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Supersymmetry: transformations of superspace

In a November 2011 article, I discussed the Grassmann anticommuting numbers that satisfy
\[ \theta_1 \theta_2 = -\theta_2 \theta_1 \] My goal was to convince you that this new kind of abstract numbers is, in some sense, equally natural if it is employed as the basic number that Nature may add, multiply, integrate over, or differentiate with respect to.

Whenever you want to calculate observable quantities including probabilities, their values are real (or complex) and therefore commuting; so if they are products involving the Grassmann numbers, there has to be an even number of Grassmann numbers in the product. However, it's important to allow the basic observables such as the quantum fields to be either commuting or anticommuting.

In this text, I would like to introduce supersymmetry as a new kind of symmetry, using the concept of a superspace.

In the 1960s, people became kind of obsessed with the notion of symmetries in physics because they were just used to deduce some amazing things about particle physics. It was realized that there is a hidden \(U(1)_{\rm em}\) gauge symmetry behind electromagnetism. Moreover, a unifying theory of electromagnetic and weak interactions – the electroweak theory – was developed. Its basic gauge group was \(SU(2)_W \times U(1)_Y\). The subscripts denote the weak interactions and the hypercharge
\[ \frac{Y}{2} = Q- T_{3,W} \] Gell-Mann has identified an \(SU(3)_f\) symmetry rotating the flavors of hadrons, something that was soon identified as a group of rotations between the \(u,d,s\) quarks, a group extending the isospin \(SU(2)_f\) group. Those groups are broken in Nature but they still approximately hold and they're useful as organizing principles for the spectrum of allowed particles.

In the early 1970s, the electroweak gauge symmetry was supplemented with \(SU(3)_c\), the gauge symmetry rotating three colors ("red", "green", "blue") of the quarks in the newly constructed theory of interactions between quarks, Quantum Chromodynamics (QCD).

Aside from these continuous groups, people already knew that the discrete \({\mathbb Z}_2\) symmetries, \(C,P,T\sim CP\), were broken by the weak interactions, while only the combined operation, the \(CPT\) symmetry, is guaranteed to be a symmetry of all microscopic laws of physics (for the evolution of pure microstates only! Discussing ensembles of states immediately introduces a time-reversal asymmetry via the logical arrow of time) because this transformation may be interpreted as a complex rotation of space, or the \(tz\)-plane, by \(\pi\).

All these symmetries were very powerful as tools to constrain the possible laws of physics. If the laws of physics are required to respect certain symmetries between objects, they inevitably contain fewer independent concepts and coefficients of various terms in the fundamental equations. People were asking: was there some greater, possibly broken, symmetry that was hiding behind all the laws of physics?




In particular, people were trying to extend the \(SU(3)\) and \(SU(2)\) and \(U(1)\) groups of various kinds in such a way that the larger group would also include the Lorentz group \(SO(3,1)\) as a subgroup. Is it possible that e.g. a group such as \(SU(9,1)\) would contain both the Lorentz group as well as the gauge groups of the Yang-Mills theories?

Various people tried all kinds of unusual models of this kind before this program was stopped by a critical 1967 paper by Sidney Coleman and Jeffrey Mandula. (The people who continue the research in various "graviweak" and similar "unifications" in the 21st century don't know what they're talking about.) The latter Gentleman is a powerful grant figure in the DoE today. These two physicists showed that the Lorentz (geometric) symmetry can't be unified with the gauge or flavor-like (internal) symmetries in any nontrivial way. In particular, if the total symmetry of your laws of physics is \(G\), it must inevitably factorize to
\[ G \sim G_{\rm geometric} \times G_{\rm internal}, \] a Cartesian product of a symmetry that acts on spacetime coordinates and a symmetry that doesn't. These two types of symmetries are therefore always segregated; you can't mix them. This conclusion is valid whenever we talk about ordinary symmetries whose parameters (such as angles or distances for translations) are commuting numbers. Later, we will discuss the only loophole – we may allow the parameters of the transformations to be the Grassmann numbers to obtain some new options, supersymmetry.

How did they prove their no-go theorem? Well, the symmetries are linked to conservation laws, via Emmy Noether's theorem: the infinitesimal generator of a Lie group which is a symmetry is inevitably a conserved observable (note that both equivalent statements may be mathematically expressed by the vanishing of the commutator between the conserved quantity and the Hamiltonian). The symmetry with respect to spacetime translations is equivalent to the conservation of the energy-momentum vector. The internal symmetries are equivalent to the conservation of quantities which are spacetime scalars. For example, the \(U(1)_{\rm em}\) symmetry of electrodynamics is equivalent to the conservation of the electric charge.

If there were some "non-segregated" symmetries that were neither the usual geometric ones nor internal ones i.e. that would be of a mixed type, then there would be corresponding conservation laws for quantities that would be spacetime vectors or tensors, different from the total energy-momentum vector. Coleman and Mandula studied the scattering of some particular particles in any hypothetical theory of this type, such as the lightest spin-0 particles or bound states. Because some new vectors or tensors would be conserved, this constrains the initial and final energy-momentum vectors in such a way that they can't change at all.

The new conservation laws would be very constraining for spin-0 particles exactly because the value of the new conserved tensor quantities in the spin-0 one-particle states would inevitably have to be a function of the energy-momentum vector of the particle. Because such a tensor quantity would be conserved, we would get new constraints on the initial and final momentum vectors in spacetime and the result would be that almost no change of the directions and energies of the particles would be allowed. Coleman and Mandula could show that any theory with such "non-segregated" symmetries would be essentially non-interacting and therefore physically uninteresting. It's not too hard to check that they were right. These symmetries are just way too big.

The supersymmetric loophole

Every theorem is only powerful if the assumptions of the theorem are weak enough. If some of the assumptions fail in an interesting theory, the theorem's assertion becomes inconsequential. Are there some interesting assumptions of the Coleman-Mandula theorem that may fail in realistic theories? You bet. The two physicists assumed that the parameters of the continuous symmetry transformations are commuting numbers and not, for example, anticommuting ones.

When you allow anticommuting parameters of transformations, you open a new world of possibilities. Well, you extend your freedom by a finite amount, but an extremely interesting finite amount.

When I discussed the ordinary "bosonic" symmetries above, I noted that they lead to new conservation laws for quantities that are spacetime tensors (scalars, vectors, tensors of higher rank). Could conserved quantities transform in a different way than tensors with 0,1,2,... Lorentz vector indices?

They could. They could also transform as spinors or spintensors. It's the object that you know from the Dirac equation. However, Pauli's spin-statistics theorem also guarantees that physical fields that transform as spinors must be fermions (anticommuting fields) and those that transform as ordinary tensors must be bosons (commuting fields). This spin-statistics link remains valid for all products of the elementary physical fields so it will be valid for the currents, too.

It means that conserved quantities that transform as spacetime spinors have to be anticommuting objects! Let's call such generators \(Q_\alpha\) where the subscript is a 2-component spinor index under \(SO(3,1)\sim SL(2,{\CC})\); the isomorphism holds locally. Now, we want to discuss symmetries that mix the geometric part and the internal part in new ways. A spinor is neither a scalar (all internal symmetries' conserved charges used to be scalars) nor a spacetime vector or 2-form (conserved quantities for the translational or Lorentz symmetry i.e. for the Poincaré symmetry). So these charges are already mixed. But we must still know what their commutators are.

Because these objects are anticommuting, the natural object we should discuss is not the ordinary commutator but rather an anticommutator
\[ \{Q_\alpha,Q_\beta\} \equiv Q_\alpha Q_\beta + Q_\beta Q_\alpha = \dots \] which would be zero if the objects \(Q_\alpha\) were just Grassmann numbers. But because they're operators, the anticommutator doesn't have to be zero anymore. It may be a nonzero expression much like the commutator \( [x,p] \) of two "bosonic" operators. OK, what the anticommutator above may be?

In the minimal four-dimensional supersymmetry, the anticommutator above is actually zero. But if we replace one of the operators by its Hermitian conjugate cousin, we obtain
\[\{Q_\alpha,\bar Q_{\bar \beta} \} \sim \gamma^\mu_{\alpha\bar \beta} P_\mu.\] Note that the indices work as you expect from a nice Lorentz-invariant equation: two spinor indices (one of which has no bar, the other one has a bar) is equivalent to one vector index; the Dirac matrices are tools to convert a pair of spinor indices to a single vector index and vice versa. So the anticommutator above transforms as the right representation of the Lorentz group so that it may be equal to the energy-momentum vector \(P^\mu\) with some properly lowered, raised, and/or contracted indices.

It's great. Spacetime translations may look like the most "elementary" kind of a symmetry you may think of. However, in a supersymmetric theories, they may be constructed out of two objects that are more elementary, the supercharges \(Q_\alpha\) or its Hermitian conjugate.

With these new fermionic conserved quantities that transform as spinors, you may avoid the Coleman-Mandula problem. The number of conserved quantities arising from a spinor is actually low enough so that you may get nice interacting theories. The minimum amount of supercharges in a 4-dimensional theory contains two complex components of a 2-component (complex Weyl) spinor of supercharges which is equivalent to 4 real supercharges (if we include the 2 complex conjugate ones, or the real parts and imaginary parts).

But even if you multiply this amount by 4, you get the so-called \({\mathcal N}=4\) supersymmetry which may still be nicely interacting even if it only uses fields of spin \(j=0,1/2,1\) i.e. scalars, spinors, and gauge bosons. And even \({\mathcal N}=8\) supersymmetry which has 32 real supercharges admits nice and interacting theories. In this case, however, the spin of elementary fields has to be allowed to go up to two and you get supersymmetric extensions of Einstein's general theory of relativity, the so-called supergravity theory (in this case, the maximally supersymmetric supergravity theory).

At any rate, we have arrived to a new interesting class of theories that should have new generators \(Q_\alpha\) and their Hermitian conjugates. The anticommutators of such supercharges and their Hermitian conjugates should be proportional to the energy-momentum vector. These spinor-valued objects are conserved quantities: they commute with the Hamiltonian i.e. the total energy. They also commute with the total momentum, all of its components. They don't commute with the generators of the Lorentz group: this commutator (and it's a commutator because the Lorentz generators are bosonic and for a fermionic-bosonic pair, one has to take a commutator, not an anticommutator) is given by the fact that the supercharges transform as spinors under the Lorentz group. So the commutator is proportional to some components of the supercharges again.

Can we visualize these abstract expressions in some way? Do they correspond to some translations or rotations of some new space? And are these really rotations or translations?

The answer to the first question is Yes, one may express supercharges as geometric operators acting on a new kind of space, a new extension of the normal spacetime we call a superspace. Aside from the \(x^\mu\) coordinates, this superspace also has new coordinates \(\theta^\alpha\) and their complex conjugates which transform as 2-component complex Weyl spinors (or the complex conjugate Weyl spinors).

The answer to the second question is that supersymmetries are actually "exactly in between" a translation and a rotation. Why is it so and what does it mean? Note that by dimensional analysis, \(Q\cdot Q\) has the same dimension as \(P_\mu\), the energy, so \(Q_\alpha\) itself has the units of the square root of energy. The dimension of this object is exactly the geometric average between a dimensionless number and an energy (or momentum). In the same way, if you create an infinitesimal supersymmetry transformation \(1+\epsilon_\alpha Q^\alpha\), then the dimension of the \(\epsilon\) object has to compensate the dimension of the supercharge. In the \(c=\hbar=1\) units, it's clear that the dimension of the \(\epsilon_\alpha\) is a square root of length; it's the geometric average of dimensionless angles and dimensionful distances expressed in meters!

This may sound even more crazy but it is not really crazy. In fact, the coordinates \(\theta^\alpha\) of the superspace have the dimension of the square root of length, too. It means that \(\theta_\alpha \theta_\beta\) has the same units as \(x^\mu\). Both of them are commuting objects, too.

Their "being in the middle" is great because the supersymmetry transformation is "in between" a translation and a rotation not only at the level of units. It's also "in between" when you study the actual expression of the supercharges and their action on the superspace. How does it work?

First, let's recall how it works for the ordinary translations and Lorentz generators. The generators of translations are simply \(P^\mu\), the energy-momentum vector. In the usual representation of wave functions, we have
\[ P_\mu = i\hbar \frac{\partial}{\partial x^\mu} .\] Note that if a raised index is in the denominator, it produces a lowered index in the numerator, and the overall sign is \(+i\) because the spatial components of the momentum must have their sign inverted in the \({+}{-}{-}{-}\) metric convention.

On the other hand, the angular momentum and other Lorentz generators are bilinear and are represented by
\[ J_{\mu\nu} = x_\mu P_\nu - x_\nu P_\mu. \] Dimensionally, they're products of a coordinate like \(x\) and a derivative with respect to \(x\). That's why these angular momenta and other Lorentz generators are dimensionless; the same must obviously hold for the parameters of the finite transformations, the angles or rapidities of the Lorentz transformations.

What about the supercharges? If we want to write \(Q_\alpha\) in terms of the superspace coordinates and perhaps ordinary spacetime coordinates (and derivatives with respect to both of those), will the expressions be linear or bilinear? Well, I have already told you what to expect. The supercharges are something in between translations and rotations so the answer must be something in between linear and bilinear. No, I don't mean a 3/2-th power. Instead, I mean an expression that has both a linear piece and a bilinear piece. When you study the possibilities, you will realize that the supercharges must have the form
\[ Q_\alpha \sim \frac{\partial}{\partial \theta^\alpha} + \theta^{\bar \beta} \gamma_{\alpha \bar\beta}^\mu \frac{\partial}{\partial x^\mu} \] This is a cute expression. It contains two terms. The first term is a derivative with respect to the new anticommuting coordinate of the superspace; this term resembles the ordinary energy-momentum which is proportional to a derivative with respect to spacetime coordinates. An ordinary "linear" expression except that we're shifting the system in the direction of the new, anticommuting coordinate.

The second term is bilinear; ignore the gamma-matrix which is just a collection of simple purely numerical constant coefficients (a better name for these matrices than gamma could have been sigma). This second term is proportional to a derivative with respect to \(x\), the ordinary spacetime coordinates, but this derivative is multiplied by some components of the new anticommuting coordinates \(\theta\). The indices are nicely contracted and dimensionally speaking, the expression above works fine. So we're mixing linear and bilinear terms in this clever way. If you calculate the anticommutator of the supercharges, you may get the momentum because in the anticommutator, you may pick the derivative with respect to \(x\) from one of the \(Q\) factors and "cancel" (by replacing the anticommutator of the following two objects by one) the corresponding \(\theta\) from the same bilinear term with the derivative with respect to \(\theta\) taken from the first, linear term of the other \(Q\).

Of course, one must spend some quality time with this abstract algebra to get familiar with the adventures that the superspace offers us. One may also add "several copies" of the new anticommuting coordinates. This corresponds to "extended supersymmetry" algebras whose existence has already been mentioned. But let's focus on the minimal supersymmetry in \(d=4\) which is directly relevant for experiments. Can we build realistic theories?

You bet. Normal quantum field theories in \(d=4\) are built by choosing an action
\[ S = \int{\rm d}^4 x\,\,{\mathcal L} \] where the integrand is the Lagrangian density. However, we have extended the spacetime into a superspace so the integral also contains
\[ \int {\rm d}^2 \theta \] i.e. the Grassmann (Berezin) integral over the two complex components of the new anticommuting spinor-like coordinates of the superspace. Well, any such integral is bound to be complex so you also need the complex Hermitian integrals which are
\[ \int {\rm d}^2 \bar \theta \] Both of the integrals above assumed that the integral only depended on \(\theta_\alpha\) or only depended on its complex conjugate (in the second case) but not both. In both cases, we obtained a nice Lorentz scalar by the integral because \(\epsilon_{01}\) of the antisymmetric tensor with two spinor indices is Lorentz invariant and the measure may be written as \[\epsilon_{\alpha\beta}{\rm d}\theta^\alpha {\rm d}\theta^\beta \] However, we may also imagine more general terms in the action which are integrated over all the four bosonic spacetime coordinates and all the four new Grassmann coordinates.
\[ \int {\rm d}^2 \theta\,\, {\rm d}^2 \bar\theta .\] Interestingly enough, realistic theories with the minimal supersymmetry in \(d=4\) contain both types of the terms.

We should say something more explicit about them. Much like fields in non-supersymmetric theories are functions of four coordinates \(x_\mu\), fields in the supersymmetric theories may be written as superfields that are functions of \(x_\mu\) and \(\theta^\alpha\) – those are known as chiral superfields. The most important ones are those whose leading \(\theta\)-free components are scalar fields. Such chiral superfields contain important terms such as
\[ \Phi(x,\theta^\alpha) = \phi(x) + \theta^\alpha \psi_\alpha (x) + \dots \] Note that this Taylor expansion (which should also include a term bilinear in \(\theta\)'s, the so-called F-term which is an auxiliary term whose equations of motion don't have any spacetime derivatives) unifies a scalar field \(\phi\) in the normal non-supersymmetric spacetime with a Weyl spinor field in the ordinary spacetime.

Or they may be functions of \(x_\mu\) and \(\bar\theta_{\bar\alpha}\) – those are antichiral fields, typically complex conjugates to the first kinds of fields. And then you can have more general superfields that are functions of \(x_\mu\) as well as \(\theta\) as well as \(\bar \theta\) variables. Recall that the Taylor expansions in the Grassmann variables only contain a finite number of terms, typically a power of two of terms.
\[ V (x,\theta,\bar\theta) = v_\mu(x) \theta \bar \theta + \psi(x) \theta\theta\bar\theta + {\rm h.c.} \] Note that the expansion unifies a gauge field \(v_\mu\) with a Majorana spinor \(\psi_\alpha\), the gaugino, which carries the same information as a Weyl spinor in \(d=4\) but in the case of these vector fields, it's better to talk about Majorana spinors because of the reality discussed in the next paragraph. I've omitted lots of other terms which may contain from 0 to 4 factors of \(\theta\) or \(\bar\theta\): all the omitted terms end up being some kind of auxiliary fields, too.

The latter fields which depend on all the new anticommuting fields may look much more complicated than the chiral superfield but there's another simplification we may do: we may demand those fields to be Hermitian. Vector superfields that contain gauge bosons are the typical examples of such fields. The Hermiticity constraint makes the a priori Dirac spinors in the expansions Majorana in character.

So you have two inequivalent types of superfields – chiral superfields only depend on one-half of the new anticommuting superspace coordinates (the antichiral ones depend on the opposite ones) while vector superfields depend on all the superspace coordinates. And you may construct a Lagrangian by either integrating an expression over one-half of the superspace; the integrand has to depend on the same coordinates only so it must be a function of the chiral superfields only. This function is known as the superpotential – and it's the "supersymmetric template" for the normal potential energy etc.
\[ S = \int{\rm d}^4 x\,\,\int {\rm d}^2 \theta\,\, W[\Phi_i] + {\rm h.c.} + \dots \] The h.c. term stands for the "Hermitian conjugate" because the action has to be real. Or you may include terms that depend on all the superspace coordinates. The integrand may depend on all the fields you consider but the dependence on the vector superfields is usually the more important one in these terms, and if you want to impose a super-generalization of gauge symmetries, this is the symmetry that constrains these terms maximally.

I obviously don't expect you to be able to make calculations in supersymmetric theories after you read this blog entry; but you may have gained some idea about where the wind blows. Because the 125 GeV Higgs has de facto been discovered, the LHC is waiting for a new beyond-the-Standard-Model discovery and supersymmetry remains the most likely thing that may be discovered by the LHC if anything new will be discovered at all.

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