One of the key insights in physics in the past century is the importance of symmetry principles in determining the fundamental laws of physics. An example from special relativity is that the laws of physics should be invariant under the spacetime transformations of boosts, rotations and translations. This mathematical structure is known as the Lorentz/Poincare group. Another example is gauge symmetry. The simplest gauge symmetry is phase invariance in quantum mechanics. Promoting phase invariance to a local symmetry provides the foundation for quantum electrodynamics-- QED.

In the early 1970's, a new type of symmetry-- supersymmetry, or SUSY for short-- was discovered. Supersymmetry turns out to be the most general extension of special relativity that is possible, while maintaining consistency with quantum mechanics. Since two supersymmetry transformations lead to a spacetime translation, supersymmetry can be thought of as the ``square root'' of a translation. In a supersymmetric theory, each fermion has a bosonic counterpart, and vice versa. However, if a spin-zero electron existed, it would certainly have been seen in experiments. Hence, supersymmetry, if it exists, must be a broken symmetry.

A wonderful thing happens if supersymmetry is promoted to a local symmetry. In much the same way as local phase invariance requires the introduction of a photon field in QED, local supersymmetry necessitates the introduction of a massless spin-2 graviton field. The classical limit of a supergravity theory is just Einstein's theory of general relativity.

In elementary particle physics, the introduction of supersymmetry has a very practical aspect. In the Standard Model (SM) of particle physics, a spin-zero Higgs field is necessary to give mass to both fermions and certain gauge bosons associated with weak interactions. Quantum corrections to Higgs boson parameters turn out to be quadratically divergent. Since the Standard Model is a renormalizable theory, these quadratic divergences manifest themselves in a fine-tuning problem. For instance, embedding the SM into a grand unified theory of strong, weak and electromagnetic interactions will necessitate fine-tuning the Higgs mass to one part in 10^29. In the supersymmetric Standard Model, all quadratic divergences miraculously cancel as a result of the enhanced symmetry of the model. Further fine-tuning is avoided if the superpartners of all matter have masses less than about 1 trillion electron volts (1 TeV/c^2). Thus, they should be accessible to future collider experiments such as those planned for the CERN Large Hadron Collider (LHC), a proton-proton collider to operate at 14 TeV. Plans are for machine turn-on in 2007.

Sometimes my experimental colleagues tease me that only half the particles of the supersymmetric Standard Model have been discovered: just the Standard Model half! However, this is easy to understand theoretically. Under exact supersymmetry and gauge symmetry, all matter and gauge boson states would be exactly massless. If supersymmetry is broken, only the superpartners of the SM particles will gain mass. Unbroken electroweak symmetry protects the quarks, leptons and gauge bosons of the SM from becoming massive, and these are exactly the states of matter that have been observed. A wonderful consequence of supersymmetry breaking is that it naturally induces the breakdown of electroweak gauge symmetry. This was found to occur in the early 1980's only if the top quark had a mass of about 100-200 GeV/c^2. In the mid-1990's, the top quark was discovered at Fermilab with a mass of about 175 GeV/c^2.

The hunt for supersymmetry is one of the main goals of the CERN LHC. But how would supersymmetric matter be revealed at a collider experiment? The answer depends on the mechanism for the breaking of supersymmetry, and how that breaking is communicated to the particle states of the supersymmetric SM. There are specific proposals for communication of SUSY breaking via gravitational effects, via gauge interactions, or involving extra dimensions. In all these models, the production rates for supersymmetric matter can be computed. Once superparticles are produced, they will decay via a cascade until the lightest SUSY particle state is reached, which is presumably absolutely stable. The lightest SUSY particle should be very weakly interacting, and escape detection much like a neutrino, giving rise to an apparent non-conservation of energy and momentum. The first realistic program for simulation of the creation of supersymmetric matter at colliding beam experiments was developed here at FSU. Detailed simulations of production and decay of supersymmetric matter in colliding beam experiments show that the spray of jets and leptons, and apparent imbalance of transverse momentum, should be distinguishable from known background processes.

Supersymmetric matter can have a profound effect on cosmology as well. Superparticles would have been thermally created and destroyed during early times in a Big Bang universe. Since the lightest SUSY particle is absolutely stable, relics from the Big Bang should pervade all space even today. Detailed calculations show that the lightest SUSY particle may in fact make up the bulk of matter in the universe. Table-top cryogenic search experiments have been constructed to search for these supposed relics of the Big Bang.

If supersymmetric matter is ever discovered, the spectrum of masses for the new matter states will likely reflect deep physical principles valid at extremely high energy scales such as those associated with grand unification and superstring theory. It will then be necessary to perform precision spectroscopic measurements of the new matter states. Such high precision measurements can be accomplished at a linear electron-positron collider operating with CM energy near the 1 TeV scale. A characterization of the entire superparticle mass spectrum will then point the way towards a more encompassing theory involving perhaps grand unification, extra dimensions and superstrings.