Standard-Model Extension
Encyclopedia
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
.
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.
Interest in Lorentz violation has grown rapidly in the last decades because it can arise in these and other candidate theories for quantum gravity
. In the early 1990s, it was shown in the context of bosonic superstrings that string interactions can also spontaneously break CPT symmetry
. This work
suggested that experiments with kaon
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
, 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 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
. 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 presented a special
isotropic limit of the SME.
Higher-order Lorentz violating terms have been studied in various contexts, including electrodynamics.
. The distinction between particle and observer transformations is essential to understanding Lorentz violation in physics.
In special relativity
, observer Lorentz transformations 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 Lorentz transformation
-- a rotation, a boost, or a combination of both. Both observers will agree on the laws of physics
, since this transformation is simply a change of coordinates. 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.
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. Notice that these are not parameters 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.
in 2004, is that explicit Lorentz violation leads to incompatibility of the Bianchi identities 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 implies that the spontaneous breaking must be accompanied by massless bosons. These modes might be identified with the photon,
the graviton,
spin-dependent interactions,
and spin-independent interactions.
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, 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.
Measurements of SME coefficients have been done with experiments involving:
All experimental results for SME coefficients are tabulated in the Data Tables for Lorentz and CPT Violation.
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
, General Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, 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
CPT symmetry
CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...
.
Development
In 1989, Alan KosteleckýAlan 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...
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.
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
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
. In the early 1990s, it was shown in the context of bosonic superstrings that string interactions can also spontaneously break CPT symmetry
CPT symmetry
CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...
. This work
suggested that experiments with 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...
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
CPT symmetry
CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...
, 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
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, 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 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
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...
gave birth to the minimal SME in flat spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. 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
Sidney Coleman
Sidney Richard Coleman was an American theoretical physicist who studied under Murray Gell-Mann.- Life and work :Sidney Coleman grew up on the Far North Side of Chicago...
and Sheldon Glashow 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
Lorentz violation implies a measurable difference between two systems differing only by a particle Lorentz transformationLorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
. The distinction between particle and observer transformations is essential to understanding Lorentz violation in physics.
In special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, observer Lorentz transformations 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 Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
-- a rotation, a boost, or a combination of both. Both observers will agree on the laws of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, since this transformation is simply a change of coordinates. 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
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
, 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 LagrangianLagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
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. Notice that these are not parameters 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 KosteleckýAlan 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...
in 2004, is that explicit Lorentz violation leads to incompatibility of the Bianchi identities 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 implies that the spontaneous breaking must be accompanied by massless bosons. These modes might be identified with the photon,
the graviton,
spin-dependent interactions,
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, 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.
Measurements of SME coefficients have been done with experiments involving:
- birefringence and dispersion from cosmological sources
- clock-comparison measurementsHughes–Drever experimentHughes–Drever experiments are testing the isotropy of mass and space. As in Michelson–Morley experiments, the existence of a preferred frame of reference, or deviations from Lorentz invariance can be tested, which also affects the validity of the equivalence principle...
- CMB polarization
- collider experiments
- electromagnetic resonant cavities
- equivalence principle
- gauge and Higgs particles
- high-energy astrophysical observations
- laboratory and gravimetric tests of gravity
- matter interferometry
- neutrino oscillationsLorentz-violating neutrino oscillationsLorentz-violating neutrino oscillation refers to the quantum phenomenon of neutrino oscillations described in a framework that allows the breakdown of Lorentz invariance...
- oscillations and decays of K, B, D mesons
- particle-antiparticle comparisonsAntimatter tests of Lorentz violationHigh-precision experiments could revealsmall previously unseen differences between the behaviorof matter and antimatter.This prospect is appealing to physicists because it mayshow that nature is not Lorentz symmetric.- Introduction :...
- post-newtonian gravity in the solar system 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.
See also
- Antimatter Tests of Lorentz ViolationAntimatter tests of Lorentz violationHigh-precision experiments could revealsmall previously unseen differences between the behaviorof matter and antimatter.This prospect is appealing to physicists because it mayshow that nature is not Lorentz symmetric.- Introduction :...
- Lorentz-violating neutrino oscillationsLorentz-violating neutrino oscillationsLorentz-violating neutrino oscillation refers to the quantum phenomenon of neutrino oscillations described in a framework that allows the breakdown of Lorentz invariance...
- Bumblebee ModelsBumblebee modelsBumblebee models are effective field theories describing a vector field with a vacuum expectation value that spontaneously breaks Lorentz symmetry....
- Tests of special relativity
- Test theories of special relativityTest theories of special relativityTest theories of special relativity give a mathematical framework for analyzing results of experiments to verify special relativity.An experiment to test the theory of relativity cannot assume the theory is true, and therefore needs some other framework of assumptions that are wider than those of...