Borel Causal Sets

**Theorem**

**1.**

**Definition**

**1.**

- ⪯ is a partial order.
- ≪ is areflexive, i.e., not $x\ll x$.
- ≪ is finer that ⪯, that is, if $x\ll y$, then $x\u2aafy$; $x\ll y\u2aafz\to x\ll z$; $x\u2aafy\ll z\to x<<z$.
- $x\to y$ iff $x\u2aafy$ and not $x\ll y$;

**Proposition**

**1.**

**Proof.**

#### 3.2. The Categorical Approach to Causality: Causal Structures as Borel Categories

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 4. Analytic Causality: Groupoids and Quantum Mechanics

#### 4.1. The Incidence Algebra of a Causal Relation and Sorkin’s Theorem

**Theorem**

**2.**

#### 4.2. Causal Structures in Groupoids and Triangular Operator Algebras

- Diffuse case. $\mathcal{A}$ is isomorphic to ${L}^{\infty}\left(\mathrm{\Omega}\right)$.
- Discrete case. $\mathcal{A}$ is isomorphic to a diagonal algebra $\mathcal{D}$ of a matrix algebra.
- Mixed case. $\mathcal{A}$ is isomorphic to $\mathcal{D}\oplus {L}^{\infty}\left(\mathrm{\Omega}\right)$.

**Theorem**

**3.**

**Theorem**

**4.**

**Proof.**

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Diagram representing a Minkowski strip space $\mathbb{M}(a,b)$ (in blue) as a subspace of Minkowski space. The causal cone $C\left(u\right)$ of an event u is marked in orange (right). Two events $x,y$ not causally related can be joined by a seesaw path $(x,{z}_{1},{z}_{2},{z}_{3},{z}_{4},{z}_{5},{z}_{6},{z}_{7},{z}_{8},{z}_{9},{z}_{10},{z}_{11},y)$, consisting of causal geodesics (dark blue) contained in $\mathbb{M}(a,b)$. Note that the set of points ${J}^{-}\left(x\right)\cap {J}^{-}\left(y\right)$ in the common causal past of $x,y$, is the causal past ${J}^{-}\left(z\right)$ of z (in red) which is out of the Minkowski strip.

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Ciaglia, F.M.; Di Cosmo, F.; Ibort, A.; Marmo, G.; Schiavone, L.; Zampini, A.
Causality in Schwinger’s Picture of Quantum Mechanics. *Entropy* **2022**, *24*, 75.
https://doi.org/10.3390/e24010075

**AMA Style**

Ciaglia FM, Di Cosmo F, Ibort A, Marmo G, Schiavone L, Zampini A.
Causality in Schwinger’s Picture of Quantum Mechanics. *Entropy*. 2022; 24(1):75.
https://doi.org/10.3390/e24010075

**Chicago/Turabian Style**

Ciaglia, Florio M., Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo, Luca Schiavone, and Alessandro Zampini.
2022. "Causality in Schwinger’s Picture of Quantum Mechanics" *Entropy* 24, no. 1: 75.
https://doi.org/10.3390/e24010075