Editing CPT symmetry

{{Short description|Invariance under simultaneous charge conjugation, parity transformation and time reversal}}
{{Redirect|CPT theorem|the album by Greydon Square|The C.P.T. Theorem}}

'''Charge, parity, and time reversal symmetry''' is a fundamental [[symmetry in physics|symmetry]] of [[physical law]]s under the simultaneous [[transformation (mathematics)|transformation]]s of [[charge conjugation]] (C), [[parity transformation]] (P), and [[T-symmetry|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.<ref name=kost>
{{cite arXiv
 |last1=Kostelecký |first1=V. A.
 |year=1998
 |title=The Status of CPT
 |eprint=hep-ph/9810365
}}</ref><ref>{{Cite news|url=https://www.forbes.com/sites/startswithabang/2020/03/25/the-one-symmetry-that-the-universe-forbids-us-from-violating/?sh=271c8cd42d4f|title = This is the One Symmetry That the Universe Must Never Violate|website = [[Forbes]] | last1=Siegel | first1=Ethan }}</ref> 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 [[Self-adjoint operator|Hermitian]] [[Hamiltonian (quantum mechanics)|Hamiltonian]] must have CPT symmetry. In layman terms, this stipulates that an [[antimatter]], mirrored, and [[T-symmetry|time reversed]] universe would behave exactly the same as our regular universe.

==History==
The CPT theorem appeared for the first time, implicitly, in the work of [[Julian Schwinger]] in 1951 to prove the [[Spin-statistics theorem|connection between spin and statistics]].<ref>
{{Cite journal
 |last = Schwinger |first = Julian
 |date = 1951
 |title =The Theory of Quantized Fields I
 |journal=[[Physical Review]]
 |volume=82 |issue=6 |pages = 914–927
 |bibcode = 1951PhRv...82..914S
 |doi = 10.1103/PhysRev.82.914
|s2cid = 121971249
 }}</ref> In 1954, [[Gerhart Lüders]] and [[Wolfgang Pauli]] derived more explicit proofs,<ref name=luders>
{{cite journal
 |last=Lüders |first=G.
 |year=1954
 |title=On the Equivalence of Invariance under Time Reversal and under Particle-Antiparticle Conjugation for Relativistic Field Theories
 |journal=[[Kongelige Danske Videnskabernes Selskab, Matematisk-Fysiske Meddelelser]]
 |volume=28 |issue=5 |pages=1–17
}}</ref><ref name=one>
{{cite book
 |editor1-last=Pauli |editor1-first=W.
 |editor2-last=Rosenfelf |editor2-first=L.
 |editor3-last=Weisskopf |editor3-first=V.
 |title=Niels Bohr and the Development of Physics
 |publisher=[[McGraw-Hill]]
 |year=1955
 |lccn=56040984
}}</ref> 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]].<ref>{{cite book |last=Whitaker |first=Andrew |title=John Stuart Bell and Twentieth-Century Physics |year=2016 |isbn=978-0198742999 |publisher=[[Oxford University Press]] |url=https://books.google.com/books?id=tDtRDAAAQBAJ&q=bell+luders+pauli+theorem&pg=PT186 }}</ref><ref>{{cite journal | author = Bell, John Stewart | year = 1955 | title = Time reversal in field theory | journal = Proc. R. Soc. Lond. A | volume = 231 | issue = 1187 | pages = 479–495 | doi = 10.1098/rspa.1955.0189| bibcode = 1955RSPSA.231..479B | s2cid = 123577175 }}</ref> 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 1958 using the framework of [[axiomatic quantum field theory]].

Efforts during the late 1950s revealed the violation of [[P-symmetry#Quantum mechanics|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 in the 1960s 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 number|imaginary]] rotation parameter. If this rotation parameter were [[real number|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 [[Antiparticle#Feynman–Stueckelberg interpretation|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:
#The theory is [[Lorentz invariant]];
#The vacuum is Lorentz invariant;
#The energy is bounded below.

When the above hold, [[quantum field theory|quantum theory]] can be extended to a Euclidean theory, defined by translating all the operators to imaginary time using the [[Hamiltonian (quantum mechanics)|Hamiltonian]]. The [[commutation relation]]s of the Hamiltonian, and the [[Poincaré group|Lorentz generator]]s, 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, [[fermion]]s change by a sign under two CPT reflections, while [[boson]]s 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 through an arbitrary point (corresponding to a [[parity (physics)|parity]] inversion), all [[momentum|momenta]] reversed (corresponding to a [[T-symmetry|time inversion]]) and with all [[matter]] replaced by [[antimatter]] (corresponding to a [[electric charge|charge]] inversion) — would evolve under exactly our physical laws. The CPT transformation turns our universe into its "mirror image" and vice versa.<ref>[https://www.livescience.com/mirror-universe-explains-dark-matter Our universe may have a twin that runs backward in time] Paul Sutter, Live Science. March 16th, 2022</ref> 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]]s.

The CPT theorem can be generalized to take into account [[pin group]]s.

In 2002 [[Oscar Greenberg]] proved that, with reasonable assumptions, CPT violation implies the breaking of [[Lorentz symmetry]].<ref name="Greenberg">
{{cite journal
 |last=Greenberg |first=O. W.
 |year=2002
 |title=CPT Violation Implies Violation of Lorentz Invariance
 |journal=[[Physical Review Letters]]
 |volume=89 |issue= 23|pages=231602
 |arxiv=hep-ph/0201258
 |bibcode=2002PhRvL..89w1602G
 |doi=10.1103/PhysRevLett.89.231602
 |pmid=12484997
|s2cid=9409237
 }}</ref>

CPT violations would be expected by some [[string theory]] models, as well as by some other models that lie outside point-particle quantum field theory. Some proposed violations of Lorentz invariance, such as a [[compact dimension]] of cosmological size, could also lead to CPT violation. Non-unitary theories, such as proposals where black holes violate unitarity, could also violate CPT. As a technical point, fields with infinite spin could violate CPT symmetry.<ref>{{cite journal |last1=Lehnert |first1=Ralf |title=CPT Symmetry and Its Violation |journal=Symmetry |date=November 2016 |volume=8 |issue=11 |pages=114 |doi=10.3390/sym8110114 |bibcode=2016Symm....8..114L |language=en |issn=2073-8994|doi-access=free }}</ref>

The overwhelming majority of [[Modern searches for Lorentz violation|experimental searches for Lorentz violation]] have yielded negative results. A detailed tabulation of these results was given in 2011 by Kostelecky and Russell.<ref name="DataTables">
{{cite journal
 |last1=Kostelecký |first1=V. A.
 |last2=Russell |first2=N.
 |title=Data tables for Lorentz and ''CPT'' violation
 |year=2011
 |journal=[[Reviews of Modern Physics]]
 |volume=83 |issue=1 |pages=11–31
 |arxiv=0801.0287
 |bibcode=2011RvMP...83...11K
 |doi=10.1103/RevModPhys.83.11
 |s2cid=3236027
 }}</ref>

==See also==

*[[Poincaré group|Poincaré symmetry]] and [[Quantum field theory]]
*[[Parity (physics)]], [[Charge conjugation]] and [[T-symmetry]]
*[[CP violation]] and [[kaon]]
*[[IKAROS]] scientific results
*{{section link|Gravitational interaction of antimatter#CPT theorem}}

==References==
{{Reflist|25em}}

==Sources==
*{{cite book|author=Sozzi, M.S.|title=Discrete symmetries and CP violation|publisher=Oxford University Press|year=2008|isbn=978-0-19-929666-8}}
*{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |publisher=Wiley, John & Sons, Inc |year=1987 |isbn=978-0-471-60386-3}}
*{{cite book |author=[[R. F. Streater]] and [[A. S. Wightman]] |title=PCT, spin and statistics, and all that|publisher=Benjamin/Cummings |year=1964 |isbn=978-0-691-07062-9}}

==External links==
*[https://web.archive.org/web/20190123122951/http://www.physics.indiana.edu/~kostelec/faq.html Background information on Lorentz and CPT violation] by [[Alan Kostelecký]] at Theoretical Physics Indiana University
*{{cite journal |arxiv=0801.0287|bibcode=2011RvMP...83...11K|title=Data tables for Lorentz and CPT violation|last1=Kostelecký|first1=V. Alan|last2=Russell|first2=Neil|journal=Reviews of Modern Physics|volume=83|issue=1|pages=11|year=2011|doi=10.1103/RevModPhys.83.11|s2cid=3236027}}
*{{cite journal |arxiv=math-ph/0012006|doi=10.1142/S0129055X01000922|title=The Pin Groups in Physics: C, P and T|year=2001|last1=Berg|first1=Marcus|last2=Dewitt-Morette|first2=Cécile|last3=Gwo|first3=Shangjr|last4=Kramer|first4=Eric|journal=Reviews in Mathematical Physics|volume=13|issue=8|pages=953–1034|s2cid=119560073}}
*[http://www.lbl.gov/abc/wallchart/chapters/05/2.html Charge, Parity, and Time Reversal (CPT) Symmetry] {{Webarchive|url=https://web.archive.org/web/20110805130543/http://www.lbl.gov/abc/wallchart/chapters/05/2.html |date=2011-08-05 }} at [[Lawrence Berkeley National Laboratory|LBL]]
*[http://pdg.lbl.gov/2006/reviews/cpt_s011254.pdf CPT Invariance Tests in Neutral Kaon Decay] at [[Lawrence Berkeley National Laboratory|LBL]]
*{{cite arXiv |eprint=hep-th/0010074|title=Space--Time Symmetry, CPT and Mirror Fermions|last1=Ying|first1=S.|year=2000}} – 8-component theory for fermions in which ''T-parity'' <!-- (''P-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.
*[https://www.youtube.com/watch?v=yArprk0q9eE This Particle Breaks Time Symmetry] – [[YouTube]] video by [[Veritasium]]
* An elementary discussion of CPT violation is given in chapter 15 of this student level textbook [https://www.routledge.com/Fundamentals-of-Molecular-Symmetry/Bunker-Jensen/p/book/9780750309417]

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