Supersymmetry

In particle physics, supersymmetry is a symmetry that relates bosons and fermions. In supersymmetric theories, every fundamental fermion has a superpartner which is a boson of equal mass, and vice versa. Although supersymmetry has yet to be observed in the real world it remains a vital part of many proposed theories of physics, including various extensions to the Standard Model as well as modern superstring theories. The mathematical structure of supersymmetry, invented in a particle-physics context, has been applied with useful results in other areas, ranging from quantum mechanics to classical statistical physics. SUSY is the acronym preferred for whichever grammatical variation of supersymmetry occurs in a sentence.

The supersymmetric standard model

Under the Standard Model all fundamental particles can be broken down into two groups, fermions that make up matter, and bosons that exchange the forces acting on matter. Due to the physics of the theory, almost all of the behaviour of the universe can be explained based on this handful of particles.

Fermions themselves further break down into three generations; that is, each fermion comes in a variety of three subtypes of increasing mass. For instance, one of the most commonly known fermions is the electron, which also has two other less-well-known subtypes, the muon and tau. Fermions also come in two versions for each generation, with differing electric charge. A graph of all the fermions in the Standard Model is quite small. It contains the three generations of quarks and leptons, each broken down into two partners with differing charge.

On the other hand the bosons come in groupings that are nowhere near as "neat", including four distinct types, with subgroups containing anywhere from one to sixteen members. In addition, there appears to be no generational structure; the photon only comes in one type, and although it has partners in the W and Z particles, they don't really match up with anything in the fermion side.

The discrepancy between the "clean" fermion side and "messy" boson side has long been one of the most bothersome points of the Standard Model.

It turns out that none of the particles in the Standard Model can be superpartners of each other, so if supersymmetry is correct there must be at least as many extra particles to discover as there are in the Standard Model. The simplest possibility consistent with the Standard Model is the Minimal Supersymmetric Standard Model (MSSM).

A possibility in some supersymmetric models is the existence of very heavy stable particles called WIMPS (weakly interacting massive particles), neutralinos or photinos which would interact very weakly with normal matter. These would be possible candidates for dark matter.

Experimental searches

At present, there is no experimental evidence that supersymmetry exists in the real world. However, there is some indirect evidence which suggests that supersymmetry may be found at energies not too far above those accessible by today's particle accelerators. The search for supersymmetry is one of the primary goals of the Large Hadron Collider (LHC) at the CERN laboratory which is due to open in 2007.

The supersymmetry algebra

Traditional symmetries in physics are generated by objects that transform under the various tensor representations of the Poincaré group. 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 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 contains two Weyl spinors with the following anti-commutation relation:

$\{Q_{\alpha },{\bar {Q}}_{\dot {\beta }}\}=2{\sigma ^{\mu }}_{\alpha {\dot {\beta }}}P_{\mu }$

and all other anti-commutation relations between the Qs and Ps vanish. In the above expression $P_{\mu }$ are the generators of translation and $\sigma ^{\mu }$ are the Pauli matrices.

Just as one can have representations of a Lie algebra, one can also have representations of a Lie superalgebra. For each Lie algebra, there exists an associated Lie group which is connected and simply connected. Unique up to isomorphism, this Lie group is canonically associated with the Lie algebra, and the algebra's representations can be extended to create group representations. In the same way, representations of a Lie superalgebra can sometimes be extended into representations of a Lie supergroup.

See supersymmetry algebra for a more detailed discussion, including a description of SUSY in Minkowski spacetime.

Supersymmetric quantum mechanics

Understanding the consequences of supersymmetry has proven mathematically daunting, and it has likewise been difficult to develop theories that could account for symmetry breaking, i.e., the lack of observed partner particles of equal mass. To make progress on these problems, physicists developed supersymmetric quantum mechanics, an application of the SUSY superalgebra to quantum mechanics as opposed to quantum field theory. It was hoped that studying SUSY's consequences in this simpler setting would lead to new understanding; remarkably, the effort created new areas of research in quantum mechanics itself.

For example, as of 2004 students are typically taught to "solve" the hydrogen atom by a laborious process which begins by inserting the Coulomb potential into the Schrödinger equation. After a considerable amount of work using many differential equations, the analysis produces a recursion relation for the Laguerre polynomials. The final outcome is the spectrum of hydrogen-atom energy states (labeled by quantum numbers n and l). Using ideas drawn from SUSY, the final result can be derived with significantly greater ease, in much the same way that operator methods are used to solve the harmonic oscillator. Oddly enough, this approach is analogous to the way Erwin Schrödinger first solved the hydrogen atom. Of course, he did not call his solution supersymmetric, as SUSY was thirty years in the future—but it is still remarkable that the SUSY approach, both older and more elegant, is taught in so few universities.

The SUSY solution of the hydrogen atom is only one example of the very general class of solutions which SUSY provides to shape-invariant potentials, a category which includes most potentials taught in introductory quantum mechanics courses.

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—but, in the relativisitic world, energy and mass are interchangeable, so we can just as easily say that the partner particles have equal mass.

SUSY concepts have provided useful extentions to the WKB approximation. In addition, SUSY has been applied to non-quantum statistical mechanics through the Fokker-Planck equation, showing that even if the original inspiration in high-energy particle physics turns out to be a blind alley, its investigation has brought about many useful benefits.

Example

Let's look at the example of a one-dimensional nonrelativistic particle with a 2D (i.e., two state) internal degree of freedom called "spin" (it's not really spin because "real" spin is for 3D particles). Let b be an operator which transforms a "spin up" particle into a "spin down" particle. Its adjoint b^† then transforms a spin down particle into a spin up particle; the operators are normalized such that the anticommutator {b,b^†}=1. And of course, b²=0. Let p be the momentum of the particle and x be its position with [x,p]=i. (Here, we use "natural units" where Planck's constant is set equal to 1.) Let W (the "superpotential") be an arbitrary differentiable function of x and let the supersymmetric operators

$Q_{1}={\frac {1}{2}}\left[(p-iW)b+(p+iW)b^{\dagger }\right]$

$Q_{2}={\frac {i}{2}}\left[(p-iW)b-(p+iW)b^{\dagger }\right]$

Note that Q₁ and Q₂ are self-adjoint. Let the Hamiltonian

$H=\{Q_{1},Q_{1}\}=\{Q_{2},Q_{2}\}={\frac {p^{2}}{2}}+{\frac {W^{2}}{2}}+{\frac {W'}{2}}(bb^{\dagger }-b^{\dagger }b)$

where W' is the derivative of W. Also note that {Q₁,Q₂}=0. This is nothing other than N = 2 supersymmetry.

Let's also call the spin down state "bosonic" and the spin up state "fermionic". This is only in analogy to quantum field theory and should not be taken literally. Then, Q₁ and Q₂ maps "bosonic" states into "fermionic" states and vice versa.

See also

• Goldstone boson
• Majoron

Related topics

• Supergroup
• Lie superalgebra
• Representation of a Lie superalgebra
• Superspace
• String theory
• Quantum group

References

• Cooper, F., A. Khare and U. Sukhatme. "Supersymmetry in Quantum Mechanics." Phys. Rep. 251 (1995) 267-85 (arXiv:hep-th/9405029).
• Junker, G. Supersymmetric Methods in Quantum and Statistical Physics, Springer-Verlag (1996).
• Kane, G. L. and Shifman, M., eds. The Supersymmetric World: The Beginnings of the Theory, World Scientific, Singapore (2000). ISBN 981-02-4522-X.
• Weinberg, Steven, The Quantum Theory of Fields, Volume 3: Supersymmetry, Cambridge University Press, Cambridge, (1999). ISBN 0-521-66000-9.
• Wess, Julius, and Jonathan Bagger, Supersymmetry and Supergravity, Princeton University Press, Princeton, (1992). ISBN 0-691-02530-4.