Supersymmetry Algebras In 1 1 Dimensions
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Supersymmetry is a theoretical framework in
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 ...
that suggests the existence of a symmetry between
particles In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
with integer
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 ...
(''
boson 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-intege ...
s'') and particles with half-integer spin (''
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s''). It proposes that for every known particle, there exists a partner particle with different spin properties. There have been multiple experiments on supersymmetry that have failed to provide evidence that it exists in
nature Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...
. If evidence is found, supersymmetry could help explain certain phenomena, such as the nature of
dark matter In astronomy, dark matter is an invisible and hypothetical form of matter that does not interact with light or other electromagnetic radiation. Dark matter is implied by gravity, gravitational effects that cannot be explained by general relat ...
and the
hierarchy problem In theoretical physics, the hierarchy problem is the problem concerning the large discrepancy between aspects of the weak force and gravity. There is no scientific consensus on why, for example, the weak force is 1024 times stronger than gravi ...
in particle physics. A supersymmetric theory is a theory in which the equations for
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
and the equations for
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...
are identical. In
theoretical A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
and
mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
, any theory with this property has the ''principle of supersymmetry'' (SUSY). Dozens of supersymmetric theories exist. In theory, supersymmetry is a type of spacetime symmetry between two basic classes of particles:
boson 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-intege ...
s, which have an integer-valued
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 ...
and follow
Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting identical particles may occupy a set of available discrete energy states at thermodynamic equilibri ...
, and
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s, which have a half-integer-valued spin and follow
Fermi–Dirac statistics Fermi–Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac distribution of part ...
. The names of bosonic partners of fermions are prefixed with ''s-'', because they are scalar particles. For example, if the
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
existed in a supersymmetric theory, then there would be a particle called a ''selectron'' (superpartner electron), a bosonic partner of the electron. In supersymmetry, each particle from the class of fermions would have an associated particle in the class of bosons, and vice versa, known as a
superpartner In particle physics, a superpartner (also sparticle) is a class of hypothetical elementary particles predicted by supersymmetry, which, among other applications, is one of the well-studied ways to extend the Standard Model of high-energy physics. ...
. The spin of a particle's superpartner is different by a half-integer. In the simplest supersymmetry theories, with perfectly " unbroken" supersymmetry, each pair of superpartners would share the same
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
and internal
quantum number In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantu ...
s besides spin. More complex supersymmetry theories have a
spontaneously broken symmetry Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or t ...
, allowing superpartners to differ in mass. Supersymmetry has various applications to different areas of physics, such as
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
,
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
,
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
,
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
,
nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies th ...
,
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
,
stochastic dynamics In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...
,
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
,
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 ...
, and
cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
. Supersymmetry has also been applied to
high-energy physics Particle physics or high-energy physics is the study of fundamental particles and forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the scale of protons and neutrons, while the stu ...
, where a supersymmetric extension of 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 ...
is a possible candidate for
physics beyond the Standard Model Physics beyond the Standard Model (BSM) refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the inability to explain the fundamental parameters of the standard model, the strong CP problem, neut ...
. However, no supersymmetric extensions of the Standard Model have been experimentally verified, and some physicists are saying the theory is dead.


History

A supersymmetry relating
mesons In particle physics, a meson () is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, the ...
and
baryon In particle physics, a baryon is a type of composite particle, composite subatomic particle that contains an odd number of valence quarks, conventionally three. proton, Protons and neutron, neutrons are examples of baryons; because baryons are ...
s was first proposed, in the context of hadronic physics, by Hironari Miyazawa in 1966. This supersymmetry did not involve spacetime, that is, it concerned internal symmetry, and was broken badly. Miyazawa's work was largely ignored at the time. J. L. Gervais and B. Sakita (in 1971), Yu. A. Golfand and E. P. Likhtman (also in 1971), and D. V. Volkov and V. P. Akulov (1972), independently rediscovered supersymmetry in the context of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, 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 microscopic phenomena. Supersymmetry with a consistent Lie-algebraic graded structure on which the Gervais−Sakita rediscovery was based directly first arose in 1971 in the context of an early version of
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 ...
by Pierre Ramond, John H. Schwarz and André Neveu. In 1974, Julius Wess and Bruno Zumino identified the characteristic renormalization features of four-dimensional supersymmetric field theories, which identified them as remarkable QFTs, and they and
Abdus Salam Mohammad Abdus Salam Salam adopted the forename "Mohammad" in 1974 in response to the anti-Ahmadiyya decrees in Pakistan, similarly he grew his beard. (; ; 29 January 192621 November 1996) was a Pakistani theoretical physicist. He shared the 1 ...
and their fellow researchers introduced early particle physics applications. The mathematical structure of supersymmetry ( graded Lie superalgebras) has subsequently been applied successfully to other topics of physics, ranging from
nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies th ...
,
critical phenomena In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the correlation length, but also the dynamics slows down. Critical phenomena include scaling relations ...
,
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
to
statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, and supersymmetry remains a vital part of many proposed theories in many branches of physics. In
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, the first realistic supersymmetric version of the Standard Model was proposed in 1977 by Pierre Fayet and is known as the
Minimal Supersymmetric Standard Model The Minimal Supersymmetric Standard Model (MSSM) is an extension to the Standard Model that realizes supersymmetry. MSSM is the minimal supersymmetrical model as it considers only "the inimumnumber of new particle states and new interactions ...
or MSSM for short. It was proposed to solve, amongst other things, the
hierarchy problem In theoretical physics, the hierarchy problem is the problem concerning the large discrepancy between aspects of the weak force and gravity. There is no scientific consensus on why, for example, the weak force is 1024 times stronger than gravi ...
. Supersymmetry was coined by Abdus Salam and John Strathdee in 1974 as a simplification of the term super-gauge symmetry used by Wess and Zumino, although Zumino also used the same term at around the same time. The term supergauge was in turn coined by Neveu and Schwarz in 1971 when they devised supersymmetry in the context of string theory.


Applications


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 group The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our unde ...
and internal symmetries and the Coleman–Mandula theorem showed that under certain assumptions, the symmetries of the
S-matrix In physics, the ''S''-matrix or scattering matrix is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...
must be a direct product of the Poincaré group with a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
internal symmetry group or if there is not any mass gap, the
conformal group In mathematics, the conformal group of an inner product space is the group (mathematics), group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometr ...
with a compact internal symmetry group. In 1971 Golfand 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–Łopuszański–Sohnius theorem In theoretical physics, the Haag–Łopuszański–Sohnius theorem states that if both commutative property, commutating and anticommutative property, anticommutating generator (mathematics), generators are considered, then the only way to nontrivi ...
analyzed all possible superalgebras in the general form, including those with an extended number of the supergenerators and
central charge In 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 element ...
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 of physics are generated by objects that transform by the
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
representations of the
Poincaré group The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our unde ...
and internal symmetries. Supersymmetries, however, are generated by objects that transform by the
spin representation In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equi ...
s. According to the spin-statistics theorem, bosonic fields commute while fermionic fields anticommute. Combining the two kinds of fields into a single
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
requires the introduction of a Z2-grading under which the bosons are the even elements and the fermions are the odd elements. Such an algebra is called a Lie superalgebra. The simplest supersymmetric extension of the Poincaré algebra is the Super-Poincaré algebra. Expressed in terms of two
Weyl spinor In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
s, has the following anti-commutation relation: :\ = 2( \sigma^ )_ P_ 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 ''σμ'' are the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
. There are representations of a Lie superalgebra that are analogous to representations of a Lie algebra. Each Lie algebra has an associated
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
and a Lie superalgebra can sometimes be extended into representations of a Lie supergroup.


Supersymmetric quantum mechanics

''Supersymmetric quantum mechanics'' adds the SUSY superalgebra to quantum mechanics as opposed to quantum field theory. Supersymmetric quantum mechanics often becomes relevant when studying the dynamics of supersymmetric
solitons In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such locali ...
, and due to the simplified nature of having fields which are only functions of time (rather than space-time), a great deal of progress has been made in this subject and it is now studied in its own right. SUSY quantum mechanics involves pairs of Hamiltonians which share a particular mathematical relationship, which are called ''partner Hamiltonians''. (The
potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
terms which occur in the Hamiltonians are then known as ''partner potentials''.) An introductory theorem shows that for every
eigenstate In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
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.


Supersymmetry in quantum field theory

In quantum field theory, supersymmetry is motivated by solutions to several theoretical problems, for generally providing many desirable mathematical properties, and for ensuring sensible behavior at high energies. Supersymmetric quantum field theory is often much easier to analyze, as many more problems become mathematically tractable. When supersymmetry is imposed as a ''local'' symmetry, Einstein's theory of
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 ...
is included automatically, and the result is said to be a theory of
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
. Another theoretically appealing property of supersymmetry is that it offers the only "loophole" to the Coleman–Mandula theorem, which prohibits spacetime and internal
symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
from being combined in any nontrivial way, for quantum field theories with very general assumptions. The
Haag–Łopuszański–Sohnius theorem In theoretical physics, the Haag–Łopuszański–Sohnius theorem states that if both commutative property, commutating and anticommutative property, anticommutating generator (mathematics), generators are considered, then the only way to nontrivi ...
demonstrates that supersymmetry is the only way spacetime and internal symmetries can be combined consistently. While supersymmetry has not been discovered at high energy, see Section Supersymmetry in particle physics, supersymmetry was found to be effectively realized at the intermediate energy of hadronic physics where
baryon In particle physics, a baryon is a type of composite particle, composite subatomic particle that contains an odd number of valence quarks, conventionally three. proton, Protons and neutron, neutrons are examples of baryons; because baryons are ...
s and
meson In particle physics, a meson () is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, the ...
s are superpartners. An exception is the
pion In particle physics, a pion (, ) or pi meson, denoted with the Greek alphabet, Greek letter pi (letter), pi (), is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the ...
that appears as a zero mode in the mass spectrum and thus protected by the supersymmetry: It has no baryonic partner. The realization of this effective supersymmetry is readily explained in quark–diquark models: Because two different
color charge Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD). Like electric charge, it determines how quarks and gluons interact through the strong force; ho ...
s close together (e.g., blue and red) appear under coarse resolution as the corresponding anti-color (e.g. anti-green), a diquark cluster viewed with coarse resolution (i.e., at the energy-momentum scale used to study hadron structure) effectively appears as an antiquark. Therefore, a baryon containing 3 valence quarks, of which two tend to cluster together as a diquark, behaves likes a meson.


Supersymmetry in condensed matter physics

SUSY concepts have provided useful
extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (proof theory) * Extension (predicate logic), the set of tuples of values t ...
to the
WKB approximation In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to Linear differential equation, linear differential equations with spatially varying coefficients. It is typically used for a Semiclass ...
. Additionally, SUSY has been applied to disorder averaged systems both quantum and non-quantum (through statistical mechanics), the
Fokker–Planck equation In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag (physi ...
being an example of a non-quantum theory. The 'supersymmetry' in all these systems arises from the fact that one is modelling one particle and as such the 'statistics' do not matter. The use of the supersymmetry method provides a mathematical rigorous alternative to the
replica trick In the statistical physics of spin glasses and other systems with quenched disorder, the replica trick is a mathematical technique based on the application of the formula: \ln Z=\lim_ or: \ln Z = \lim_ \frac where Z is most commonly the partit ...
, but only in non-interacting systems, which attempts to address the so-called 'problem of the denominator' under disorder averaging. For more on the applications of supersymmetry in
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
see Efetov (1997). In 2021, a group of researchers showed that, in theory, N=(0,1) SUSY could be realised at the edge of a Moore–Read quantum Hall state. However, to date, no experiments have been done yet to realise it at an edge of a Moore–Read state. In 2022, a different group of researchers created a computer simulation of atoms in 1 dimensions that had supersymmetric
topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
quasiparticles In condensed matter physics, a quasiparticle is a concept used to describe a collective behavior of a group of particles that can be treated as if they were a single particle. Formally, quasiparticles and collective excitations are closely relate ...
.


Supersymmetry in optics

In 2013, integrated optics was found to provide a fertile ground on which certain ramifications of SUSY can be explored in readily-accessible laboratory settings. Making use of the analogous mathematical structure of the quantum-mechanical
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
and the
wave equation The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...
governing the evolution of light in one-dimensional settings, one may interpret the
refractive index In optics, the refractive index (or refraction index) of an optical medium is the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refrac ...
distribution of a structure as a potential landscape in which optical wave packets propagate. In this manner, a new class of functional optical structures with possible applications in
phase matching Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typicall ...
, mode conversion and space-division multiplexing becomes possible. SUSY transformations have been also proposed as a way to address inverse scattering problems in optics and as a one-dimensional
transformation optics Transformation optics is a branch of optics which applies metamaterials to produce spatial variations, derived from coordinate transformations, which can direct chosen bandwidths of electromagnetic radiation. This can allow for the construction o ...
.


Supersymmetry in dynamical systems

All stochastic (partial) differential equations, the models for all types of continuous time dynamical systems, possess topological supersymmetry. In the operator representation of stochastic evolution, the topological supersymmetry is the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
which is commutative with the stochastic evolution operator defined as the stochastically averaged
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
induced on
differential forms In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
by SDE-defined
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...
s of the
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
. The topological sector of the so-emerging supersymmetric theory of stochastic dynamics can be recognized as the Witten-type topological field theory. The meaning of the topological supersymmetry in dynamical systems is the preservation of the phase space continuity—infinitely close points will remain close during continuous time evolution even in the presence of noise. When the topological supersymmetry is broken spontaneously, this property is violated in the limit of the infinitely long temporal evolution and the model can be said to exhibit (the stochastic generalization of) the
butterfly effect In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. The term is closely associated w ...
. From a more general perspective, spontaneous breakdown of the topological supersymmetry is the theoretical essence of the ubiquitous dynamical phenomenon variously known as
chaos Chaos or CHAOS may refer to: Science, technology, and astronomy * '' Chaos: Making a New Science'', a 1987 book by James Gleick * Chaos (company), a Bulgarian rendering and simulation software company * ''Chaos'' (genus), a genus of amoebae * ...
,
turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
,
self-organized criticality Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phas ...
etc. The Goldstone theorem explains the associated emergence of the long-range dynamical behavior that manifests itself as noise,
butterfly effect In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. The term is closely associated w ...
, and the scale-free statistics of sudden (instantonic) processes, such as earthquakes, neuroavalanches, and solar flares, known as the
Zipf's law Zipf's law (; ) is an empirical law stating that when a list of measured values is sorted in decreasing order, the value of the -th entry is often approximately inversely proportional to . The best known instance of Zipf's law applies to the ...
and the
Richter scale The Richter scale (), also called the Richter magnitude scale, Richter's magnitude scale, and the Gutenberg–Richter scale, is a measure of the strength of earthquakes, developed by Charles Richter in collaboration with Beno Gutenberg, and pr ...
.


In finance

In 2021, supersymmetric quantum mechanics was applied to
option pricing In finance, a price (premium) is paid or received for purchasing or selling options. The calculation of this premium will require sophisticated mathematics. Premium components This price can be split into two components: intrinsic value, and ...
and the analysis of markets in
finance Finance refers to monetary resources and to the study and Academic discipline, discipline of money, currency, assets and Liability (financial accounting), liabilities. As a subject of study, is a field of Business administration, Business Admin ...
, and to financial networks.


Supersymmetry in 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 model A toy or plaything is an object that is used primarily to provide entertainment. Simple examples include toy blocks, board games, and dolls. Toys are often designed for use by children, although many are designed specifically for adults and ...
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 monopoles. The proof of the
Atiyah–Singer index theorem In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space ...
is much simplified by the use of supersymmetric quantum mechanics.


Supersymmetry in string theory

Supersymmetry is an integral part of
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 ...
, a possible
theory of everything A theory of everything (TOE), final theory, ultimate theory, unified field theory, or master theory is a hypothetical singular, all-encompassing, coherent theoretical physics, theoretical framework of physics that fully explains and links togeth ...
. There are two types of string theory, supersymmetric string theory or
superstring theory 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 t ...
, and non-supersymmetric string theory. By definition of superstring theory, supersymmetry is required in superstring theory at some level. However, even in non-supersymmetric string theory, a type of supersymmetry called misaligned supersymmetry is still required in the theory in order to ensure no physical
tachyons A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists posit that faster-than-light particles cannot exist because they are inconsistent with the known laws of physics. If such particles ...
appear. Any string theories without some kind of supersymmetry, such as
bosonic string theory Bosonic string theory is the original version of string theory, developed in the late 1960s. It is so called because it contains only bosons in the spectrum. In the 1980s, supersymmetry was discovered in the context of string theory, and a new ve ...
and the E_7 \times E_7, SU(16), and E_8 heterotic string theories, will have a tachyon and therefore the
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 ...
vacuum A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
itself would be unstable and would decay into some tachyon-free string theory usually in a lower spacetime dimension. There is no experimental evidence that either supersymmetry or misaligned supersymmetry holds in our universe, and many physicists have moved on from supersymmetry and string theory entirely due to the non-detection of supersymmetry at the LHC. Despite the null results for supersymmetry at the LHC so far, some
particle physicists Particle physics or high-energy physics is the study of fundamental particles and forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the scale of protons and neutrons, while the stud ...
have nevertheless moved to string theory in order to resolve the naturalness crisis for certain supersymmetric extensions of the Standard Model. According to the particle physicists, there exists a concept of "stringy naturalness" in
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 ...
, where the
string theory landscape In string theory, the string theory landscape (or landscape of vacua) is the collection of possible false vacua,The number of metastable vacua is not known exactly, but commonly quoted estimates are of the order 10500. See M. Douglas, "The stat ...
could have a power law statistical pull on soft SUSY breaking terms to large values (depending on the number of hidden sector SUSY breaking fields contributing to the soft terms). If this is coupled with an anthropic requirement that contributions to the weak scale not exceed a factor between 2 and 5 from its measured value (as argued by Agrawal et al.), then the Higgs mass is pulled up to the vicinity of 125 GeV while most sparticles are pulled to values beyond the current reach of LHC. (The Higgs was determined to have a mass of 125 GeV ±0.15 GeV in 2022.) An exception occurs for higgsinos which gain mass not from SUSY breaking but rather from whatever mechanism solves the SUSY mu problem. Light higgsino pair production in association with hard initial state jet radiation leads to a soft opposite-sign dilepton plus jet plus missing transverse energy signal.


Supersymmetry in particle physics

In particle physics, a supersymmetric extension of the Standard Model is a possible candidate for undiscovered particle physics, and seen by some physicists as an elegant solution to many current problems in particle physics if confirmed correct, which could resolve various areas where current theories are believed to be incomplete and where limitations of current theories are well established. In particular, one supersymmetric extension of 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 ...
, the
Minimal Supersymmetric Standard Model The Minimal Supersymmetric Standard Model (MSSM) is an extension to the Standard Model that realizes supersymmetry. MSSM is the minimal supersymmetrical model as it considers only "the inimumnumber of new particle states and new interactions ...
(MSSM), became popular in theoretical particle physics, as the Minimal Supersymmetric Standard Model is the simplest supersymmetric extension of the Standard Model that could resolve major hierarchy problems within the Standard Model, by guaranteeing that quadratic divergences of all orders will cancel out in
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
. If a supersymmetric extension of the Standard Model is correct, superpartners of the existing
elementary particles In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a con ...
would be new and undiscovered particles and supersymmetry is expected to be spontaneously broken. There is no experimental evidence that a supersymmetric extension to the Standard Model is correct, or whether or not other extensions to current models might be more accurate. It is only since around 2010 that
particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel electric charge, charged particles to very high speeds and energies to contain them in well-defined particle beam, beams. Small accelerators are used for fundamental ...
s specifically designed to study physics beyond the Standard Model have become operational (i.e. the
Large Hadron Collider The Large Hadron Collider (LHC) is the world's largest and highest-energy particle accelerator. It was built by the CERN, European Organization for Nuclear Research (CERN) between 1998 and 2008, in collaboration with over 10,000 scientists, ...
(LHC)), and it is not known where exactly to look, nor the energies required for a successful search. However, the negative results from the LHC since 2010 have already ruled out some supersymmetric extensions to the Standard Model, and many physicists believe that the
Minimal Supersymmetric Standard Model The Minimal Supersymmetric Standard Model (MSSM) is an extension to the Standard Model that realizes supersymmetry. MSSM is the minimal supersymmetrical model as it considers only "the inimumnumber of new particle states and new interactions ...
, while not ruled out, is no longer able to fully resolve the hierarchy problem.


Supersymmetric extensions of the Standard Model

Incorporating supersymmetry into the Standard Model requires doubling the number of particles since there is no way that any of the particles in the Standard Model can be superpartners 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 Model (MSSM) which can include the necessary additional new particles that are able to be superpartners of those in the Standard Model. One of the original motivations for the Minimal Supersymmetric Standard Model came from the
hierarchy problem In theoretical physics, the hierarchy problem is the problem concerning the large discrepancy between aspects of the weak force and gravity. There is no scientific consensus on why, for example, the weak force is 1024 times stronger than gravi ...
. Due to the quadratically divergent contributions to the Higgs mass squared in the Standard Model, 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 greatest scale possible. Furthermore, the electroweak scale receives enormous Planck-scale quantum corrections. The observed hierarchy between the electroweak scale and the Planck scale must be achieved with extraordinary fine tuning. This problem is known as the hierarchy problem. Supersymmetry close to the electroweak scale, such as in the Minimal Supersymmetric Standard Model, would solve the
hierarchy problem In theoretical physics, the hierarchy problem is the problem concerning the large discrepancy between aspects of the weak force and gravity. There is no scientific consensus on why, for example, the weak force is 1024 times stronger than gravi ...
that afflicts the Standard Model. It would reduce the size of the quantum corrections by having automatic cancellations between fermionic and bosonic Higgs interactions, and Planck-scale quantum corrections cancel between partners and superpartners (owing to a minus sign associated with fermionic loops). The hierarchy between the electroweak scale and the Planck scale would be achieved in a
natural Nature is an inherent character or constitution, particularly of the ecosphere or the universe as a whole. In this general sense nature refers to the laws, elements and phenomena of the physical world, including life. Although humans are part ...
manner, without extraordinary fine-tuning. If supersymmetry were restored at the weak scale, then the Higgs mass would be 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. Another motivation for the Minimal Supersymmetric Standard Model comes from
grand unification A Grand Unified Theory (GUT) is any model in particle physics that merges the electromagnetic, weak, and strong forces (the three gauge interactions of the Standard Model) into a single force at high energies. Although this unified force ha ...
, the idea that the gauge symmetry groups should unify at high-energy. In the Standard Model, however, the
weak Weak may refer to: Songs * Weak (AJR song), "Weak" (AJR song), 2016 * Weak (Melanie C song), "Weak" (Melanie C song), 2011 * Weak (SWV song), "Weak" (SWV song), 1993 * Weak (Skunk Anansie song), "Weak" (Skunk Anansie song), 1995 * "Weak", a son ...
,
strong Strong may refer to: Education * The Strong, an educational institution in Rochester, New York, United States * Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas * Strong School, New Haven, Connecticut, United ...
and
electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
gauge couplings fail to unify at high energy. In particular, the
renormalization group In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying p ...
evolution of the three gauge
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between tw ...
s of the Standard Model 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 Model. After incorporating minimal SUSY at the electroweak scale, the running of the gauge couplings are modified, and joint convergence of the gauge coupling constants is projected to occur at approximately 1016  GeV. The modified running also provides a natural mechanism for radiative electroweak symmetry breaking. In many supersymmetric extensions of the Standard Model, such as the Minimal Supersymmetric Standard Model, there is a heavy stable particle (such as the
neutralino In supersymmetry, the neutralino is a hypothetical particle. In the Minimal Supersymmetric Standard Model (MSSM), a popular model of realization of supersymmetry at a low energy, there are four neutralinos that are fermions and are electrically ...
) which could serve as a
weakly interacting massive particle Weakly interacting massive particles (WIMPs) are hypothetical particles that are one of the proposed candidates for dark matter. There exists no formal definition of a WIMP, but broadly, it is an elementary particle which interacts via gravity an ...
(WIMP)
dark matter In astronomy, dark matter is an invisible and hypothetical form of matter that does not interact with light or other electromagnetic radiation. Dark matter is implied by gravity, gravitational effects that cannot be explained by general relat ...
candidate. The existence of a supersymmetric dark matter candidate is related closely to R-parity. Supersymmetry at the electroweak scale (augmented with a discrete symmetry) typically provides a candidate dark matter particle at a mass scale consistent with thermal relic abundance calculations. 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 spontaneously. 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 matter. In order to parameterize the relevant features of supersymmetry breaking, arbitrary soft SUSY breaking terms are added to the theory which temporarily break SUSY explicitly but could never arise from a complete theory of supersymmetry breaking.


Searches and constraints for supersymmetry

SUSY extensions of 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 ...
are constrained by a variety of experiments, including measurements of low-energy observables – for example, the anomalous magnetic moment of the muon at
Fermilab Fermi National Accelerator Laboratory (Fermilab), located in Batavia, Illinois, near Chicago, is a United States Department of Energy United States Department of Energy National Labs, national laboratory specializing in high-energy particle phys ...
; the
WMAP The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP and Explorer 80), was a NASA spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic mic ...
dark matter density measurement and direct detection experiments – for example,
XENON Xenon is a chemical element; it has symbol Xe and atomic number 54. It is a dense, colorless, odorless noble gas found in Earth's atmosphere in trace amounts. Although generally unreactive, it can undergo a few chemical reactions such as the ...
-100 and
LUX The lux (symbol: lx) is the unit of illuminance, or luminous flux per unit area, in the International System of Units (SI). It is equal to one lumen per square metre. In photometry, this is used as a measure of the irradiance, as perceived by ...
; and by particle collider experiments, including B-physics, Higgs phenomenology and direct searches for superpartners (sparticles), at the
Large Electron–Positron Collider The Large Electron–Positron Collider (LEP) was one of the largest particle accelerators ever constructed. It was built at CERN, a multi-national centre for research in nuclear and particle physics near Geneva, Switzerland. LEP collided electr ...
,
Tevatron The Tevatron was a circular particle accelerator (active until 2011) in the United States, at the Fermilab, Fermi National Accelerator Laboratory (called ''Fermilab''), east of Batavia, Illinois, and was the highest energy particle collider unt ...
and the
LHC The Large Hadron Collider (LHC) is the world's largest and highest-energy particle accelerator. It was built by the European Organization for Nuclear Research (CERN) between 1998 and 2008, in collaboration with over 10,000 scientists, and ...
. In fact, CERN publicly states that if a supersymmetric model of the Standard Model "is correct, supersymmetric particles should appear in collisions at the LHC." Historically, the tightest limits were from direct production at colliders. The first mass limits for squarks and gluinos were made at
CERN The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in Meyrin, western suburb of Gene ...
by the UA1 experiment and the UA2 experiment at the
Super Proton Synchrotron The Super Proton Synchrotron (SPS) 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. History The SPS was d ...
. LEP later set very strong limits, which in 2006 were extended by the D0 experiment at the Tevatron. From 2003 to 2015, WMAP's and
Planck Max Karl Ernst Ludwig Planck (; ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
's dark matter density measurements have strongly constrained supersymmetric extensions of the Standard Model, which, if they explain dark matter, have to be tuned to invoke a particular mechanism to sufficiently reduce the
neutralino In supersymmetry, the neutralino is a hypothetical particle. In the Minimal Supersymmetric Standard Model (MSSM), a popular model of realization of supersymmetry at a low energy, there are four neutralinos that are fermions and are electrically ...
density. Prior to the beginning of the LHC, in 2009, fits of available data to CMSSM and NUHM1 indicated that squarks and gluinos were most likely to have masses in the 500 to 800 GeV range, though values as high as 2.5 TeV were allowed with low probabilities. Neutralinos and sleptons were expected to be quite light, with the lightest neutralino and the lightest stau most likely to be found between 100 and 150 GeV. The first runs of the LHC surpassed existing experimental limits from the Large Electron–Positron Collider and Tevatron and partially excluded the aforementioned expected ranges. In 2011–12, the LHC discovered a
Higgs boson The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...
with a mass of about 125 GeV, and with couplings to fermions and bosons which are consistent with the Standard Model. The MSSM predicts that the mass of the lightest Higgs boson should not be much higher than the mass of the
Z boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , and ...
, and, in the absence of fine tuning (with the supersymmetry breaking scale on the order of 1 TeV), should not exceed 135 GeV. The LHC found no previously unknown particles other than the Higgs boson which was already suspected to exist as part of the Standard Model, and therefore no evidence for any supersymmetric extension of the Standard Model. Indirect methods include the search for a permanent electric dipole moment (EDM) in the known Standard Model particles, which can arise when the Standard Model particle interacts with the supersymmetric particles. The current best constraint on the
electron electric dipole moment The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: : U = - \mathbf d_ \cdot \mathbf E. The electron's electric dipole moment (EDM ...
put it to be smaller than 10−28 e·cm, equivalent to a sensitivity to new physics at the TeV scale and matching that of the current best particle colliders. A permanent EDM in any fundamental particle points towards time-reversal violating physics, and therefore also
CP-symmetry In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (Charge (physics), charge conjugation symmetry) and Parity (physics), P-symmetry (Parity (physics), parity sym ...
violation via the CPT theorem. Such EDM experiments are also much more scalable than conventional particle accelerators and offer a practical alternative to detecting physics beyond the standard model as accelerator experiments become increasingly costly and complicated to maintain. The current best limit for the electron's EDM has already reached a sensitivity to rule out so called 'naive' versions of supersymmetric extensions of the Standard Model. Research in the late 2010s and early 2020s from experimental data on the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is a coefficient that Albert Einstein initially added to his field equations of general rel ...
,
LIGO The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a large-scale physics experiment and observatory designed to detect cosmic gravitational waves and to develop gravitational-wave observations as an astronomical tool. Prior to LIG ...
noise Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
, and
pulsar timing Methods of detecting exoplanets usually rely on indirect strategies – that is, they do not directly Astrophotography, image the planet but deduce its existence from another signal. Any planet is an extremely faint light source compared to its ...
, suggests it's very unlikely that there are any new particles with masses much higher than those which can be found in the standard model or the LHC. However, this research has also indicated that
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 ...
or
perturbative In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which ...
quantum field theory will become strongly coupled before 1 PeV, leading to other new physics in the TeVs.


Current status

The negative findings in the experiments disappointed many physicists, who believed that supersymmetric extensions of the Standard Model (and other theories relying upon it) were by far the most promising theories for "new" physics beyond the Standard Model, and had hoped for signs of unexpected results from the experiments. In particular, the LHC result seems problematic for the Minimal Supersymmetric Standard Model, as the value of 125 GeV is relatively large for the model and can only be achieved with large radiative loop corrections from top squarks, which many theorists consider to be "unnatural" (see naturalness and fine tuning). In response to the so-called "naturalness crisis" in the Minimal Supersymmetric Standard Model, some researchers have abandoned naturalness and the original motivation to solve the hierarchy problem naturally with supersymmetry, while other researchers have moved on to other supersymmetric models such as split supersymmetry. Still others have moved to string theory as a result of the naturalness crisis. Former enthusiastic supporter Mikhail Shifman went as far as urging the theoretical community to search for new ideas and accept that supersymmetry was a failed theory in particle physics. However, some researchers suggested that this "naturalness" crisis was premature because various calculations were too optimistic about the limits of masses which would allow a supersymmetric extension of the Standard Model as a solution.


General supersymmetry

Supersymmetry appears in many related contexts of theoretical physics. 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 are the field content and interactions. Typically the number of copies of a supersymmetry is a power of 2 (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 is due to the Weinberg–Witten theorem. This corresponds to an ''N'' = 8 supersymmetry theory. Theories with 32 supersymmetries automatically have a
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 ...
. For four dimensions there are the following theories, with the corresponding multipletsPolchinski, J. ''String Theory. Vol. 2: Superstring theory and beyond'', Appendix B (CPT adds a copy, whenever they are not invariant under such symmetry): :


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 approximately 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.


Fractional supersymmetry

Fractional supersymmetry is a generalization of the notion of supersymmetry in which the minimal positive amount of spin does not have to be but can be an arbitrary for integer value of ''N''. Such a generalization is possible in two or fewer
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 ...
dimensions.


See also

* 4D N = 1 global supersymmetry *
Anyon In physics, an anyon is a type of quasiparticle so far observed only in two-dimensional physical system, systems. In three-dimensional systems, only two kinds of elementary particles are seen: fermions and bosons. Anyons have statistical proper ...
* Next-to-Minimal Supersymmetric Standard Model *
Quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...
* Split supersymmetry *
Supercharge In theoretical physics, a supercharge is a generator of supersymmetry transformations. It is an example of the general notion of a charge (physics), charge in physics. Supercharge, denoted by the symbol Q, is an operator which transforms bosons in ...
*
Supermultiplet In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra, possibly with extended supersymmetry. Then a superfield is a field on superspace which is valued in such a representation. Naïvely, or when considering ...
*
Supergeometry Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field the ...
*
Supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
* Supergroup *
Superpartner In particle physics, a superpartner (also sparticle) is a class of hypothetical elementary particles predicted by supersymmetry, which, among other applications, is one of the well-studied ways to extend the Standard Model of high-energy physics. ...
*
Superspace Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann num ...
* Supersplit supersymmetry *
Supersymmetric gauge theory In theoretical physics, there are many theories with supersymmetry (SUSY) which also have internal gauge symmetries. Supersymmetric gauge theory generalizes this notion. Gauge theory A gauge theory is a field theory with gauge symmetry. Rough ...
*
Supersymmetry nonrenormalization theorems In theoretical physics a nonrenormalization theorem is a limitation on how a certain quantity in the classical description of a quantum field theory may be modified by renormalization in the full quantum theory. Renormalization theorems are common ...
* Wess–Zumino model


References


Further reading


Supersymmetry and Supergravity
page i
String Theory Wiki
lists more books and reviews.


Theoretical introductions, free and online

* . * * * . * * * .


Monographs

* Baer, H., and Tata, X., ''Weak Scale Supersymmetry'', Cambridge University Press, Cambridge, (1999). . * Binetruy, P., ''Supersymmetry: Theory, Experiment, and Cosmology'', Oxford University Press, Oxford, (2012). . * Cecotti, S., ''Supersymmetric Field Theories: Geometric Structures and Dualities'', Cambridge University Press, Cambridge, (2015). . * Drees, M., Godbole, R., and Roy, P., ''Theory & Phenomenology of Sparticles'', World Scientific, Singapore (2005). . * Dreiner, H.K.,
Haber Haber is a surname of German origin. The meaning in old German is "oat". The cereal is now in German called . Notable people with this surname include: * Alan Haber, American student activist * Alessandro Haber, Italian actor, director, and singer ...
, H.E., Martin, S.P., ''From Spinors to Supersymmetry'', Cambridge University Press, Cambridge, (2023). . * Duplij, S., Siegel, W., and Bagger, J.,''Concise Encyclopedia of Supersymmetry'', Springer, (2003). . * Freud, P.G.O., ''Introduction to Supersymmetry'', Cambridge University Press, Cambridge, (1988). . * Junker, G., ''Supersymmetric Methods in Quantum and Statistical Physics'', Springer, (2011). . * Kane, G.L., ''Supersymmetry: Unveiling the Ultimate Laws of Nature'', Basic Books, New York (2001). . * Kane, G.L., and Shifman, M., eds. ''The Supersymmetric World: The Beginnings of the Theory'', World Scientific, Singapore (2000). . * Müller-Kirsten, H.J.W., and Wiedemann, A., ''Introduction to Supersymmetry'', 2nd ed., World Scientific, Singapore (2010). . * Nath, P., ''Supersymmetry, Supergravity and Unification'', Cambridge University Press, Cambridge, (2016). . * Raby, S., ''Supersymmetric Grand Unified Theories'', Springer, (2017). . * Tachikawa, Y., ''N=2 Supersymmetric Dynamics for Pedestrians'', Springer, (2014). . * Terning, J., ''Modern Supersymmetry: Dynamics and Duality'', Oxford University Press, Oxford, (2009). . * Wegner, F., ''Supermathematics and its Applications in Statistical Physics'', Springer, (2016). . * Weinberg, S., ''The Quantum Theory of Fields, Volume 3: Supersymmetry'', Cambridge University Press, Cambridge, (1999). . *
Wess Wesley Johnson known by his stage name Wess (August 13, 1945 — September 21, 2009) was an American-born Italian singing, singer and bass guitarist, perhaps mostly known for representing Italy along with Dori Ghezzi at the Eurovision Song Conte ...
, J. and Bagger, J., ''Supersymmetry and Supergravity'', Princeton University Press, Princeton, (1992). .


On experiments

* * Brookhaven National Laboratory (Jan 8, 2004).
New g−2 measurement deviates further from Standard Model
'' Press Release. * Fermi National Accelerator Laboratory (Sept 25, 2006).

' Press Release.


External links


Supersymmetry
European Organization for Nuclear Research (CERN)
The status of supersymmetry
– Symmetry Magazine (Fermilab/SLAC), January 12, 2021
As Supersymmetry Fails Tests, Physicists Seek New Ideas
Quanta Magazine ''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. History ''Quanta Magazine'' was initially launched as ''Simons Science ...
, November 20, 2012
What is Supersymmetry?
Fermilab Fermi National Accelerator Laboratory (Fermilab), located in Batavia, Illinois, near Chicago, is a United States Department of Energy United States Department of Energy National Labs, national laboratory specializing in high-energy particle phys ...
, May 21, 2013
Why Supersymmetry?
– Fermilab, May 31, 2013
The Standard Model and Supersymmetry
World Science Festival The World Science Festival is an annual science festival hosted by the World Science Foundation, a 501(c)(3) nonprofit organization based in New York City. There is also an Asia-Pacific event held in Brisbane, Australia. The foundation's go ...
, March 4, 2015
SUSY running out of hiding places
– BBC, December 11, 2012 {{Authority control Concepts in physics Concepts in the philosophy of science History of science Physics beyond the Standard Model Quantum field theory Theoretical physics