A Short Course on Quantum Mechanics and Methods of Quantization 25Notice that, ifAandBare Hermitean, their Jordan product and their Lie bracket
will be Hermitean as well. Hence, the set of Hermitean operators onHR, equipped
with the binary operations (1.104) and (1.105), becomes aLie-Jordan algebra^15
with the associative product:
(A, B)≡^1
2(
[A, B]++ı[A, B]−)
=AB. (1.106)
Going back to quadratic functions, it is not hard to check that:
{fA,fB}H=2fAB, (1.107)leading to:
{{fA,fB}H,fC}H={fA,{fB,fC}H}H=4fABC,∀A, B, C∈gl(H). (1.108)It is also clear thatfAwill be a real function iffAis Hermitean. Thus, the Jordan
and Poisson brackets will define a Lie-Jordan algebra structure on the set of real,
quadratic functions, and, according to Eq. (1.108), the bracket{·,·}Hwill be an
associative bracket.
We are ready now to define, for any functionf, two vector fields, thegradient
∇foffand theHamiltonian vector fieldXfdefined by:
g(∇f,·)=df
ω(Xf,·)=df orG(·,df)=∇f
Λ(·,df)=Xf , (1.109)that allow to rewrite the Jordan and the Poisson brackets as:
{f,h}g=g(∇f,∇h), (1.110)
{f,h}ω=ω(Xf,Xh). (1.111)Explicitly, in coordinates:
∇f=∂f
∂qk∂
∂qk+∂f
∂pk∂
∂pk=2(
∂f
∂zk∂
∂ ̄zk+∂f
∂z ̄k∂
∂zk)
, (1.112)
Xf=∂f
∂pk∂
∂qk−∂f
∂qk∂
∂pk=2i(
∂f
∂zk∂
∂z ̄k−∂f
∂ ̄zk∂
∂zk)
. (1.113)
Also:J(∇f)=XfandJ(Xf)=−∇f.
In particular, if we start from a linear Hermitean operatorA:H→Hto
which we can associate: (i) the quadratic functionfAas in Eq. (1.101) and (ii)
(^15) We remark parenthetically that all this extends without modifications to the infinite-
dimensional case, if we assume:A, Bto be bounded self-adjoint operators on the Hilbert space
H.