Quantum Mechanics for Mathematicians

derivation property analogous to the derivation property of the usual Poisson
bracket is, for monomialsF 1 ,F 2 ,F 3 ,

{F 1 F 2 ,F 3 }+=F 1 {F 2 ,F 3 }++ (−1)|F^2 ||F^3 |{F 1 ,F 3 }+F 2

where|F 2 |and|F 3 |are the degrees ofF 2 andF 3. It will also have the symmetry
property
{F 1 ,F 2 }+=−(−1)|F^1 ||F^2 |{F 2 ,F 1 }+

and these properties can be used to compute the fermionic Poisson bracket for
arbitrary functions in terms of the relations for generators.
Theξjcan be thought of as the “anticommuting coordinate functions” with
respect to a basisejofV =Rn. We have seen that the symmetric bilinear
forms onRnare classified by a choice of positive signs for some basis vectors,
negative signs for the others. So, on generatorsξjone can choose

{ξj,ξk}+=±δjk

with a plus sign forj=k= 1,···,rand a minus sign forj=k=r+ 1,···,n,
corresponding to the possible inequivalent choices of non-degenerate symmetric
bilinear forms.
Taking the case of a positive-definite inner product for simplicity, one can
calculate explicitly the fermionic Poisson brackets for linear and quadratic com-
binations of the generators. One finds

{ξjξk,ξl}+=ξj{ξk,ξl}+−{ξj,ξl}+ξk=δklξj−δjlξk (30.1)

and

{ξjξk,ξlξm}+={ξjξk,ξl}+ξm+ξl{ξjξk,ξm}+

=δklξjξm−δjlξkξm+δkmξlξj−δjmξlξk (30.2)

The second of these equations shows that the quadratic combinations of the
generatorsξjsatisfy the relations of the Lie algebra of the group of rotations in
ndimensions (so(n) =spin(n)). The first shows that theξkξlacts on theξjas
infinitesimal rotations in theklplane.
In the case of the conventional Poisson bracket, the antisymmetry of the
bracket and the fact that it satisfies the Jacobi identity imply that it is a Lie
bracket determining a Lie algebra (the infinite dimensional Lie algebra of func-
tions on a phase spaceR^2 d). The fermionic Poisson bracket provides an example
of something called a Lie superalgebra. These can be defined for vector spaces
with some usual and some fermionic coordinates:

Definition(Lie superalgebra).A Lie superalgebra structure on a real or com-
plex vector spaceV is given by a Lie superbracket[·,·]±. This is a bilinear
map onVwhich on generatorsX,Y,Z(which may be usual or fermionic ones)
satisfies
[X,Y]±=−(−1)|X||Y|[Y,X]±