# CPT Symmetry and Its Violation

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## Abstract

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## 1. Introduction

## 2. The CPT Theorem

#### 2.1. Proof Based on Lagrangian Field Theory

**Antilinearity of T.**The time-reversal transformation is an antilinear operation [11]; it complex conjugates complex numbers z: $T\phantom{\rule{0.166667em}{0ex}}z\phantom{\rule{0.166667em}{0ex}}{T}^{-1}={z}^{*}$. The other two transformations, charge conjugation C and parity inversion P, are both linear. This means that Θ, just like T, is also antilinear:

**Connection between Spin and Statistics.**The proof of the CPT theorem proceeds by using the connection between spin and statistics in the following way [12]. Since the Lagrangian density is a Lorentz scalar, spinor indices must be contracted, so that spinors always come in pairs. These spinor bilinears are formed such that they transform like Lorentz tensors, i.e., that they possess zero, one, or two Lorentz indices: $\overline{\psi}\psi $, $\overline{\chi}{\gamma}_{5}{\gamma}^{\mu}\psi $, $\overline{\psi}{\sigma}^{\mu \nu}\psi $, etc. This allows a straightforward Lorentz-invariant coupling to other tensors, like ${A}_{\mu}$, ϕ, and ${\partial}_{\mu}$. A CPT transformation yields $\overline{\chi}\left(x\right)\psi \left(x\right)\to \Theta \phantom{\rule{0.166667em}{0ex}}\overline{\chi}\left(x\right){\Theta}^{\u2020}\phantom{\rule{0.166667em}{0ex}}\Theta \phantom{\rule{0.166667em}{0ex}}\psi \left(x\right)\phantom{\rule{0.166667em}{0ex}}{\Theta}^{\u2020}={\chi}^{T}(-x){\gamma}_{5}{\gamma}^{0*}{\gamma}_{5}{\psi}^{\u2020T}(-x)=-{\chi}^{T}(-x){\gamma}^{0*}{\psi}^{\u2020T}(-x)=-{\chi}^{\u2020T\u2020}{\gamma}^{0*}{\psi}^{\u2020T}(-x)=-{\left[{\psi}^{T}(-x){\gamma}^{0T}{\chi}^{T\u2020}(-x)\right]}^{\u2020}=+{\left[{\left\{{\chi}^{\u2020}(-x){\gamma}^{0}\psi (-x)\right\}}^{T}\right]}^{\u2020}$. Here, we used that ${\gamma}^{0}$ and ${\gamma}_{5}$ anticommute, and that ${\gamma}_{5}{\gamma}_{5}=1$. The crucial step is the last one, in which we simplified the spinor-space transpose; it requires reversing the order of χ and ψ. It is this step that uses the connection between spin and statistics: fermion fields anticommute. The term in the curly brackets is a spinor-space scalar, so the spinor transposition may be left out and we have $\overline{\chi}\left(x\right)\psi \left(x\right)\to {\left[\overline{\chi}(-x)\psi (-x)\right]}^{\u2020}$. Extension of this reasoning including fermion anticommutation to the other Dirac bilinears shows that they also follow the general rule (9) established above for dynamical tensor densities.

**Lorentz invariance.**We have already used Lorentz symmetry implicitly in the above individual ingredients for the construction of field-theory Lagrangian densities: the Minkowski position x, the scalar, spinor, and vector fields in Equations (5)–(8), and the general tensor densities in Equation (9) are all realizations of the Lorentz group, i.e., rotations and boosts of these objects are implemented by Lorentz transformations. In Lagrangian field theory in Minkowski space, Lorentz invariance is guaranteed if the action, and thus the Lagrangian density, are both Lorentz scalars. This implies that all fields and derivatives in a Lagrangian density must be combined such that not only the spinor indices, but also all Lorentz indices are properly contracted. Through this pairwise contraction, the total number of Lorentz indices in each Lagrangian term of field products must be even $n=2k$, $k\in \mathbb{N}$. According to Equation (9), this yields for the Lagrangian density

**Unitarity.**The next ingredient for the CPT theorem is a Hermitian Lagrangian density $\mathcal{L}={\mathcal{L}}^{\u2020}$, so that with Equation (13), we have

**Point interactions.**To conclude the Lagrangian version of the proof of the CPT theorem, we show that the action $S=\int {d}^{4}x\phantom{\rule{0.166667em}{0ex}}\mathcal{L}\left(x\right)$ remains invariant under Θ. With the above result (14), we write

#### 2.2. Proof Based on Axiomatic Field Theory

**(1)****Lorentz- and translation-covariant Hilbert space $\mathbb{H}$.**This assumption essentially states that we consider a relativistic version of quantum theory in which the usual rules of quantum mechanics apply. In particular, there are unitary operators $U(\mathsf{\Lambda},a)$ that implement Lorentz transformations Λ and spacetime translations a. The unitarity of these transformations ensures that under Λ and a states in $\mathbb{H}$ transform to other states in $\mathbb{H}$ such that all transition amplitudes remain unchanged.**(2)****Vacuum state.**The Hilbert space contains a unique state, called the vacuum $|0\rangle $, that remains invariant under both the Lorentz transformations and the translations up to a phase $U(\mathsf{\Lambda},a)|0\rangle \sim |0\rangle $. In particular, the vacuum can neither have a nonzero four-momentum nor a nonzero angular momentum, as these quantities would change under $U(\mathsf{\Lambda},a)$. Together with axiom (4) below, the vacuum needs to be the state with lowest energy. These requirements are intuitively reasonable: the flat-spacetime vacuum looks the same to all inertial observers. An additional, more technical assumption is that $|0\rangle $ be cyclic. This essentially means that all other physical states in $\mathbb{H}$ can be constructed by acting with the field operators of axiom (3) on the vacuum. This property is akin to that of the usual quantum harmonic oscillator, where excited states can be reached from the ground state with the creation operator.**(3)****Field operators.**Physical quantities are represented by polynomials of field operators $\varphi \left(x\right)$ acting on this Hilbert space. These field operators transform under the Lorentz transformations as scalars, vectors, tensors, spinors, etc. Moreover, these fields are set up such that each corresponds to a definite finite spin and mass allowing the usual particle interpretation. It turns out that field operators are mathematically not well-defined at a spacetime point, so there is the technical assumption of them being tempered distributions “smeared out” with test functions. This assumption can, for example, be used to establish continuity properties as the field operators vary with spacetime, but otherwise this level of rigor will be unnecessary for our present purposes.**(4)****Energy positivity.**Translation invariance leads to a conserved four-momentum operator ${P}^{\mu}$. Its zeroth component ${P}^{0}$, the energy operator or Hamiltonian, is postulated to have non-negative eigenvalues ${p}^{0}\ge 0$. Together with the condition of Lorentz symmetry, this implies that the four-momentum eigenvalues ${p}^{\mu}$ are lightlike or timelike four-vectors ${p}^{\mu}{p}_{\mu}\ge 0$. In other words, ${p}^{\mu}$ must lie in the forward momentum-space lightcone. This property is closely tied to the requirement of stability: if there were no lowest-energy state, it would seem difficult to prevent the system from filling an infinite number of pairs of positive- and negative-energy states.**(5)****Microscopic causality.**Many textbooks seems to suggest that the property of causality is automatically contained in a Lorentz-symmetric theory. However, consider a model with spacelike particle four-velocities ${u}^{\mu}$. Being a four-vector, ${u}^{\mu}$ transforms covariantly under the Lorentz transformations compatible with Lorentz symmetry. However, a spacelike four-velocity is associated with superluminal particle speeds and thus acausalities. For this reason, causality is imposed separately as follows. Field operators ϕ commute or anticommute if they cannot be connected by light signals: ${[\varphi \left(x\right),\varphi \left(y\right)]}_{\pm}=0$ for $({x}^{\mu}-{y}^{\mu})({x}_{\mu}-{y}_{\mu})<0$. In the mathematical-physics literature, this requirement is sometimes also called locality. The microcauslity condition may be understood intuitively by recalling the usual quantum-mechanical uncertainty relation ${\mathrm{\Delta}}_{\psi}A\phantom{\rule{0.166667em}{0ex}}{\mathrm{\Delta}}_{\psi}B\ge \frac{1}{2}|\langle \psi |[A,B]|\psi \rangle |$ for two Hermitian operators A and B, with the usual definition of the uncertainty ${\mathrm{\Delta}}_{\psi}\mathcal{A}=\sqrt{\langle \psi |{\mathcal{A}}^{2}|\psi \rangle -{\langle \psi \left|\mathcal{A}\right|\psi \rangle}^{2}}$ for any Hermitian operator $\mathcal{A}$ with respect to the state $|\psi \rangle $. Here, one measurement generally affects the other measurement because their uncertainties are not independent unless the commutator $[A,B]$ vanishes. A careful reasoning in the present context shows that with the above microcausality condition, the physics at x cannot affect the physics at y and vice versa if the separation between x and y is spacelike [17]. We remark in passing that in this axiomatic framework the spin–statistics theorem follows rigorously, so that the above choice between commutators and anticommutators is actually fixed: commutators for integer-spin fields and anticommutators for half-integer spin fields.

**Physical Lorentz symmetry.**Axioms (1) and (3) imply invariance of the Wightman function under the usual physical Lorentz transformation Λ, so we can write:

**Energy positivity.**The next idea is to resolve $\mathcal{W}$ into its Fourier components. These will contain plane-wave 4-momenta, for which we can use Axiom (4). As an example, let us sketch this idea for the particular Wightman function $\mathcal{W}(y-x)=\langle 0\left|\varphi \right(x\left)\varphi \right(y\left)\right|0\rangle $ involving two field operators. Inserting a complete set of momentum eigenstates $\int {d}^{4}p\phantom{\rule{0.166667em}{0ex}}|p\rangle \langle p|$ and employing the translation operator on both fields $\varphi \left(x\right)=\mathrm{exp}(iP\xb7x)\varphi \left(0\right)\mathrm{exp}(-iP\xb7x)$ and $\varphi \left(y\right)=\mathrm{exp}(iP\xb7y)\varphi \left(0\right)\mathrm{exp}(-iP\xb7y)$ yields

**Complex Lorentz transformation.**A theorem by Bargmann, Hall, and Wightman [19] now states that Equation (24) remains valid for an even larger set of complex $\mathrm{\Delta}z$ and also for complex Lorentz transformations. This larger set consists of all the original $\mathrm{\Delta}{z}_{k}$ that have their imaginary part in the forward cone and also all $\mathrm{\Delta}{z}_{k}^{\prime}$ that can be generated from $\mathrm{\Delta}{z}_{k}$ with complex Lorentz transformations ${\mathsf{\Lambda}}_{c}$: $\mathrm{\Delta}{z}_{k}^{\prime}={\mathsf{\Lambda}}_{c}\phantom{\rule{0.166667em}{0ex}}\mathrm{\Delta}{z}_{k}$. This set is sometimes called the “extended tube.” Equation (24) therefore takes the form

**Microscopic causality and spin–statistics.**Translating Equation (28) back to vacuum expectation values gives

## 3. Some Physical Consequences of CPT Symmetry

#### 3.1. CPT-Symmetry Implications for Eigenstates

#### 3.2. CPT-Symmetry Implications for the Dynamics of Nonstationary States

#### 3.3. CPT-Symmetry Implications for Couplings

## 4. CPT Violation

## Acknowledgments

## Conflicts of Interest

## References and Notes

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Lehnert, R.
CPT Symmetry and Its Violation. *Symmetry* **2016**, *8*, 114.
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Lehnert R.
CPT Symmetry and Its Violation. *Symmetry*. 2016; 8(11):114.
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Lehnert, Ralf.
2016. "CPT Symmetry and Its Violation" *Symmetry* 8, no. 11: 114.
https://doi.org/10.3390/sym8110114