

This page answers some frequently asked questions about Lorentz and CPT violation.
The focus below is mainly on the global Lorentz and CPT properties of the Standard Model of particle physics in flat spacetime. Applications of these ideas also exist to the local Lorentz and diffeomorphism properties of Einstein's theory of gravity, General Relavitity, including its coupling to the Standard Model.
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Answering this question requires understanding what is meant by "Lorentz transformations" and the "CPT transformation."
Lorentz transformations come in two basic types, rotations and boosts.
The CPT transformation is formed by combining three transformations: charge conjugation (C), parity inversion (P), and time reversal (T).
A physical system is said to have "Lorentz symmetry" if the relevant laws of physics are unaffected by Lorentz transformations (rotations and boosts). Similarly, a system is said to have "CPT symmetry" if the physics is unaffected by the combined transformation CPT. These symmetries are the basis for Einstein's relativity.
Experiments show to exceptionally high precision that all the basic laws of nature seem to have both Lorentz and CPT symmetry. Experimental results are compiled in the Data Tables for Lorentz and CPT Violation.
Note: CPT is the only combination of C, P, T that is presently observed to be an exact symmetry of nature.
The CPT theorem is a very general theoretical result linking Lorentz and CPT symmetry. Roughly, it states that certain theories (local quantum field theories) with Lorentz symmetry must also have CPT symmetry. These theories include all the ones used to describe known particle physics (for example, electrodynamics or the Standard Model) and many proposed theories (for example, Grand Unified Theories).
The CPT theorem can be used to show that a particle and its antiparticle must have certain identical properties, including mass, lifetime, and size of charge and magnetic moment.
Many texts discuss the CPT theorem and its implications. See, for example, R.G. Sachs, The Physics of Time Reversal (University of Chicago Press, Chicago, 1987).
The existence of highprecision experimental tests together with the general proof of the CPT theorem for Lorentzsymmetric theories implies that the observation of Lorentz or CPT violation would be a sensitive signal for unconventional physics. This means it's interesting to consider possible theoretical mechanisms through which Lorentz or CPT symmetry might be violated.
It is relatively easy to write down a phenomenological description of Lorentz or CPT violation, without attempting to create a consistent theory for the effects. However, without an underlying theory one cannot know whether the phenomenology could be relevant to nature or how plausible it really is.
In contrast, it is relatively difficult to find a theoretically compelling description of Lorentz or CPT violation because Lorentz and CPT symmetry is deeply ingrained into the structure of modern theories of nature. Most published suggestions for a theory of Lorentz or CPT violation either have physical features that seem unlikely to be realized in nature or involve radical revisions of conventional quantum field theory, or both.
In a series of books and articles since the 1980s, we have developed a theoretical framework to describe Lorentz and CPT violation that is compatible both with experimental constraints and with established quantum field theory.
The theory suggests that apparent breaking of CPT and Lorentz symmetry might be observable in existing or feasible experiments, and it leads to a general phenomenology for violations of spacetime symmetries at the level of the Standard Model of particle physics and General Relativity. Other standard theories such as Quantum Electrodynamics are recovered as special cases.
All elementary particles and their nongravitational interactions are very successfully described by a theory called the Standard Model of particle physics. At the classical level, gravity is well described by Einstein's General Relativity.
We have constructed a generalization of the usual Standard Model and General Relativity that has all the conventional desirable properties but that allows for violations of Lorentz and CPT symmetry. This theory is called the StandardModel Extension, or SME.
The StandardModel Extension provides a quantitative description of Lorentz and CPT violation, controlled by a set of coefficients whose values are to be determined or constrained by experiment. The theory of General Relativity coupled to the Standard Model is the special limit in which all the coefficients are zero.
A type of converse to the CPT theorem has been proved under mild assumptions: if CPT is violated, then Lorentz symmetry is too. This implies observable CPT violation is described by the StandardModel Extension. A constructive proof is given in V.A. Kostelecky and D. Colladay, Phys. Rev. D 55, 6760 (1997), and a formal proof in O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002), archived preprint.
Violations of all the symmetries C, P, T, CP are predicted by the Standard Model of particle physics and are observed in experiments. Only the combination CPT is required by the Standard Model to be a symmetry of nature. For example, processes are known in nature that violate C but not CPT.
The StandardModel Extension allows for violations of spacetime symmetries, including Lorentz and CPT symmetry, that cannot occur in the usual Standard Model or Einstein's General Relativity.;
We have shown, for instance, that it allows for CPT violation unaccompanied by C or P or CP violation, causing effects such as a difference between the spectra of hydrogen and antihydrogen. Similarly, it allows for CPT violation unaccompanied by P or T or PT violation, producing effects such as modifications of the behavior of kaons and antikaons. All these effects are forbidden in conventional theories obeying the CPT theorem.
A particularly interesting and conceivably physical source of Lorentz and CPT violation is spontaneous symmetry breaking. This common physical effect occurs when a symmetry of the dynamics is not respected by the solutions of the theory.
For example, the dynamical forces controlling the interactions between planets in Newtonian gravity have rotational symmetry, but the solution of the theory representing our solar system exhibits a definite orientation in space given by the plane of the solar system. Another example is the spontaneous breaking of the electroweak gauge symmetry in the Standard Model.
We have proposed that, even if the underlying theory of nature has Lorentz and CPT symmetry, the vacuum solution of the theory could spontaneously violate these symmetries. This is an attractive way of breaking Lorentz and CPT symmetry because the dynamics remains symmetric and so desirable features of the symmetry are preserved.
The usual Standard Model doesn't have the dynamics necessary to cause spontaneous Lorentz and CPT violation. However, spontaneous breaking could occur in more complicated theories. These may include ones based on extended objects like strings, some of which are known to have dynamics of the necessary type.
As in the usual Standard Model, spontaneous breaking of Lorentz and CPT symmetry is triggered by interactions destabilizing the empty vacuum. In the usual case, the vacuum fills with quantities that are symmetric under Lorentz and CPT transformations (but that violate other symmetries). Here, the vacuum fills instead with quantities that are oriented in the fourdimensional sense and hence break spacetime symmetries, including Lorentz invariance and (under some circumstances) CPT.
In this scenario, CPT breaking always implies Lorentz breaking, but not vice versa. The CPT theorem is bypassed because Lorentz symmetry is broken. The underlying theory would then produce the StandardModel Extension with Lorentz and CPT violation instead of the usual Standard Model.
A technical question sometimes asked is: what happened to the NambuGoldstone bosons? For a discrete symmetry like CPT, Goldstone's theorem doesn't apply. For global Lorentz symmetry, it implies that spontaneous breaking must be accompanied by massless bosons. These modes might be identified with the photon or the graviton. If gravity is included a priori, then Lorentz symmetry becomes local. In gauge theories the NambuGoldstone bosons could be absorbed to generate masses for the gauge bosons according to the Higgs mechanism, but we have shown the analogue of this effect doesn't occur for gravity. Instead, in some gravitational theories the massless bosons might again be identified with the photon, while in others (with propagating spin connection) the massless bosons can be absorbed to generate mass terms in analogy with the usual Higgs mechanism.
The quick answer is that the StandardModel Extension has the usual properties of the Standard Model and General Relativity, except that spacetime symmetries such as Lorentz and CPT invariance can be violated. The violations can depend on the type of particle or field involved.
In the StandardModel Extension, one type of Lorentz symmetry remains valid: the theory transforms normally under rotations or boosts of the observer's inertial frame (observer Lorentz transformations). The apparent Lorentz violations appear only when the particle fields are rotated or boosted (particle Lorentz transformations) relative to the vacuum tensor expectation values.
More technically: the full StandardModel Extension contains all possible coordinateinvariant operators formed by combining StandardModel and gravitational fields with couplings having spacetime indices. For most situations at energies well below the scale of the underlying theory, it suffices to study the subset of the full StandardModel Extension for which the gauge structure and the powercounting renormalizability of the usual Standard Model are unchanged and for which energy and momentum are conserved. The usual quantization methods then apply.
Here is a table listing some of the usual and unusual properties of the StandardModel Extension in this limit.
USUAL  UNUSUAL 
SU(3) x SU(2) x U(1)
gauge structure 
. 
Powercounting renormalizability  . 
Energy and momentum conservation  . 
SU(2) x U(1) breaking  . 
Quantization  . 
Microcausality  . 
Spinstatistics  . 
Observer Lorentz covariance  Particle Lorentz violation 
.  CPT violation 
As an analogy, consider a conventional particle moving inside a crystal. This is similar to a particle moving in a vacuum with spontaneous Lorentz violation. In a crystal, the particle's behavior typically appears to break both rotations and boost symmetry. However, instead of leading to fundamental problems, the lack of Lorentz symmetry merely results from the presence of the background crystal fields.
It turns out that this analogy is realized in certain condensedmatter systems. For example, the StandardModel Extension can be used to describe general deviations from emergent Lorentz symmetry in Dirac and Weyl semimetals.
The StandardModel Extension provides a quantitative theoretical framework within which various experimental tests of spacetime symmetries can be studied and compared. Potentially observable signals can be deduced in some cases.
Without an explicit fundamental theory, it's very difficult to make any estimates of the size of possible effects. Highprecision tests have found no compelling evidence for the violation of Lorentz or CPT symmetry as yet, so any effects must be minuscule.
A very crude estimate of the suppression of possible effects might be made by comparing presently attainable energy scales to the natural scale of an underlying theory including gravity, which involves 1723 orders of magnitude. Additional suppressions from dimensionless couplings could also appear. Although most experimental tests would lack the necessary sensitivity to such signals, a few special ones can already place useful constraints on some of the unconventional terms in the StandardModel Extension.
In the context of the StandardModel Extension, one cannot identify a single best test for Lorentz or CPT symmetry because the theory contains different kinds of coefficients to which only certain experiments may be sensitive. For example, the bounds on CPT violation from the measurement of the fractional mass difference between a kaon and an antikaon and those from comparisons of hydrogen and antihydrogen are sensitive to completely different coefficients in the StandardModel Extension.
We have performed theoretical studies of several kinds of experiments to date, including:
See the bibliography of
books and articles
for details of our theoretical analyses.
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