CPT symmetry

Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T that is observed to be an exact symmetry of nature at the fundamental level.[1] The CPT theorem says that CPT symmetry holds for all physical phenomena, or more precisely, that any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry.


The CPT theorem appeared for the first time, implicitly, in the work of Julian Schwinger in 1951 to prove the connection between spin and statistics.[2] In 1954, Gerhart Lüders and Wolfgang Pauli derived more explicit proofs,[3][4] 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.[5] These proofs are based on the principle of Lorentz invariance and the principle of locality in the interaction of quantum fields. Subsequently, Res Jost gave a more general proof in the framework of axiomatic quantum field theory.

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 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 as well.

Derivation of the CPT theorem

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 rotation parameter. If this rotation parameter were real, 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, 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 can be extended to a Euclidean theory, defined by translating all the operators to imaginary time using the Hamiltonian. The commutation relations of the Hamiltonian, and the Lorentz generators, guarantee that Lorentz invariance implies rotational invariance, so that any state can be rotated by 180 degrees.

Since a sequence of two CPT reflections is equivalent to a 360-degree rotation, fermions change by a sign under two CPT reflections, while bosons do not. This fact can be used to prove the spin-statistics theorem.

Consequences and implications

The implication of CPT symmetry is that a "mirror-image" of our universe — with all objects having their positions reflected by an arbitrary plane (corresponding to a parity inversion), all momenta reversed (corresponding to a time inversion) and with all matter replaced by antimatter (corresponding to a charge 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 violations.

The CPT theorem can be generalized to take into account pin groups.

In 2002 Oscar Greenberg published an apparent proof that CPT violation implies the breaking of Lorentz symmetry.[6] If correct, this would imply that any study of CPT violation also includes Lorentz violation. However, Chaichian et al later disputed the validity of Greenberg's result.[7] Greenberg replied that the model used in their paper meant that their "proposed objection was not relevant to my result".[8]

The overwhelming majority of experimental searches for Lorentz violation have yielded negative results. A detailed tabulation of these results was given in 2011 by Kostelecky and Russell.[9]

See also


  1. ^ Kostelecký, V. A. (1998). "The Status of CPT". arXiv:hep-ph/9810365.
  2. ^ Schwinger, Julian (1951). "The Theory of Quantized Fields I". Physical Review. 82 (6): 914–927. Bibcode:1951PhRv...82..914S. doi:10.1103/PhysRev.82.914.
  3. ^ Lüders, G. (1954). "On the Equivalence of Invariance under Time Reversal and under Particle-Antiparticle Conjugation for Relativistic Field Theories". Kongelige Danske Videnskabernes Selskab, Matematisk-Fysiske Meddelelser. 28 (5): 1–17.
  4. ^ Pauli, W.; Rosenfelf, L.; Weisskopf, V., eds. (1955). Niels Bohr and the Development of Physics. McGraw-Hill. LCCN 56040984.
  5. ^ Whitaker, Andrew (2016). John Stuart Bell and Twentieth-Century Physics. Oxford University Press. ISBN 978-0198742999.
  6. ^ Greenberg, O. W. (2002). "CPT Violation Implies Violation of Lorentz Invariance". Physical Review Letters. 89 (23): 231602. arXiv:hep-ph/0201258. Bibcode:2002PhRvL..89w1602G. doi:10.1103/PhysRevLett.89.231602. PMID 12484997.
  7. ^ Chaichian, M.; Dolgov, A. D.; Novikov, V. A.; Tureanu, A. (2011). "CPT Violation Does Not Lead to Violation of Lorentz Invariance and Vice Versa". Physics Letters B. 699 (3): 177–180. arXiv:1103.0168. Bibcode:2011PhLB..699..177C. doi:10.1016/j.physletb.2011.03.026.
  8. ^ Greenberg, O. W. (4 May 2011). "Remarks on a challenge to the relation between CPT and Lorentz violation". arXiv:1105.0927. Bibcode:2011arXiv1105.0927G. The objection [arXiv:1103.0168] to my theorem [arXiv:hep-ph/0201258] that violation of CPT symmetry implies violation of Lorentz covariance is based on a nonlocal model in which time-ordered products are not well defined. I used covariance of time-ordered products as the condition for Lorentz covariance; therefore the proposed objection is not relevant to my result.
  9. ^ Kostelecký, V. A.; Russell, N. (2011). "Data tables for Lorentz and CPT violation". Reviews of Modern Physics. 83 (1): 11–31. arXiv:0801.0287. Bibcode:2011RvMP...83...11K. doi:10.1103/RevModPhys.83.11.


  • Sozzi, M.S. (2008). Discrete symmetries and CP violation. Oxford University Press. ISBN 978-0-19-929666-8.
  • Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, Inc. ISBN 978-0-471-60386-3.
  • R. F. Streater and A. S. Wightman (1964). PCT, spin and statistics, and all that. Benjamin/Cummings. ISBN 978-0-691-07062-9.

External links


Antihydrogen (H) is the antimatter counterpart of hydrogen. Whereas the common hydrogen atom is composed of an electron and proton, the antihydrogen atom is made up of a positron and antiproton. Scientists hope studying antihydrogen may shed light on the question of why there is more matter than antimatter in the observable universe, known as the baryon asymmetry problem. Antihydrogen is produced artificially in particle accelerators. In 1999, NASA gave a cost estimate of $62.5 trillion per gram of antihydrogen (equivalent to $94 trillion today), making it the most expensive material to produce. This is due to the extremely low yield per experiment, and high opportunity cost of using a particle accelerator.


The antiproton,
, (pronounced p-bar) is the antiparticle of the proton. Antiprotons are stable, but they are typically short-lived, since any collision with a proton will cause both particles to be annihilated in a burst of energy.

The existence of the antiproton with −1 electric charge, opposite to the +1 electric charge of the proton, was predicted by Paul Dirac in his 1933 Nobel Prize lecture. Dirac received the Nobel Prize for his previous 1928 publication of his Dirac equation that predicted the existence of positive and negative solutions to the Energy Equation () of Einstein and the existence of the positron, the antimatter analog to the electron, with positive charge and opposite spin.

The antiproton was first experimentally confirmed in 1955 at the Bevatron particle accelerator by University of California, Berkeley physicists Emilio Segrè and Owen Chamberlain, for which they were awarded the 1959 Nobel Prize in Physics. In terms of valence quarks, an antiproton consists of two up antiquarks and one down antiquark (uud). The properties of the antiproton that have been measured all match the corresponding properties of the proton, with the exception that the antiproton has electric charge and magnetic moment that are the opposites of those in the proton. The questions of how matter is different from antimatter, and the relevance of antimatter in explaining how our universe survived the Big Bang, remain open problems—open, in part, due to the relative scarcity of antimatter in today's universe.


Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection). Symmetry is an important property of both physical and abstract systems and it may be displayed in precise terms or in more aesthetic terms. The absence of or violation of symmetry that are either expected or desired can have important consequences for a system.


In physical cosmology, baryogenesis is the hypothetical physical process that took place during the early universe that produced baryonic asymmetry, i.e. the imbalance of matter (baryons) and antimatter (antibaryons) in the observed universe.

One of the outstanding problems in modern physics is the predominance of matter over antimatter in the universe. The universe, as a whole, seems to have a nonzero positive baryon number density – that is, matter exists. Since it is assumed in cosmology that the particles we see were created using the same physics we measure today, it would normally be expected that the overall baryon number should be zero, as matter and antimatter should have been created in equal amounts. This has led to a number of proposed mechanisms for symmetry breaking that favour the creation of normal matter (as opposed to antimatter) under certain conditions. This imbalance would have been exceptionally small, on the order of 1 in every 10000000000 (1010) particles a small fraction of a second after the Big Bang, but after most of the matter and antimatter annihilated, what was left over was all the baryonic matter in the current universe, along with a much greater number of bosons. Experiments reported in 2010 at Fermilab, however, seem to show that this imbalance is much greater than previously assumed. In an experiment involving a series of particle collisions, the amount of generated matter was approximately 1% larger than the amount of generated antimatter. The reason for this discrepancy is yet unknown.Most grand unified theories explicitly break the baryon number symmetry, which would account for this discrepancy, typically invoking reactions mediated by very massive X bosons (X) or massive Higgs bosons (H0). The rate at which these events occur is governed largely by the mass of the intermediate X or H0 particles, so by assuming these reactions are responsible for the majority of the baryon number seen today, a maximum mass can be calculated above which the rate would be too slow to explain the presence of matter today. These estimates predict that a large volume of material will occasionally exhibit a spontaneous proton decay.

Baryogenesis theories are based on different descriptions of the interaction between fundamental particles. Two main theories are electroweak baryogenesis (standard model), which would occur during the electroweak epoch, and the GUT baryogenesis, which would occur during or shortly after the grand unification epoch. Quantum field theory and statistical physics are used to describe such possible mechanisms.

Baryogenesis is followed by primordial nucleosynthesis, when atomic nuclei began to form.

Baryon asymmetry

In physics, the baryon asymmetry problem, also known as the matter asymmetry problem or the matter-antimatter asymmetry problem, is the observed imbalance in baryonic matter (the type of matter experienced in everyday life) and antibaryonic matter in the observable universe. Neither the standard model of particle physics, nor the theory of general relativity provides a known explanation for why this should be so, and it is a natural assumption that the universe be neutral with all conserved charges. The Big Bang should have produced equal amounts of matter and antimatter. Since this does not seem to have been the case, it is likely some physical laws must have acted differently or did not exist for matter and antimatter.

Several competing hypotheses exist to explain the imbalance of matter and antimatter that resulted in baryogenesis. However, there is as of yet no consensus theory to explain the phenomenon. As remarked in a 2012 research paper, "The origin of matter remains one of the great mysteries in physics."


Charge conjugation is a transformation that switches all particles with their corresponding antiparticles, and thus changes the sign of all charges: not only electric charge but also the charges relevant to other forces. 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.

CP violation

In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge conjugation symmetry) and P-symmetry (parity symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C symmetry) while its spatial coordinates are inverted ("mirror" or P symmetry). The discovery of CP violation in 1964 in the decays of neutral kaons resulted in the Nobel Prize in Physics in 1980 for its discoverers James Cronin and Val Fitch.

It plays an important role both in the attempts of cosmology to explain the dominance of matter over antimatter in the present Universe, and in the study of weak interactions in particle physics.

Experimental testing of time dilation

Time dilation as predicted by special relativity is often verified by means of particle lifetime experiments. According to special relativity, the rate of a clock C traveling between two synchronized laboratory clocks A and B, as seen by a laboratory observer, is slowed down relative to the laboratory clock rates. Since any periodic process can be considered a clock, the lifetimes of unstable particles such as muons must also be affected, so that moving muons should have a longer lifetime than resting ones. A variety of experiments confirming this effect have been performed both in the atmosphere and in particle accelerators. Another type of time dilation experiments is the group of Ives–Stilwell experiments measuring the relativistic Doppler effect.

Group field theory

Group field theory (GFT) is a quantum field theory in which the base manifold is taken to be a Lie group. It is closely related to background independent quantum gravity approaches such as loop quantum gravity, the spin foam formalism and causal dynamical triangulation. It can be shown that its perturbative expansion can be interpreted as spin foams and simplicial pseudo-manifolds (depending on the representation of the fields). Thus, its partition function defines a non-perturbative sum over all simplicial topologies and geometries, giving a path integral formulation of quantum spacetime.

Loschmidt's paradox

Loschmidt's paradox, also known as the reversibility paradox, irreversibility paradox or Umkehreinwand, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behaviour of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict; hence the paradox.

Michael Cates

Michael Elmhirst Cates (born 5 May 1961) is a British physicist. He is the 19th Lucasian Professor of Mathematics at the University of Cambridge and has held this position since 1 July 2015.

He was previously Professor of Natural Philosophy at the University of Edinburgh, and has held a Royal Society Research Professorship since 2007.His work focuses on the theory of soft matter, such as polymers, colloids, gels, liquid crystals, and granular material. A recurring goal of his research is to create a mathematical model that predicts the stress in a flowing material as a functional of the flow history of that material. Such a mathematical model is called a constitutive equation. He has worked on theories of active matter, particularly dense suspensions of self-propelled particles which can include motile bacteria. His interests also include fundamental field theories of active systems in which time-reversal symmetry (T-symmetry, and more generally, CPT symmetry) is absent. Such theories are characterised by nonzero steady-state Entropy production.

At Edinburgh, Cates was the Principal Investigator of an EPSRC Programme Grant, awarded in 2011, entitled Design Principles for New Soft Materials. On his departure for Cambridge, Cait MacPhee took over as Principal Investigator. Cates remains an Honorary Professor at Edinburgh.

Michael Holzscheiter

Michael Holzscheiter is a German-born professor at the University of New Mexico.Holzscheiter received his PhD from the University of Mainz in 1978. He has conducted numerous experiments with low energy antiprotons at CERN for over 30 years.

After initial work on antiprotons traps he initiated the ATHENA collaboration on forming antihydrogen for precision studies of CPT symmetry. Then he turned his interest to medical physics and led the AD-4/ACE (Antiproton Cell Experiment) collaboration on biological effects of antiprotons from 2003 to 2013.

Modern searches for Lorentz violation

Modern searches for Lorentz violation are scientific studies that look for deviations from Lorentz invariance or symmetry, a set of fundamental frameworks that underpin modern science and fundamental physics in particular. These studies try to determine whether violations or exceptions might exist for well-known physical laws such as special relativity and CPT symmetry, as predicted by some variations of quantum gravity, string theory, and some alternatives to general relativity.

Lorentz violations concern the fundamental predictions of special relativity, such as the principle of relativity, the constancy of the speed of light in all inertial frames of reference, and time dilation, as well as the predictions of the standard model of particle physics. To assess and predict possible violations, test theories of special relativity and effective field theories (EFT) such as the Standard-Model Extension (SME) have been invented. These models introduce Lorentz and CPT violations through spontaneous symmetry breaking caused by hypothetical background fields, resulting in some sort of preferred frame effects. This could lead, for instance, to modifications of the dispersion relation, causing differences between the maximal attainable speed of matter and the speed of light.

Both terrestrial and astronomical experiments have been carried out, and new experimental techniques have been introduced. No Lorentz violations could be measured thus far, and exceptions in which positive results were reported have been refuted or lack further confirmations. For discussions of many experiments, see Mattingly (2005). For a detailed list of results of recent experimental searches, see Kostelecký and Russell (2008–2013). For a recent overview and history of Lorentz violating models, see Liberati (2013).

Semileptonic decay

In particle physics the semileptonic decay of a hadron is a decay caused by the weak force in which one lepton (and the corresponding neutrino) is produced in addition to one or more hadrons. An example for this can be

K0 → e− + νe + π+This is to be contrasted with purely hadronic decays, such as K0 → π+ + π−, which are also mediated by the weak force.

Semileptonic decays of neutral kaons have been used to study kaon oscillations.

Spin foam

In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. Also, see loop quantum gravity.

Standard-Model Extension

Standard-Model Extension (SME) is an effective field theory that contains the Standard Model, general relativity, and all possible operators that break Lorentz symmetry.

Violations of this fundamental symmetry can be studied within this general framework. CPT violation implies the breaking of Lorentz symmetry,

and the SME includes operators that both break and preserve CPT symmetry.


T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal:

T-symmetry implies the conservation of entropy. Since the second law of thermodynamics means that entropy increases as time flows toward the future, the macroscopic universe does not in general show symmetry under time reversal. In other words, time is said to be non-symmetric, or asymmetric, except for special equilibrium states when the second law of thermodynamics predicts the time symmetry to hold. However, quantum noninvasive measurements are predicted to violate time symmetry even in equilibrium, contrary to their classical counterparts, although this has not yet been experimentally confirmed.

Time asymmetries are generally distinguished as among those...

Time reversibility

A mathematical or physical process is time-reversible if the dynamics of the process remain well-defined when the sequence of time-states is reversed.

A deterministic process is time-reversible if the time-reversed process satisfies the same dynamic equations as the original process; in other words, the equations are invariant or symmetrical under a change in the sign of time. A stochastic process is reversible if the statistical properties of the process are the same as the statistical properties for time-reversed data from the same process.

C, P, and T symmetries

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