# Phenomenological Effects of CPT and Lorentz Invariance Violation in Particle and Astroparticle Physics

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## Abstract

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## 1. Introduction

## 2. CPT Theorem

#### 2.1. Wightman Axioms

**Poincaré covariance of the Hilbert space where the theory is set.**This means that unitary operators $U(\Lambda ,\phantom{\rule{0.166667em}{0ex}}a)$ exist that implement Lorentz transformations and space-time translations;**Existence of a vacuum state $|0\rangle $**.There is a unique state, the vacuum, that is unaffected by Poincaré transformations up to a phase: $\left|U(\Lambda ,\phantom{\rule{0.166667em}{0ex}}a)\right|0\rangle ={e}^{i\varphi}|0\rangle $. This implies that this state can only have null four-momentum and angular momentum, since these quantities change under Lorentz transformations. The vacuum state must have even the lowest allowed energy and must be cyclic, which is acting on it via the creation operators every Hilbert space state can be constructed;**Fields constructed via operators.**All physical quantities can be constructed using polynomials of fields acting on the Hilbert space. The fields transform under Poincaré symmetry as scalars, spinors, and tensors. The fields are defined in such a way that each one corresponds to a definite physical state, i.e., a particle with defined physical quantities and quantum numbers, such as mass, spin etc.;**Energy positivity.**The Hamiltonian operator is supposed to have non negative eigenvalues. This property together with covariance under the action of Lorentz group implies that the physical four-momentum belongs to the light cone;**Microscopic causality.**Causality is imposed in the meaning of locality, that is, the field operators can commute/anticommute only if they are defined on points separated by space-like vectors: ${[\varphi \left(x\right),\phantom{\rule{0.166667em}{0ex}}\varphi \left(y\right)]}_{\pm}=0$ for all space-time points such that $({x}^{\mu}-{y}^{\mu}){\eta}_{\mu \nu}({x}^{\nu}-{y}^{\nu})<0$.

#### 2.2. Complex Lorentz Group

#### 2.3. Axiomatic CPT Theorem Demonstration

#### 2.4. Lagrangian Field Theory CPT Theorem Demonstration

- Scalar:$${\varphi}_{CPT}\left(x\right)=\theta \varphi \left(x\right){\theta}^{\u2020}={\varphi}^{\u2020}(-x);$$
- Fermionic field with spin $=\frac{1}{2}$:$${\psi}_{CPT}\left(x\right)=\theta \psi \left(x\right){\theta}^{\u2020}=-{\gamma}_{5}{\psi}^{\u2020T}(-x);$$
- Bosonic fields with spin $=1$$${A}_{\mu CPT}\left(x\right)=\theta {A}_{\mu}\left(x\right){\theta}^{\u2020}=-{A}_{\mu}^{\u2020}(-x).$$

## 3. CPT Violation Implies Lorentz Invariance Violation

## 4. Consequences of CPT Symmetry

## 5. CPT Violation Motivations

## 6. CPT Theorem in Curved Spacetime

**Locality and covariance.**The indices must transform under the action of a generic $\rho $ transformation as:$$({\rho}^{*}\times \dots {\rho}^{*}\times {\rho}_{*}^{-1}){C}_{\left({j}^{\prime}\right)}^{({i}_{1}^{\prime},\dots ,{i}_{n}^{\prime})}\left[{M}^{\prime}\right]={C}_{\left(j\right)}^{({i}_{1},\dots ,{i}_{n})}\left[M\right]$$**Identity element..**$${C}_{\left(j\right)}^{{i}_{1},\dots \mathbf{i},\dots ,{i}_{n}}({x}_{1},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)={C}_{\left(j\right)}^{{i}_{1},\dots ,{i}_{k-1},\phantom{\rule{0.166667em}{0ex}}{i}_{k+1},\dots ,{i}_{n}}({x}_{1},\dots {x}_{k-1},\phantom{\rule{0.166667em}{0ex}}{x}_{k+1},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)$$**Compatibility with ∗**$${C}_{\left(j\right)}^{{i}_{1},\dots ,{i}_{n}}({x}_{1},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)\sim {C}_{\left({j}^{*}\right)}^{{i}_{1}^{*},\dots ,{i}_{n}^{*}}{\pi}_{0}({x}_{1},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)$$**Commutativity-anticommutativity.**$${C}_{\left(j\right)}^{{i}_{1},\dots ,{i}_{k+1},\phantom{\rule{0.166667em}{0ex}}{i}_{k},\dots ,{i}_{n}}({x}_{1},\dots ,{x}_{k+1},\phantom{\rule{0.166667em}{0ex}}{x}_{k},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)=-{(-1)}^{F\left({i}_{k}\right)F\left({i}_{k+1}\right)}{C}_{\left(j\right)}^{{i}_{1},\dots ,{i}_{n}}({x}_{1},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)$$**Scaling degree.**$$sd\{{C}_{\left(k\right)}^{\left(i\right)\left(j\right)}\}\le dim\left(i\right)+dim\left(j\right)-dim\left(k\right);$$**Asymptotic positivity.**$dim\left(i\right)\ge 0$ and $dim\left(i\right)=0$ if and only if $i=\mathbb{I}$;**Spectrum condition.**The singularities on the field product must have a positive frequency;**Associativity.**An opportunely defined notion of associativity is required;**Analytic dependence upon the metric.**The OPE coefficients must be regular functionals of the space-time metric.

#### Construction of a QFT from OPE Coefficients

**Linearity.**$\forall {a}_{i}$ and $\forall {f}_{i}$ must be true that:$${\varphi}^{i}(\sum _{j}{a}_{j}{f}_{j})=\sum _{j}{a}_{j}\varphi \left({f}_{j}\right);$$**Existence of ∗ operator.**$${\left[{\varphi}^{\left(i\right)}\left(f\right)\right]}^{*}={\varphi}^{i*}\left(\overline{f}\right);$$**Relations arising from OPE.**If $O(\xb7)$ is a smeared quantum field, then $O\left(y\right)$ is a well defined algebra element for every point y, that is if $\langle O\left(y\right)|O{\left(y\right)}^{*}\rangle =0$ then $O\left(y\right)=0$ as an algebra element;**Anticommutation Relation.**$${\varphi}^{{i}_{1}}\left({f}_{1}\right){\varphi}^{{i}_{1}}\left({f}_{2}\right)={(-1)}^{K\left({i}_{1}{i}_{2}\right)}{\varphi}^{{i}_{1}}\left({f}_{2}\right){\varphi}^{{i}_{1}}\left({f}_{1}\right)$$**Positivity.**$$\langle {A}^{*}A\rangle \ge 0\phantom{\rule{2.em}{0ex}}\forall A\in \mathcal{A}\left(M\right);$$**OPE series expansion.**$$\langle {\varphi}^{{i}_{1}}\left({x}_{1}\right),\dots ,{\varphi}^{{i}_{n}}\left({x}_{n}\right)\rangle \sim \sum _{j}{C}_{j}^{({i}_{1},\dots ,{i}_{n})}({x}_{1},\dots {x}_{n},\phantom{\rule{0.166667em}{0ex}}y)\langle {\varphi}^{j}\left(y\right)\rangle ;$$**Spectrum condition.**The previously cited spectrum condition can be written in the following form:$$WF\left(\langle {\varphi}^{{i}_{1}}({x}_{1},\dots {\varphi}^{{i}_{n}}\left({x}_{n}\right)\rangle \right)\subset {\Gamma}_{n}\left(M\right)$$

**Lemma**

**1.**

**Theorem**

**1.**

## 7. CPT and Gravity

## 8. CPT Violation and LIV Research

#### 8.1. Very Special Relativity

#### 8.2. Standard Model Extension

#### 8.3. Theories Preserving Covariance

#### 8.4. CPT Violation and LIV Geometry Framework

## 9. Search for CPT and LIV Violation in Astroparticle Physics

#### 9.1. Ultra High Energy Cosmic Rays

#### 9.2. Time Delays

#### 9.3. CPT and LIV in Neutrino Physics

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Antonelli, V.; Miramonti, L.; Torri, M.D.C.
Phenomenological Effects of CPT and Lorentz Invariance Violation in Particle and Astroparticle Physics. *Symmetry* **2020**, *12*, 1821.
https://doi.org/10.3390/sym12111821

**AMA Style**

Antonelli V, Miramonti L, Torri MDC.
Phenomenological Effects of CPT and Lorentz Invariance Violation in Particle and Astroparticle Physics. *Symmetry*. 2020; 12(11):1821.
https://doi.org/10.3390/sym12111821

**Chicago/Turabian Style**

Antonelli, Vito, Lino Miramonti, and Marco Danilo Claudio Torri.
2020. "Phenomenological Effects of CPT and Lorentz Invariance Violation in Particle and Astroparticle Physics" *Symmetry* 12, no. 11: 1821.
https://doi.org/10.3390/sym12111821