Chapter 32

A Summary: Parallels

Between Bosonic and

Fermionic Quantization

To summarize much of the material we have covered, it may be useful to con-
sider the following table, which explicitly gives the correspondence between the
parallel constructions we have studied in the bosonic and fermionic cases.

Bosonic Fermionic

Dual phase spaceM=R^2 d Dual phase spaceV=Rn

Non-degenerate antisymmetric

bilinear form Ω(·,·) onM

Non-degenerate symmetric

bilinear form (·,·) onV

Poisson bracket{·,·}on

functions onM=R^2 d

Poisson bracket{·,·}+on

anticommuting functions onV=Rn

Lie algebra of polynomials of

degree 0, 1 , 2

Lie superalgebra of anticommuting

polynomials of degree 0, 1 , 2

Coordinatesqj,pj, basis ofM Coordinatesξj, basis ofV

Quadratics inqj,pj, basis forsp(2d,R) Quadratics inξj, basis forso(n)

Sp(2d,R) preserves Ω(·,·) SO(n,R) preserves (·,·)

Weyl algebra Weyl(2d,C) Clifford algebra Cliff(n,C)

Momentum, position operatorsPj,Qj Clifford algebra generatorsγj

Quadratics inPj,Qjprovide

representation ofsp(2d,R)

Quadratics inγjprovide

representation ofso(2d)

Metaplectic representation Spinor representation