From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 77

simply assumed that the time evolution of states is again described by the equation
above whereHis replaced byH(t):

i

dψt

dt

=H(t)ψt.

Again, this equation permits one to define a two-parametergroupoidof unitary
operatorsU(t 2 ,t 1 ), wheret 2 ,t 1 ∈R, such that

ψt 2 =U(t 2 ,t 1 )ψt 1 ,t 2 ,t 1 ∈R.

The groupoid structure arises from the following identities: U(t, t)=I and
U(t 3 ,t 2 )U(t 2 ,t 1 )=U(t 3 ,t 2 )andU(t 2 ,t 1 )−^1 =U(t 2 ,t 1 )†=U(t 1 ,t 2 ).

Remark 2.1.8. In our elementary case whereHis finite dimensional,Dyson’s
formulaholds with the simple hypothesis that the mapRt→Ht∈L(H)is
continuous (adopting any topology compatible with the vector space structure of
L(H))[ 5 ]

U(t 2 ,t 1 )=

+∑∞

n=0

(−i)n

n!

∫t 2

t 1

···

∫t 2

t 1

T[H(τ 1 )···H(τn)]dτ 1 ···dτn.

Above, we defineT[H(τ 1 )···H(τn)] =H(τπ(1))···H(τπ(n)), where the bijective
functionπ:{ 1 ,...,n}→{ 1 ,...,n}is any permutation withτπ(1)≥···≥τπ(n).

2.1.4 Composite systems

If a quantum systemSis made of two parts,S 1 andS 2 , respectively described in
the Hilbert spacesH 1 andH 2 , it is assumed that the whole system is described
in the spaceH 1 ⊗H 2 equipped with the unique Hermitian scalar product〈·,·〉
such that〈ψ 1 ⊗ψ 2 ,φ 1 ⊗φ 2 〉=〈ψ 1 ,φ 1 〉 1 〈ψ 2 ,φ 2 〉 2 (in the infinite dimensional case
H 1 ⊗H 2 is the Hilbert completion of the afore-mentioned algebraic tensor product).
IfH 1 ⊗H 2 is the space of a composite systemSas before andA 1 represents an
observable for the partS 1 , it is naturally identified with the selfadjoint operator
A 1 ⊗I 2 defined inH 1 ⊗H 2. A similar statement holds when swapping 1 and 2.
Notice thatσ(A 1 ⊗I 2 )=σ(A 1 ) as one easily proves. (The result survives the
extension to the infinite dimensional case.)

Remark 2.1.9.
(a)Composite systems are in particular systems made of many (either identical
or not) particles. If we have a pair of particles respectively described in the Hilbert
spaceH 1 andH 2 , the full system is described inH 1 ⊗H 2. Notice that the dimension
of the final space is theproductof the dimension of the component spaces. In CM,
the system would instead be described in a space of phases which is the Cartesian