In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners. In a theory with unbroken supersymmetry, for every type of boson there exists a corresponding type of fermion with the same mass and internal quantum numbers, and viceversa.
So far, there is only indirect evidence for the existence of supersymmetry.^{[1]} Since the superpartners of the Standard Model particles have not been observed, supersymmetry, if it exists, must be a broken symmetry, allowing the superparticles to be heavier than the corresponding Standard Model particles.
If supersymmetry exists close to the TeV energy scale, it allows for a solution of the hierarchy problem of the Standard Model, i.e., the fact that the Higgs boson mass is subject to quantum corrections which — barring extremely finetuned cancellations among independent contributions — would make it so large as to undermine the internal consistency of the theory. In supersymmetric theories, on the other hand, the contributions to the quantum corrections coming from Standard Model particles are naturally canceled by the contributions of the corresponding superpartners. Other attractive features of TeVscale supersymmetry are the fact that it allows for the highenergy unification of the weak interactions, the strong interactions and electromagnetism, and the fact that it provides a candidate for Dark Matter and a natural mechanism for electroweak symmetry breaking.
Another advantage of supersymmetry is that supersymmetric quantum field theory can sometimes be solved. Supersymmetry is also a feature of most versions of string theory, though it can exist in nature even if string theory is incorrect.
The Minimal Supersymmetric Standard Model is one of the best studied candidates for physics beyond the Standard Model.
Is supersymmetry a symmetry of Nature? If so, how is supersymmetry broken, and why? Can the new particles predicted by supersymmetry be detected? 
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A supersymmetry relating mesons and baryons was first proposed, in the context of hadronic physics, by Hironari Miyazawa in 1966, but his work was ignored at the time.^{[2]}^{[3]}^{[4]}^{[5]} In the early 1970s, J. L. Gervais and B. Sakita (in 1971), Yu. A. Golfand and E.P. Likhtman (also in 1971), D.V. Volkov and V.P. Akulov (in 1972) and J. Wess and B. Zumino (in 1974) independently rediscovered supersymmetry, a radically new type of symmetry of spacetime and fundamental fields, which establishes a relationship between elementary particles of different quantum nature, bosons and fermions, and unifies spacetime and internal symmetries of the microscopic world. Supersymmetry first arose in the context of an early version of string theory by Pierre Ramond, John H. Schwarz and Andre 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 2009 there is no irrefutable experimental evidence that supersymmetry is a symmetry of nature. In 2010 the Large Hadron Collider at CERN is scheduled to produce the world's highest energy collisions and offers the best chance at discovering superparticles for the foreseeable future.
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 Smatrix 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 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 HaagLopuszanskiSohnius theorem analyzed all possible superalgebras in the general form, including those with an extended number of the supergenerators and central charges. This extended superPoincaré algebra paved the way for obtaining a very large and important class of supersymmetric field theories.
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 spinstatistics theorem, bosonic fields commute while fermionic fields anticommute. Combining 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, expressed in terms of two Weyl spinors, has the following anticommutation relation:
and all other anticommutation relations between the Qs and commutation relations between the Qs and Ps vanish. In the above expression are the generators of translation and σ^{μ} 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.
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 main motivations for SUSY comes from the quadratically divergent contributions to the Higgs mass squared. The quantum mechanical interactions of the Higgs boson causes a large renormalization of the Higgs mass and unless there is an accidental cancellation, the natural size of the Higgs mass is the highest scale possible. This problem is known as the hierarchy problem. Supersymmetry reduces the size of the quantum corrections by having automatic cancellations between fermionic and bosonic Higgs interactions. If supersymmetry is restored at the weak scale, then the Higgs mass is related to supersymmetry breaking which can be induced from small nonperturbative effects explaining the vastly different scales in the weak interactions and gravitational interactions.
In many supersymmetric Standard Models there is a heavy stable particle (such as neutralino) which could serve as a Weakly interacting massive particle (WIMP) dark matter candidate. The existence of a supersymmetric dark matter candidate is closely tied to Rparity.
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 currently 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.
One piece of evidence for supersymmetry existing is gauge coupling unification. The renormalization group evolution of the three gauge coupling constants 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.^{[1]} With the addition of minimal SUSY joint convergence of the coupling constants is projected at approximately 10^{16} GeV.^{[1]}
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 spacetime), 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 nonquantum statistical mechanics through the FokkerPlanck equation.
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 Sduality in four dimensional gauge theories that interchanges particles and monopoles.
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.
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
It is possible to have supersymmetry in dimensions other than four. Because the properties of spinors change drastically between different dimensions, each dimension has its characteristic. In d dimensions, the size of spinors is roughly 2^{d/2} or 2^{(d − 1)/2}. Since the maximum number of supersymmetries is 32, the greatest number of dimensions in which a supersymmetric theory can exist is eleven.
Supersymmetry can be reinterpreted in the language of noncommutative geometry and quantum groups. In particular, it involves a mild form of noncommutativity, namely supercommutativity. See the main article for more details.
Supersymmetry is part of a larger enterprise of theoretical physics to unify everything we know about the physical world into a single fundamental framework of physical laws, known as the quest for a Theory of Everything (TOE). A significant part of this larger enterprise is the quest for a theory of quantum gravity, which would unify the classical theory of general relativity and the Standard Model, which explains the other three basic forces in physics (electromagnetism, the strong interaction, and the weak interaction), and provides a palette of fundamental particles upon which all four forces act. Two of the most active approaches to forming a theory of quantum gravity are string theory and loop quantum gravity (LQG), although in theory, supersymmetry could be a component of other theoretical approaches as well.
For string theory to be consistent, supersymmetry appears to be required at some level (although it may be a strongly broken symmetry). In particle theory, supersymmetry is recognized as a way to stabilize the hierarchy between the unification scale and the electroweak scale (or the Higgs boson mass), and can also provide a natural dark matter candidate. String theory also requires extra spatial dimensions which have to be compactified as in KaluzaKlein theory.
Loop quantum gravity (LQG), in its current formulation, predicts no additional spatial dimensions, nor anything else about particle physics. These theories can be formulated in three spatial dimensions and one dimension of time, although in some LQG theories dimensionality is an emergent property of the theory, rather than a fundamental assumption of the theory. Also, LQG is a theory of quantum gravity which does not require supersymmetry. Lee Smolin, one of the originators of LQG, has proposed that a loop quantum gravity theory incorporating either supersymmetry or extra dimensions, or both, be called "loop quantum gravity II".
If experimental evidence confirms supersymmetry in the form of supersymmetric particles such as the neutralino that is often believed to be the lightest superpartner, some people believe this would be a major boost to string theory. Since supersymmetry is a required component of string theory, any discovered supersymmetry would be consistent with string theory. If the Large Hadron Collider and other major particle physics experiments fail to detect supersymmetric partners or evidence of extra dimensions, many versions of string theory which had predicted certain low mass superpartners to existing particles may need to be significantly revised. The failure of experiments to discover either supersymmetric partners or extra spatial dimensions, as of 2008, has encouraged loop quantum gravity researchers. However, the likelihood of detecting supersymmetric particles in the near future remains a controversial issue. Recently prediction markets like intrade offered scientific contracts that may give estimates for that probability.

This book on supersymmetry intends to present it WITHOUT the use of Grassmann variables, prefering to use instead the formalism of Z_{2} grading.
A Z_{2}graded vector space is a vector space together with an assignment of an even (bosoninc) (corresponding to 0 of Z_{2}) and an odd (fermionic) (corresponding to 1) subspace such that the vector space is the direct sum of the even and odd subspaces.
An even vector is an element of the even subspace and an odd vector is an element of the odd subspace. A pure vector is either an even or an odd vector. Any vector can be decomposed uniquely as the sum of an even and an odd vector.
The tensor product of two Z_{2}graded vector spaces is another Z_{2}graded vector space.
In fact, in this book, we will take the stronger point of view that it makes no physical sense to add even and odd vectors together. From this point of view, we might as well view a Z_{2}graded vector space as an ordered pair <V_{0},V_{1}> where V_{0} is the even space and V_{1} is the odd space.
Similarly, a Z_{2}graded algebra is an algebra A with a direct sum decomposition into an even and an odd part such that the product of two pure elements obeys the Z_{2} relations. Alternatively, we can think of it as <A_{0},A_{1}>.
A Lie superalgebra is a Z_{2}graded algebra whose product [·, ·], called the Lie superbracket or supercommutator, satisfies
and
where x, y, and z are pure in the Z_{2}grading. Here, x denotes the degree of x (either 0 or 1).
Lie superalgebras are a natural generalization of normal Lie algebras to include a Z_{2}grading. Indeed, the above conditions on the superbracket are exactly those on the normal Lie bracket with modifications made for the grading. The last condition is sometimes called the super Jacobi identity.
Note that the even subalgebra of a Lie superalgebra forms a (normal) Lie algebra as all the funny signs disappear, and the superbracket becomes a normal Lie bracket.
One way of thinking about a Lie superalgebra  it's not the most symmetric way of looking at it  is to consider its even and odd parts, L_{0} and L_{1} separately. Then, L_{0} is a Lie algebra, L_{1} is a linear rep of L_{0}, and there exists a symmetric L_{0}intertwiner such that for all x,y and z in L_{1},
A supermanifold is a concept in noncommutative geometry. Recall that in noncommutative geometry, we don't look at point set spaces but instead, the algebra of functions over them. If M is a (differential) manifold and H is an (smooth) algebra bundle over M with a Grassmann algebra as the fiber, then the space of (smooth) sections of M forms a supercommutative algebra under pointwise multiplication. We say that this algebra defines the supermanifold (which isn't a point set space).
If M is a real manifold and we define an involution * over the fiber turning it into a * algebra, then the resulting algebra would define a real supermanifold.
Supersymmetry is a of theory (commonly found in some forms of string theory) that says that when the universe was made, there was also the same number of theoretical "superparticles" created. If this theory is true, it would at least double the number of particles in the universe. Supersymmetry may create more than one copy, since there are many dimensions (some string theories predict up to 11). Supersymmetry is believed to exist by many scientists because it solves many inconsistencies in the Standard Model of physics.
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A superparticle is the supersymmetric copy of its counterpart (that is, regular matter). A superparticle also has a slightly different spin than its counterpart, by 1/2. Many scientists believe that in the Big Bang, superparticles were created for an almost immeasurably short amount of time (a ten trillionth of a ten trillionth of nanosecond) before they effectively froze into space (this is their way of saying we don't really know where they went). Note that these "copies" of what we would call normal matter may or may not be identical copies (except for the fact that it will always have a half of a spin difference), depending on the correctness of "unbroken supersymmetry," which simply states that supersymmetric particles are copies of matter with a 1/2 spin difference.
If some of the principles of supersymmetry are correct, it is possible to recreate these superparticles with particle accelerators. This attempt could prove or disprove the ideas of supersymmetry.
In outer space, scientists have seen areas of matter that are completely dark. This is not normal, as all known matter reflects light–or absorbs it and reflects it in a different color. Light is a form of electromagnetism (the carrier particle of electromagnetism is a photon). Some scientists suspect that this means that the matter that is not giving off light may not even interact with electromagnetism. This would explain why they don't emit light, but not why no light would pass through them, as they simply don't interact with electromagnetism.
Many people wonder what "dark matter" is, and scientists do not really have an answer for them. However, a simple answer for this is that dark matter is matter that does not emit light. From now on, the abovementioned matter will now be referred to as dark matter.
Around the mid 1930's, scientists predicted the existence of dark matter for variance reasons. For example, the spin of galaxies was nothing like how the visible matter would suggest it should. Also, some scientists noticed gravity fields from something that was not visible. They labeled it "dark matter." Many scientists accept that dark matter exists, but not all accept the connection between dark matter and supersymmetry.
If scientists could somehow make a link between supersymmetry and dark matter, this would raise many new questions. Electromagnetism is one of the four fundemental forces, as is strong force.
Matter as we know it is held together by strong force (aka nuclear force). Strong force holds the protons and neutrons together in an atom, and is thought to be a result of the lighter mesons for the interaction between protons. Protons (and neutrons) themselves are made of quarks, which are in turn thought to be held together by gluons. As a result of all of this, an atom nucleus is formed.
Strong force is necessary for an atom nucleus to hold together because apart from the fact that the protons wouldn't stick together, protons actually repel each other because of their electromagnetic charge. (The same statement applies to quarks, but it is due to both charge and chromodynamic forces which are governed by W and Z bosons).
However, if a superparticle does not interact with electromagnetism, strong force would not be so necessary because the superprotons do not repel each other. However, this lack of interaction with electromagnetism would not allow electrons to orbit an atom nucleus. This poses many new questions about superparticles, as electrons are the reason that you cannot push your hand through a wall without the wall breaking; electrons are negatively charged and repel each other. In addition, electrons are fermions, which basically means that two electrons can't exist in the same place at the same time. Since no electrons would orbit the nucleus, it would be much easier to push two superparticle nuclei together.
It is difficult to imagine how a superparticle could exist if it was only held together by strong force. This is one of the many problems with the supersymmetry theory.
Close, Frank (2004). Particle Physics. Oxford. ISBN 0192804340.
"Dark matter". Wikipedia. 15 December 2010. http://en.wikipedia.org/wiki/Dark_matter. Retrieved 16 December 2010.
"Supersymmetry". Wikipedia. 2 December 2010. http://en.wikipedia.org/wiki/Supersymmetry. Retrieved 16 December 2010.
