Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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the third postulate, which connects the mathemati-

cal content of the theory with the measurement

process.

The third axiom, while crucial for the connection

of the mathematical formalism with the experimen-

tal data, contains the seed of the conceptual

difficulties which plague QM and have not been

cured so far.

Indeed, the third axiom indicates that the process

of measurement is described by laws that are

intrinsically different from the laws that rule the

evolution without measurement. This privileged role

of the changing by effect of a measurement leads to

serious conceptual difficulties since the changing is

independent of whether or not the result is recorded

by some observer; one should therefore have a way

to distinguish between measurements and generic

interactions with the environment.

A related problem that is originated by Axiom III

is that the formulation of this axiom refers implicitly

to the presence of a classical observer that certifies

the outcomes of measurements and is allowed to

make use of classical probability theory. This

observer is not subjected therefore to the laws

of QM.

These two aspects of the conceptual difficulties

have their common origin in the separation of the

measuring device and of the measured systems into

disjoint entities satisfying different laws. The diffi-

culties in the theory of measurement have not yet

received a satisfactory answer, but various attempts

have been made, with various degree of success, and

some of them are described briefly in the section

‘‘Interpretationproblems.’’Itappearsthereforethat

QM in its present formulation is a refined and

successful instrument for the description of the

nonrelativistic phenomena at the Planck scale, but

its internal consistency is still standing on shaky

ground.

Returning to the axioms, it is worth remarking

explicitly that according to Axiom II a state is a

linear functional over the observables, but it is

represented by a sesquilinear function on the

complex Hilbert space H. Since Axiom II states

that any normalized element of H represents a state

(and elements that differ only by a phase represent

the same state) together with ,  also   a þ

b , jaj

þjbj

= 1 represent a state superposition of

 and (superposition principle).

But for an observable A, one has in general





(A) 6¼jaj





(A) þjbj



(A), due to the cross-terms

in the scalar product. The superposition principle is

one of the characteristic features of QM. The

superposition of the two pure states  and has

properties completely different from those of a

statistical mixture of the same two states, defined

by the density matrix  = jaj





þjbj



, where we

have denoted by 



the orthogonal projection onto

the normalized vector . Therefore, the search for

these interference terms is one of the means to verify

the predictions of QM, and their smallness under

given conditions is a sign of quasiclassical behavior

of the system under study.

Strictly connected to superposition are entangle-

ment and the partial trace operation. Suppose that

one has two systems which when considered

separately are described by vectors in two Hilbert

spaces H

, i = 1, 2, and which have observables A

B(H

). When we want to study their mutual

interaction, it is natural to describe both of them in

the Hilbert space H

H

and to consider the

observables A

 I and I  A

When the systems interact, the interaction will not

in general commute with the projection operator 

onto H

. Therefore, even if the initial state is of the

form 

 

, 

, the final state (after the

interaction) is a vector  2H

H

which cannot

be written as  = 

 

with 

. It can be

shown, however, that there always exist two

orthonormal family vectors 

and

such that  =





for suitable c

2 C,

= 1 (this decomposition is not unique in

general).

Recalling that 



 I) = 



), one can write





ðA

 IÞ¼





ðA

Þ¼



ðA

 





The map 

: 



!



is called reduction or also

conditioning) with respect to H

; it is also called

‘‘partial trace’’ with respect to H

. The first notation

reflects the analogy with conditioning in classical

probability theory.

The map 

can be extended by linearity to a map

from normal states (density matrices) on B(H

H

)

to normal states on B(H

) and gives rise to a

positivity-preserving and trace-preserving map.

One can in fact prove (Takesaki 1971) that any

conditioning for normal states of a von Neumann

algebra M is completely positive in the sense that it

remains positive after tensorization of M with B(K ),

where K is an arbitrary Hilbert space.

It can also be proved that a partial converse is

true, that is, that every completely positive trace-

preserving map  on normal states of a von

Neumann algebra AB(H) can be written, for a

suitable choice of a larger Hilbert space K and

partial isometries V

, in the form (Kraus form)

(a) =



Introductory Article: Quantum Mechanics 115

But it must be remarked that, if U(t) is a one-

parameter group of unitary operators on H

H

and  is a density matrix, the one-parameter family

of maps (t)   !

(U(t)U



(t)) does not, in

general, have the semigroup property (t þ s) =

(t)   (s) s, t > 0 and therefore there is in general

no generator (of a reduced dynamics) associated

with it. Only in special cases and under very strong

hypothesis and approximations is there a reduced

dynamics given by a semigroup (Markov property).

Since entanglement and (nontrivial) conditioning are

marks of QM, and on the other side the Markov

property described above is typical of conditioning in

classical mechanics, it is natural to search for condi-

tions and approximations under which the Markov

property is recovered, and more generally under which

the coherence properties characteristic of QM are

suppressed (decoherence). We shall discuss briefly this

probleminthesection‘‘Interpretationproblems,’’

devoted to the attempts to overcome the serious

conceptual difficulties that descend from Axiom III.

It is seen from the remarks and definitions above

that normal states (density matrices) play the role

that in classical mechanics is attributed to measures

over phase space, with the exception that pure states

in QM do not correspond to Dirac measures (later

on we shall discuss the possibility of describing a

quantum-mechanical states with a function (Wigner

function) on phase space).

In this correspondence, evaluation of an observa-

ble (a measurable function over phase space) over a

state (a normalized, positive measure) is related to

finding the (Hilbert space) trace of the product of an

operator in B(H) with a density matrix. Notice that

the trace operation shares some of the properties of

the integral, in particular tr AB = tr BA if A is in

trace class and B 2B(H) (cf. g 2 L

and f 2 L

)

and tr AB > 0ifA is a density matrix and B is a

positive operator. This suggests to define functions

over the density matrices that correspond to quan-

tities which are important in the theory of dynami-

cal systems, in particular the entropy.

This is readily done if the Hilbert space is finite

dimensional, and in the infinite-dimensional case if

one takes as observables all Hermitian bounded

operators. In quantum statistical mechanics one is

led to consider an infinite collection of subsystems,

each one described with a Hilbert space (finite or

infinite dimensional) H

, i = 1, 2, ..., the space of

representation is a subspace K of H

H

,

and the observables are a (weakly closed) subalgebra

A of B(K) (typically constructed as an inductive

limit of elements of the form I  I A

 I ).

In this context one also considers normal states on A

and defines a trace operation, with the properties

described above for a trace. Most of the definitions

(e.g., of entropy) can be given in this enlarged

context, but differences may occur, since in general

A does not contain finite-dimensional projections,

and therefore the trace function is not the trace

commonly defined in a Hilbert space. We shall not

describe further this very interesting and much

developed theory, of major relevance in quantum

statistical mechanics. For a thorough presentation

see Ohya and Petz (1993).

The simplest and most-studied example is the

case when each Hilbert space H

is a complex

two-dimensional space. The resulting system is

constructed in analogy with the Ising model of

classical statistical mechanics, but in contrast to that

system it possesses, for each value of the index i,

infinitely many pure states. The corresponding

algebra of observables is a closed subalgebra of

C

)

Z

and generically does not contain any

finite-dimensional projection.

This model, restricted to the case (C

C

)

, K a

finite integer, has become popular in the study of

quantum information and quantum computation, in

which case a normalized element of H

is called a q-bit

(in analogy with the bits of information in classical

information theory). It is clear that the unit sphere in

C

) contains many more than four points, and

this gives much more freedom for operations on the

system. This is the basis of quantum computation and

quantum information, a very interesting field which

has received much attention in recent years.

Quantization and Dynamics

The evolution in nonrelativistic QM is described by

the Schro¨ dinger equation in the representation in

which for an N-particle system the Hilbert space is

 C

, where C

is a finite-dimensional space

which accounts for the fact that some of the

particles may have a spin content.

Apart from (often) inessential parameters, the

Schro¨ dinger equation for spin-0 particles can be

written typically as

ih

@

¼ H

H 

k¼1

ðihr

þ A

k¼1

ðx

Þþ

i6¼k;1

i;k

ðx

 x

Þ½9

where h is Planck’s constant, A

are vector-valued

functions (vector potentials), and V

and V

i, k

are

scalar-valued function (scalar potentials) on R

116 Introductory Article: Quantum Mechanics

If some particles have of spin 1/2, the correspond-

ing kinetic energy term should read (ih r)

where 

, k = 1, 2, 3, are the Pauli matrices and one

must add a term W(x) which is a matrix field with

values in C

 C

and takes into account the

coupling between the spin degrees of freedom.

Notice that the local operator i r is a ‘‘square

root’’ of the Laplacian.

A relativistic extension of the Schro¨ dinger equa-

tion for a free particle of mass m  0 in dimension

3 was obtained by Dirac in a space of spinor-

valued functions

(x, t), k = 0, 1, 2, 3, which carries

an irreducible representation of the Lorentz group.

In analogy with the electromagnetic field, for which

a linear partial differential equation (PDE) can be

written using a four-dimensional representation of

the Lorentz group, the relativistic Dirac equation is

the linear PDE

k¼0



¼ m ; x

 ct

where the 

generate the algebra of a representation

of the Lorentz group. The operator

(@=@x

)

is a

local square root of the relativistically invariant

d’Alembert operator @

=@x

þ   m  I.

When one tries to introduce (relativistically

invariant) local interactions, one faces the same

problem as in the classical mechanics, namely one

must introduce relativistically covariant fields (e.g.,

the electromagnetic field), that is, systems with an

infinite number of degrees of freedom. If this field is

considered as external, one faces technical problems,

which can be overcome in favorable cases. But if one

tries to obtain a fully quantized theory (by also

quantizing the field) the obstacles become unsur-

mountable, due also to the nonuniqueness of the

representation of the canonical commutation rela-

tions if these are taken as the basis of quantization,

as in the finite-dimensional case.

In a favorable case (e.g., the interaction of a

quantum particle with the quantized electromagnetic

field) one can set up a perturbation scheme in a

parameter  (the physical value of  in natural units

is roughly 1/137). We shall come back later to

perturbation schemes in the context of the Schro¨-

dinger operator; in the present case one has been

able to find procedures (renormalization) by which

the series in  that describe relevant physical

quantities are well defined term by term. But even

in this favorable case, where the sum of the first few

terms of the series is in excellent agreement with the

experimental data, one has reasons to believe that

the series is not convergent, and one does not even

know whether the series is asymptotic.

One is led to wonder whether the structure of

fields (operator-valued elements in the dual of

compactly supported smooth functions on classical

spacetime), taken over in a simple way from the

field structure of classical electromagnetism, is a

valid instrument in the description of phenomena

that take place at a scale incomparably smaller than

the scale (atomic scale) at which we have reasons to

believe that the formalisms of Schro¨ dinger and

Heisenberg provide a suitable model for the descrip-

tion of natural phenomena.

The phenomena which are related to the interac-

tion of a quantum nonrelativistic particle interacting

with the quantized electromagnetic field take place

at the atomic scale. These phenomena have been the

subject of very intense research in theoretical

physics, mostly within perturbation theory, and the

analysis to the first few orders has led to very

spectacular results (although there is at present no

proof that the perturbation series are at least

asymptotic).

In this field rigorous results are scarce, but

recently some progress has been made, establishing,

among other things, the existence of the ground

state (a nontrivial result, because there is no gap

separating the ground-state energy from the con-

tinuous part of the spectrum) and paving the way

for the description of scattering phenomena; the

latter result is again nontrivial because the photon

field may lead to an anomalous infrared (long-

range) behavior, much in the same way that the

long-range Coulomb interaction requires a special

treatment in nonrelativistic scattering theory.

This contribution to the Encyclopedia is meant to

be an introduction to QM and therefore we shall

limit ourselves to the basic structure of nonrelativis-

tic theory, which deals with systems of a finite

number of particles interacting among themselves

and with external (classical) potential fields, leaving

for more specialized contributions a discussion of

more advanced items in QM and of the successes

and failures of a relativistically invariant theory of

interaction between quantum particles and quan-

tized fields.

We shall return therefore to basics.

One may begin a section on dynamics in QM by

discussing some properties of the solutions of the

Schro¨ dinger equation, in particular dispersive effects

and the related scattering theory, the problem of

bound states and resonances, the case of time-

dependent perturbation and the ionization effect,

the binding of atoms and molecules, the Rayleigh

scattering, the Hall effect and other effects in

nanophysics, the various multiscale and adiabatic

limits, and in general all the physical problems that

Introductory Article: Quantum Mechanics 117

have been successfully solved by Schro¨ dinger’s QM

(as well as the very many interesting and unsolved

problems).

We will consider briefly these issues and the

approximation schemes that have been developed in

order to derive explicit estimates for quantities of

physical interest. Since there are very many excellent

reviews of present-day research in QM (e.g., Araki

and Ezawa (2004), Blanchard and Dell’Antonio

(2004), Cycon et al. (1986), Islop and Sigal (1996),

Lieb (1990), Le Bris (2005), Simon (2002), and

Schlag (2004)) we refer the reader to the more

specialized contributions to this Encyclopedia for a

detailed analysis and precise statements about the

results.

We prefer to come back first to the foundations of

the theory; we shall take the point of view of

Heisenberg and start discussing the mapping proper-

ties of the algebra of observables and of the states.

Since transition probabilities play an important role,

we consider only transformations  which are such

that, for any pair of pure states 

and 

, one has

<(

), (

)> = <

, 

>. We call these maps

Wigner automorphisms.

A result of Wigner (see Weyl (1931)) states that if

 is a Wigner automorphism then there exists a

unique operator U



, either unitary or antiunitary,

such that (P) = U





for all projection operators.

If there is a one-parameter group of such auto-

morphisms, the corresponding operators are all

unitary (but they need not form a group).

A generalization of this result is due to Kadison.

Denoting by I

1, þ

the set of density matrices, a

Kadison automorphism  is, by definition, such that

for all 

, 

2 I

1, þ

and all 0 < s <1 one has (s

(1  s)

) = s(

) þ (1  s)(

). For Kadison auto-

morphisms the same result holds as for Wigner’s.

A similar result holds for automorphisms of the

observables. Notice that the product of two Hermi-

tian operators is not Hermitian in general, but

Hermiticity is preserved under Jordan’s product

defined as A  B  (1=2)[AB þ BA].

A Segal automorphism is, by definition, an

automorphism of the Hermitian operators that

preserves the Jordan product structure. A theorem

of Segal states that  is a Segal automorphism if and

only if there exist an orthogonal projector E,a

unitary operator U in EH, and an antiunitary

operator V in (I  E)H such that (A) = WAW



where W  U  V.

We can study now in more detail the description

of the dynamics in terms of automorphism of

Wigner or Kadison type when it refers to states

and of Segal type when it refers to observables. We

require that the evolution be continuous in suitable

topologies. The strongest result refers to Wigner’s

case. One can prove that if a one-parameter group

of Wigner automorphism 

is measurable in the

weak topology (i.e., 

(A) is measurable in t for

every choice of A and ) then it is possible to choose

the U(t) provided by Wigner’s theorem in such a

way that they form a group which is continuous in

the strong topology. Similar results are obtained for

the cases of Kadison and Segal automorphism, but

in both cases one has to assume continuity of 

in a

stronger topology (the strong operator topology in

the Segal case, the norm topology in Kadison’s).

Weak continuity is sufficient if the operator product

is preserved (in this case one speaks of automorph-

isms of the algebra of bounded operators). The

existence of the continuous group U(t) defines a

Hamiltonian evolution. One has indeed:

Theorem 1 (Stone). The map t !U(t), t 2 Risa

weakly continuous representation of R in the set of

unitary operators in a Hilbert space H if and only if

there exists a self -adjoint operator H on (a dense set

of) H such that U(t) = e

itH

and therefore

 2 DðHÞ!i

dUðtÞ

 ¼ HUðtÞ ½10

The operator H is called generator of the dynamics

described by U(t).

Note In Schro¨ dinger’s approach the operator

described in Stone’s theorem is called Hamiltonian,

in analogy with the classical case. In the case of one

particle of mass m in R

subject to a conservative

force with potential energy V(x) it has the following

form, in units in which h = 1:

H ¼

 þ VðxÞ;  ¼

½11

If the potential V depends on time, Stone’s theorem

is not directly applicable but still the spectral

properties of the self-adjoint operators H

and of

the Kernel of the group  !e



are essential to

solve the (time-dependent) Schro¨ dinger equation.

The semigroup t !e

tH

is usually a positivity-

preserving semigroup of contractions and defines a

Markov process; in favorable cases, the same is true

of t !e

tH

(Feynmann–Kac formula).

There is an analogous situation in the general

theory of dynamical systems on a von Neumann

algebra; in analogy with the case of elliptic

operators, one defines as ‘‘dissipation’’ a map  on

a von Neumann algebra M which satisfies (a



a) 



(a) þ (a



)a for all a 2M. The positive dissipa-

tion  is called completely positive if it remains

positive after tensorization with B(K) for any

118 Introductory Article: Quantum Mechanics

Hilbert space K. Notice that according to this

definition every



-derivation is a completely positive

dissipation. For dissipations there is an analog of the

theorem of Stinespring, and often bounded dissipa-

tion can be written as

ðaÞ= i½h; aþ









; a}

for a 2M

(the symbols {. , .} denote the anticommutator).

In general terms, by quantization is meant the

construction of a theory by deforming a commutative

algebra of functions on a classical phase X in such a

way that the dynamics of the quantum system can be

derived from the prescription of deformation, usually

by deforming the Poisson brackets if X is a cotangent

bundle T



M (Halbut 2002, Landsman 2002). We

shall discuss only the Weyl quantization (Weyl 1931)

that has its roots in Heisenberg’s formulation of QM

and refers to the case in which the configuration space

is R

, or, with some variant (Floquet–Zak) the

N-dimensional torus. We shall add a few remarks

on the Wick (anti-Weyl) quantization. More general

formulations are needed when one tries to quantize a

classical system defined on the cotangent bundle of

a generic variety and even more so if it defined on a

generic symplectic manifold.

The Weyl quantization is a mathematically accu-

rate rendering of the essential content of the

procedure adopted by Born and Heisenberg to

construct dynamics by finding operators which

play the role of symplectic coordinates.

Consider a system with one degree of freedom.

The first naive attempt would be to find operators

p that satisfy the relation

piI ½12

and to construct the Hamiltonian in analogy with

the classical case. To play a similar role, the

operators

q and

p must be self-adjoint and satisfy

[12] at least in a weak sense. If both are bounded,

[12] implies e

ib

q þ bI (the exponential is

defined through a convergent series) and therefore

the spectrum of

q is the entire real line, a contra-

diction. Therefore, that inclusion sign in [12] is strict

and we face domain problems, and as a consequence

[12] has many inequivalent solutions (‘‘equivalence’’

here means ‘‘unitary equivalence’’).

Apart from ‘‘pathological’’ ones, defined on

-spaces over multiple coverings of R, there are

inequivalent solutions of [12] which are effectively

used in QM.

The most common solution is on the Hilbert space

(R) (with Lebesgue measure), with

x defined as

the essentially self-adjoint operator that acts on the

smooth functions with compact support as multi-

plication by the coordinate x and

p is defined

similarly in Fourier space. This representation can

be trivially generalized to construct operators

and

in L

Another frequently used representation of [12] is

on L

) (and when generalized to N degrees of

freedom, on T

). In this representation, the operator

p is defined by c

!kc

on functions f () =

k =  M

ik=2

,0 M, N < 1. In this case the

operator

q is defined as multiplication by the angle

coordinate . It is easy to check that this representa-

tion is inequivalent to the previous one and that [12]

is satisfied (as an identity) on the (dense) set of

vectors which are in the domain both of

q and

p. But notice that the domain of essential self-

adjointness of

p is not left invariant by the action of

q (f () is a function on S

only if f (2) = 0).

We shall denote

p in this representation by the

symbol @=@

per

and refer to it as the Bloch

representation. It can be modified by setting the

action of

p as c

!nc

þ ,0<<2, and this

gives rise to the various Bloch–Zak and magnetic

representations.

The Bloch representation can be extended to

periodic functions on R

noticing that L

(R) =

)  l

(N); similarly, the Bloch–Zak and the

magnetic representation can be extended to L

The difference between the representations can be

seen more clearly if one considers the one-parameter

groups of unitary operators generated by the

‘‘canonical operators’’

q and

p. In the Schro¨ dinger

representation on L

(R), these groups satisfy

UðaÞVðbÞ¼e

iab

VðbÞUðaÞ

UðaÞ¼e

; VðbÞ¼e

and therefore, setting z = a þ ib and W(z) 

iab=2

V(b)U(a) one has

WðzÞWðz

Þ¼e

i!ðz;z

Þ=2

Wðz þ z

z 2 C;!ðz; z

Þ¼Imð



z; z

½13

The unitary operators W(z) are therefore projective

representations of the additive group C. This

generalizes immediately to the case of N degrees

of freedom; the representation is now of the

additive group C

and ! is the standard symplectic

form on C

In the Bloch representation, the unitaries

U(a)V(b)U



(a)V



(b) are not multiples of the iden-

tity, and have no particularly simple form. The map

3 z !W(z) with the structure [13] is called Weyl

system; it plays a major role in QM. The following

Introductory Article: Quantum Mechanics 119

theorem has therefore a major importance in the

mathematical theory of QM.

Theorem 2 (von Neumann 1965). There exists

only one, modulo unitary equivalence, irr educible

representation of the Weil system.

The proof of this theorem follows a general

pattern in the theory of group representations. One

introduces an algebra W

(N)

of operators



f ðzÞWðzÞdz; f 2 L

ðC

called Weyl algebra.

It easy to see that jW

j= jf j

and that f !W

is a

linear isomorphism of algebras if one considers W

(N)

with its natural product structure and L

as a

noncommutative algebra with product structure

f  g 

f ðz  z

Þgðz

Þexp

!ðz; z

Þ½14

So far the algebra W

(N)

is a concrete algebra of

bounded operators on L

). But it can also be

considered an abstract C



-algebra which we still

denote by W

(N)

It is easy to see that, according to [14],iff

chosen to be a suitable Gaussian, then W

is a

projection operator which commutes with all the

’s. Moreover, W

= 

f , g

f g

for a suitable

phase factor . Considering the Gelfand–Neumark–

Segal construction for the C



-algebra W

(N)

, one

finds that these properties lead to a decomposition

of any representation in cyclic irreducible equivalent

ones, completing the proof of the theorem.

The Weyl system has a representation (equivalent

to the Schro¨ dinger one) in the space L

, g),

where g is Gauss’s measure. This allows an exten-

sion in which C

is replaced by an infinite-

dimensional Banach space equipped with a Gauss

measure (weak distribution (Segal 1965, Gross

1972, Wiener 1938)). Uniqueness fails in this more

general setting (uniqueness is strictly connected with

the compactness of the unit ball in C

). Notice that

in the Schro¨ dinger representation (and, therefore, in

any other representation) the Hamiltonian for the

harmonic oscillator defines a positive self-adjoint

operator

N ¼

; N

¼

þ x

 1

The spectrum of each of the commuting operators

consists of the positive integers (including 0) and

is therefore called number operator for the kth

degree of freedom. The operator N

can be written

as N

= a



, where a

= (1=

ﬃﬃﬃ

)(x

þ @=@x

)anda



is the formal adjoint of a

in L

(R). One has

þ 1)

1=2

j< 1. In the domain of N these

operators satisfy the following relations (canonical

commutation relations)

½a

; a



¼

k;h

; ½a

; a

¼0

½N

; a

¼a



h;k

; ½N

; a



¼a





h;k

½15

In view of the last two relations, the operator a

called the annihilation operator (relative to the kth

degree of freedom) and its formal adjoint is called

the creation operator. The operators a

have as

spectrum the entire complex plane, the operators a



have empty spectrum; the eigenvectors of N

are the

Hermite polynomials in the variable x

. The

eigenvectors of a

(i.e., the solutions in L

(R)of

the equation a





= 



,  2 C) are called coherent

states; they have a major role in the Bargmann–

Fock–Segal quantization and in general in the

semiclassical limit.

The operators {N

} generate a maximal abelian

system and therefore the space L

) has a natural

representation as the symmetrized subspace of



)

(Fock representation). In this representa-

tion, a natural basis is given by the common

eigenvectors 

}

, k = 1, ..., N, of the operators N

A generic vector can be written as



;

and therefore can be represented by the sequence c

}

Notice that the creation operators do not create

particles in R

but rather act as a shift in the basis

of the Hermite polynomials.

It is traditional to denote by (L

)) the Fock

representation (also called second quantization

because for each degree of freedom the wave

function is written in the quantized basis of the

harmonic oscillator) and to denote by (A) the lift

of a matrix A 2 B(C

). These notations are espe-

cially used if C

is substituted with a Banach space

X. This terminology was introduced by Segal in his

work on quantization of the wave equation; it is

used ever since, mostly in a perturbative context.

In the theory of quantized fields, the space C

substituted with a Banach space, X, of functions.

In this setting, ‘‘second quantization’’ (Segal 1965,

Nelson 1974) considers the state 

}

as represent-

ing a configuration of the system in which there are

precisely n

particles in the kth physical state (this

presupposes having chosen a basis in the space of

distribution on R

). There is no problem in doing

this (Gross 1972) and one can choose for X a

suitable Sobolev space (which one depends on the

Gaussian measure given in X) if one wants that the

120 Introductory Article: Quantum Mechanics

generalization of the commutation relations [15] be

of the form [a



(f ), a(g)] =< f , g> with a suitable

scalar product <, > in X. The problem with

quantization of relativistic fields is that, in order to

ensure locality, one is forced to use a Sobolev space

of negative index (depending on the dimension of

physical space), and this gives rise to difficulties in

the definition of the dynamics for nonlinear vector

fields.

One should notice that in the work of Segal

(1965), and then in Constructive field theory

(Nelson 1974), the Fock representation is placed in

a Schro¨ dinger context exhibiting the relevant opera-

tors as acting on a space L

(X, g), where X is a

subspace of the space of Schwartz distributions on

the physical space of the particles one wants to

describe and g is a suitably defined Gauss measure

on X.

The Fock representation is related to the Bargmann–

Fock–Segal representation (Bargmann 1967), a repre-

sentation in a space of holomorhic functions on C

square integrable with respect to a Gaussian measure.

For its development, this representation relies on the

properties of Toeplitz operators and on Tauberian

estimates. It is much used in the study of the

semiclassical limit and in the formulation of QM in

systems for which the classical version has, for phase

space, a manifold which is not a cotangent bundle

(e.g., the 2-sphere).

Remark The Fock representation associated with

the Weyl system in the infinite-dimensional context

can describe only particles obeying Bose–Einstein

statistics; indeed, the states are qualified by their

particle content for each element of the basis chosen

and there is no possibility of identifying each

particle in an N-particle state. This is obvious in

the finite-dimensional case: the Hermite polynomial

of order 2 cannot be seen as ‘‘composed’’ of two

polynomials of order 1.

In the infinite-dimensional context, if one wants

to treat particles which obey Fermi–Dirac statistics,

one must rely on the Pauli exclusion principle (Pauli

1928), which states that two such particles cannot

be in the same configuration; to ensure this, the

wave function must be antisymmetric under permu-

tation of the particle symbols. It is a matter of fact

(and a theorem in relativistic quantum field theory

which follows in that theory from covariance,

locality and positivity of the energy (Streater and

Wightman 1964) that particles with half-integer spin

obey the Fermi–Dirac statistics. Therefore, to quan-

tize such systems, one must introduce (commuta-

tion) relations different from those of Weyl. Since it

must now be that (a



)

= 0, due to antisymmetry, it

is reasonable to introduce the following relations

(canonical anticommutation relations:

; a



g¼

k;h

; fa

; a

g¼0

½N

; a

¼a



h;k

; fA; BgAB  BA

½16

The Hilbert space is now 

, where H

is a

two-dimensional complex Hilbert space. Notice that

carries an irreducible two-dimensional represen-

tation of sU(2)  o(3) (spin representation) so that

this quantization associates spin 1/2 and

antisymmetry.

The operators in [16] are all bounded (in fact

bounded by 1 in norm). The Fock representation is

constructed as in the case of Weyl (see Araki

(1988)), with n

equal 0 or 1 for each index k.

The infinite-dimensional case is defined in the same

way, and leads to inequivalent irreducible represen-

tations (Araki 1988); only in one of them is the

number operator defined and bounded below. Some

of these representations can be given a Schro¨ dinger-

like form, with the introduction of a gauge and an

integration formalism based on a trace (Gross

1972). This system is much used in quantum

statistical mechanics because it deals with bounded

operators and can take advantage of strong results

in the theory of C



-algebras. In the finite-dimensional

case (and occasionally also in the general case) it is

used in quantum information (the space H

is the

space of a quantum bit).

Returning to the Weyl system, we now introduce

the strictly related Wigner function which plays an

important role in the analysis of the semiclassical

limit and in the discussion of some scaling limits, in

particular the hydrodynamical limit and the Bose–

Einstein condensation when N !1.

The Wigner function W



for a pure state  is a

real-valued function on the phase space of the

classical system which represents the state faithfully.

It is defined as

ðx;Þ¼ð2Þ

n

ið;xÞ

x þ





x 



The Wigner function is not positive in general (the

only exceptions are those Gaussian states that satisfy

(x)  (p)  h). But is has the interesting property

that its marginals reproduce correctly the Born rule.

In fact, one has



(x, )dx = j

()j

. If the func-

tion (t, x) x 2 R

is a solution of the free Schro¨dinger

equation ih@=@t = h

 then its Wigner function

satisfies the Liouville (transport) equation @W



=@t þ

 rW = 0.

The Wigner function is strictly linked with the

Weyl quantization. This quantization associates

with every function (p, x) in a given regularity

Introductory Article: Quantum Mechanics 121

class an operator (D, x) (the Weyl symbol of the

function ) defined by

ððD; xÞf ; gÞ

ð; xÞWðf; gÞð; xÞd dx

Wðf ; gÞð; xÞ

ið;pÞ

fxþ

; x 



It can be verified that the action of F preserves the

Schwartz classes S and S

and is unitary in L

Moreover, one has (D, x)



=(D, x).

The relation between Weyl’s quantization and

Wigner functions can be readily seen from the

natural duality between bounded operators and

pure states:

trð

Þ

aðp; qÞðp; qÞdp dq

ðp; qÞ¼

iðp;q

ðq

; qÞdq

We give now a brief discussion of the general

structure of a quantization, and apply it to the

Weyl quantization. By quantization of a Hamilto-

nian system we mean a correspondence, parame-

trized by a small parameter h, between classical

observables (real functions on a phase space F) and

quantum observables (self-adjoint operators on a

Hilbert space H) with the property that the

corresponding structures coincide in the limit h !0

and the difference for h 6¼0 can be estimated in a

suitable topology.

This last requirement is important for the applica-

tions and, from this point of view, Weyl’s quantiza-

tion gives stronger results than the other formalisms

of quantization.

We limit our analysis to the case FT



X, with

X  R

, and we make use of the realization of H as

Let {x

} be Cartesian coordinates in R

and

consider a correspondence A !

A that satisfies the

following requirements:

1. A $

A is linear;

2. x

where

is multiplication by x

;

3. p

$ih@=@x

;

4. if f is a continuous function in R

, one has

f (x) $f (

x) and

f (p) = (Ff )(

x), where F denotes a

Fourier transform;

5. L



,   (, ), ,  2 R

, where L



is the

generator of the translations in phase space in

the direction  and



is the generator of the one-

parameter group t !W(t) associated with  by

the Weyl system.

Note that (1) and (4) imply (2) and (3) through a

limit procedure.

Under the correspondence A $

A, linear symplec-

tic maps correspond to unitary transformations.

This is not in general the case for nonlinear maps.

One can prove that conditions (1)–(5) give

a complete characterization of the map A $

Moreover, the correspondence cannot be extended

to other functions in phase space. Indeed, one has:

Theorem 3 (van Hove). Let G be the class of

functions C

on R

which are generators of global

symplectic flows. For g 2 G let 

(t) be the

corresponding group. There cannot exist for every

g a correspondence g $

g, with

g self-adjoint, such

that

g(x, p) = g(

p).

We described the Weyl quantization as a corre-

spondence between functions in the Schwartz class S

and a class of bounded operators. Weyl’s quantiza-

tion can be extended to a much wider class of

functions. Operators that can be so constructed are

called Fourier integral operators. One uses the

notation

  (D, x).

We have the following useful theorems (Robert

1987):

Theorem 4 Let l

, ..., l

be linear funct ions on R

such that {l

} = 0. Let P be a polynomial and let

(, x)  P[l

(, x), l

(, x)]. Then

(i) (D, x) maps S in L

) and self-adjoint;

(ii) if g is continuous, then (g()(D, x) = g((D, x)).

One proves that (D, x) extends to a continuous

map S

(X) !S

(X) and, moreover,

Theorem 5 (Calderon–Vaillancourt). If 



jjþjj2Nþ1







j < 1 the norm of the opera-

tor (D, x) is bounded by 

Any operator obtained from a suitable class of

functions through Weyl’s quantization is called a

pseudodifferential operator. If (q, p) = P(p), where

P is a polynomial,

(p, q) is a differential operator.

Moreover, if (p, x) 2 L

then (D, x )isa

Hilbert–Schmidt operator and

jðD; xÞj

¼ð2hÞ

n=2

jAðzÞj



1=2

Pseudodifferential operators turn out to be very

important in particular in the quantum theory of

molecules (Le Bris 2003), where adiabatic analysis

and Peierls substitution rules force the use of

pseudodifferential operators.

The next important problem in the theory of

quantization is related to dynamics.

Let  be a quantization procedure and let H(p, q)

be a classical Hamiltonian on phase space. Let A

122 Introductory Article: Quantum Mechanics

the evolution of a classical observable A under the

flow defined by H and assume that (A

) is well

defined or all t.

Is there a self-adjoint operator

H such that

(A

) = e

(A)e

it

? If so, can one estimate

H  (H)j? Conversely, if the generator of the

quantized flow is, by definition,

H (as is usually

assumed), is it possible to give an estimate of the

difference j(A

)  ((A))

j for a dense set of  2

H, where A

 e

it

, or to estimate j

 A

where

is defined by (

) = ((A))

. Is it possible

to write an asymptotic series in h for the differences?

For the Weyl quantization some quantitative

results have been obtained if one makes use of the

semiclassical observables (Robert 1987). We shall

not elaborate further on this point.

For completeness, we briefly mention another

quantization procedure which is often used in

mathematical physics.

Wick Quantization

This quantization assigns positive operators to

positive functions, but does not preserve polynomial

relations. It is strictly related to the Bargmann–

Fock–Segal representation.

Call coherent state centered in the point (y, )of

phase space the normalized solution of (i

p þ

x 

i þ x)

y, 

(x) = 0.

Wick’s quantization of the classical observable A

is by definition the map A !Op

(A), where

ðAÞ ð2hÞ

n

Aðy;Þ



ð ;

y;

Þ

y;

dy d

One can prove, either directly or going through

Weyl’s representation, that

1. if A  0 then Op

h

(A)  0;

2. the Weyl symbol of the operator Op

h

(A)is

ðhÞ

n

Aðy;Þe



h

½ðxyÞ

þðÞ



dy d

3. for every A 2 O(0) one has kOp

h

(A) 

Ak=

O(h).

Wick’s quantization associates with every vector

 2Ha positive Radon measure 



in phase space,

called Husimi measure. It is defined by

A d

(Op

h

(A)  ), A 2 S(z). Wick’s quantization is less

adapted to the treatment of nonrelativistic particles,

in particular Eherenfest’s rule does not apply, and

the semiclassical propagation theorem has a more

complicated formulation. It is very much used for

the analysis in Fock space in the theory of quantized

relativistic fields, where a special role is assigned to

Wick ordering, according to which the polynomials

and

are reordered in terms of creation and

annihilation operators by placing all creation opera-

tors to the left.

We now come back to Schro¨ dinger’s equation and

notice that it can be derived within Heisenberg’s

formalism and Weyl’s quantization scheme from the

Hamiltonian of an N-particle system in Hamiltonian

mechanics (at least if one neglects spin, which has

no classical analog).

Apart from (often) inessential parameters, the

Schro¨ dinger equation for N scalar particles in R

can be written as

ih

@

k¼1

ðihr

þ A

 þ V   H

 2 L

ðR

½17

where A

are vector-valued functions (vector poten-

tials) and V = V

) þ V

i, k

 x

) are scalar-

valued function (scalar potentials) on R

Typical problems in Schro¨ dinger’s quantum

mechanics are:

1. Self-adjointness of H, existence of bound states

(discrete spectrum of the operator), their number

and distribution, and, in general, the properties

of the spectrum.

2. Existence, completeness, and continuity proper-

ties of the wave operators



 s  lim

1

itH

itH

½18

and the ensuing existence and properties of the

S-matrix and of the scattering cross sections. In

[18] H

is a suitable reference operator, usually

 (with periodic boundary conditions if the

potentials are periodic in space), for which

Schro¨ dinger’s equation can be somewhat analy-

tically controlled.

3. Existence and property of a semiclassical limit.

In [17] and [18] we have implicitly assumed that H

is time independent; very interesting problems arise

when H depends on time, in particular if it is

periodic or quasiperiodic in time, giving rise to

ionization phenomena. In the periodic case, one is

helped by Floquet’s theory, but even in this case

many interesting problems are still unsolved.

If the potentials are sufficiently regular, the

spectrum of H consists of an absolutely continuous

part (made up of several bands in the space-periodic

case) and a discrete part, with few accumulation

points.

On the contrary, if V(x, !) is a measurable

function on some probability space , with a

suitable distribution (e.g., Gaussian), the spectrum

may have totally different properties almost surely.

Introductory Article: Quantum Mechanics 123

For example, in the case N = 1 (so that the terms V

i, j

are absent) in one and two spatial dimensions the

spectrum is pure point and dense, with eigenfunctions

which decrease at infinity exponentially fast (although

not uniformly); as a consequence, the evolution group

does not give rise to a dispersive motion. The same is

true in three dimensions if the potential is sufficiently

strong and the kinetic energy content of the initial state

is sufficiently limited. This very interesting behavior is

due roughly to the randomness of the ‘‘barriers’’

generated by the potential and is also present, to a

large extent, for potentials quasiperiodic in space

(Pastur and Figotin 1992).

In these as well as in most problems related

to Schro¨ dinger’s equation, a crucial role is taken

by the resolvent operator (H  I)

1

, where  is

any complex number outside the spectrum of H;

many of the results are obtained when the difference

(H  I)

1

 (H

 I)

1

is a compact operator.

Problems of type (1) and (2) are of great physical

interest, and are of course common with theoretical

physics and quantum chemistry (Le Bris 2003),

although the instruments of investigation are some-

what different in mathematical physics. The semi-

classical limit is often more of theoretical interest,

but its analysis has relevance in quantum chemistry

and its methods are very useful whenever it is

convenient to use multiscale methods, as in the

study of molecular spectra.

We start with a brief description of point (3); it

provides a valid instrument in the description of

quantum-mechanical systems at a scale where it is

convenient to use units in which the physical

constant h has a very small value (h ’ 10

27

CGS units). From Heisenberg’s commutation rela-

tions, [

p]  hI, it follows that the product of the

dispersion (uncertainty) of the position and momen-

tum variables is proportional to h and therefore at

least one of these two quantities must have very

large values (compared to h). One considers usually

the case in which these dispersions have comparable

values, which is therefore very small, of the order of

magnitude h

1=2

(but very large as compared with h).

In order to make connection with the Hamilton–

Jacobi formalism of classical mechanics one can also

consider the case in which the dispersion in

momentum is of the order h (the WKB method).

The semiclassical limit takes advantage mathema-

tically from the fact that the parameter h is very

small in natural units, and performs an asymptotic

analysis, in which the terms of ‘‘lowest order’’ are

exactly described and the difference is estimated.

The problem one faces is that the Schro¨ dinger

equation becomes, in the ‘‘mathematical limit’’

h !0, a very singular PDE (the coefficients of the

differential terms go to zero in this limit).

Dividing each term of the equation by h (because

we do not want to change the scale of time) leads, in

the case of one quantum particle in R

in potential

field V(x) (we treat, for simplicity, only this case), to

the equation

@ðx; tÞ

¼hðx; tÞþh

1

VðxÞðx; tÞ½19

It is convenient therefore to ‘‘rescale’’ the spatial

variables by a factor h

1=2

(i.e., choose different

units) setting x =

ﬃﬃﬃ

h

X and look for solutions of [19]

which remain regular in the limit h !0 as functions

of the rescaled variable X. One searches therefore

for solutions that on the ‘‘physical scale’’ have

support that becomes ‘‘vanishingly small’’ in the

limit. It is therefore not surprising that, in the limit,

these solutions may describe point particles; the

main result of semiclassical analysis is that he

coordinates of these particles obey Hamilton’s laws

of classical mechanics.

This can be roughly seen as follows (accurate

estimates are needed to make this empirical analysis

precise). Using multiscale analysis, one may write the

solution in the form (X, x, t) and seek solutions

which are smooth in X and x. Both terms on the right-

hand side of [19] contain contributions of order 2

and 1in

ﬃﬃﬃ

h

and in order to have regular solutions

one must have cancellations between equally singular

contributions. For this, one must perform an expan-

sion to the second order of the potential (assumed at

least twice differentiable) around a suitable trajectory

q(t), q 2 R

, and choose this trajectory in such a way

that the cancellations take place.

A formal analysis shows that this is achieved only

if the trajectory chosen is precisely a solution of the

classical Lagrange equations. Of course, a more

refined analysis and good estimates are needed to

make this argument precise, and to estimate the

error that is made when one neglects in the resulting

equation terms of order

ﬃﬃﬃ

h

; in favorable cases, for

each chosen T the error in the solution for most

initial conditions of the type described is of order

ﬃﬃﬃ

h

for jtj < T.

This semiclassical result is most easily visualized

using the formalism of Wigner functions (the

technical details, needed to to make into a proof

the formal arguments, take advantage of regularity

estimates in the theory of functions).

In natural units, one defines

h;

ðx;;t Þ¼

2







h

; t



124 Introductory Article: Quantum Mechanics