Feynman Path Integral Formulation

34 1 Continuum Formulation
under the Poincaré group, whose generators include P μ for space translations, and
Σ μν = −Σ νμ for Lorentz transformations. Their algebra
[P μ ,P ν ]=0
[Σ μν ,P λ ]=P μ η νλ − P ν η μλ
[Σ μν ,Σ λσ ]=Σ μσ η νλ − Σ νσ η μλ − Σ μλ η νσ + Σ νλ η μσ ,
(1.163)
with infinitesimal group element
and infinitesimal spacetime transformation
has therefore the general structure
U(ω,ε) =1 + 1 2 Σ μνω μν + iP μ ε μ , (1.164)
x μ → x μ + ω μν x ν + ε μ , (1.165)
[P,P] =0 [P,Σ] ≃ P [Σ,Σ] ≃ Σ . (1.166)
The first relationship implies that translations commute with each other, the second
one that translations transform under the Lorentz group as four-vectors, and the third
one that the Lorentz generators transform under the Lorentz group as antisymmetric
tensors.
Supersymmetry generalizes the Poincaré group by adding Grassman-valued
(fermionic) generators Q α , whose most important property is to transform bosons
into fermions
Q|boson〉 = |fermion〉
Q|fermion〉 = |boson〉 . (1.167)
The new fermionic operators are such that their anti-commutator is an operator proportional
to the Hamiltonian, so that it automatically commutes with it; but actually
in a Lorentz-invariant theory the anticommutator should be proportional to the total
energy-momentum P μ . Consequently the new fermion operators need to satisfy a
set of mixed commutation and anti-commutation relations of the type
[P μ ,Q α ]=[P μ , ¯Q α ]=0
{Q α ,Q β } = { ¯Q α , ¯Q β } = 0
{Q α , ¯Q β } = 2γ μ αβ P μ
[Σ μν ,Q α ]= i ( )
σμν
2 αβ Q β , (1.168)
with the Dirac spin matrix σ μν ≡ 1 2i [γ μ,γ ν ]; for a more complete discussion see for
example (Fayet and Ferrara, 1977; Ferrara, 1984). The superalgebra of the P’s, Q’s