Generally, this is a transformation performed on an object (model), which leaves the object (model) unchanged. For example, the rotation of an unmarked perfect sphere by an arbitrary degree about its center is a symmetry of the sphere; the rotation of a beach ball about the axis that passes through its 'top' and 'bottom' by 180° is a symmetry of a beach ball, whereas a rotation by an arbitrary angle is not, since the stripes will in general reveal the change. Similarly, shifting the days of a week by seven days is unnoticeable (unless we also track the days of the month).
Some physical quantities have the property of direction. For example, velocity has both a magnitude (speed) but also a direction; so does force, angular momentum, etc. Unlike scalar quantities, such as temperature or pressure which (at any given point in space) have a single value, velocities, forces and momenta (linear or angular) are well specified only by providing three values (at every given point in space). Such quantities are called vectors. More precisely, scalars and vectors (more generally, tensors) are defined through their behavior during a change of coordinates. For example, if the coordinate system used to specify the position of an observation point is rotated about that point, the value of a scalar (temperature, pressure) at that point remains unchanged. In sharp contrast, the value(s) of a vector (velocity, force, momentum) will appear different in the rotated coordinate system. For example, a force which in one coordinate system is acting in the x-direction and so may be represented by its triple of values, (F,0,0); upon rotating the coordinate system about the z-axis by 90° counter-clockwise, the 'new' y-axis will take the place of the 'old' x-axis, and the same force will be represented by the triple (0,F,0) in the 'new' coordinate system. The 'new' triple, |F>' = (0,F,0), can always be obtained from the 'old' one, |F> = (F,0,0) through the equation |x>' = M|x>, where M is a 3x3 matrix of 'directional (co)sines'. Vectors are objects which transform in this way. Scalars are those which do not transform. These two are the simplest instances of tensors: scalars are rank-0 tensors, vectors are rank-1 tensors.
As coordinates are changed by rotating through 360°, all tensors (or arbitrarily high rank) transform right back into themselves. In fundamental physics, only quantities representing interactions (forces) transform as tensors: gravitation is mediated by the rank-2 (metric) tensor and all other fundamental forces are mediated by rank-1 (vector) potentials.
By contrast, the most fundamental (as known to date) 'building blocks' of matter (electrons, neutrini and quarks) exhibit a peculiar property: when coordinates are rotated through 360°, the mathematical functions representing these object return not to them selves, but to their own negatives. Clearly, a rotation of 720° does then return them to precisely their former selves. Objects that transform in this peculiar way are called spinors (more precisely for all those appearing in Nature, spin-1/2 objects).
Theoretical elementary particle physics attempts to describe the interaction and dynamics of these matter particles, and in doing so constructs models which involve all or a subset of them, and possibly additional as yet undiscovered particles. Supersymmetry involves transformations which swap spinors with (some) tensors. Moreover, for such a transformation to be a supersymmetry, it clearly must leave a supersymmetric model unchanged, but it is also required that its two-fold iteration produce a simple translation. In this sense, the supersymmetry transformation is the square root of translation.
Supersymmetry transformations are the only ones to swap matter (spinors) with forces (tensors). Only supersymmetry can therefore truly unify both fundamental concepts of modern physics: matter (‘objects’ in events, akin to nouns in sentences) and interaction (‘agents’ in events, akin to verbs in sentences). The apparent distinction between the two in the ‘real world’ is then a measure of the brokenness of supersymmetry. As an analogy, consider the randomly moving water molecules in a cup, at room temperature. The average distance between two molecules is a function of the temperature alone, and remains the same in all directions: one says that their distribution is isotropic (symmetric with respect to all rotations). However, below the freezing point, the molecules assemble themselves into ice, and settle in the ‘nodes’ of a lattice. The isotropy is broken: the average distance between two nearest molecules in the direction along a lattice axis is smaller than that in the direction slightly “off”, just as the distance between two nearest sites on a board for the game of Go is much smaller in the vertical or horizontal direction than it is, say, in the direction of 45° (1-right, 1-up), 26.57° (2-right, 1-up), 18.43° (3-right, 1-up), ...
As with all symmetries, systems that exhibit supersymmetry are technically easier to analyze. ‘Real world’ phenomena however do not exhibit supersymmetry, and the mismatch between the (rest-)masses of supersymmetric particle partners is then a measure of the brokenness of supersymmetry. While several possible models exhibiting supersymmetry breaking are known and well studied, the precise details of how this happens ‘in the real world’ are far less well known than in cases of other symmetries of the ‘real world;’ see "Through Strings to Cosmic Strings, and Why" and the references therein.
Unlike in higher dimensions, the Lorentz symmetry in 1+1-dimensional spacetime is abelian (commutative). Therefore, all representations (spinors and tensors) are decomposable in a Lorentz-invariant way into 1-dimensional objects. For example, the position 2-vector (s,t) may be replaced by s++=(s+t)/2 and s--=(s-t)/2, the characteristic coordinates of the (flat space) Laplacian. Functions that depend only on s++ are called left-movers as their pattern moves to the left along the s-axis as time passes; functions of only s-- are then right-movers. There exists no Lorentz transformation that would turn one into another. Thus, left-moving spinors and right-moving spinors can be introduced independently, and with then also left-moving supersymmetries and right-moving supersymmetries. (p,q)-supersymmetry then simply denotes p left-moving and q right-moving supersymmetries.
© Tristan Hübsch, 2008