From Classical Mechanics to Quantum Field Theory

8 From Classical Mechanics to Quantum Field Theory. A Tutorial

We remark that linearity and complexfield are needed in order to accommodate
all interference phenomena that radiation and matter display in the quantum
realm, while a scalar product is necessary to define the notion of probability. The
vectors are to be normalized to one since their norm gives the total probability.
Also, multiplying by an overall pure phase factor does not change the physical
content. Separability is necessary to have a finite or denumerable complete set of
orthonormal vectors, according to which we can decompose any state as a linear
superposition of such possible alternatives.
In the following, we will use Dirac notation and represent a vectorψ∈Hwith
the “ket”|ψ〉, while its dualψ∗∈H∗is represented through the “bra”〈ψ|,insuch
a way that the scalar product〈ψ,φ〉between any two vectors can be represented
(according to Riesz theorem) as the “bracket”:〈ψ|φ〉.
ArayinHcan be univocally determined by the projection operator:

ρψ≡|ψ〉〈ψ|, (1.5)

wherewehavesupposed(aswewilldointhefollowing, if not specified differently)
that the vector is normalized. We recall thatρψ is a bounded, positive semi-
definite, trace-one operator such thatρ^2 ψ =ρψ. This also means that it is a
rank-one projector.
In many experimental and theoretical problems, it is interesting to consider
the possibility that a system might be prepared not in a unique state, but in a
statistical mixture or mixed state, i.e. a collection of states{|ψ 1 〉,|ψ 2 〉,···,|ψn〉}
with, respectively, probabilitiesp 1 ,p 2 ,···,pn. Such a state is represented by a
so-called density matrix operator, defined as:

ρ≡

∑n

j=1

pj|ψj〉〈ψj|. (1.6)

Again,ρis bounded, positive semi-definite and trace-one, butρ^2 =ρ(forn>1).
Notice that nowρis a ranknoperator.

Example 1.2.1. Explicit realizations.
The simplest example one can consider isthat of a two-level system, able to de-
scribe, e.g., the polarization degrees of freedom of a photon or the spin of an
electron. In this caseH=C^2.
More generally, one can consider ann-level system, whose space of states is
given byCn.
In the case of infinite dimensions,His often realized as the space of square-
integrable functions over an open domainD∈Rn, the latter usually representing