Supersymmetry

Supersymmetry is a theoretical framework in physics that suggests the existence of a symmetry between particles with integer spin and particles with half-integer spin. 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. If evidence is found, supersymmetry could help explain certain phenomena, such as the nature of dark matter and the hierarchy problem in particle physics.
A supersymmetric theory is a theory in which the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry. Dozens of supersymmetric theories exist. In theory, supersymmetry is a type of spacetime symmetry between two basic classes of particles: bosons, which have an integer-valued spin and follow Bose–Einstein statistics, and fermions, which have a half-integer-valued spin and follow Fermi–Dirac statistics. The names of bosonic partners of fermions are prefixed with s-, because they are scalar particles. For example, if the electron existed in a supersymmetric theory, then there would be a particle called a selectron, 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. 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 and internal quantum numbers besides spin. More complex supersymmetry theories have a spontaneously broken symmetry, allowing superpartners to differ in mass.
Supersymmetry has various applications to different areas of physics, such as quantum mechanics, statistical mechanics, quantum field theory, condensed matter physics, nuclear physics, optics, stochastic dynamics, astrophysics, quantum gravity, and cosmology. Supersymmetry has also been applied to high-energy physics, where a supersymmetric extension of the Standard Model is a possible candidate for physics beyond the Standard Model. 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 and baryons 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, Yu. A. Golfand and E. P. Likhtman, and D. V. Volkov and V. P. Akulov, independently rediscovered supersymmetry in the context of quantum field theory, 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 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 and their fellow researchers introduced early particle physics applications. The mathematical structure of supersymmetry has subsequently been applied successfully to other topics of physics, ranging from nuclear physics, critical phenomena, quantum mechanics to statistical physics, and supersymmetry remains a vital part of many proposed theories in many branches of physics.
In particle physics, 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 or MSSM for short. It was proposed to solve, amongst other things, the hierarchy problem.
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 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 compact internal symmetry group or if there is not any mass gap, the conformal group 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, which later became known as supercharges. In 1975, the Haag–Łopuszański–Sohnius theorem analyzed all possible superalgebras in the general form, including those with an extended number of the supergenerators and central charges. This extended super-Poincaré algebra paved the way for obtaining a very large and important class of supersymmetric field theories.

Supersymmetry algebra

Traditional symmetries of physics are generated by objects that transform by the tensor representations of the Poincaré group and internal symmetries. Supersymmetries, however, are generated by objects that transform by the spin representations. According to the spin-statistics theorem, bosonic fields commute while fermionic fields anticommute. Combining the two kinds of fields into a single algebra 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 spinors, has the following anti-commutation relation:
and all other anti-commutation relations between the s and commutation relations between the s and s vanish. In the above expression are the generators of translation and are the Pauli matrices.
There are representations of a Lie superalgebra 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.

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, and due to the simplified nature of having fields which are only functions of 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. 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.

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 is included automatically, and the result is said to be a theory of supergravity. 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 theories with very general assumptions. The Haag–Łopuszański–Sohnius theorem 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 baryons and mesons are superpartners. An exception is the pion 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 charges close together appear under coarse resolution as the corresponding anti-color, a diquark cluster viewed with coarse resolution 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.