24 From Classical Mechanics to Quantum Field Theory. A Tutorial
with∂z∂k≡^12
(∂
∂qk−i
∂
∂pk
)
,∂∂ ̄zk≡^12
(∂
∂qk+i
∂
∂pk
)
.Also:
J=−i
(
dzk⊗
∂
∂zk
−dz ̄k⊗
∂
∂z ̄k
)
. (1.98)
1.2.2.4 Geometric structures on the space of functions and operators
The geometric structures just examined allow us to introduce two (non-associative)
real brackets on smooth, real-valued functions onHR:
- the (symmetric) Jordan bracket{f,h}g≡G(df , d h);
- the (antisymmetric) Poisson bracket{f,h}ω≡Λ(df , d h).
By extending both these brackets to complex functions via complex linearity, we
eventually obtain a complex bracket{., .}Hdefined as:
{f,h}H=〈df , d h〉H∗R≡{f,h}g+i{f,h}ω. (1.99)
Explicitly, in complex coordinates, we can write:
{f,h}g=2
(
∂f
∂zk
∂h
∂z ̄k+
∂h
∂zk
∂f
∂ ̄zk
)
,{f,h}ω=
2
i
(
∂f
∂zk
∂h
∂z ̄k−
∂h
∂zk
∂f
∂z ̄k
)
.
(1.100)
In particular, if we associate to any operatorA∈gl(H) the quadratic function:
fA(x)=
1
2
〈x, Ax〉=
1
2
z†Az (1.101)
(
wherezis the column vector
(
z^1 , ..., zn
))
, it follows immediately from Eq. (1.100)
that, for anyA, B∈gl(H), we have:
{fA,fB}g=fAB+BA, (1.102)
{fA,fB}ω=fAB−iBA. (1.103)
This means that the Jordan bracket of any two quadratic functionsfAandfBis
related to the (commutative) Jordan bracket ofAandB,[A, B]+, defined as:
[A, B]+≡AB+BA, (1.104)
while their Poisson bracket is related to the commutator product (the Lie bracket)
[A, B]−defined as:
[A, B]−≡^1
i