In

particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

,

**supersymmetry** (often abbreviated

**SUSY**) is a

symmetryIn physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...

that relates

elementary particleIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

s of one

spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

to other particles that differ by half a unit of spin and are known as

superpartnerIn particle physics, a superpartner is a hypothetical elementary particle. Supersymmetry is one of the synergistic theories in current high-energy physics which predicts the existence of these "shadow" particles....

s. In a theory with unbroken supersymmetry, for every type of

bosonIn particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

there exists a corresponding type of

fermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

with the same mass and internal quantum numbers, and vice-versa.

There is no direct evidence for the existence of supersymmetry. It is motivated by possible solutions to several theoretical problems. Since the superpartners of the

Standard ModelThe 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...

particles have not been observed, supersymmetry, if it exists, must be a broken symmetry, allowing the

superparticlesIn particle physics, a superpartner is a hypothetical elementary particle. Supersymmetry is one of the synergistic theories in current high-energy physics which predicts the existence of these "shadow" particles....

to be heavier than the corresponding Standard Model particles.

If supersymmetry exists close to the

TeVIn physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

energy scale, it allows for a solution of the

hierarchy problemIn theoretical physics, a hierarchy problem occurs when the fundamental parameters of some Lagrangian are vastly different than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization...

of the Standard Model, i.e., the fact that the

Higgs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

mass is subject to quantum corrections which — barring extremely

fine-tunedFine Tuning was the name of XM Satellite Radio's eclectic music channel. The program director for Fine Tuning was Ben Smith and the tag line was, "The World's Most Interesting Music"....

cancellations among independent contributions — would make it so large as to undermine the internal consistency of the theory. In supersymmetric theories, on the other hand, the contributions to the quantum corrections coming from Standard Model particles are naturally canceled by the contributions of the corresponding superpartners. Other attractive features of TeV-scale supersymmetry are the fact that it allows for the high-energy

unificationThe term Grand Unified Theory, often abbreviated as GUT, refers to any of several similar candidate models in particle physics in which at high-energy, the three gauge interactions of the Standard Model which define the electromagnetic, weak, and strong interactions, are merged into one single...

of the weak interactions, the strong interactions and

electromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, and the fact that it provides a candidate for

Dark MatterIn astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...

and a natural mechanism for electroweak symmetry breaking. Therefore, scenarios where supersymmetric partners appear with masses not much greater than 1 TeV are considered the most well-motivated by theorists. These scenarios would imply that experimental traces of the superpartners should begin to emerge in high-energy collisions at the

LHCLHC may refer to:* Large Hadron Collider, a particle accelerator and collider located on the Franco-Swiss border near Geneva, SwitzerlandLHC also may refer to:* La hora Chanante, a Spanish comedy television show...

relatively soon. As of September 2011, no meaningful signs of the superpartners have been observed, which is beginning to significantly constrain the most popular incarnations of supersymmetry. However, the total parameter space of consistent supersymmetric extensions of the Standard Model is extremely diverse and can not be definitively ruled out at the LHC.

Another theoretically appealing property of supersymmetry is that it offers the only "loophole" to the Coleman–Mandula theorem, which prohibits spacetime and internal symmetries from being combined in any nontrivial way, for

quantum field theoriesQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

like the Standard Model under very general assumptions. The

Haag-Lopuszanski-Sohnius theoremIn theoretical physics, the Haag–Lopuszanski–Sohnius theorem shows that the possible symmetries of a consistent 4-dimensional quantum field theory do not only consist of internal symmetries and Poincaré symmetry, but can also include supersymmetry as a nontrivial extension of the Poincaré algebra...

demonstrates that supersymmetry is the only way spacetime and internal symmetries can be consistently combined.

In general, supersymmetric

quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

is often much easier to work with, as many more problems become exactly solvable. Supersymmetry is also a feature of most versions of

string theoryString 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...

, though it may exist in nature even if string theory is incorrect.

The

Minimal Supersymmetric Standard ModelThe Minimal Supersymmetric Standard Model is the minimal extension to the Standard Model that realizes supersymmetry, although non-minimal extensions do exist. Supersymmetry pairs bosons with fermions; therefore every Standard Model particle has a partner that has yet to be discovered...

is one of the best studied candidates for physics beyond the Standard Model. Theories of gravity that are also invariant under supersymmetry are known as

supergravityIn theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry...

theories.

## History

A supersymmetry relating mesons and baryons was first proposed, in the context of hadronic physics, by

Hironari Miyazawa in 1966, but his work was ignored at the time.

In the early 1970s, J. L. Gervais and

B. Sakitawas a Japanese-American theoretical physicist who made important contributions in quantum field theory, superstring theory and discovered supersymmetry in 1971. He was a Distinguished Professor of Physics at the City College of New York.-Early years:...

(in 1971),

Yu. A. GolfandYuri Golfand was a Russian and Israeli physisist known, in particular, for his 1971 paper where they proposed supersymmetry between bosonic and ferminoic particles by extending the Poincare algebra with anticommuting spinor generators. The algebra they constructed is also called a Super-Poincaré...

and E.P. Likhtman (also in 1971), D.V. Volkov and V.P. Akulov (in 1972) and

J. WessJulius Wess was an Austrian theoretical physicist noted as the co-inventor of the Wess–Zumino model and Wess–Zumino–Witten model in the field of supersymmetry...

and

B. ZuminoBruno Zumino is an Italian theoretical physicist and emeritus faculty at the University of California, Berkeley. He got his bachelor degree from the University of Rome in 1945...

(in 1974) independently rediscovered supersymmetry, a radically new type of symmetry of spacetime and fundamental fields, which establishes a relationship between elementary particles of different quantum nature, bosons and fermions, and unifies spacetime and internal symmetries of the microscopic world. Supersymmetry first arose in 1971 in the context of an early version of

string theoryString 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...

by

Pierre RamondPierre Ramond is a Distinguished Professor of Physics at University of Florida in Gainesville, Florida...

, John H. Schwarz and

Andre NeveuAndré Neveu is a French physicist working on string theory and quantum field theory who coinvented the Neveu-Schwarz algebra and the Gross-Neveu model.Neveu studied in Paris at the École normale supérieure...

, but the mathematical structure of supersymmetry has subsequently been applied successfully to other areas of physics; firstly by Wess, Zumino, and

Abdus SalamMohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk (Urdu: محمد عبد السلام, pronounced , (January 29, 1926– November 21, 1996) was a Pakistani theoretical physicist and Nobel laureate in Physics for his work on the electroweak unification of the...

and their fellow researchers to particle physics, and later to a variety of fields, ranging from

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

to

statistical physicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

. It remains a vital part of many proposed theories of physics.

The first realistic supersymmetric version of the Standard Model was proposed in 1981 by

Howard GeorgiHoward Mason Georgi III, born January 6, 1947 in San Bernardino, California, is Harvard College Professor and Mallinckrodt Professor of Physics at Harvard University...

and

Savas DimopoulosSavas Dimopoulos is a Greek particle physicist at Stanford University. He was born in Istanbul, Turkey and later moved to Athens due to ethnic tensions in Turkey during the 1950s and 1960s. Dimopoulos studied as an undergraduate at the University of Houston...

and is called the

Minimal Supersymmetric Standard ModelThe Minimal Supersymmetric Standard Model is the minimal extension to the Standard Model that realizes supersymmetry, although non-minimal extensions do exist. Supersymmetry pairs bosons with fermions; therefore every Standard Model particle has a partner that has yet to be discovered...

or MSSM for short. It was proposed to solve the

hierarchy problemIn theoretical physics, a hierarchy problem occurs when the fundamental parameters of some Lagrangian are vastly different than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization...

and predicts superpartners with masses between 100 GeV and 1 TeV.

As of 2009 there is no irrefutable experimental evidence that supersymmetry is a symmetry of nature. Since 2010, the

Large Hadron ColliderThe Large Hadron Collider is the world's largest and highest-energy particle accelerator. It is expected to address some of the most fundamental questions of physics, advancing the understanding of the deepest laws of nature....

at

CERNThe European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...

is producing the world's highest energy collisions and offers the best chance at discovering superparticles for the foreseeable future.

### Extension of possible symmetry groups

One reason that physicists explored supersymmetry is because it offers an extension to the more familiar symmetries of quantum field theory. These symmetries are grouped into the

Poincaré groupIn physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...

and internal symmetries and the Coleman–Mandula theorem showed that under certain assumptions, the symmetries of the S-matrix must be a direct product of the Poincaré group with a

compactIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

internal symmetry group or if there is no

mass gapIn quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest...

, the

conformal groupIn mathematics, the conformal group is the group of transformations from a space to itself that preserve all angles within the space. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important:*...

with a compact internal symmetry group.

In 1971

GolfandYuri Golfand was a Russian and Israeli physisist known, in particular, for his 1971 paper where they proposed supersymmetry between bosonic and ferminoic particles by extending the Poincare algebra with anticommuting spinor generators. The algebra they constructed is also called a Super-Poincaré...

and Likhtman were the first to show that the Poincaré algebra can be extended through introduction of four

anticommuting spinor generators (in four dimensions), which later became known as supercharges.

In 1975 the

Haag-Lopuszanski-Sohnius theoremIn theoretical physics, the Haag–Lopuszanski–Sohnius theorem shows that the possible symmetries of a consistent 4-dimensional quantum field theory do not only consist of internal symmetries and Poincaré symmetry, but can also include supersymmetry as a nontrivial extension of the Poincaré algebra...

analyzed all possible superalgebras in the general form, including those with an extended number of the supergenerators and

central chargeIn theoretical physics, a central charge is an operator Z that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group -- the subgroup of elements that commute with all other elements of the original group—or to the center of a Lie algebra...

s.

This extended super-Poincaré algebra paved the way for obtaining a very large and important class of supersymmetric field theories.

#### The supersymmetry algebra

Traditional symmetries in physics are generated by objects that transform under the

tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

representations of the

Poincaré groupIn physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...

and internal symmetries. Supersymmetries, on the other hand, are generated by objects that transform under the

spinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

representations. According to the

spin-statistics theoremIn quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...

,

bosonIn particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

ic fields commute while

fermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

ic fields

anticommuteIn mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...

. Combining the two kinds of fields into a single

algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

requires the introduction of a

**Z**_{2}-gradingIn mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....

under which the bosons are the even elements and the fermions are the odd elements. Such an algebra is called a

Lie superalgebraIn mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...

.

The simplest supersymmetric extension of the Poincaré algebra is the Super-Poincaré algebra. Expressed in terms of two Weyl spinors, has the following

anti-commutationIn mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...

relation:

and all other anti-commutation relations between the

*Q*s and commutation relations between the

*Q*s and

*P*s vanish. In the above expression

are the generators of translation and

are the

Pauli matricesThe Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...

.

There are

representations of a Lie superalgebraIn the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra L on a Z2-graded vector space V, such that if A and B are any two pure elements of L and X and Y are any two pure elements of V, then[X]=c_1 A[X] + c_2 B[X]\,A[c_1 X + c_2...

that are analogous to representations of a Lie algebra. Each Lie algebra has an associated Lie group and a Lie superalgebra can sometimes be extended into representations of a Lie supergroup.

### The Supersymmetric Standard Model

Incorporating supersymmetry into the

Standard ModelThe 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...

requires doubling the number of particles since there is no way that any of the particles in the Standard Model can be

superpartnerIn particle physics, a superpartner is a hypothetical elementary particle. Supersymmetry is one of the synergistic theories in current high-energy physics which predicts the existence of these "shadow" particles....

s of each other. With the addition of new particles, there are many possible new interactions. The simplest possible supersymmetric model consistent with the Standard Model is the

Minimal Supersymmetric Standard ModelThe Minimal Supersymmetric Standard Model is the minimal extension to the Standard Model that realizes supersymmetry, although non-minimal extensions do exist. Supersymmetry pairs bosons with fermions; therefore every Standard Model particle has a partner that has yet to be discovered...

(MSSM) which can include the necessary additional new particles that are able to be

superpartnerIn particle physics, a superpartner is a hypothetical elementary particle. Supersymmetry is one of the synergistic theories in current high-energy physics which predicts the existence of these "shadow" particles....

s of those in the

Standard ModelThe 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...

.

One of the main motivations for SUSY comes from the quadratically divergent contributions to the Higgs mass squared. The quantum mechanical interactions of the Higgs boson causes a large renormalization of the Higgs mass and unless there is an accidental cancellation, the natural size of the Higgs mass is the highest scale possible. This problem is known as the

hierarchy problemIn theoretical physics, a hierarchy problem occurs when the fundamental parameters of some Lagrangian are vastly different than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization...

. Supersymmetry reduces the size of the quantum corrections by having automatic cancellations between fermionic and bosonic Higgs interactions. If supersymmetry is restored at the weak scale, then the Higgs mass is related to supersymmetry breaking which can be induced from small non-perturbative effects explaining the vastly different scales in the weak interactions and gravitational interactions.

In many supersymmetric Standard Models there is a heavy stable particle (such as

neutralinoIn particle physics, the neutralino is a hypothetical particle predicted by supersymmetry. There are four neutralinos that are fermions and are electrically neutral, the lightest of which is typically stable...

) which could serve as a Weakly interacting massive particle (WIMP)

dark matterIn astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...

candidate. The existence of a supersymmetric dark matter candidate is closely tied to

R-parityR-parity is a concept in particle physics. In the supersymmetric extension of the Standard Model, baryon number and lepton number are no longer conserved by all of the renormalizable couplings...

.

The standard paradigm for incorporating supersymmetry into a realistic theory is to have the underlying dynamics of the theory be supersymmetric, but the ground state of the theory does not respect the symmetry and supersymmetry is

broken spontaneouslySpontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....

. The supersymmetry break can not be done permanently by the particles of the MSSM as they currently appear. This means that there is a new sector of the theory that is responsible for the breaking. The only constraint on this new sector is that it must break supersymmetry permanently and must give superparticles TeV scale masses. There are many models that can do this and most of their details do not currently matter. In order to parameterize the relevant features of supersymmetry breaking, arbitrary

soft SUSY breakingIn theoretical physics, soft SUSY breaking is type of supersymmetry breaking that does not cause ultraviolet divergences to appear in scalar masses. These terms are relevant operators—i.e. operators whose coefficients have a positive dimension of mass—though there are some exceptions.A model...

terms are added to the theory which temporarily break SUSY explicitly but could never arise from a complete theory of supersymmetry breaking.

#### Gauge Coupling Unification

One piece of evidence for supersymmetry existing is gauge coupling unification.

The

renormalization groupIn theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...

evolution of the three gauge

coupling constantIn physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...

s of the

Standard ModelThe 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...

is somewhat sensitive to the present particle content of the theory. These coupling constants do not quite meet together at a common energy scale if we run the renormalization group using the

Standard ModelThe 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...

. With the addition of minimal SUSY joint convergence of the coupling constants is projected at approximately 10

^{16} GeVGEV or GeV may stand for:*GeV or gigaelectronvolt, a unit of energy equal to billion electron volts*GEV or Grid Enabled Vehicle that is fully or partially powered by the electric grid, see plug-in electric vehicle...

.

### Supersymmetric quantum mechanics

*Supersymmetric quantum mechanics* adds the SUSY superalgebra to

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

as opposed to

quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

. Supersymmetric quantum mechanics often comes up when studying the dynamics of supersymmetric solitons and due to the simplified nature of having fields only functions of time (rather than space-time), a great deal of progress has been made in this subject and is now studied in its own right.

SUSY quantum mechanics involves pairs of

HamiltoniansIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

which share a particular mathematical relationship, which are called

*partner Hamiltonians*. (The

potential energyIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

terms which occur in the Hamiltonians are then called

*partner potentials*.) An introductory theorem shows that for every eigenstate of one Hamiltonian, its partner Hamiltonian has a corresponding eigenstate with the same energy. This fact can be exploited to deduce many properties of the eigenstate spectrum. It is analogous to the original description of SUSY, which referred to bosons and fermions. We can imagine a "bosonic Hamiltonian", whose eigenstates are the various bosons of our theory. The SUSY partner of this Hamiltonian would be "fermionic", and its eigenstates would be the theory's fermions. Each boson would have a fermionic partner of equal energy.

SUSY concepts have provided useful extensions to the

WKB approximationIn mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear partial differential equations with spatially varying coefficients...

. In addition, SUSY has been applied to non-quantum

statistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

through the

Fokker-Planck equationThe Fokker–Planck equation describes the time evolution of the probability density function of the velocity of a particle, and can be generalized to other observables as well.It is named after Adriaan Fokkerand Max Planck...

.

### Mathematics

SUSY is also sometimes studied mathematically for its intrinsic properties. This is because it describes complex fields satisfying a property known as holomorphy, which allows holomorphic quantities to be exactly computed. This makes supersymmetric models useful

toy modelIn physics, a toy model is a simplified set of objects and equations relating them that can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory....

s of more realistic theories. A prime example of this has been the demonstration of S-duality in four-dimensional gauge theories that interchanges particles and

monopoleMonopole may refer to:*Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole, or related concepts in physics and mathematics:...

s.

## General supersymmetry

Supersymmetry appears in many different contexts in theoretical physics that are closely related. It is possible to have multiple supersymmetries and also have supersymmetric extra dimensions.

### Extended supersymmetry

It is possible to have more than one kind of supersymmetry transformation. Theories with more than one supersymmetry transformation are known as

extended supersymmetric theories. The more supersymmetry a theory has, the more constrained the field content and interactions are. Typically the number of copies of a supersymmetry is a power of 2, i.e. 1, 2, 4, 8. In four dimensions, a spinor has four degrees of freedom and thus the minimal number of supersymmetry generators is four in four dimensions and having eight copies of supersymmetry means that there are 32 supersymmetry generators.

The maximal number of supersymmetry generators possible is 32. Theories with more than 32 supersymmetry generators automatically have massless fields with spin greater than 2. It is not known how to make massless fields with spin greater than two interact, so the maximal number of supersymmetry generators considered is 32. This corresponds to an

*N* = 8 supersymmetry theory. Theories with 32 supersymmetries automatically have a

gravitonIn physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...

.

In four dimensions there are the following theories, with the corresponding multiplets(CPT adds a copy, whenever they are not invariant under such symmetry)

Chiral multiplet:

(0,)

Vector multiplet:

Gravitino multiplet:

(1,)

Graviton multiplet:

hypermultiplet:

(-,0

^{2},)

vector multiplet:

(0,

^{2},1)

supergravity multiplet:

(1,

^{2},2)

Vector multiplet:

(-1,-

^{4},0

^{6},

^{4},1)

Supergravity multiplet:

(0,

^{4},1

^{6},

^{4},2)

Supergravity multiplet:

(-2,-

^{8},-1

^{28},-

^{56},0

^{70},

^{56},1

^{28},

^{8},2)

### Supersymmetry in alternate numbers of dimensions

It is possible to have supersymmetry in dimensions other than four. Because the properties of spinors change drastically between different dimensions, each dimension has its characteristic. In

*d* dimensions, the size of spinors is roughly 2

^{d/2} or 2

^{(d − 1)/2}. Since the maximum number of supersymmetries is 32, the greatest number of dimensions in which a supersymmetric theory can exist is eleven.

## Supersymmetry as a quantum group

Supersymmetry can be reinterpreted in the language of

noncommutative geometryNoncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...

and

quantum groupIn mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...

s. In particular, it involves a mild form of noncommutativity, namely supercommutativity. See the main article for more details.

## Supersymmetry in quantum gravity

Supersymmetry is part of a larger enterprise of theoretical physics to unify everything we know about the physical world into a single fundamental framework of physical laws, known as the quest for a

Theory of EverythingA theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle....

(TOE). A significant part of this larger enterprise is the quest for a theory of

quantum gravityQuantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

, which would unify the classical theory of

general relativityGeneral 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 the

Standard ModelThe 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...

, which explains the other

three basic forcesIn particle physics, fundamental interactions are the ways that elementary particles interact with one another...

in physics (

electromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, the

strong interactionIn particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...

, and the

weak interactionWeak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...

), and provides a palette of fundamental particles upon which all four forces act. Two of the most active approaches to forming a theory of quantum gravity are

string theoryString 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...

and

loop quantum gravityLoop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...

(LQG), although in theory, supersymmetry could be a component of other theoretical approaches as well.

For

string theoryString 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...

to be consistent, supersymmetry appears to be required at some level (although it may be a strongly broken symmetry). In particle theory, supersymmetry is recognized as a way to stabilize the

hierarchyIn theoretical physics, a hierarchy problem occurs when the fundamental parameters of some Lagrangian are vastly different than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization...

between the unification scale and the electroweak scale (or the

Higgs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

mass), and can also provide a natural

dark matterIn astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...

candidate. String theory also requires extra spatial dimensions which have to be compactified as in Kaluza-Klein theory.

Loop quantum gravityLoop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...

(LQG), in its current formulation, predicts no additional spatial dimensions, nor anything else about particle physics. These theories can be formulated in three spatial dimensions and one dimension of time, although in some LQG theories dimensionality is an emergent property of the theory, rather than a fundamental assumption of the theory. Also, LQG is a theory of quantum gravity which does not require supersymmetry.

Lee SmolinLee Smolin is an American theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics, and an adjunct professor of physics at the University of Waterloo. He is married to Dina Graser, a communications lawyer in Toronto. His brother is David M...

, one of the originators of LQG, has proposed that a loop quantum gravity theory incorporating either supersymmetry or extra dimensions, or both, be called "loop quantum gravity II".

If experimental evidence confirms supersymmetry in the form of supersymmetric particles such as the

neutralinoIn particle physics, the neutralino is a hypothetical particle predicted by supersymmetry. There are four neutralinos that are fermions and are electrically neutral, the lightest of which is typically stable...

that is often believed to be the lightest

superpartnerIn particle physics, a superpartner is a hypothetical elementary particle. Supersymmetry is one of the synergistic theories in current high-energy physics which predicts the existence of these "shadow" particles....

, some people believe this would be a major boost to

string theoryString 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...

. Since supersymmetry is a required component of string theory, any discovered supersymmetry would be consistent with string theory. If the

Large Hadron ColliderThe Large Hadron Collider is the world's largest and highest-energy particle accelerator. It is expected to address some of the most fundamental questions of physics, advancing the understanding of the deepest laws of nature....

and other major particle physics experiments fail to detect supersymmetric partners or evidence of extra dimensions, many versions of

string theoryString 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...

which had predicted certain low mass superpartners to existing particles may need to be significantly revised. The failure of experiments to discover either supersymmetric partners or extra spatial dimensions, , has encouraged

loop quantum gravityLoop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...

researchers.

## Current Limits

The tightest limits will of course come from direct production at colliders. Both the Large Electron–Positron Collider and

TevatronThe Tevatron is a circular particle accelerator in the United States, at the Fermi National Accelerator Laboratory , just east of Batavia, Illinois, and is the second highest energy particle collider in the world after the Large Hadron Collider...

had set limits for specific models which have now been exceeded by the

Large Hadron ColliderThe Large Hadron Collider is the world's largest and highest-energy particle accelerator. It is expected to address some of the most fundamental questions of physics, advancing the understanding of the deepest laws of nature....

. Searches are only applicable for a finite set of tested points because simulation using the

Monte Carlo methodMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

must be made so that limits for that particular model can be calculated. This complicates matters because different experiments have looked at different sets of points. Some extrapolation between points can be made within particular models but it is difficult to set general limits even for the

Minimal Supersymmetric Standard ModelThe Minimal Supersymmetric Standard Model is the minimal extension to the Standard Model that realizes supersymmetry, although non-minimal extensions do exist. Supersymmetry pairs bosons with fermions; therefore every Standard Model particle has a partner that has yet to be discovered...

.

The first mass limits for squarks and gluinos were made at

CERNThe European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...

by the UA1 experiment and the UA2 experiment at the

Super Proton SynchrotronThe Super Proton Synchrotron is a particle accelerator of the synchrotron type at CERN. It is housed in a circular tunnel, in circumference, straddling the border of France and Switzerland near Geneva, Switzerland. The SPS was designed by a team led by John Adams, director-general of what was...

. LEP later set very strong limits. In 2006 these limits were extended by the D0 experiment The LHC has now extended these limits, by extending the search area, with no sign of supersymmmetry.

## Further reading

- A Supersymmetry Primer by S. Martin, 2011
- Introduction to Supersymmetry By Joseph D. Lykken
Joseph David Lykken is a theoretical physicist at the Fermi National Accelerator Laboratory.-Background and education:...

, 1996
- An Introduction to Supersymmetry By Manuel Drees, 1996
- Introduction to Supersymmetry By Adel Bilal, 2001
- An Introduction to Global Supersymmetry by Philip Arygres, 2001
- Weak Scale Supersymmetry by Howard Baer and Xerxes Tata, 2006.
- Cooper, F., A. Khare and U. Sukhatme. "Supersymmetry in Quantum Mechanics." Phys. Rep. 251 (1995) 267-85 (arXiv:hep-th/9405029).
- Junker, G.
*Supersymmetric Methods in Quantum and Statistical Physics*, Springer-Verlag (1996).
- Gordon L. Kane.
*Supersymmetry: Unveiling the Ultimate Laws of Nature* Basic Books, New York (2001). ISBN 0-7382-0489-7.
- Gordon L. Kane and Shifman, M., eds.
*The Supersymmetric World: The Beginnings of the Theory,* World Scientific, Singapore (2000). ISBN 981-02-4522-X.
- D.V. Volkov, V.P. Akulov, Pisma Zh.Eksp.Teor.Fiz. 16 (1972) 621; Phys.Lett. B46 (1973) 109.
- V.P. Akulov, D.V. Volkov, Teor.Mat.Fiz. 18 (1974) 39.
- Weinberg, Steven,
*The Quantum Theory of Fields, Volume 3: Supersymmetry*, Cambridge University Press, Cambridge, (1999). ISBN 0-521-66000-9.
- Wess, Julius, and Jonathan Bagger,
*Supersymmetry and Supergravity*, Princeton University Press, Princeton, (1992). ISBN 0-691-02530-4.
- Brookhaven National Laboratory (Jan. 8, 2004).
*New g−2 measurement deviates further from Standard Model.* Press Release.
- Fermi National Accelerator Laboratory (Sept 25, 2006).
*Fermilab's CDF scientists have discovered the quick-change behavior of the B-sub-s meson.* Press Release.

## External links