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In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that interchanges bosons and fermions. In supersymmetric theories, every fundamental fermion has a bosonic superpartner and vice versa. A supersymmetric quantum field theory tames quantum mechanical dynamics and sometimes allows the theory to be solved. If supersymmetry is applied to the Standard Model of particle physics, the hierarchy problem can be solved. The minimal supersymmetric Standard Model is one of the best studied candidates for physics beyond the Standard Model. Supersymmetry was originally proposed in 1973 by Julius Wess and Bruno Zumino. Earlier, the supersymmetry algebra had been discovered in the late 1960's by Soviet theorists Gol'fand and Likhtman, but was not applied by them directly to the then outstanding problems in elementary particle physics. Supersymmetry first arose in the context of an early version of string theory by Ramond, Schwarz and Neveu, but the mathematical structure of supersymmetry has subsequently been applied successfully to other areas of physics; firstly by Wess, Zumino, and Abdus Salam and their fellow researchers to particle physics, and later to a variety of fields, ranging from quantum mechanics to statistical physics. 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 Georgi and Savas Dimopoulos and is called the minimal supersymmetric Standard Model or MSSM for short. It was proposed to solve the hierarchy problem and predicts superpartners with masses between 100 GeV and 1 TeV. As of 2006 there is no irrefutable experimental evidence that supersymmetry is a symmetry of nature. In 2008 the Large Hadron Collider at CERN is scheduled to produce the worlds 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 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 no mass gap, the conformal group with a compact internal symmetry group. In 1975, the Haag-Lopuszanski-Sohnius theorem showed that considering symmetry generators which satisfy anticommutation relations allows for such nontrivial extensions of space-time symmetry. This extension to the Coleman-Mandula theorem prompted some physicists to study this wider class of theories. The supersymmetry algebra Main article: Supersymmetry algebra Traditional symmetries in physics are generated by objects that transform under the tensor representations of the Poincaré group and internal symmetries. Supersymmetries, on the other hand, are generated by objects that transform under the spinor representations. According to the spin-statistics theorem, bosonic fields commute while fermionic fields anticommute. In order to combine the two kinds of fields into a single algebra requires the introduction of a '''Z'''2-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 contains two Weyl spinors with the following anti-commutation relation: and all other anti-commutation relations between the Qs and Ps 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. The Supersymmetric Standard Model
Gauge Coupling Unification Main article: gauge coupling unification One piece of evidence for supersymmetry existing at the weak scale is gauge coupling unification. The renormalization group evolution of the three gauge coupling constants of the Standard Model is sensitive to the particle content of the theory. If the Standard Model do not quite meet together at a common energy scale if we run the renormalization group using the Standard Model. With the addition of SUSY, the match is within the ability that theory is able to predict the values. Supersymmetric quantum mechanics Main article: supersymmetric quantum mechanics Supersymmetric quantum mechanics adds the SUSY superalgebra to quantum mechanics as opposed to quantum field theory. 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 Hamiltonians which share a particular mathematical relationship, which are called partner Hamiltonians. (The potential energy 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 approximation. In addition, SUSY has been applied to non-quantum statistical mechanics through the Fokker-Planck equation. See supersymmetric quantum mechanics for a more detailed discussion. 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 models 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. 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 Main article: 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 graviton. In four dimensions there are the following theories Supersymmetry in Alternate Numbers of Dimensions It is possible to have supersymmetry in alternate dimensions. Because the properties of spinors change drastically between different dimensions, each dimension has its characteristic. In dimensions, the size of spinors is roughly or . Since the maximum number of supersymmetries is 32, the greatest number of dimensions a supersymmetric theory can exist in is eleven dimensions. See also | |||||||||
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