Standard-Model Extension (SME) is an

effective field theory In physics, an effective field theory is a type of approximation, or effective theory, for an underlying physical theory, such as a quantum field theory or a statistical mechanics model. An effective field theory includes the appropriate degrees ...

that contains the

Standard Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

, and all possible operators that break

Lorentz symmetry In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the sam ...

. 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 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 ...

Development

In 1989, Alan Kostelecký and Stuart Samuel proved that interactions in string theories could lead to the spontaneous breaking of Lorentz symmetry. Later studies have indicated that loop-quantum gravity, non-commutative field theories, brane-world scenarios, and random dynamics models also involve the breakdown of

Lorentz invariance In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. While ...

. Interest in Lorentz violation has grown rapidly in the last decades because it can arise in these and other candidate theories for

quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...

. In the early 1990s, it was shown in the context of bosonic

superstrings Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string th ...

that string interactions can also spontaneously break

. This work suggested that experiments with

kaon In particle physics, a kaon, also called a K meson and denoted , is any of a group of four mesons distinguished by a quantum number called strangeness. In the quark model they are understood to be bound states of a strange quark (or antiquark ...

interferometry would be promising for seeking possible signals of CPT violation due to their high sensitivity. The SME was conceived to facilitate experimental investigations of Lorentz and CPT symmetry, given the theoretical motivation for violation of these symmetries. An initial step, in 1995, was the introduction of effective interactions. Although Lorentz-breaking interactions are motivated by constructs such as

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

, the low-energy effective action appearing in the SME is independent of the underlying theory. Each term in the effective theory involves the expectation of a

tensor field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...

in the underlying theory. These coefficients are small due to Planck-scale suppression, and in principle are measurable in experiments. The first case considered the mixing of neutral mesons, because their interferometric nature makes them highly sensitive to suppressed effects. In 1997 and 1998, two papers by Don Colladay and Alan Kostelecký gave birth to the minimal SME in flat

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

. This provided a framework for Lorentz violation across the spectrum of standard-model particles, and provided information about types of signals for potential new experimental searches. In 2004, the leading Lorentz-breaking terms in curved spacetimes were published, thereby completing the picture for the minimal SME. In 1999, Sidney Coleman and

Sheldon Glashow Sheldon Lee Glashow (, ; born December 5, 1932) is a Nobel Prize-winning American theoretical physicist. He is the Metcalf Professor of Mathematics and Physics at Boston University, and a Eugene Higgins Professor of Physics, emeritus, at Harv ...

presented a special isotropic limit of the SME. Higher-order Lorentz violating terms have been studied in various contexts, including electrodynamics.

Lorentz transformations: observer vs. particle

The distinction between particle and observer transformations is essential to understanding Lorentz violation in physics because Lorentz violation implies a measurable difference between two systems differing only by a particle

Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...

. In

special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...

, observer

Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...

relate measurements made in reference frames with differing velocities and orientations. The coordinates in the one system are related to those in the other by an observer

—a rotation, a boost, or a combination of both. Each observer will agree on the laws of

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, since this transformation is simply a

change of coordinates In mathematics, an ordered basis of a vector space of finite dimension (vector space), dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a finite sequence, sequence of scalar (mathematics), ...

. On the other hand, identical experiments can be rotated or boosted relative to each other, while being studied by the same inertial observer. These transformations are called particle transformations, because the matter and fields of the experiment are physically transformed into the new configuration. In a conventional vacuum, observer and particle transformations can be related to each other in a simple way—basically one is the inverse of the other. This apparent equivalence is often expressed using the terminology of active and passive transformations. The equivalence fails in Lorentz-violating theories, however, because fixed background fields are the source of the symmetry breaking. These background fields are tensor-like quantities, creating preferred directions and boost-dependent effects. The fields extend over all space and time, and are essentially frozen. When an experiment sensitive to one of the background fields is rotated or boosted, i.e. particle transformed, the background fields remain unchanged, and measurable effects are possible. Observer Lorentz symmetry is expected for all theories, including Lorentz violating ones, since a change in the coordinates cannot affect the physics. This invariance is implemented in field theories by writing a scalar lagrangian, with properly contracted spacetime indices. Particle Lorentz breaking enters if the theory includes fixed SME background fields filling the universe.

Building the SME

The SME can be expressed as a Lagrangian with various terms. Each Lorentz-violating term is an observer scalar constructed by contracting standard field operators with controlling coefficients called coefficients for Lorentz violation. These are not parameters, but rather predictions of the theory, since they can in principle be measured by appropriate experiments. The coefficients are expected to be small because of the Planck-scale suppression, so perturbative methods are appropriate. In some cases, other suppression mechanisms could mask large Lorentz violations. For instance, large violations that may exist in gravity could have gone undetected so far because of couplings with weak gravitational fields. Stability and causality of the theory have been studied in detail.

Spontaneous Lorentz symmetry breaking

In field theory, there are two possible ways to implement the breaking of a symmetry: explicit and spontaneous. A key result in the formal theory of Lorentz violation, published by Kostelecký in 2004, is that explicit Lorentz violation leads to incompatibility of the

Bianchi identities In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algebra ...

with the covariant conservation laws for the energy–momentum and spin-density tensors, whereas spontaneous Lorentz breaking evades this difficulty. This theorem requires that any breaking of Lorentz symmetry must be dynamical. Formal studies of the possible causes of the breakdown of Lorentz symmetry include investigations of the fate of the expected Nambu–Goldstone modes.

Goldstone's theorem In physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu within the context of the BCS superc ...

implies that the spontaneous breaking must be accompanied by massless

bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-integer ...

. These modes might be identified with the photon, the

graviton In theories of quantum gravity, the graviton is the hypothetical elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with re ...

,V.A. Kostelecký and R. Potting, ''Gravity from Local Lorentz Violation'', Gen. Rel. Grav. 37, 1675 (2005). spin-dependent interactions,N. Arkani-Hamed, H.C. Cheng, M. Luty, and J. Thaler, ''Universal dynamics of spontaneous Lorentz violation and a new spin-dependent inverse-square law force'', JHEP 0507, 029 (2005). and spin-independent interactions.

Experimental searches

The possible signals of Lorentz violation in any experiment can be calculated from the SME. It has therefore proven to be a remarkable tool in the search for Lorentz violation across the landscape of experimental physics. Up until the present, experimental results have taken the form of upper bounds on the SME coefficients. Since the results will be numerically different for different inertial reference frames, the standard frame adopted for reporting results is the Sun-centered frame. This frame is a practical and appropriate choice, since it is accessible and inertial on the time scale of hundreds of years. Typical experiments seek couplings between the background fields and various particle properties such as

spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...

, or propagation direction. One of the key signals of Lorentz violation arises because experiments on Earth are unavoidably rotating and revolving relative to the Sun-centered frame. These motions lead to both annual and sidereal variations of the measured coefficients for Lorentz violation. Since the translational motion of the Earth around the Sun is nonrelativistic, annual variations are typically suppressed by a factor 10⁻⁴. This makes sidereal variations the leading time-dependent effect to look for in experimental data.
Lorentz symmetry stays intact
', Physics World, Feb 25, 2003. Measurements of SME coefficients have been done with experiments involving: *

birefringence Birefringence, also called double refraction, is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are described as birefrin ...

and dispersion from cosmological sources * clock-comparison measurements * CMB polarization * collider experiments * electromagnetic resonant cavities *

equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...

* gauge and Higgs particles * high-energy astrophysical observations * laboratory and gravimetric tests of gravity * matter interferometry *

neutrino oscillations Neutrino oscillation is a quantum mechanical phenomenon in which a neutrino created with a specific lepton family number ("lepton flavor": electron, muon, or tau) can later be measured to have a different lepton family number. The probability o ...

* oscillations and decays of K, B, D mesons * particle-antiparticle comparisons * Post-newtonian gravity in the

Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...

and beyond * second- and third-generation particles * space-based missions * spectroscopy of hydrogen and antihydrogen * spin-polarized matter. All experimental results for SME coefficients are tabulated in the Data Tables for Lorentz and CPT Violation.

References

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External links

Background information on Lorentz and CPT violation

Data Tables for Lorentz and CPT Violation
Physics beyond the Standard Model