Feynman Path Integral Formulation

1.8 Supersymmetry 35
and Σ’s has therefore the general structure
{Q,Q} = 0 [P,Q] =0 [Q,Σ] ≃ Q {Q, ¯Q} ≃P . (1.169)
The first relationship implies that Q is Grassmann-valued, the second that Q commutes
with spacetime translations (including therefore the generator of time translations
H), the third one that Q transforms under Lorentz transformations as a twocomponent
Weyl spinor, and finally the last one that two supersymmetry transformations
together give a spacetime translation. Physically, the first and second identities
imply that there are pairs of fermion-boson states which are degenerate, while the
last equality implies that the supersymmetry charge Q can in some sense be considered
as the “square root” of the Hamiltonian operator P 0 ≡ H. The fact that the
supersymmetry generator Q is tied with the translation generator P causes some
problems when trying to implement supersymmetry on a lattice, since the latter is
generally only translationally invariant by an integer multiple of the lattice spacing
(although one can find ways around it, as shown in Curci and Veneziano, 1987).
One of the remarkable properties of supersymmetry is that it predicts that every
bosonic state be paired with a fermionic state of the same energy, and vice versa.
Furthermore the supersymmetric vacuum has zero energy, since zero momentum
implies P i |0〉 >= 0, while supersymmetry gives
and therefore from Eq. (1.168)
Q α |0〉 > = ¯Q β |0〉 > = 0 , (1.170)
〈0|H |0〉 > = 0 . (1.171)
The last result is particularly interesting in the case of gravity, since it would tend
to generally imply, for unbroken supersymmetry, a vanishing vacuum energy, and
therefore a vanishing cosmological constant (until recently it was in fact believed
that observationally the cosmological constant was consistent with zero, and therefore
in good agreement with the predictions from supersymmetry; this has changed
in the last decade since the distant supernovae surveys find that the cosmological
constant is non-zero and positive). One more consequence of supersymmetry is that
in a relativistic quantum field theory mass renormalization effects are expected to be
identical for particles belonging to the same supersymmetric pair. A more general
feature of supersymmentric theories is that they give rise to what are often referred
to as non-renormalization theorems: if a particular type of bare coupling is omitted,
it cannot be generated by radiative corrections. But since no supersymmetric partners
of the standard model particles have been observed so far, supersymmetry must
be rather far from an exact symmetry of the real world, at least at ordinary energies.
In the original formulation of supersymmetry there is only one fermionic generator
which is a Majorana spinor. But it is in fact possible to have more than one
supersymmetric charge, so that a more general form for the anti-commutation relation
for the Q’s is of the type