CPT symmetry
Encyclopedia
CPT symmetry is a fundamental symmetry
Symmetry in physics
In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...

of physical law
Physical law
A physical law or scientific law is "a theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." Physical laws are typically conclusions...

s under transformation
Transformation (mathematics)
In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.Examples include...

s that involve the inversions of charge
Charge (physics)
In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...

, parity
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...

, and time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

simultaneously.

## History

Efforts during the late 1950s revealed the violation of P-symmetry by phenomena that involve the weak force, and there were well-known violations of C-symmetry
C-symmetry
In physics, C-symmetry means the symmetry of physical laws under a charge-conjugation transformation. Electromagnetism, gravity and the strong interaction all obey C-symmetry, but weak interactions violate C-symmetry.-Charge reversal in electromagnetism:...

as well. For a short time, the CP-symmetry was believed to be preserved by all physical phenomena, but that was later found to be false too, which implied, by CPT invariance, violations of T-symmetry
T-symmetry
T Symmetry is the symmetry of physical laws under a time reversal transformation: T: t \mapsto -t.Although in restricted contexts one may find this symmetry, the observable universe itself does not show symmetry under time reversal, primarily due to the second law of thermodynamics.Time asymmetries...

as well. The CPT theorem requires the preservation of CPT symmetry by all physical phenomena. It assumes the correctness of quantum
Quantum
In physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...

laws and Lorentz invariance. Specifically, the CPT theorem states that any Lorentz invariant local quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

with a Hermitian
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

must have CPT symmetry.

The CPT theorem appeared for the first time, implicitly, in the work of Julian Schwinger
Julian Schwinger
Julian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...

in 1951 to prove the connection between spin and statistics
Spin-statistics theorem
In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...

. In 1954, Gerhart Lüders
Gerhart Lüders
Gerhart Lüders was a German theoretical physicist who worked mainly in quantum field theory and was well known for the discovery and a general proof of the CPT theorem...

and Wolfgang Pauli
Wolfgang Pauli
Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...

derived more explicit proofs, so this theorem is sometimes known as the Lüders-Pauli theorem. At about the same time, and independently, this theorem was also proved by John Stewart Bell
John Stewart Bell
John Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...

. These proofs are based on the validity of Lorentz invariance and the Principle of locality
Principle of locality
In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must violate either the principle of locality or the form of philosophical realism known as counterfactual...

in the interaction of quantum fields. Subsequently Res Jost
Res Jost
Res Jost in Zurich) was a Swiss theoretical physicist, who worked mainly in constructive quantum field theory.-Life and work:...

gave a more general proof in the framework of axiomatic quantum field theory
Axiomatic quantum field theory
Axiomatic quantum field theory is a mathematical discipline which aims to describe quantum field theory in terms of rigorous axioms. It is strongly associated with functional analysis and operator algebras, but has also been studied in recent years from a more geometric and functorial...

.

## Derivation

Consider a Lorentz boost in a fixed direction z. This can be interpreted as a rotation of the time axis into the z axis, with an imaginary
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...

rotation parameter. If this rotation parameter were real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

, it would be possible for a 180° rotation to reverse the direction of time and of z. Reversing the direction of one axis is a reflection of space in any number of dimensions. If space has 3 dimensions, it is equivalent to reflecting all the coordinates, because an additional rotation of 180° in the x-y plane could be included.

This defines a CPT transformation if we adopt the Feynman-Stueckelberg interpretation of antiparticles as the corresponding particles traveling backwards in time. This interpretation requires a slight analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

, which is well-defined only under the following assumptions:
1. The theory is Lorentz invariant;
2. The vacuum is Lorentz invariant;
3. The energy is bounded below.

When the above hold, quantum theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

can be extended to a Euclidean theory, defined by translating all the operators to imaginary time using the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

. The commutation relations of the Hamiltonian, and the Lorentz generator
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...

s, guarantee that Lorentz invariance implies rotational invariance
Rotational invariance
In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument...

, so that any state can be rotated by 180 degrees.

Since a sequence of two CPT reflections is equivalent to a 360-degree rotation, fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s change by a sign under two CPT reflections, while boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s do not. This fact can be used to prove the spin-statistics theorem
Spin-statistics theorem
In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...

.

## Consequences and implications

A consequence of this derivation is that a violation of CPT automatically indicates a Lorentz violation.

The implication of CPT symmetry is that a "mirror-image" of our universe — with all objects having their positions reflected by an imaginary plane (corresponding to a parity
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...

inversion), all momenta
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

reversed (corresponding to a time inversion
T-symmetry
T Symmetry is the symmetry of physical laws under a time reversal transformation: T: t \mapsto -t.Although in restricted contexts one may find this symmetry, the observable universe itself does not show symmetry under time reversal, primarily due to the second law of thermodynamics.Time asymmetries...

) and with all matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

replaced by antimatter
Antimatter
In particle physics, antimatter is the extension of the concept of the antiparticle to matter, where antimatter is composed of antiparticles in the same way that normal matter is composed of particles...

(corresponding to a charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

inversion)— would evolve under exactly our physical laws. The CPT transformation turns our universe into its "mirror image" and vice versa. CPT symmetry is recognized to be a fundamental property of physical laws.

In order to preserve this symmetry, every violation of the combined symmetry of two of its components (such as CP) must have a corresponding violation in the third component (such as T); in fact, mathematically, these are the same thing. Thus violations in T symmetry are often referred to as CP violation
CP violation
In particle physics, CP violation is a violation of the postulated CP-symmetry: the combination of C-symmetry and P-symmetry . CP-symmetry states that the laws of physics should be the same if a particle were interchanged with its antiparticle , and left and right were swapped...

s.

The CPT theorem can be generalized to take into account pin group
Pin group
In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group....

s.

## CPT violation

In 2002 Oscar Greenberg proved that CPT violation implies the breaking of Lorentz symmetry. This implies that any study of CPT violation includes also Lorentz violation. Several [experimental searches] of such violations have been performed during the last few years and recently there has been some strong evidence for a violation of charge symmetry in that antineutrinos seem to have a different mass than neutrinos. A detailed list of results of these experimental searches are summarized in the Data Tables for Lorentz and CPT Violation.

• Poincaré symmetry
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...

and Quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

• Parity (physics)
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...

, Charge conjugation and Time reversal symmetry
• CP violation
CP violation
In particle physics, CP violation is a violation of the postulated CP-symmetry: the combination of C-symmetry and P-symmetry . CP-symmetry states that the laws of physics should be the same if a particle were interchanged with its antiparticle , and left and right were swapped...

and kaon
Kaon
In particle physics, a kaon is any one of a group of four mesons distinguished by the fact that they carry a quantum number called strangeness...

• Background information on Lorentz and CPT violation by Alan Kostelecký
Alan Kostelecký
Alan Kostelecký is a theoretical physicist who is currently a distinguished professor of physics at Indiana University, Bloomington. He is noted for his work on Lorentz symmetry breaking in particle physics...

at Theoretical Physics Indiana University
• Data Tables for Lorentz and CPT Violation at the arXiv
ArXiv
The arXiv |Chi]], χ) is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance which can be accessed online. In many fields of mathematics and physics, almost all...

• The Pin Groups in Physics: C, P, and T at the arXiv
ArXiv
The arXiv |Chi]], χ) is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance which can be accessed online. In many fields of mathematics and physics, almost all...

• Charge, Parity, and Time Reversal (CPT) Symmetry at LBL
Lawrence Berkeley National Laboratory
The Lawrence Berkeley National Laboratory , is a U.S. Department of Energy national laboratory conducting unclassified scientific research. It is located on the grounds of the University of California, Berkeley, in the Berkeley Hills above the central campus...

• CPT Invariance Tests in Neutral Kaon Decay at LBL
Lawrence Berkeley National Laboratory
The Lawrence Berkeley National Laboratory , is a U.S. Department of Energy national laboratory conducting unclassified scientific research. It is located on the grounds of the University of California, Berkeley, in the Berkeley Hills above the central campus...

• Space--Time Symmetry, CPT and Mirror Fermions at the arXiv
ArXiv
The arXiv |Chi]], χ) is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance which can be accessed online. In many fields of mathematics and physics, almost all...

8-component theory for fermions in which T-parity can be a complex number with unit radius. The CPT invariance is not a theorem but a better to have property in these class of theories.