what is supersymmetry and supergravity?

[Lawrence Crowell]

Let me see if I can illuminate a bit of this. Just what is supersymmetry? Before it is condemned and reduced to rubbish I think people need to have some idea of just what it is. I try to illustrate this with a very bare bones set of physics.

I will appeal to some standard quantum mechanics. In particular we have the boson operators a and a^† and a standard rule is that a|n> = √n|n-1> and a^†|n> = √(n+1)|n+1>. It is not hard so see that a^†a|n) = n|n) and this defines the Hamiltonian H = ħωa^†a. The Hamiltonian is really ½ħω(a^†a + aa†) and the commutator results in  factor of ½ from the unit commutator, and is what gives the zero point energy or vacuum energy. Boson operators obey a commutation rule [a, a^†] = 1 We also have the fermion operators  b^† and b these obey an anticommutation rule {b^†, b} = 1. Also the Pauli exclusion principle gives b^2 = (b^†)^2 = 0, which is a topological principle of boundary of boundary is zero. 

Now let me consider the operator Q = a^†b, which removes a fermion state and replaces it with a boson. Similarly we then have Q^† = b^†a that removes a boson and replaces it with a fermion. Now compute the anti-commutator

{Q^†, Q} =  Q^†Q +  QQ^† =  b^†aa^†b +  a^†bb^†a 

I can pass the a and b operators passed each other and so

{Q^†, Q} = b^†baa^† +  bb^†a^†a = {b^†, b}a^†a + b^†b[a, a^†],

which equals the Hamiltonian for the boson field times 1 due to the anticommutator and so we have

{Q^†, Q} = H = a^†a 

Well this is a neat result, which says that if we toggle between a fermion and boson state we get a time translation, where the Hamiltonian is the generator of time translations. The two fermion and boson states are doublets of supersymmetric pairs. This with further study and work is generalized to a case where

{Q^†, Q} = iσ^μp_μ,

which tells us that a transformation between bosons and fermions so they anticommute to define a Lorentz boost with the momentum-energy generator p_μ. I have suppressed a lot of spinor indices and stuff. It must be pointed out that in natural units this anti-commutator has units of inverse length ℓ^{-1}

This is basic supersymmetry and there is tons more to think about, but I am going to skip to supergravity. Let's think of the graviton, the putative quantum unit of a gravitational wave, as an entanglement between two spin 1 bosons. This is what Berg and Dixon do, and this has a certain economy. This is because gravitational waves have two helicity states, or two polarizations, and if we think of the graviton as an entanglement between two gauge bosons in a colorless or chargeless configuration. So we can write G = aa and G^† = a^†a^† as graviton operators. These gauge bosons in this SUSY picture have their corresponding fermion with operators b and b^†. Let us now do the same thing as above with Gb^† = aab^† and G^†b = a^†a^†b. Now

{Gb^†,  G^†b} = aaa^†a^†b^†b + a^†a^†aabb^†

= aa^†(aa^† + 1) b^†b + a^†a(a^†a – 1)bb^†

=  (a^†aa^†a + terms linear in aa^†) {b^†, b} 

where the anti-commutator is one and there is this  a^†aa^†a with dimension ℓ^{-2} and terms with dimension ℓ^{-1}. This is now a curvature and a translation. This should not be too surprising because a gravitation Lagrangian is the Ricci curvature R plus quantum corrections of the form kR^{αβμν}R_{αβμν} that in field components is going to appear as G^†G. The standard Ricci term can be seen to be a quartic term of fermions. See Feynman and Weinberg's little tribute book to Dirac for some of this. This means we have particles on the more classical-like solution given by fermions, which notoriously dislike being in the same state, while the quantum corrections are given by bosons! That quartic term of fermions is similar to the Thirring fermion that is S-dual to the Sine-Gordon equation for a classical soliton.

So there is something very deep here, and the lesson is that transformations between quantum statistics is correlated with spacetime symmetry, and entanglements of YM particles as a graviton has very deep structure with transformations between fermions and bosons.I want to get to where problems might lie. I am trying to distill things down to certain essentials without all the mathematical filigree so common in physics literature. 

Before then consider my operators  Gb^† and G^†b. These are for the Rarita-Schwinger field and corresponds to the gravitino. This is a spin 3/2 particle. It obeys an equation similar to the Dirac equation for spin ½ fields. The Ω baryon is a Rarita-Schwinger particle, but it turns out to be composite with spin ½ quarks. So this is not a fundamental. So far no fundamental RS field or particle has been found. Yet Gb^† = aab^† can be thought of as replacing two spin 1 fields, or operators that lower such a field, in an entanglement and replacing it with a spin ½ particle. Similarly G^†b replaces a fermion with two spin 1 fields or a graviton. We might further think of the gravitational action as a quartic term in fermions (Feynman and Weinberg above), say thinking of the spin 1 particle as built from fermions, where we in effect remove one of the fermions. Finkelstein wrote a paper titled Ω where he considered the world as composite from fermion fields. 

This appears to be a topology changing operation. Fermions with b^2 = (b^†)^2 = 0 mean we have a rule “boundary of a boundary is zero.” Apply a fermion operator twice and you get zero, which physically means the fermion can only exist in one state. Yet the Ricci curvature as a quartic term in fermions means we can violate this. Clearly there is condensate physics here, similar to superconductivity. So pulling out a fermion, or equivalently removing a boson (maybe composite of fermions) and replacing it with a fermion breaks what might be called Cooper pairs and changes topology. There is only one arena where this can happen without blasting physics to pieces and that is on a 2-dimensional surface with anyons. The YM fields a and a^† are gauge-like fields for nonabelian anyons. In gravitation there is only one arena where this can occur; it is on the horizon or stretched horizon of a black hole. That this occurs in higher dimensions is a manifestation of holography.

Now to get to problems here. The LHC has found nothing of SUSY partners of known particles. This is a big difficulty. The argument that Kane and others made was that with the elementary ½(Q^†^2 + Q^2) = H_{susy}. and the Hamiltonian as the mass states, then symmetry breaking of SUSY would be a weak field that splits the degeneracy of these masses, similar to Zeeman splitting. The earliest ideas had SUSY partners of particles with 10GeV masses. The Tevatron and LEP failed to find them. So work the phenomenology some more, keeping the Higgs mass “moderate” so the quartic potential of the Higgs field does not send it to the Planck mass. The then proposed multi-100GeV masses of gluinos and neutralinos were not found. There is some hope that longer LHC runs might bear these out, but the problem is that hope can turn into hopium. Will another collider bear fruit for SUSY? It is not impossible, but it means there is a lot of strange “tuning” of the Higgs, and if masses are too high the Higgs field “explodes.”

So why is SUSY failing us? It turns out the total Hamilonian for SUSY is zero. Symmetry breaking raises the energy of the SUSY so H > 0. Yet we have some evidence for inflation, where the vacuum energy was huge, and currently the cosmological constant is small but nonzero. So inflation would have violently broken SUSY, and it is not clear whether SUSY recovered partial symmetry at the end with reheating. Think of water in the liquid state at temperatures far below freezing; without nucleation particles the water can't solidify. Something similar may have occurred here. We may have with particle physics to say adieu SUSY.

What about supergravity? Without going into great depth, I have commented on how the de Sitter vacuum may have a correspondence with the anti-de Sitter vacuum. This may serve to “protect” the SUSY sector involved with gravity. My simple argument above suggests that without SUSY there is no quantum gravity. With nonabelian anyons there may emerge a form of supersymmetry (http://math.ucr.edu/home/baez/qg-winter2005/group.pdf), and this may occur with fields on the stretched horizon of a black hole.

LC