# Consistent Theories of Free Dirac Particle without Singular Predictions

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## Abstract

**:**

## 1. Introduction

## 2. General Implications of Poincaré Invariance

- -
- A unitary representation of the universal covering group ${\tilde{\mathcal{P}}}_{+}^{\uparrow}$ of the proper orthochronous Poincaré group ${\mathcal{P}}_{+}^{\uparrow}$ that realizes the quantum transformations of quantum observables, implied by transformations in ${\mathcal{P}}_{+}^{\uparrow}$;
- -
- Two operators ${}^{\u25c3}\phantom{\rule{-1.5pt}{0ex}}\mathtt{T}$ and ${}_{\u25c3}\phantom{\rule{-2.0pt}{0ex}}\mathtt{S}$ that realize the quantum transformations implied by time reversal and space inversion.

#### 2.1. Basic Prerequisites

#### 2.1.1. The Formalism of a Quantum Theory

- -
- The set $\Omega \left(\mathcal{H}\right)$ of all self-adjoint operators representing observables;
- -
- The set $\mathcal{S}\left(\mathcal{H}\right)$ of all density operators $\rho $ identified with quantum states; a quantum state $\rho $ is pure if and only if it is a one-dimensional projection operator, i.e., if $\rho =|\psi \rangle \langle \psi |$, where $\psi \in \mathcal{H}$ and $\parallel \psi \parallel =1$; in this case, $\psi $ is called the state vector of the system;
- -
- The group $\mathcal{U}\left(\mathcal{H}\right)$ of all unitary operators;
- -
- The larger set $\mathcal{V}\left(\mathcal{H}\right)$ of all unitary or antiunitary operators.

#### 2.1.2. Poincaré Group

#### 2.2. Derivation of the Theory of Isolated System

- $\mathcal{IP}$
- The theory of an isolated system is invariant with respect to changes of frames within the class $\mathcal{F}$.

- –
- A continuous unitary representation U of ${\tilde{\mathcal{P}}}_{+}^{\uparrow}$ exists such that ${S}_{\tilde{g}}\left[A\right]={U}_{\tilde{g}}A{U}_{\tilde{g}}^{-1}$.
- –
- Two operators ${}_{\u25c3}\phantom{\rule{-2.0pt}{0ex}}\mathtt{S}$ and ${}^{\u25c3}\phantom{\rule{-1.5pt}{0ex}}\mathtt{T}$, each of them unitary or antiunitary, exist such that (The unitarity of ${}_{\u25c3}\phantom{\rule{-2.0pt}{0ex}}\mathtt{S}$ and ${}^{\u25c3}\phantom{\rule{-1.5pt}{0ex}}\mathtt{T}$ cannot be proven in general because ${}_{\u25c3}\phantom{\rule{-2.2pt}{0ex}}\mathtt{s}$ and ${}^{\u25c3}\phantom{\rule{-3.2pt}{0ex}}\mathtt{t}$ ar not connected with the identity element of $\mathcal{P}$ [19].)

- ($\mathcal{FI}$)
- In the quantum theory of an isolated system, a triplet $(U,{}_{\u25c3}\phantom{\rule{-2.0pt}{0ex}}\mathtt{S},{}^{\u25c3}\phantom{\rule{-1.5pt}{0ex}}\mathtt{T})$ must exist, called the transformer triplet of the theory, formed by a continuous representation U of ${\tilde{\mathcal{P}}}_{+}^{\uparrow}$ and by two operators ${}_{\u25c3}\phantom{\rule{-2.0pt}{0ex}}\mathtt{S},{}^{\u25c3}\phantom{\rule{-1.5pt}{0ex}}\mathtt{T}\in \mathcal{V}\left(\mathcal{H}\right)$ such that$${S}_{\tilde{g}}\left[A\right]={U}_{\tilde{g}}A{U}_{\tilde{g}}^{-1}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{}_{\u25c3}\phantom{\rule{-2.0pt}{0ex}}\mathtt{S}A{{}_{\u25c3}\phantom{\rule{-2.0pt}{0ex}}\mathtt{S}}^{-1}={S}_{{}_{\u25c3}\phantom{\rule{-2.2pt}{0ex}}\mathtt{s}}\left[A\right]\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{}^{\u25c3}\phantom{\rule{-1.5pt}{0ex}}\mathtt{T}A{{}^{\u25c3}\phantom{\rule{-1.5pt}{0ex}}\mathtt{T}}^{-1}={S}_{{}^{\u25c3}\phantom{\rule{-3.2pt}{0ex}}\mathtt{t}}\left[A\right],for\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{4.pt}{0ex}}A\in \Omega \left(\mathcal{H}\right).\phantom{\rule{2.em}{0ex}}$$

#### 2.3. Constraints for the Transformer Triplet

## 3. Theories of Elementary Free Particles

- (Q.1)
- $[{Q}_{j},{Q}_{k}]=\mathrm{I}\phantom{\rule{-4.99997pt}{0ex}}\mathrm{O}$, for all $j,k=1,2,3$. This condition establishes that a measurement of position yields all three values of the coordinates of the same specimen of the system.
- (Q.2)
- For every $g\in \mathcal{P}$, the three-operator $({Q}_{1},{Q}_{2},{Q}_{3})\equiv \mathbf{Q}$ and the transformed position operator ${S}_{g}\left[\mathbf{Q}\right]\equiv ({S}_{g}\left[{\mathbf{Q}}_{1}\right],{S}_{g}\left[{\mathbf{Q}}_{2}\right],{S}_{g}\left[{\mathbf{Q}}_{3}\right])$ satisfy the specific relations implied by the transformation properties of position with respect to g.

#### 3.1. The First Step for Positive-Mass Elementary Particle

**Theorem**

**1.**

#### 3.1.1. The Classes ${\mathcal{I}}^{+}(\mu ,s)$ and ${\mathcal{I}}^{-}(\mu ,s)$

- –
- The Hilbert space is $\mathcal{H}={L}_{2}({\mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{R}}^{3},{\mathrm{I}\phantom{\rule{-4.99997pt}{0ex}}\mathrm{C}}^{2s+1},d\nu )$, where $d\nu \left(\underline{p}\right)=\frac{d{p}_{1}d{p}_{2}d{p}_{3}}{\sqrt{{\mu}^{2}+{\mathbf{p}}^{2}}}$;
- –
- The generators are defined by $\left({P}_{j}\psi \right)\left(\mathbf{p}\right)={p}_{j}\psi \left(\mathbf{p}\right)$, $\left({P}_{0}\psi \right)\left(\underline{p}\right)={p}_{0}\psi \left(\underline{p}\right)$,

- –
- The space-inversion and time-reversal operators are ${}_{\u25c3}\phantom{\rule{-2.0pt}{0ex}}\mathtt{S}={\rm Y}$, ${}^{\u25c3}\phantom{\rule{-1.5pt}{0ex}}\mathtt{T}=\tau \mathcal{K}{\rm Y}$,

#### 3.1.2. The Class ${\mathcal{I}}^{-+}(\mu ,s)$

#### 3.2. Second Step

## 4. Dirac’s Theory

#### 4.1. Dirac Theory and Zitterbewegung

- –
- The state vectors are four-component complex functions on ${\mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{R}}^{3}$: $\phi \left(\mathbf{x}\right)=\left[\begin{array}{c}{\phi}^{\left(1\right)}\left(\mathbf{x}\right)\\ {\phi}^{\left(2\right)}\left(\mathbf{x}\right)\end{array}\right]$, with ${\phi}^{\left(n\right)}\left(\mathbf{x}\right)\in {L}_{2}({\mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{R}}^{3},{\mathrm{I}\phantom{\rule{-4.99997pt}{0ex}}\mathrm{C}}^{2})$.
- –
- The position operator ${\mathbf{Q}}^{D}$ is defined by ${Q}_{j}^{D}\phi \left(\mathbf{x}\right)={x}_{j}\phi \left(\mathbf{x}\right)$, i.e., ${\mathbf{Q}}^{D}=\mathbf{x}$;
- –
- The self-adjoint generators relative to spatial translations are ${P}_{j}^{D}=-i\frac{\partial}{\partial {x}_{j}}$;
- –
- The operator ${P}_{0}^{D}=\mu \beta +{\alpha}_{1}{P}_{1}^{D}+{\alpha}_{2}{P}_{2}^{D}+{\alpha}_{3}{P}_{3}^{D}\equiv (\mu \beta +\alpha \xb7{\mathbf{P}}^{D})$ is the self-adjoint generator relative to time translation, where $\beta $ and ${\alpha}_{j}$ are the Dirac matrices [2].

#### 4.2. The Class of Dirac’s Theory in the Present Approach

**Definition**

**1.**

**Remark**

**1.**

## 5. Consistent Alternatives to Dirac Theory

#### 5.1. A Special Class of Theories

- (JC)
- In the Hilbert space ${L}_{2}({\mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{R}}^{3},{\mathrm{I}\phantom{\rule{-4.99997pt}{0ex}}\mathrm{C}}^{2s+1};d\nu )$, let us consider the operators ${J}_{j}$ defined by (14). A commutative three-operator $\mathbf{R}=({R}_{1},{R}_{2},{R}_{3})$ of self-adjoint operators of ${L}_{2}({\mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{R}}^{3},{\mathrm{I}\phantom{\rule{-4.99997pt}{0ex}}\mathrm{C}}^{2s+1};d\nu )$ satisfies $[{R}_{j},{p}_{k}]=i{\delta}_{jk}$ and $[{J}_{j},{R}_{k}]=i{\widehat{\u03f5}}_{jkl}{R}_{l}$ if and only if$${R}_{j}=\eta ({p}_{0},\widehat{z})\mathbf{r}\wedge (\mathbf{r}\wedge \mathbf{S})+\gamma ({p}_{0},\widehat{z})\mathbf{r}\wedge \mathbf{S},\phantom{\rule{2.em}{0ex}}$$where $\eta ({p}_{0},\widehat{z})$ and $\gamma ({p}_{0},\widehat{z})$ are self-adjoint operators of ${L}_{2}({\mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{R}}^{3},{I\phantom{\rule{-5.99997pt}{0ex}}C}^{2s+1},d\nu )$ functions of ${p}_{0}$ and of the “reduced” helicity $\widehat{z}=\mathbf{r}\xb7\mathbf{S}$, with $\mathbf{r}=\frac{\mathbf{p}}{\sqrt{{p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{2}}}$.

- (i) $\phantom{\rule{0.277778em}{0ex}}\tau \overline{{S}_{k}}{\tau}^{-1}=-{S}_{k},\hspace{1em}$ (ii) $\phantom{\rule{0.277778em}{0ex}}{\rm Y}\mathbf{r}=-\mathbf{r}{\rm Y};\hspace{1em}$ (iii) $\phantom{\rule{0.277778em}{0ex}}{\rm Y}\mathbf{S}=\mathbf{S}{\rm Y};\hspace{1em}$ (iv) $\phantom{\rule{0.277778em}{0ex}}{\rm Y}\mathbf{F}=-\mathbf{F}{\rm Y}$;
- (v) $\phantom{\rule{0.277778em}{0ex}}{\rm Y}{\eta}_{n}({p}_{0},\widehat{z})={\eta}_{n}({p}_{0},-\widehat{z}){\rm Y},\hspace{1em}$$\phantom{\rule{0.277778em}{0ex}}{\rm Y}{\gamma}_{n}({p}_{0},\widehat{z})={\gamma}_{n}({p}_{0},-\widehat{z}){\rm Y};\hspace{1em}$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(26\right)$
- (vi) $\phantom{\rule{0.277778em}{0ex}}\mathcal{K}\mathbf{F}=-\mathbf{F}\mathcal{K};\hspace{1em}$(vii) $\phantom{\rule{0.277778em}{0ex}}\tau \mathcal{K}{\rm Y}\mathbf{F}=\mathbf{F}\tau \mathcal{K}{\rm Y};\hspace{1em}$ (viii) $\phantom{\rule{0.277778em}{0ex}}\tau \mathcal{K}\mathbf{S}=-\mathbf{S}\tau \mathcal{K};\hspace{1em}$
- (ix) $\phantom{\rule{0.277778em}{0ex}}\tau \mathcal{K}\widehat{z}=-\widehat{z}\tau \mathcal{K};\hspace{1em}$ (x) $\phantom{\rule{0.277778em}{0ex}}\tau \mathcal{K}{\eta}_{n}={\eta}_{n}\tau \mathcal{K},\hspace{1em}$ $\phantom{\rule{0.277778em}{0ex}}\tau \mathcal{K}{\gamma}_{n}={\gamma}_{n}\tau \mathcal{K}$.

**Theorem**

**2.**

- (i)
- ${\eta}_{1}={\eta}_{2}=0$ and ${\gamma}_{1}={\gamma}_{2}=\gamma $ for $n=1,2,3,4$;
- (ii)
- ${\eta}_{1}={\eta}_{2}=\eta $, and ${\gamma}_{1}={\gamma}_{2}=\gamma $, where η and γ are, respectively, odd and even with respect to $\widehat{z}$, for $n=5,6$.

#### 5.2. No Singular Features

**Remark**

**2.**

## 6. Characterization of Dirac’s Theory by Peculiar Transformation Properties

**Example**

**1.**

## 7. The Problem of the Experimental Comparison of the Theories

#### 7.1. The Formalism for the Different Predictions

#### 7.2. The Different Ideally Observable Predictions

- (
**a**) - To implement the experimental conditions such that the state vector of the particle is ${\psi}^{D}$ according to Dirac’s theory;
- (
**b**) - To perform measurements of the particle coordinate ${x}_{1}$ at extremely small, fixed times ${t}_{n}$ to evaluate the expectation values, without introducing a noninstantaneous interaction.

## 8. Discussion

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Nisticò, G.
Consistent Theories of Free Dirac Particle without Singular Predictions. *Particles* **2023**, *6*, 1-16.
https://doi.org/10.3390/particles6010001

**AMA Style**

Nisticò G.
Consistent Theories of Free Dirac Particle without Singular Predictions. *Particles*. 2023; 6(1):1-16.
https://doi.org/10.3390/particles6010001

**Chicago/Turabian Style**

Nisticò, Giuseppe.
2023. "Consistent Theories of Free Dirac Particle without Singular Predictions" *Particles* 6, no. 1: 1-16.
https://doi.org/10.3390/particles6010001