From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 193

dress the underpinnings of the Casimir effect calculations, the theory of functional
Gaussian measures and an introduction to the fairly unknown Peierls approach to
the canonical formulation of classical field theories.

3.2 QuantumMechanicsandRelativity.................

3.2.1 Quantummechanics.....................

A quantum theory is defined by a space of states which are projective rays of vec-
tors|ψ〉of a Hilbert spaceH^2. The physical observables are Hermitian operators in
this Hilbert space (see Section 2.2 of this volume). In any quantum system, there
is a special observable, the HamiltonianH(t), which governs the time evolution of
the quantum states by the first order differential equation

i∂t|ψ(t)〉=H(t)|ψ(t)〉.

The symmetries of a quantum system are unitary operatorsU which commute
with the Hamiltonian of the system. In the particular case that the Hamiltonian
H(t) is time independent the unitary group defined by

U(t)=eitH

is a symmetry group, i.e. [U(t),H] = 0, and defines the dynamics of the quantum
system,

|ψ(t)〉=U(t)|ψ(t)〉.
Some interesting cases of quantum systems are those who arise from the canon-
ical quantization of classical mechanical systems. The archetype of those systems
is the harmonic oscillator. Let us analyze this case in some detail because it will
be useful to understand its generalization to field theory.

Classical Harmonic Oscillator

The Lagrangian of a harmonic oscillator is

L=

1

2 mx ̇

(^2) −^1
2 mω
(^2) x (^2).
The Euler-Lagrange equations give rise to the classical Newton’s equation of
motion
̈x=−ω^2 x. (3.4)
(^2) The foundations of quantum mechanics are summarized in the first and second parts of this
volume.