Modern theoretical physics is rooted in the study of symmetry, that is, the study of the transformations we can perform on a system and get "the same physics".  The central concepts of freshman physics, those of conservation of energy and linear and angular momentum, we find arise from the invariance of physical law under time and space translation and spatial rotation, respectively.   The particle content of the Standard Model is defined by the symmetry group SU(3) x SU(2) x U(1).  The purpose of this page is to present my lecture on a new, speculative symmetry that has gained a very prominent and fashionble status among possible new physics to be sought out at the LHC.   This is supersymmetry("SUSY"): a fundamental symmetry in nature between bosons and fermions, and what I set out to accomplish is to present how it works, its mathematical foundations and why it is so important.


You can find the full presentation, with words added to make it more readable here.(best if viewed by Internet Explorer)  What follows is a brief synopsis of the ideas that I used.  I have relegated the discussion of the MSSM and SUSY QED to the above link, as well as the full explanation of any potentially unfamiliar terminology.





This basic symmetry between bosons and fermions can be seen most simply by considering a free theory with a single scalar and a single Weyl fermion.


We want to impose a symmetry that rotates the bosons into fermions.
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where in order for the expression to be consistent, epsilon must be a Grassman parameter and have mass dimention -1/2.
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Equivalently, the symmetry turns fermions back into scalars.
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We show we can perform this transformation and get back the same physics.  By substituting these transformed fields back into our Lagrangian and utilizing the fundamental relation between the Pauli matrices,
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the lagrangian is changed in total by an amount
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a total derivative!  Because the edges of space are fixed in our consideration of the variational principle, the action is unchanged by Gauss' theorem, and we get back the exact same physics.  This is the behavior of a symmetry of nature, and we have just demonatrated the simplest example of a supersymmetric theory. Let's try to rewrite this theory in a more general and applicable way.
We want to write these infinitessimal changes in the fields in the same way we write changes in the fields over any other symmetry transformation: as being acted on by an infinitessimally-generated group.  So we write
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These generators Q generally speaking, are defined by special commutation relations among themselves.   We try to find some such relation with the current model that is independent of the actual model itself.
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From this, we can pick out the fundamental relations between the supersymmetry generators.  Using the standard quantum mechanical definition of momentum, we have
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(The curly brackets denote anti-commutation.)  As we will see, thse define supersymmetry as displayed in more general models.
Similarly, for fermions, we have

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The red terms vanish when the particle obeys the Dirac equation and the same algebraic relation holds for when these transformations are applied to fermions, but if we have fermions propagating off mass-shell, this is not the case due to quantum effects. This can be remedied by adding a new scalar field of mass dimension 2 to the theory.
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This transforms in such a way as to cancel the extra terms that show up above. Note that F has a trivial field equation.  All in all, the fields present in the model transform as
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In general, supersymmetry is defined by the anticommutation relations between each other and momentum, the generators of translations in space-time.  For our present purposes, supersymmetry is global, and we have as our defining relations
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These also have nontrivial relations with the generators of the Lorenz group.  I want to in particular call attention to the commutation relations between supersymmetry and angular momentum along the z-axis, for example.
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These are exactly the same as the commutation relations for angular momentum raising and lowering operators in usual quantum mechanics, except for an important factor is 1/2.  These generators commute with linear momentum along the z-axis  What these facts mean is that if the Q's act on a helicity state, then they act as raising or lowering operators on that state.  The helicity is raised or lowered by 1/2. Bosons are really turned into fermions and vice-versa.   We have gone backwards; in defining a set of (anti-)commutation relations like the ones above, we have found that the generators, how we defined them, actually do impose a symmetry of nature under turning bosons into fermions.
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We can simply impose this symmetry on top of any field theory, with these symmetry generators defined above in order to make it supersymmetric.  The generators of a symmetry group are generally Lie algrbra-valued and are therefore defined by commutation relations among themselves.   These supersymmetry generators are defined by anticommutation relations.  Also note that these transformations mix internal (boson vs. fermion) and spacetime symmetries with the fundamental anticommutation relation.   This is in apparent violation of what is known as the Coleman-Mandula theorem, which states that all internal symmetries in a theory can only transform trivially under the Lorenz group. Because supersymmetry is defined by an anticommutation relation, it is not subject to this theorem.  In fact, supersymmetry puts what is called a grading on the Lie algebra that generates the known spacetime and internal symmetries.  A later theorem by Haag, Sohnius and Liupusanski demonstrates that supersymmetry is the only possible nontrivial extension of the known spacetime symmetries.


If we want to treat SUSY like any other symmetry, we should put boson and fermion degrees of freedom as components of one big field and write our Lagrangian in terms of that. Such a field is known as a superfield. It is written as an expansion in the Grassman paramaters we discussed above:


This expansion bust terminate at fourth power because we have assumed only one set of SUSY generators( known as N=1 SUSY )in 4 spacetime dimensions, and so these theta's have two complex components that, when squared, give zero.  We saw that supersymmetry, even though global, is intimately connected with spacetime translation, so we define the Grassman parameter along with the spacetime coordinates to define what is called superspace.


A general supersymmetry transformation looks like
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and this performs a transformation of superspace coordinates (using the commutation relations and the Baker-Hausdorff lemma)
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This gives us our supersymmetry generators in the form of differential operators
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If we were to write an action that depends on superfields, it would need to be an integral over all of the superspace degrees of freedom for each term, and in general such an action would be of the form
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This integral yields an interesting result; because of the properties of two-component Grassman numbers
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If we perform a supersymmetry transformation on this action, using the differential operators given above, the only terms that are nonzero are those that have two and all four possible factors of theta and thata-bar.  From the form of the operators, the change in the action is a total derivative plus zero.  This means that any action written in terms of superfields in superspace is automatically supersymmetric.   This makes the superfield formalism very elegant and useful.

A specific kind of superfield gives us back the model we started with, and since that is a basic way of going back and forth between fermions and their scalar partners, we would like to recover it. Such a field is known as a chiral superfield, for reasons beyond our current scope, and are defined, along with the SUSY-covariant derivative, as follows
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We expand this superfield explicity in components, and recover what would be the kinetic terms we started with (after integrating the action by parts).
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We have not talked about the interactions between fields. We write the interactions of superfields in third-order (at most, if our theory is renormalizable) polynomials of the various superfields.
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(the first-order term does not usually show up in theories and is commonly neglected.)  By looking at the expansions of each of these fields, along with the proper power of fermion parameters, we see that we recover all the usual Yukawa couplings that would already be there without supersymmetry. Also, the field equations for the F's would be nontrivial,
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and we would recover all the scalar^4 interactions needed for renormalizability.


How we impose already-existent gauge symmetry into the theory is by making it invariant under the usual transformations.  We work in abelian U(1) for simplicity, and our results generalize.  We want our action to be invariant under the transformation
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So, we include an exponent to the theory to cancel that in
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So our kinetic energy becomes
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We examine this V, known as the vector superfield. We know it must be real, and its most general expansion that performs this calcellation is of the form
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Most of the fields here are nonphysical( as they must be for N = 1), and we can choose an additional Lambda above to cancel them, or gauge them away. This is what is known as the Wess-Zumino gauge. Explicity, we can choose a gauge so that
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Note that we have recovered the original symmetry over the general gauge transformation imposed independent of SUSY. In the Wess-Zumino gauge, our vector superfield is
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The v is the original 4-vector potential in the non-SUSY model, and the lambdas are its fermionic partners, known as gauginos. If we write out the components of the kinetic part of the Lagrangian with vector superfields in components, we recover the original kinetic energies in terms of gauge covariant derivatives of every particle plus new interactions unseen before.
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The supersymmetric extension of the field strength tensor is
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Aside from its aesthetic value, there are practical problems that SUSY answers, and that is why it is at the top of the list of things to find experimentally.

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The Higgs and all scalar particles have the problem that if we try to compute the radiative corrections to their amplitudes in loop diagrams, we find corrections that are quadratically divergent.
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Fermion and vector boson masses are protected by symmetry already in the theory, and their radiative corrections are only logarithmically divergent, which we have methods of absorbing cleanly into the original mass to create a well-behaved mass correction. No such symmetry exists when this is done for scalars. There is no way to get rid of that infinity, and the Higgs mass must depend on an energy scale higher than we can do experiments in or even talk about physical theories. This is what is called the hierarchy problem. The lion's share of this problem comes from the loops with virtual top quarks, as that is the most massive known fermion. In the MSSM, this problem is solved by the top squark (or stop), which has quartic couplings to the Higgs, and the same diagram with a stop loop cancels the original that gave us our troublesome dependence on higher energy scales.


The cancellation is of the form
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Also important is that SUSY solves a problem in grand unification. We would like, at some energy scale, to unify the three SM forces.


See Zee's text for an excellent introduction to grand unification. In order for this to happen, the gauge fields must couple to a particle each with the same strength in order to act as one element from a single group. If we plot the dependence of the coupling constants as a function of the scale we use, the three charges in the Standard Model do not all unify anywhere. However, in the Minimal Supersymmetric Standard Model, they actually do unify. This is one of the most exciting aspects of supersymmetry's predictions.


Although supersymmetry remains completely experimentally unverified, it is of the utmost relevance to current phenomenological work and future experimental searches. This was conceived originally for string theory in 1+1 dimensions, and now it is among the most important new physics to be sought out at the LHC. This is much like gauge theories and the Higgs mechanism, which were researched for their mathematical beauty, and only later formed the basis of the Standard Model of particle physics. It seems impossible that nature would not make use of a perfectly good symmetry that comes from such basic mathematical consideration and an extension of known symmetries. Physics grows and thrives because of experiment, though. That must have the last word. Supersymmetry is very beautiful and convenient, but we are on the verge of actually putting these ideas to the test, and we are due for nature to once again surprise us all.
The focus of this page is to present the basics of what supersymmetry is and how to describe supersymmetric theories. I refer anyone interested in learning more to the revised slideshow at the top of the page and the references therein. The revised slideshow was going to be the webpage, but the page produced would not come together correctly, and so I wrote this instead. I'm also linking to my paper, which is much more detailed and covers many more topics.