8. Rigorous Finite-Dimensional Perturbation
Theory

Perturbation theory is the most important method in modern physics.
Folklore

8.1 Renormalization
In quantum field theory, a crucial role is played by renormalization. Let us now
study this phenomenon in a very simplified manner.
• We want to show how mathematical difficulties arise if nonlinear equations are
linearized in the incorrect place.
• Furthermore, we will discuss how to overcome these difficulties by using the
methods of bifurcation theory.
The main trick is to replace the original problem by an equivalent one by introducing
so-called regularizing terms. We have to distinguish between
• the non-resonance case (N) (or regular case), and
• the resonance case (R) (or singular case).
In celestial mechanics, it is well-known that resonance may cause highly complicated
motions of asteroids.1
In rough terms, the complexity of phenomena in quantum field theory is
caused by resonances.
In Sect. 7.16, the non-resonance case and the resonance case were studied for linear
operator equations. We now want to generalize this to nonlinear problems.

8.1.1 Non-Resonance
Consider the nonlinear operator equation
H0 ϕ + κ(v0 + V (ϕ)) = Eϕ,

ϕ ∈ X.

(8.1)

We make the following assumptions.
(A1) The complex Hilbert space X has the finite dimension N = 1, 2, . . .
1

This is described mathematically by KAM theory (Kolmogorov–Arnold–Moser
theory). As an introduction, we recommend Scheck (2000), Vol. 1, and Thirring
(1997).


498

8. Rigorous Finite-Dimensional Perturbation Theory

(A2) The operator H0 : X → X is linear and self-adjoint. Furthermore,
H0 |Ej0 = Ej0 |Ej0 ,

j = 1, . . . , N.

0
Here, the energy eigenstates |E10 , . . . , |EN
form a complete orthonormal system of X.
(A3) We set V (ϕ) := W (ϕ, ϕ, ϕ) for all ϕ ∈ X, where the given operator W :
X × X × X → X is linear in each argument. For example, we may choose
V (ϕ) := ϕ|ϕ ϕ.
(A4) We are given the complex constant κ called the coupling constant, and we
are given the fixed element v0 of the space X.

We are looking for an element ϕ of X.
Theorem 8.1 Suppose that we are given the complex number E different from the

0
. Then, there exist positive numbers κ0 and r0 such that,
energy values E10 , . . . , EN
for each given coupling constant κ with |κ| ≤ κ0 , equation (8.1) has precisely one
solution ϕ ∈ X with ||ϕ|| ≤ r0 .
Proof. Equation (8.1) is equivalent to
ϕ = −κ(H0 − EI)−1 (v0 + V (ϕ)),

ϕ ∈ X.

The statement follows now from the Banach fixed-point theorem in Sect. 7.13.

✷
In particular, the solution ϕ can be computed by using the following iterative
method
ϕn+1 = −κ(H0 − EI)−1 (v0 + V (ϕn )),

n = 0, 1, . . .

(8.2)

with ϕ0 := 0. This method converges to ϕ as n → ∞ in the Hilbert space X. For
the first approximation, we get
ϕ1 = −κ(H0 − EI)−1 v0 = κ

N
X
|Ej0 Ej0 |v0
.
E − Ej0

j=1

(8.3)

Let us discuss this.
(N) The non-resonance case (regular case). The expression (8.3) makes sense, since
0
.
we assume that the parameter E is different from the eigenvalues E10 , . . . , EN
0
.
We say that the value E is not in resonance with the eigenvalues E10 , . . . , EN
Then, the Green’s operator (H0 − EI)−1 is well-defined. Explicitly,
(H0 − EI)−1 =

N
X
|Ej0 Ej0 |
.
Ej0 − E
j=1

(R) The resonance case (singular case). The situation changes completely if we
choose
E := E10 .
Here, we say that the value E is in resonance with the eigenvalue E10 . Then,
the Green’s operator (H0 − E10 I)−1 does not exist, and the iterative method
(8.2) above fails completely. As a rule, ϕ1 is an infinite quantity. Furthermore,
if we set
ε = 0,

E := E10 + ε,


8.1 Renormalization
then we obtain
(H0 − EI)−1 =

N
X
j=1

499

|Ej0 Ej0 |
.
Ej0 − E10 − ε

Since

1
1
= − lim
= −∞,
ε→+0 ε
E10 − E
some of the expressions arising from perturbation theory become very large if
the perturbation ε is very small.
lim

ε→+0


Summarizing, it turns out that
Naive perturbation theory fails completely in the resonance case.
This situation is typical for the naive use of perturbation theory in quantum field
theory. In what follows, we will show how to obtain a rigorous result. To this
end, we will replace the naive iterative method (8.2) above by the rigorous, more
sophisticated iterative method (8.12) below.

8.1.2 Resonance, Regularizing Term, and Bifurcation
Set E := E10 + ε. Consider the nonlinear operator equation
H0 ϕ + κV (ϕ) = Eϕ,

ϕ ∈ X.

(8.4)

In addition to (A1) through (A4) above, we assume that the energy eigenvalue
E10 is simple, that is, the eigenvectors to E10 have the form |E10 where is an
arbitrary nonzero complex number. We are looking for a solution (ϕ, E) of (8.4)
with ϕ ∈ X and E ∈ C. The proof of the following theorem will be based on the
use of regularizing terms.
Theorem 8.2 There exist positive constants κ0 , s0 , η0 and r0 such that for given
complex parameters κ and s with
|κ| ≤ κ0 ,

0 < |s| ≤ s0 ,

equation (8.4) has precisely one solution ϕ, E which satisfies the normalization condition
E10 |ϕ = s
and the smallness conditions |E − E10 | ≤ η0 and ||ϕ|| ≤ r0 .

Before proving this, let us discuss the physical meaning of this result. We will
show below that the zeroth approximation of the solution looks like
ϕ = s|E10 ,

E = E10 .

The first approximation of the energy is given by
E = E10 + κs2 Greg V (ψ1 )|ψ1
where we set ψ1 := |E10 . Observe the following point which is crucial for understanding the phenomenon of renormalization in physics.
From the mathematical point of view, we obtain a branch of solutions which
depends on the parameter s.
The free parameter s has to be determined by physical experiments.


500

8. Rigorous Finite-Dimensional Perturbation Theory

Let us discuss this. Suppose that we measure the
• energy E and
• the running coupling constant κ.
We then obtain the approximation
κ=

E − E10
.
s2 Greg V (ψ1 )|ψ1

This tells us the value of the parameter s. This phenomenon is typical for renormalization in quantum field theory. The energy E10 is called the bare energy. However,
this bare energy is not a relevant physical quantity. In a physical experiment we

do not measure the bare energy E10 , but the energy E and the running coupling
constant κ. In elementary particle physics, this corresponds to the fact that the rest
energy of an elementary particle (e.g., an electron) results from complex interaction processes. Therefore, the rest energy E differs from the bare energy E10 . In the
present simple example, interactions are modelled by the nonlinear term κV (ϕ).
Proof of Theorem 8.2. (I) The resonance condition. To simplify notation, set
ψj := |Ej0 , j = 1, . . . , N. For given χ ∈ X, consider the linear operator equation
H0 ϕ − E10 ϕ = χ,

ϕ ∈ X.

(8.5)

By Theorem 7.15 on page 376, this problem has a solution iff the so-called resonance
condition
ψ1 |χ = 0
is satisfied. The general solution is then given by
ϕ = sψ1 +

N
X
j=2

ψj |χ
ψj
Ej0 − E10

(8.6)

where s is an arbitrary complex parameter.
(II) The regularized Green’s operator Greg . Set P ϕ := ψ1 |ϕ ψ1 . The operator

P : X → span(ψ1 ) projects the Hilbert space X orthogonally onto the 1-dimensional
eigenvector space to the energy eigenvalue E10 . We now consider the modified equation
H0 ϕ + P ϕ − E10 ϕ = χ,

ϕ ∈ X.

(8.7)

Theorem 7.16 on page 377 tells us that, for each given χ ∈ X, equation (8.7) has
the unique solution
ϕ = (H0 + P − E10 I)−1 χ.
We define Greg := (H0 + P − E10 I)−1 . Explicitly,2
ϕ = Greg χ = ψ1 |χ ψ1 +

N
X
j=2

ψj |χ
ψj .
Ej0 − E10

In particular, Greg ψ1 = ψ1 . The term P ϕ = ψ1 |ϕ ψ1 in (8.7) is called regularizing
term.
(III) The trick of regularizing term. The original equation (8.4) on page 499 can
be written equivalently as
P
2
In fact, (H0 + P − E10 I)ϕ is equal to ψ1 |χ ψ1 + N
j=2 ψj |χ ψj = χ.



8.1 Renormalization
H0 ϕ − E10 ϕ + κV (ϕ) + ψ1 |ϕ ψ1 = sψ1 + εϕ,

ϕ∈X

501

(8.8)

along with the normalization condition
ψ1 |ϕ = s.

(8.9)

By (II), this is equivalent to the operator equation
ϕ = Greg (sψ1 − κV (ϕ) + εϕ)
along with (8.9). Finally, since Greg ψ1 = ψ1 , we obtain the equivalent operator
equation
ϕ = sψ1 − κGreg V (ϕ) + εGreg ϕ

(8.10)

along with (8.9). We have to solve the system (8.9), (8.10). To this end, we will use
both a rescaling and the Banach fixed-point theorem.
(IV) Rescaling. Set ϕ := s(1 + ε)ψ1 + sχ. Equation (8.9) yields
s ψ1 |ψ1 + εψ1 + χ = s.
Since s = 0 and ψ1 |ψ1 = 1, we get ε = − ψ1 |χ . Furthermore, it follows from
(8.10) that

s(1 + ε)ψ1 + sχ = sψ1 − κs3 Greg V ((1 + ε)ψ1 + χ)
+sε(1 + ε)ψ1 + sεGreg χ.
Consequently, the system (8.9), (8.10) corresponds to the following equivalent system
χ = A(χ, ε, κ, s),
ε = − ψ1 |A(χ, ε, κ, s) ,

χ ∈ X, ε ∈ C

(8.11)

along with
A(χ, ε, κ, s) := −κs2 Greg V ((1 + ε)ψ1 + χ) + εGreg χ + ε2 ψ1 .
(V) The Banach fixed-point theorem. The system (8.11) represents an operator
equation on the Banach space X × C with the norm
||(χ, ε)|| := ||χ|| + |ε|.
We are given the complex parameters s and κ with 0 < |s| ≤ s0 and |κ| ≤ κ0
where s0 > 0 and κ0 > 0 are sufficiently small numbers. By the Banach fixed-point
theorem in Sect. 7.13 on page 366, there exists a small closed ball B about the origin
in the Banach space X × C such that the operator equation (8.11) has a unique
solution in the closed ball B.
(V) Iterative method. By the Banach fixed-point theorem, the solution (χ, ε) of
(8.11) can be computed by using the following iterative method
χn+1 = A(χn , εn , κ, s),
εn+1 = − ψ1 |A(χn , εn , κ, s) ,

n = 0, 1, 2, . . .

(8.12)

with χ0 := 0 and ε0 := 0. This method converges in the Banach space X × C. In

particular, we get
χ1 = −κs2 Greg V (ψ1 ),

ε1 = κs2 ψ1 |Greg V (ψ1 ) .

✷
Bifurcation. On the product space X × C, the original nonlinear problem (8.4)
on page 499 has two different solution curves, namely,


502

8. Rigorous Finite-Dimensional Perturbation Theory

• the trivial solution curve ϕ = 0, E = arbitrary complex number,
• and the nontrivial solution curve (ϕ = ϕ(s, κ), E = E(s, κ)) given by Theorem
8.2 on page 499.
The two curves intersect each other at the point ϕ = 0, E = E10 . Therefore, we
speak of bifurcation. The nontrivial solution branch of equation (8.4) represents a
perturbation of the curve
ϕ = sψ1 ,

E = E10 ,

s∈C

which corresponds to the linearized problem H0 ϕ = E10 ϕ. Bifurcation theory is part
of nonlinear functional analysis. A detailed study of the methods of bifurcation
theory along with many applications in mathematical physics and mathematical
biology can be found in Zeidler (1986).


8.1.3 The Renormalization Group
The method of renormalization group plays a crucial role in modern physics.
Roughly speaking, this method studies the behavior of physical effects under
the rescaling of typical parameters.
We are going to study a very simplified model for this. Let (ϕ(s, κ), E(s, κ)) be the
solution of the original equation (8.4) on page 499, that is,
H0 ϕ(s, κ) + κV [ϕ(s, κ)] = E(s, κ)ϕ(s, κ)
along with ψ1 |ϕ(s, κ) = s. Choose the fixed real number λ > 0. Replacing s → λs
and κ → λκ2 , we get
“
“
κ”
κ ”i
κ h “
κ” “
κ”
= E λs, 2 ϕ λs, 2
H0 ϕ λs, 2 + 2 V ϕ λs, 2
λ
λ
λ
λ
λ
´
`
κ
along with ψ1 |ϕ λs, λ2 = λs. Define
ψ(s, κ) :=


“
1
κ”
· ϕ λs, 2 .
λ
λ

Noting that V (λψ) = λ3 V (ψ), we obtain
“
κ”
H0 ψ(s, κ) + κV [ψ(s, κ)] = E λs, 2 ψ(s, κ)
λ
along with ψ1 |ψ(s, κ) = s. By the uniqueness statement from Theorem 8.2 on
page 499, we get
“
κ”
1
· ϕ λs, 2 = ϕ(s, κ)
λ
λ

(8.13)

“
κ”
E λs, 2 = E(s, κ)
λ

(8.14)


along with

for all nonzero complex parameters s and κ in a sufficiently small neighborhood of
the origin.


8.1 Renormalization

503

Summarizing, the homogeneity of the potential, V (λϕ) = λ3 V (ϕ), implies
the symmetries (8.13), (8.14) of the solution branch.
Differentiating equation (8.13) with respect to the parameter λ, and setting λ = 1,
we obtain
ϕ(s, κ) − sϕs (s, κ) + 2κϕκ (s, κ) = 0.

(8.15)

In our model, the differential equation (8.15) can be regarded as a simplified version
of the Callan–Szymanzik equation in quantum field theory.
×
Let R×
+ denote the set of all positive real numbers; that is, x ∈ R+ iff x > 0.
2
2
,
define
the
map
T

:
C
→
C
given
by
For each parameter λ ∈ R×
λ
+
“
κ”
Tλ (s, κ) := λs, 2 .
λ
For all parameters λ, µ ∈ Rì
+,
Tà = T Tà .
Therefore, the family {T }Rì of all operators Tλ forms a group. This group is
+

called the renormalization group of the original operator equation (8.4) on page
499.

8.1.4 The Main Bifurcation Theorem
Let us now study the general case of the nonlinear equation
H0 ϕ + κV (ϕ) = Eϕ,

ϕ∈X

(8.16)


where the eigenvalue E10 of the linearized problem
H0 ϕ = E10 ϕ
is not simple as in Sect. 8.1.2, but it has general multiplicity. To this end, we will
reduce the problem to the nonlinear system (8.17) below. We make the following
assumptions.
(A1) The complex Hilbert space X has the finite dimension N = 1, 2, . . .
(A2) Linear operator: The operator H0 : X → X is linear and self-adjoint. Furthermore,
j = 1, . . . , N.
H0 |Ej0 = Ej0 |Ej0 ,
0
Here, the energy eigenstates |E10 , . . . , |EN
form a complete orthonormal system of X.
(A3) Multiplicity: The eigenvalue E10 has the multiplicity m, that is, the eigenvec0
form a basis of the eigenspace of H0 to the eigenvalue E10 .
tors |E10 , . . . , |Em
Let 1 ≤ m < N. To simplify notation, set ψj := |Ej0 . Define the orthogonal
projection operator P : X → X by setting

P ϕ :=

m
X

ψj |ϕ ψj

for all

ϕ ∈ X.

j=1


(A4) Nonlinearity: We set V (ϕ) := W (ϕ, ϕ, ϕ) for all ϕ ∈ X, where the given
operator W : X × X × X → X is linear in each argument. For example,
V (ϕ) := ϕ|ϕ ϕ.


504

8. Rigorous Finite-Dimensional Perturbation Theory

(A5) Resonance condition: The nonlinear equation3
σ ∈ P X, κ ∈ C

σ = κP V (σ),

(8.17)

has a solution (σ0 , κ0 ) where σ0 = 0 and κ0 = 0. This solution is regular, that
is, the linearized equation
h = κ0 P · V (σ0 )h,

h∈X

(8.18)

has only the trivial solution h = 0.
Theorem 8.3 There exists a number α0 > 0 such that for each given complex
number α with |α| ≤ α0 , the nonlinear problem (8.16) with the coupling constant
κ0 has a solution
ϕ = ασ0 + O(α2 ),


α → 0.

E = E10 + α2 ,

Proof. (I) The regularized Green’s operator. For all χ ∈ X, define
Greg χ :=

m
X

N
X

ψj |χ ψj +

j=1

j=m+1

ψj |χ
ψj .
Ej0 − E10

Suppose that we are given χ ∈ X with P χ = 0. By Theorem 7.16 on page 377, the
equation
H0 − E10 = χ
has precisely one solution

∈ X with P = 0. This solution is given by

N
X

= Greg χ =

j=m+1

ψj |χ
ψj .
Ej0 − E10

(II) Equivalent system. Set E := E10 +ε, and introduce the orthogonal projection
operator Q := I − P. Explicitly,
Qϕ =

N
X

ψj |ϕ ψj

for all

ϕ ∈ X.

j=m+1

Then, the original nonlinear problem (8.16) is equivalent to
Q(H0 ϕ − (E10 + ε)ϕ + κV (ϕ)) = 0,
P (H0 ϕ − (E10 + ε)ϕ + κV (ϕ)) = 0.


(8.19)

The idea of the following proof is
3

Set σ := s1 ψ1 + . . . + sm ψm . Equation (8.17) is then equivalent to the system
gj (s1 , . . . , sm ; κ) = 0

j = 1, . . . , m,

s1 , . . . , sm ∈ C.

Here, gj (s1 , . . . , sm ; κ) := sj − κ · ψj |V (s1 ψ1 + . . . + sm ψm ) . Condition (8.18)
means that
«
„
∂gj (s01 , . . . , s0m ; κ0 )
|j,k=1,...,m = 0.
det
∂sk


8.1 Renormalization

505

(i) to solve the first equation from (8.19) by the Banach fixed-point theorem,
(ii) to insert the solution from (i) into the second equation from (8.19), and
(iii) to solve the resulting equation by using the implicit function theorem near the
solution (σ0 , κ0 ) of equation (8.17).

For each ϕ ∈ X, define χ := P ϕ and

:= Qϕ. Then

ϕ=χ+ ,

χ ∈ P X,

∈ QX.

Therefore, system (8.19) is equivalent to
Q{H0 (χ + ) − (E10 + ε)(χ + ) + κV (χ + )} = 0,
P {H0 (χ + ) − (E10 + ε)(χ + ) + κV (χ + )} = 0.
Observe that ψj |H0 ϕ −

E10 ϕ

= H0 ψj −

E10 ψj |ϕ

(8.20)

= 0 for j = 1, . . . , m. Hence

P (H0 − E10 I) = 0.
Furthermore, P χ = χ, Qχ = 0 and Q = , P = 0. Thus, choosing the coupling
constant κ := κ0 and recalling that Q := I − P , the system (8.20) is equivalent to
H0 − E10 = ε − κ0 QV (χ + ),
εχ = κ0 P V (χ + ).


(8.21)

Finally, using the regularized Green’s operator, this system is equivalent to the
equation
= εGreg − κ0 Greg QV (χ + )

(8.22)

along with
εχ = κ0 P V (χ + ).

(8.23)

(III) Rescaling. We set χ := ασ and ε := α2 . Equation (8.22) passes then over
to
= α2 Greg − κ0 Greg QV (ασ + ).

(8.24)

(IV) The Banach fixed-point theorem. By Theorem 7.12 on page 367, there
exist positive parameters α0 , β0 and r0 such that for given α ∈ C and σ ∈ P X with
|α| ≤ α0 ,

||σ|| ≤ β0 ,

equation (8.24) has precisely one solution ∈ QX with || || ≤ r0 . This solution
will be denoted by
= (α, σ).
By the analytic form of the implicit function theorem,4 the components of (α, σ)

depend holomorphically on the complex parameter α. The iterative method
n+1

with
4

0

= α2 Greg

n

− κ0 Greg QV (ασ +

n ),

:= 0 (or comparison of coefficients) shows that

This can be found in Zeidler (1986), Vol. I, Sect. 8.3.

n = 0, 1, . . .


506

8. Rigorous Finite-Dimensional Perturbation Theory
(α, σ) = −α3 κ0 Greg QV (σ) + O(α4 ),

α → 0.


(V) The bifurcation equation. Inserting (α, σ) into equation (8.23), we get
α3 σ = κ0 P V (ασ + (α, σ)).
Dividing this by α3 , we obtain the so-called bifurcation equation
α → 0.

σ = κ0 P V (σ) + O(α),

(8.25)

For α = 0, this equation has the solution σ = σ0 , by assumption (A5). Choose
h ∈ X. Differentiating the equation
σ0 + th = κ0 P V (σ0 + th)
with respect to the real parameter t at t = 0, we get
h = κ0 P · V (σ0 )h.

(8.26)

This is the linearization of (8.25) at the point σ = σ0 , α = 0. By assumption (A5),
equation (8.26) has only the trivial solution h = 0. By the implicit function theorem,
the bifurcation equation (8.25) has a solution of the form
σ = σ0 + O(α),

α → 0.

✷
Modification. If the resonance condition (A5) above is satisfied for the modified equation
σ = −κP V (σ),
σ ∈ P X, κ ∈ C,
then Theorem 8.3 remains true if we replace E = E10 + α2 by E = E10 − α2 .


8.2 The Rellich Theorem
Let X be a complex Hilbert space of finite dimension N = 1, 2, . . . . Consider the
eigenvalue equation
Aϕ = λϕ,

λ ∈ R, ϕ ∈ X \ {0}

along with the perturbed problem
A(ε)ϕ(ε) = λ(ε)ϕ(ε),

λ(ε) ∈ R, ϕ(ε) ∈ X \ {0}

where ε is a small real perturbation parameter, and A(0) = A. We assume the
following.
(H1) The linear operator A : X → X is self-adjoint.
(H2) There exists an open neighborhood U (0) of the origin of the real line such
that for each ε ∈ U (0), the operator A(ε) : X → X is linear and self-adjoint,
and it depends holomorphically on the parameter ε. Explicitly,
A(ε) = A + εA1 + ε2 A2 + . . .
This means that for each arbitrary, but fixed basis |1 , . . . , |N of the space X,
all of the matrix elements m|A(ε)|n are power series expansions which are
convergent for all real parameters ε ∈ U (0).


8.3 The Trotter Product Formula

507

By the principal axis theorem, each operator A(ε) with ε ∈ U (0) possesses a complete orthonormal system of eigenvectors with real eigenvalues.
Theorem 8.4 There exists a small neighborhood of the origin V (0) of the real line

such that the eigenvalues and eigenvectors of the operator A(ε) depend holomorphically on the real parameter ε ∈ V (0).
Explicitly, this means the following. Let ϕ be an eigenvector of multiplicity m
of the operator A with eigenvalue λ ∈ R. Then, there exist power series expansions
λj (ε) = λ + ελj1 + ε2 λj2 + . . . ,
ϕj (ε) = ϕ + εϕj1 + ε2 ϕj2 + . . . ,

j = 1, 2, . . . , m

which converge for all ε ∈ V (0), and which are eigensolutions of A(ε).5 In addition,
there exists a number δ > 0 such that λ1 (ε), . . . , λm (ε) are the only eigenvalues of
A(ε) which lie in the interval ]λ − δ, λ + δ[.
Theorem 8.4 is the special case of a general theorem due to Rellich which is
valid for a broad class of self-adjoint operators in Hilbert spaces. The proof can be
found in Riesz and Nagy (1978), Sect. 136.

8.3 The Trotter Product Formula
Theorem 8.5 Let A, B : X → X be linear operators on the finite-dimensional
Hilbert space X. Then
eA+B = lim (eA/N eB/N )N .
N→∞

Proof. Set C := e(A+B)/N and D := eA/N eB/N . Then
C−D =

∞
1
1 X
2
N m=2 N m−2


X Ak B l
(A + B)m
−
m!
k!l!
k+l=m

!
,

since the terms for m = 1 cancel each other. Using
||(A + B)m || ≤ (||A|| + ||B||)m ,
we get

„
||C − D|| = O

1
N2

||Ak B l || ≤ ||A||k ||B||l ,
«
N → ∞.

Moreover, max{||C||, ||D||} ≤ e(||A||+||B||)/N . It follows from the identity
C N − DN =

N−1
X


C k (C − D)DN−1−k

k=0

that ||C

N

− D || ≤ N ||C − D|| · e
N

(||A||+||B||)(N−1)/N

||C N − DN || = O

„

1
N

. Hence

«
,

N → ∞.

This implies limN→+∞ C N − DN = 0.
5


The convergence of ϕj (ε) refers to the components.

✷


508

8. Rigorous Finite-Dimensional Perturbation Theory

8.4 The Magic Baker–Campbell–Hausdorff Formula
Let A, B : X → X be linear operators on the complex finite-dimensional Hilbert
space X with AB = BA. Then
eA eB = eA+B .
However, the commutation relation AB = BA is frequently violated in mathematics
and physics. We then have to use the following Baker–Campbell–Hausdorff formula
1

eA eB = eA+B+ 2 [A,B]− +r(A,B)

(8.27)

if the operator norms ||A|| and ||B|| are sufficiently small. Again, we see the Lie
product [A, B]− := AB − BA. The remainder r(A, B) has the form
r(A, B) =

∞
X

pk (A, B)


(8.28)

k=3

where pk (A, B) is a polynomial of order k of the variables A, B with respect to the
Lie product [., .]− . The coefficients of pk (A, B) are rational numbers. For example,
12p3 (A, B) = A · (A · B) + B · (B · A),
24p4 (A, B) = B · (A · (B · A))
where we write A · B instead of [A, B]− . The point is that the exponent
A + B + 12 [A, B]− + r(A, B)
lies in the Lie algebra generated by the operators A and B. Thus, the generalized
addition theorem (8.27) for the exponential function leads us in a natural way to
the concept of Lie algebra.
Theorem 8.6 Let A, B : X → X be linear operators on the finite-dimensional
Hilbert space X. Then there exists a number r > 0 such that (8.27), (8.28) hold true
if ||A|| ≤ r and ||B|| ≤ r.
Formula (8.27) is named after contributions made independently by Campbell,
Baker, and Hausdorff around 1900. In 1950 Dynkin discovered the following explicit
formula:
r(A, B) =

∞
X
(−1)k X
1
k+1
l1 + . . . + l k + 1
k=2
ô


(ad A)lk (ad B)mk
(ad A)l1 (ad B)m1
(A).
ì
ÃÃÃ
l1 !
m1 !
lk !
mk !

Here, we use the operator ad : L(X) → L(X) given on the space L(X) of linear
operators on X. Explicitly, for each C ∈ L(X), the linear operator ad C : X → X
is given by
for all D ∈ L(X).
(ad C)D := [C, D]−
P
The sum
refers to all integers l1 , . . . , lk ≥ 0 and m1 , . . . , mk ≥ 0 with mj +lj > 0
for all j. The proof can be found in Duistermaat and Kolk (2000), p. 30.


8.5 Regularizing Terms

509

8.5 Regularizing Terms
The naive use of perturbation theory in quantum field theory leads to divergent
mathematical expressions. In order to extract finite physical information from this,
physicists use the method of renormalization. In Volume II we will study quantum electrodynamics. In this setting, renormalization can be understood best by
proceeding as follows.

(i) Put the quantum system in a box of finite volume V .
(ii) Consider a finite lattice in momentum space of grid length ∆p and maximal
momentum Pmax .
The maximal momentum corresponds to the choice of a maximal energy, Emax . We
then have to carry out the limits
V → +∞,

Emax → +∞,

∆p → 0.

Unfortunately, it turns out that the naive limits do not always exist. Sometimes
divergent expressions arise.
The idea of the method of regularizing terms is to force convergence of divergent expressions by introducing additional terms. This technique is well-known in
mathematics. In what follows we will study three prototypes, namely,
• the construction of entire functions via regularizing factors (the Weierstrass product theorem),
• the construction of meromorphic functions via regularizing summands (the
Mittag–Leffler theorem), and
• the regularization of divergent integrals by adding terms to the integrand via
Taylor expansion.
In this monograph, we distinguish between
• regularizing terms and
• counterterms.
By convention, regularizing terms are mathematical objects which give divergent
expressions a well-defined rigorous meaning. Counterterms are added to Lagrangian
densities in order to construct regularizing terms. Roughly speaking, this allows
us a physical interpretation of the regularizing terms. In quantum field theory,
renormalization theory is based on counterterms.

8.5.1 The Weierstrass Product Theorem

Recall that by an entire function, we mean a function f : C → C which is holomorphic on the complex plane. The entire function f has no zeros iff there exists an
entire function g : C → C such that
f (z) = eg(z)

for all

z ∈ C.

Suppose that the function f is a polynomial which has the zeros z0 , z1 , . . . zm with
the multiplicities n0 , . . . , nm , respectively, where z0 := 0 and zj = 0 if j = 1, . . . , m.
Then
«nk
m „
Y
z
f (z) = az n0
for all z ∈ C.
(8.29)
1−
zk
k=1


510

8. Rigorous Finite-Dimensional Perturbation Theory

Here, a is a complex number. If z = 0 is not a zero of f , then the factor z n0 drops
out. Now consider the case where the function f has an infinite number of zeros.
The key formula reads as

g(z) n0

f (z) = e

z

∞ „
Y
k=1

z
1−
zk

«nk

epk (z)

for all

z ∈ C.

(8.30)

The point is that the naive generalization of (8.29) fails, but we have to add the
regularizing factors epk (z) which force the convergence of the product.
Theorem 8.7 Let f : C → C be an entire function which has an infinite number
of zeros z0 , z1 , . . . ordered by modulus, |z0 | < |z1 | < . . . with z0 := 0. Let nk be
the multiplicity of the zero zk . Then, there exist polynomials p1 , p2 , .. and an entire
function g such that the product formula (8.30) holds true.

This classical theorem is due to Weierstrass (1815–1897). The proof can be found
in Remmert (1998), Sect. 3.1.

8.5.2 The Mittag–Leffler Theorem
We want to generalize the decomposition into partial fractions from rational functions to meromorphic functions. As prototypes, let us consider the two functions
f (z) :=

2z
A−
A+
=
+
(z − i)(z + i)
z−i
z+i

with A± = limz→±i f (z)(z ± i) = 1, and
π cot πz =

π cos πz
.
sin πz

The function z → sin πz has precisely the zeros zk := k with k = 0, ±1, ±2, . . .
Since limz→zk π(z − zk ) cot πz = 1, we get the representation
π cot πz =

1
+ gk (z),
z − zk


k = 0, ±1, ±2, . . .

for all z different from zk in a sufficiently small neighborhood of the point zk .
The function gk is locally holomorphic at the point zk . Thus, the given function
z → cot πz has a pole of first order at each point zk with the principal part 1/(z−zk ).
Motivated by the decomposition into partial fractions of the function f , we make
the ansatz6
∞
X
1
π cot πz =
.
z
−
zk
k=−∞
However, this ansatz does not work, since the series is not convergent. We have to
pass to the modified sum
π cot πz =

∞
X
k=−∞

6

The sum

P∞

k=−∞

. . . stands for

P∞
k=0

1
+ Ck
z − zk

... +

P−∞
k=−1

...

(8.31)


8.5 Regularizing Terms

511

with the so-called regularizing terms Ck := 1/k for k = ±1, ±2, . . . and C0 := 0.
These regularizing terms force the convergence of the series from (8.31) for all
complex points z different from the critical points zk with k = 0, ±1, ±2, . . . In
1748 Euler incorporated this formula in his Introductio.7 Interestingly enough, the
regularizing terms cancel if we combine the right terms with each other. Explicitly,

π cot πz = z

∞
X
k=−∞

1 X 2z
1
= +
.
2
− zk
z
z 2 − zk2
∞

z2

k=1

A similar cancellation was observed by Brown and Feynman in 1952 when computing radiative corrections to Compton scattering.8 Generally, such cancellations
occur in renormalization theory for low energies if one takes the full set of possible
Feynman diagrams into account.
Theorem 8.8 Let f : C → C be a meromorphic function on the complex plane
which has an infinite number of poles z0 , z1 , . . . ordered by modulus, |z0 | < |z1 | < . . .
Let fk denote the principal part of the function f at the pole zk . Then, there exist
polynomials p0 , p1 , .. and an entire function g such that
f (z) = g(z) +

∞

X

fk (z) − pk (z)

k=0

for all complex numbers z different from z0 , z1 , . . .
The polynomials pk are called regularizing terms. This classical theorem is due to
Mittag–Leffler (1846–1927). The proof can be found in Remmert (1998), Sect. 6.1.

8.5.3 Regularization of Divergent Integrals
Let f : R → R be a continuous function, and let be a real number. Consider the
integral
Z R
f (x)dx.
E(R) :=
Suppose that
E(R) = a ln R + g(R)

(8.32)

for a fixed nonzero real number number a and all sufficiently large real real numbers
R. In addition, suppose that the finite limit limR→+∞ g(R) exists. In the classical
sense,
Z
Z
∞

R


f (x)dx =

lim

R→+∞

f (x)dx = (sgn a) ∞.

The regularized integral is defined by
Z
reg
7
8

∞

f (x)dx :=

lim g(R).

R→+∞

(8.33)

A proof of this formula can be found in Remmert (1991), Sect. 11.2.
L. Brown and R. Feynman, Radiative corrections to Compton scattering, Phys.
Rev. 85(2) (1952), 231–244.


512


8. Rigorous Finite-Dimensional Perturbation Theory

This value is well-defined. In fact, suppose that there exists a second decomposition
Z R
f (x)dx = a1 ln R + g1 (R)
where a = a1 . Then, (a − a1 ) ln R + g(R) − g1 (R) = 0. Letting R → +∞, we get a
contradiction. Therefore, a = a1 and g = g1 .
Regularizing terms. Suppose that the function f behaves asymptotically like
„ «
1
a
,
x → +∞
(8.34)
f (x) = + O
x
x2
where a is a nonzero real number. Let > 0. Then
Z R
Z R“
a”
f (x)dx = a ln R − a ln +
dx.
f (x) −
x
This implies
Z

∞


reg

Z
f (x)dx = −a ln +

∞

“
a”
dx.
f (x) −
x

(8.35)

The second integral is finite. The term − xa is called regularizing term.
Example. It follows from
Z R
R+1
dx
= ln(R + 1) − ln 3 = ln R + ln
− ln 3
x+1
R
2
Z

that
reg


2

∞

dx
= − ln 3.
x+1

Physical interpretation. Regard E(R) above as the energy of a quantum
system on the interval [ , R]. This energy is very large if the size of the system is
very large. Such extremely large energies are not observed in physical experiments.
Physicists assume that we only measure relative energies with respect to the ground
state. In our model above, we measure E(R) − a ln R. In the limit R → +∞, we
get the regularized value reg E(∞).
R∞
f (x)dx
The method of subtracting infinities. Suppose that the integral
is finite. For given nonzero real number b,
Z ∞
(b + f (x))dx = (sgn b) ∞.
Since

„Z
lim

R→+∞

we define


Z

∞

Z

R

(b + f (x))dx −
Z
(b + f (x))dx −

«

R

bdx
Z

∞

bdx :=

Z
=

R

f (x)dx,


∞

f (x)dx.

This procedure helps to cancel infinities in renormalization theory.


8.5 Regularizing Terms

513

8.5.4 The Polchinski Equation
Suppose again that the function f has the asymptotic behavior given in (8.34).
Then
lim Rf (R) = a.
R→+∞

Consequently, the coefficient a of the regularizing term can be uniquely determined
by using the equation
lim R

R→+∞

d
dR

Z

R


“

f (x) −

a”
dx = 0.
x

(8.36)

This is the prototype of the so-called Polchinski equation which plays an important
role in modern renormalization theory based on the renormalization group. We will
study this in a later volume. We also refer to J. Polchinski, Renormalization and
effective Lagrangians, Nucl. Phys. B 231 (1984), 269–295.


9. Fermions and the Calculus for Grassmann
Variables

In 1844, Hermann Grassmann (1809–1877) emphasized the importance
of the wedge product (Grassmann product) for geometry in higher dimensions. But his contemporaries did not understand him. Nowadays the
wedge product is fundamental for modern mathematics (cohomology) and
physics (fermions and supersymmetry).
Folklore
Recall that we distinguish between bosons (elementary particles with integer spin
like photons or mesons) and fermions (elementary particles with half-integer spin
like electrons and quarks). The rigorous finite-dimensional approach from the preceding Chap. 7 refers to bosons. However, it is possible to extend this approach to
fermions by replacing complex numbers by Grassmann variables. In this chapter,
we are going to discuss this.


9.1 The Grassmann Product
Vectors. Let X be a complex linear space. For two elements ϕ and ψ of X, we
define the Grassmann product ϕ ∧ ψ by setting
(ϕ ∧ ψ)(f, g) := f (ϕ)g(ψ) − f (ψ)g(ϕ)

for all

f, g ∈ X d .

Recall that the dual space X d consists of all linear functionals f : X → C. The map
ϕ ∧ ψ : Xd × Xd → C
is bilinear and antisymmetric. Explicitly, for all f, g, h ∈ X d and all complex numbers α, β, we have
• (ϕ ∧ ψ)(f, g) = −(ϕ ∧ ψ)(g, f );
• (ϕ ∧ ψ)(f, αg + βh) = α(ϕ ∧ ψ)(f, g) + β(ϕ ∧ ψ)(f, h).
The two crucial properties of the Grassmann product are
• ϕ ∧ ψ = −ψ ∧ ϕ (anticommutativity), and
• (αϕ + βχ) ∧ ψ = αϕ ∧ ψ + βχ ∧ ψ (distributivity)
for all ϕ, ψ, χ ∈ X and all complex numbers α, β. If we write briefly ϕψ instead of
the wedge product ϕ ∧ ψ, then
• ϕψ = −ψϕ, and
• (αϕ + βχ)ψ = αϕψ + βχψ.


516

9. Fermions and the Calculus for Grassmann Variables

This implies the key relation
ϕ2 = 0


ϕ ∈ X.

for all

Functionals. Dually, for f, g ∈ X d , we define
(f ∧ g)(ϕ, ψ) := f (ϕ)g(ψ) − f (ψ)g(ϕ)

for all

ϕ, ψ ∈ X.

The map f ∧ g : X × X → C is bilinear and antisymmetric.

9.2 Differential Forms
Dual basis. Let b1 , . . . , bn be a basis of the complex linear space X. We define the
linear functional bi : X → C by setting
bi (β 1 b1 + . . . + β n bn ) := β i ,
1

i = 1, . . . , n

for all complex numbers β , . . . , β . We call b , . . . , bn the dual basis to b1 , . . . , bn .
The expressions
n
X

n

αi bi


1

and

i=1

n
1 X
αij bi ∧ bj
2 i,j=1

(9.1)

are called 1-forms and 2-forms on X, respectively. Here, the coefficients α1 , . . . , αn
and α12 , . . . are complex numbers with αij = −αji for all i, j.
Terminology. In modern mathematics, one writes
dxi

instead of

bi .

Using this convention, the differential forms (9.1) are written as
n
X

αi dxi

n
1 X

αij dxi ∧ dxj .
2 i,j=1

and

i=1

(9.2)

9.3 Calculus for One Grassmann Variable
Consider the set of all formal power series expansions
α + βη + γη 2 + . . .

(9.3)

with respect to the variable η and complex coefficients α, β, . . . . Add the relations
η2 = 0
and αη = ηα for all complex numbers α. This way, the expansion (9.3) reduces to
α + βη. This procedure allows us to define functions of the Grassmann variable η.
For example, for each complex number α, we define
eαη := 1 + αη.
This is motivated by the formal power series expansion
eαη = 1 + αη + 12 (αη)2 + . . . = 1 + αη.
For all complex numbers α, β, we define


9.4 Calculus for Several Grassmann Variables

517


d
(α + βη) := β;
• derivative: dη
R
• integral: (α + βη)dη := β.
The reader should note that in the case of Grassmann variables, derivative and
integral coincide.

9.4 Calculus for Several Grassmann Variables
We now consider the set of all formal power series expansions
α0 + α1 η1 + . . . + αn ηn + α12 η1 η2 + . . .
with respect to the variables η1 , . . . , ηn and complex coefficients α1 , α2 , . . . . We add
the relations
ηi ηj = −ηj ηi ,
This implies

ηi2

αηi = ηi α,

i, j = 1, . . . , n,

α ∈ C.

= 0 for all i.

∂l
f (η1 , . . . , ηn ) is performed after moving the variable
• The left partial derivative ∂η
k

ηk to the left. For example,
∂l
(η2 η1 ) = η1
∂η2
and
∂l
∂l
(η1 η2 ) =
(−η2 η1 ) = −η1 .
∂η2
∂η2
In general we have the anticommutativity property

∂l2
∂l2
=−
.
∂η1 ∂η2
∂η2 ∂η1

(9.4)

∂r
f (η1 , . . . , ηn ) is performed after moving
• Similarly, the right partial derivative ∂η
k
the variable ηk to the right. For example,

∂r
∂r

(η2 η1 ) =
(−η1 η2 ) = −η1 .
∂η2
∂η2
As for the left partial derivative, we have the anticommutativity property
∂r2
∂r2
=−
.
∂η1 ∂η2
∂η2 ∂η1
• By definition, the integral
Z
f (η1 , . . . , ηn )dη1 · · · dηn := c
is equal to the coefficient c of ηn ηn−1 · · · η1 in the expansion of f ,
f (η1 , . . . , ηn ) = c(ηn ηn−1 · · · η1 ) + . . . .
For example, η1 η2 = −η2 η1 implies
Z
η1 η2 dη1 dη2 = −1.

(9.5)


518

9. Fermions and the Calculus for Grassmann Variables

9.5 The Determinant Trick
Gaussian integrals play a fundamental role in the functional integral approach to
the Standard Model in particle physics. We want to generalize the formula

Z
eζαη dηdζ = α
for all α ∈ C
with respect to the Grassmann variables η, ζ to 2n variables. To this end, let
η1 , . . . , ηn , ζ1 , . . . , ζn variables which satisfy the following relations
ηi ηj = −ηj ηi ,

ζi ζj = −ζj ζi ,

ηi ζj = −ζj ηi + γδij

for all i, j = 1, . . . , n and fixed complex number γ.
Theorem 9.1 For each complex (n × n)-matrix A = (aij ),
! n
Z
n
X
Y
ζi aij ηj
dηi dζi = det A.
exp
i,j=1

(9.6)

i=1

Proof. (I) Let n = 2. We have to compute the coefficient c of η2 ζ2 η1 ζ1 in the
expansion of the integrand,
!2

2
2
X
1 X
ζi aij ηj +
ζi aij ηj
= c(η2 ζ2 η1 ζ1 ) + . . .
1+
2 i,j=1
i,j=1
The dots denote the remaining terms. It turns out that
c = a11 a22 − a12 a21 = det(A).
Let us show this. Since ζ2 η2 = −η2 ζ2 + γ, we get
ζ1 η1 ζ2 η2 = −ζ1 η1 η2 ζ2 + γζ1 η1 .
Furthermore, it follows from η1 η2 = −η2 η1 and ζ1 η2 = −η2 ζ1 that
ζ1 η1 ζ2 η2 = −η2 ζ1 η1 ζ2 + . . . = −η2 ζ2 ζ1 η1 + . . . = η2 ζ2 η1 ζ1 + . . . .
The dots denote terms that contain less than four factors. This implies
1
1
(ζ1 a11 η1 )(ζ2 a22 η2 ) = a11 a22 (η2 ζ2 η1 ζ1 ) + . . .
2
2
The same expression is obtained for

1
(ζ a η )(ζ1 a11 η1 ).
2 2 22 2

Similarly, we get


1
1
(ζ1 a12 η2 )(ζ2 a21 η1 ) = − a12 a21 (η2 ζ2 η1 ζ1 ) + . . . ,
2
2
and the same expression is obtained for 12 (ζ2 a21 η1 )(ζ1 a12 η2 ).
(II) For n = 3, 4, . . ., the proof proceeds analogously.
✷
Remark. Observe that Theorem 9.1 is related to the classical Gaussian integral
!
Z
n
dx1
1 X i
dxn
1
j
√ ··· √
(9.7)
exp −
x aij x
= √
2
n
2π
2π
det A
R
i,j=1
for all real symmetric (n × n)-matrices A = (aij ) whose eigenvalues are positive.



9.7 The Fermionic Response Model

519

The Grassmannian-Gaussian integral (9.6) has the advantage over the classical Gaussian integral (9.7) that the determinant det A appears in the numerator.
This simplifies computations in physics. As a typical application, we will introduce
the Faddeev–Popov trick in Sect. 16.6 on page 889. This trick introduces ghosts
into gauge field theories in order to guarantee the crucial unitarity of the S-matrix.

9.6 The Method of Stationary Phase
We want to compute the following Grassmann integral
Z
eΦ(ψ,ψ,J,J )

W (J, J) :=

N
Y

dψ(k)dψ(k)

(9.8)

k=1

with the phase function
Φ(ψ, ψ, J, J ) := iψAψ + iψJ + iJ ψ.
Let us first explain the notation.

• For fixed N = 1, 2, . . . , the complex (N × N )-matrix A is invertible.
• The quantities ψ(k), ψ(k), J(k), J(k) with k = 1, . . . N form a sequence χ1 , . . . χ4N
of Grassmann variables, that is,
χi χj = −χj χi ,
i, j = 1, . . . , 4N.
QN
In particular, the symbol
k=1 dψ(k)dψ(k) stands for the ordered product
dψ(1)dψ(1) · · · dψ(N )dψ(N ).
• We use the following matrices
0
1
0
1
J(1)
ψ(1)
B . C
B . C
C
C
J =B
ψ=B
@ .. A
@ .. A ,
J(N )
ψ(N )
along with ψ = (ψ(1), . . . , ψ(N )) and J = (J (1), . . . , J (N )).
The proof of the following theorem proceeds analogously to the proof of Theorem
7.36 on page 438.
−1


Theorem 9.2 W (J, J) = e−iJ A

J

W (0, 0).

9.7 The Fermionic Response Model
The global quantum action principle. Parallel to Sect. 7.26 on page 477, we
study the generating functional
Z
Z(J, J ) = N

eiS[ψ,ψ,J,J ]/ DψDψ


520

9. Fermions and the Calculus for Grassmann Variables

along with the action functional
S[ψ, ψ, J, J] := ψ|(D + iεI)ψ + κ 1|Lint (ψ, ψ) +
+ J|ψ + ψ|J .
The detailed notation can be found in (7.161) on page 477. In contrast to Sect.
7.26, we now assume that the quantities
ψ(x), ψ(x), J(x), J (x)
are not complex numbers, but independent Grassmann variables. Here, the index
x denotes an arbitrary discrete space-time point, that is, x ∈ M. Explicitly, the
functional integral is to be understood as the following integral
Z

Y
eiS[ψ,ψ,J,J ]/
dψ(x)dψ(x)
Z(J, J ) = N
x∈M

with respect to Grassmann variables. The normalization
factor N has to be chosen
Q
in such a way that Z(0, 0) = 1. The symbol x∈M dψ(x)dψ(x) stands for the
product
dψ(1)dψ(1) · · · dψ(N )dψ(N )
where the discrete space-time points x ∈ M are numbered in a fixed order.
The magic quantum action reduction formula. The point is that this
formula reads as in the case of the extended response model considered in Sect.
7.26. Explicitly,
(
Z(J, J ) = exp

iκ X

„
∆ x Lint
4

x∈M

δ
δ
,

i δJ(x) i δJ(x)

«)
Zfree (J, J )

along with
Zfree (J, J) := ei

J |Rε J /

where Rε := −(D + εI)−1 . Here, we use the partial functional derivatives
δ
1
∂l
:= 4
,
δJ(x)
∆ x ∂J(x)

1
δ
∂l
:= 4
.
∆ x ∂J(x)
δJ (x)

The proof proceeds analogously to the proof given in Sect. 7.24 on page 438. Observe
that, in the present case, we have to use the principle of stationary phase for
integrals with Grassmann variables from Sect. 9.6. The different determinant tricks

for Gaussian and Grassmannian–Gaussian integrals do not matter, since the choice
of the normalization factor N always cancels the different determinants.
The magic LSZ reduction formula. From the quantum action reduction
formula above, we get the LSZ reduction formula as in Sect. 7.26. We will come
back to this in connection with the Standard Model in particle physics. We also
refer to Faddeev and Slavnov (1980).


10. Infinite-Dimensional Hilbert Spaces

Quantum fields possess an infinite number of degrees of freedom. This
causes a lot of mathematical trouble.
Folklore
Smooth functions. Let Ω be an open subset of RN , N = 1, 2, . . . The function
f : Ω → C is called smooth iff it is continuous and the partial derivatives of f
of arbitrary order are also continuous on Ω. For the theory of infinite-dimensional
Hilbert spaces and its applications in physics, it is important to use not only smooth
functions, but also reasonable discontinuous functions which are limits of smooth
functions. Here, we use pointwise limits, lim n→∞ fn (x) = f (x) for all x ∈ Ω, and
more general limits in the sense of the averaging over integrals, for example,
Z
|fn (x) − f (x)|2 dN x = 0.
lim
n→∞

Ω

Furthermore, it is important to replace the classical Riemann integral by the modern
Lebesgue integral.


10.1 The Importance of Infinite Dimensions in
Quantum Physics
We want to discuss why an infinite number degrees of freedom for quantum physics
is inevitable. To this end, we will show that the Heisenberg uncertainty relation
cannot be realized in a finite-dimensional Hilbert space.

10.1.1 The Uncertainty Relation
Before you start to axiomatize things, be sure that you first have something
of mathematical substance.
Hermann Weyl (1885–1955)
In 1927 Heisenberg (1901–1976) discovered that in contrast to Newton’s classical
mechanics, it is impossible to measure precisely position and momentum of a quantum particle at the same time. Heisenberg based his mathematical argument on the
commutation relation
QP − P Q = i I

(10.1)

for the position operator Q and the momentum operator P , along with the Schwarz
inequality.


522

10. Infinite-Dimensional Hilbert Spaces

Finite-dimensional Hilbert spaces fail. Observe first that the fundamental
commutation relation (10.1) cannot be realized for observables Q and P living in
a nontrivial finite-dimensional Hilbert space X if the Planck constant is different
from zero. 1 Indeed, suppose that there exist two self-adjoint linear operators
Q, P : X → X

such that (10.1) holds true. By Proposition 7.11 on page 364, tr(QP ) = tr(P Q).
This implies
0 = tr(QP − P Q) = i · tr I = i dim X.
Thus, relation (10.1) forces the vanishing of the Planck constant in the setting of
a nontrivial finite-dimensional Hilbert space.
A nontrivial mathematical model. Our goal is to construct a nontrivial
model which realizes the commutation relation (10.1). To this end, we choose the
which consists of all continuous
functions ψ : R → C with the property
space
R ∞ C2 (R)
R∞
2
†
|ψ(x)|
dx
<
∞.
Define
ϕ|ψ
:=
ϕ(x)
ψ(x)dx.
−∞
−∞
Proposition 10.1 For all functions ϕ, ψ ∈ C2 (R), the integral ϕ|ψ is finite.
Z

Proof. Set


n

an :=

ϕ(x)† ψ(x)dx,

n = 1, 2, . . .

−n

By the classical Schwarz inequality for integrals,
Z n
Z n
|ϕ(x)|2 dx
|ψ(x)|2 dx.
|an |2 ≤
−n

−n

This implies
Z
|an |2 ≤

∞

−∞

Z
|ϕ(x)|2 dx


∞
−∞

|ψ(x)|2 dx,

n = 1, 2, . . .

Thus, the sequence a1 , a2 , . . . is increasing and bounded. Consequently, the finite
✷
limit limn→∞ an exists.
Proposition 10.2 The space C2 (R) is a complex pre-Hilbert space.
Proof. It can be checked easily that ϕ|ψ possesses the properties (P1) through
(P5) from page 337. In particular, for a given continuous function ψ : R → C, we
have ψ|ψ = 0, that is,
Z
R

|ψ(x)|2 dx = 0

iff the function ψ vanishes identically.
✷
Consider now the Schwartz space D(R) which consists of all smooth functions
ψ : R → C that vanish outside some finite interval. For all functions ψ ∈ D(R), we
define the so-called position operator Q,
(Qψ)(x) := xψ(x)

for all

x ∈ R,


and the so-called momentum operator P ,
1

In the trivial Hilbert space {0}, relation (10.1) is obviously true.