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Chapter 8
General Theory of Quantum Fields
General mathematical aspects of quantum field theory are described.
8.1
Introduction
From this chapter, we discuss mathematical theory of quantum fields. As
mentioned in the Preface of this book, there are two approaches to study
quantum field theory (QFT) mathematically, namely, axiomatic approach
and constructive one. In this chapter we first look at what axiomatic QFT
is like briefly. But, as for axiomatic QFT, we restrict ourselves to giving
a minimum that a beginner in mathematical QFT should know, since the
present book is mainly intended to give mathematical bases for constructive
approach in QFT.
A standard heuristic approach to obtain a notion of quantum field is
to apply the canonical quantization scheme to a classical field as is usually
done in textbooks on QFT in physics. From the view-point which regards
QFT as a fundamental theory in physics, however, the concept of a quantum
field (if it exists) should be found as one independent of classical fields. In
the framework of such a QFT, a classical field should be positioned as a
classical limit of a quantum field where the reduced Planck constant (the
Dirac constant) ~ tends to 0 in a suitable sense. The purpose of this chapter
is to define a general concept of quantum field in the sense just mentioned
and to investigate its basic properties.
425
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8.2
Operator-valued Distributions
8.2.1
Operator-valued functions
Let F be a Hilbert space and L(F ) be the space of linear operators on F
(see Subsection 1.1.1).
A mapping A : Rd → L(F ); Rd ∋ x 7→ A(x) ∈ L(F ) is called an
operator-valued function on Rd with values in L(F ).
Example 8.1. Let fj : Rd → C (j = 1, . . . , n, n ∈ N) and Aj ∈ L(F ).
Pn
d
Then A : Rd → L(F ) defined by A(x) :=
j=1 fj (x)Aj , x ∈ R is an
operator-valued function on Rd .
Example 8.2. Let (A1 , . . . , Ad ) be a d-tuple of self-adjoint operators on
F . Then U : Rd → B(F ) (the Banach space of bounded linear operators
d
1
on F ) defined by U (x) = eix A1 · · · eix Ad , x = (x1 , . . . , xd ) ∈ Rd is an
operator-valued function on Rd . Note that, for all x ∈ Rd , U (x) is unitary.
8.2.2
Operator-valued distributions
We can also define an operator-valued functional as an extension of the
concept of operator-valued function. Let V be a complex vector space.
Then a mapping φ : V → L(F );V ∋ v 7→ φ(v) ∈ L(F ) is called an
operator-valued functional on V with values in L(F ).
We denote by S (Rd ) the Schwartz space of rapidly decreasing functions
on Rd (see Appendix D).
Example 8.3. Let A : Rd → L(F ) be an operator-valued function such
that, for all x ∈ Rd , A(x) ∈ B(F ) and A is Borel measurable with
Z
|f (x)| kA(x)Ψkdx < ∞, f ∈ S (Rd ), Ψ ∈ F .
Rd
For each f ∈ S (Rd ), one can define a linear operator φA (f ) on F as
follows:
Z
D(φA (f )) := Ψ ∈ F |
|f (x)| kA(x)Ψkdx < ∞ ,
Rd
Z
f (x)A(x)Ψdx, Ψ ∈ D(φA (f )),
φA (f )Ψ :=
Rd
where the integral on the right hand side is taken in the sense of Bochner
integral (see Appendix E). The mapping φA : f 7→ φA (f ) is an operatorvalued functional on S (Rd ) with values in L(F ).
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In analogy with the case of usual functions and distributions (see Appendix D), the concept of operator-valued function can be extended to a
concept of operator-valued distribution:
Definition 8.1. Let d ∈ N and D be a dense subspace in F . A mapping
ϕ : S (Rd ) → L(F ); S (Rd ) ∋ f 7→ ϕ(f ) ∈ L(F ) is called an operatorvalued distribution on Rd with values in L(F ) if the following (ϕ.1)–
(ϕ.3) hold:
(ϕ.1) For all f ∈ S (Rd ), ϕ(f ) is densely defined with D ⊂ D(ϕ(f )) ∩
D(ϕ(f )∗ ).
(ϕ.2) (linearity in test functions) For all f, g ∈ S (Rd ) and α, β ∈ C,
ϕ(αf + βg) = αϕ(f ) + βϕ(g) on D.
(ϕ.3) (continuity) For all Ψ, Φ ∈ D, the correspondence:S (Rd ) ∋ f 7→
hΨ, ϕ(f )Φi is continuous in S (Rd ). Namely, the linear functional
FΨ,Φ : S (Rd ) → C defined by FΨ,Φ (f ) := hΨ, ϕ(f )Φi , f ∈ S (Rd )
is a tempered distribution on Rd : FΨ,Φ ∈ S ′ (Rd ).
The subspace D is called a common domain of ϕ.
We denote by S ′ (Rd , L(F )) the set of operator-valued distributions
on Rd with values in L(F ).
Remark 8.1. In Definition 8.1, one can replace S (Rd ) by D(Rd ), the
space of test functions with compact support (see Appendix D). In this
case FΨ,Φ ∈ D ′ (Rd ). But, in this book, we consider only operator-valued
distributions in the above sense.
Remark 8.2. By condition (ϕ.1), D(ϕ(f )∗ ) is dense. Hence ϕ(f ) is closable.
Let ϕ1 and ϕ2 be in S ′ (Rd , L(F )) with common domains D1 and D2
respectively and D be a subspace of F such that D ⊂ D1 ∩ D2 . Then
we say that ϕ1 is equal to ϕ2 on D if, for all Ψ ∈ D and f ∈ S (Rd ),
ϕ1 (f )Ψ = ϕ2 (f )Ψ. In this case we write “ϕ1 = ϕ2 on D”.
As in the case of usual distributions, for an operator-valued distribution
ϕ, a fictitious symbol ϕ(x) is sometimes used in the form
Z
ϕ(x)f (x)dx.
ϕ(f ) =
Rd
The symbol ϕ(x) is called the operator-valued distribution kernel of
ϕ. In reference to this form, one says that ϕ(f ) is obtained by “smearing”
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ϕ(x) with f . But this expression can have a meaning only if ϕ(x) is an
operator-valued function on Rd with suitable properties.
Example 8.4. Let S1 , . . . , Sn (n ∈ N) be densely defined closed operators
on F and φ1 , . . . , φn ∈ S ′ (Rd ). Suppose that there exists a dense subspace
D such that D ⊂ ∩nj=1 D(Sj ) ∩ D(Sj∗ ). Then the mapping ϕ : S (Rd ) →
L(F ) defined by
ϕ(f ) :=
n
X
φj (f )Sj ,
j=1
f ∈ S (Rd )
is in S ′ (Rd , L(F )).
Note that, if n = 1 and S1 = I (identity), then ϕ(f ) = φ1 (f ) ( a scalar
multiplication operator). Hence each element in S ′ (Rd ) can be regarded
as a special case of operator-valued distributions on Rd .
Example 8.5. Let us consider the case where F = Fb (L2 (Rd )), the boson
Fock space over L2 (Rd ) (see Chapter 5). We denote by a(f ) the boson
annihilation operator on Fb (L2 (Rd )) with test vector f ∈ L2 (Rd ). Let Nb
be the boson number operator on Fb (L2 (Rd )). Then, as we have seen in
1/2
Chapter 5, D(Nb ) ⊂ D(a(f )) ∩ D(a(f )∗ ) for all f ∈ L2 (Rd ) and
1/2
ka(f )Ψk ≤ kf k kNb Ψk,
∗
(8.1)
1/2
ka(f ) Ψk ≤ kf k k(Nb + 1)
1/2
Ψk, Ψ ∈ D(Nb ), f ∈ L2 (Rd ).
(8.2)
We denote by f ∗ the complex conjugate of f . It is easy to see that, for all
1/2
Ψ ∈ D(Nb ), a(f ∗ )Ψ is linear in f ∈ L2 (Rd ). Moreover, for all Ψ, Φ ∈
1/2
D(Nb ),
| hΨ, a(f ∗ )Φi | ≤ kΨk ka(f ∗)Φk ≤ CΨ,Φ kf k,
1/2
where CΨ,Φ := kΨk kNb Φk. Let f ∈ S (Rd ) and 2n > d. Then
Z
kf k2 =
(1 + |x|)−2n (1 + |x|)2n |f (x)|2 dx
Rd
≤ c2d,n kf k2n,0 ,
where kf km,α (m ∈ Z+ , α ∈ Zd+ ) is defined by (D.3) in Appendix D and
qR
−2n dx < ∞. Hence
cd,n :=
Rd (1 + |x|)
| hΨ, a(f ∗ )Φi | ≤ kΨk ka(f ∗)Φk ≤ CΨ,Φ cd,n kf kn,0 .
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S
Hence, if fn −→ f (n → ∞) (fn , f ∈ S (Rd )), then limn→∞ hΨ, a(fn∗ )Φi =
hΨ, a(f ∗ )Φi. Therefore S (Rd ) ∋ f 7→ hΨ, a(f ∗ )Φi is a tempered distribution on Rd . Thus the mapping : f 7→ a(f ∗ ) is an operator-valued dis1/2
tribution on Rd with values in L(Fb (L2 (Rd ))) and D(Nb ) is a common
domain of this operator-valued distribution.
Similarly one can show that the mapping f 7→ a(f )∗ (the boson creation
operator with test vector f ∈ S (Rd )) is in S ′ (Rd , Fb (L2 (Rd )) with a
1/2
common domain D(Nb ).
8.2.3
Transformations of operator-valued distributions
Let ϕ ∈ S ′ (Rd , L(F )) with a common domain D and T be a continuous
linear mapping on S (Rd ). Then one can define a mapping ϕT : S (Rd ) →
L(F ) by
ϕT (f ) := ϕ(T f ),
f ∈ S (Rd ).
(8.3)
Then it is easy to see that ϕT ∈ S ′ (Rd , L(F )). We call ϕT the T transform of ϕ. If there exists a unitary operator U on F such that
U ϕ(f )U −1 = ϕT (f ) = ϕ(T f ),
f ∈ S (Rd ),
then ϕT is unitarily implementable by U .
One can also consider the case where T is a continuous anti-linear mapping on S (Rd ). In this case too, ϕT (f ) is defined by (8.3). But, in this
case, by the anti-linearity of T , the mapping:f 7→ ϕT (f )∗ is an operatorvalued distribution on Rd . If there exists an anti-unitary operator1 U ′ on
F such that
U ′ ϕ(f )U ′
−1
= ϕ(T f ),
f ∈ S (Rd ),
then ϕT is anti-unitarily implementable by U ′ .
8.2.4
Partial derivatives of operator-valued distributions
Let ϕ ∈ S ′ (Rd , L(F )) with a common domain D. Then, for each multiindex α ∈ Zd+ , one can define a mapping Dα ϕ : S (Rd ) → L(F ) by
(Dα ϕ)(f ) = (−1)|α| ϕ(∂ α f ),
f ∈ S (Rd ).
Then, as in the case of usual distributions (see Section D.1.4 in Appendix
D), one can see that Dα ϕ ∈ S ′ (Rd , L(F )) with D being a common domain.
1 See Section 4.9.
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We call the operator-valued distribution Dα ϕ a partial derivative of ϕ
of order |α|.
For a non-negative integer αj (j = 1, . . . , d), we write
α
Dj j := Dα
jth
with α = (0, . . . , 0, αj , 0, . . . , 0). Then, for every multi-index α,
Dα ϕ = D1α1 · · · Ddαd ϕ,
8.3
8.3.1
ϕ ∈ S ′ (Rd , L(F )).
General Concept of Quantum Field
Time-translation covariant quantum field theory
We consider a quantum system in the d-dimensional space Rd with d ∈
N. In the usual physical sense, the space dimension d is taken to be 3.
But we do not restrict ourselves to this case for the following reasons: (1)
mathematical generality; (2) we want to clarify the dependence of theories
on the space dimension d and to know what are essential differences of
theories with d = 3 from those with d 6= 3. This may bring us insights
about the “meaning” of d = 3.
The space-time is given by
R1+d := R × Rd = {(x0 , x)|x0 ∈ R, x ∈ Rd }
= {(x0 , x1 , . . . , xd )|xµ ∈ R, µ = 0, . . . , d},
where xµ (µ = 0, 1, . . . , d) denotes the µth coordinate of the point x ∈ R1+d ,
in particular, x0 and x = (x1 , . . . , xd ) are the time coordinate and the space
component of x respectively.2
The vector space R1+d has many metrics.3 But, for the time being,
we proceed without specifying a metric of R1+d , treating R1+d only as a
2 We take the upper index notation xµ (µ = 0, 1, . . . , d), which is convenient in relativistic QFT discussed later.
3 Let V be a real vector space. A metric of V is a bilinear form g : V ×V → R satisfying
the following (i) and (ii): (i) (non-degeneracy) if a vector u ∈ V satisfies g(u, v) = 0 for
all v ∈ V , then u = 0; (ii) (symmetry) g(u, v) = g(v, u) for all u, v ∈ V . In one word, g
is a non-degenerate symmetric covariant tensor of degree 2 on V . A metric g of V is said
to be positive (resp. negative) definite if g(u, u) ≥ 0 (resp. g(u, u) ≤ 0) for all u. In
the both cases, one can show that g(u, u) = 0 if and only if u = 0. Hence a metric g of
V is an inner product of V if and only if g is positive definite. On the other hand, g is
said to be indefinite if there exist points u, v ∈ V such that g(u, u) > 0 and g(v, v) < 0.
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topological space with the standard topology.4 This approach will clarify
aspects of QFT independent of metrics of R1+d .
In general, in the Hilbert space formalism of quantum mechanics which
is due to von Neumann, a state of a quantum system is described by a
non-zero vector in a Hilbert space H and a physical quantity is given by
a self-adjoint operator on H . Among others, the self-adjoint operator H
which describes the total energy of a quantum system is called the Hamiltonian of the system. According to an axiom of quantum mechanics, the
strongly continuous one-parameter unitary group {e−itH/~ }t∈R governs the
time development of the quantum system in the way that, for any state vector Ψ ∈ H at time 0, the state vector at time t ∈ R is given by e−itH/~ Ψ,
provided that no measurement is made on the quantum system in the time
interval [0, t].
As suggested by heuristic arguments made in textbooks of QFT in
physics, a quantum field may be an object which has the function that
creates or annihilates elementary particles in the space-time. A priori a
quantum field itself may not be necessarily a physical quantity, but it should
be a linear operator acting in a Hilbert space of quantum states and related to other physical quantities in a harmonious way with laws. Thus it
is suggested that a general concept of a quantum field may not be defined
without reference to other physical quantities. We first define a class of
QFT with a Hamiltonian.
For a function f on R1+d and a = (a0 , a) ∈ R1+d , we define the function
fa on R1+d by
fa (x) := f (x − a) = f (x0 − a0 , x − a),
x = (x0 , x) ∈ R1+d .
(8.4)
The function fa is called the space-time translation (or simply translation) of f by the vector a. In particular, for each time t ∈ R, f(t,0) is
called the time-translation of f by t, and f(0,a) with a ∈ Rd is called the
space-translation of f by a.
It is easy to see that, for all f ∈ S (R1+d ), fa ∈ S (R1+d ) (Problem 2).
Definition 8.2. A time-translation covariant QFT on the space-time
R1+d is a quartet (F , D, H, (ϕr )sr=1 ) (s ∈ N) consisting of a Hilbert space
F , a dense subspace D ⊂ F , a self-adjoint operator H on F and an s-tuple
(ϕr )sr=1 with ϕr ∈ S ′ (R1+d , L(F )) (r = 1, . . . , s) such that the following
(QF.1) and (QF.2) hold:
4 It is important to distinguish the concept of metric from that of topology. In the
physics literature, confusions are sometimes seen on this aspect.
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(QF.1) For all f ∈ S (R1+d ), ϕr (f ) and ϕr (f )∗ leave D invariant:
ϕr (f )D ⊂ D, ϕr (f )∗ D ⊂ D.
(QF.2) (time-translation covariance) For all t ∈ R and f ∈ S (R1+d ),
ϕr (f(t,0) ) = eitH/~ ϕr (f )e−itH/~ on D (r = 1, . . . , s).
(8.5)
The s-tuple (ϕr )sr=1 of operator-valued distributions is called an scomponent quantum field and H is called the Hamiltonian. Also
each component ϕr is called a quantum field.
Remark 8.3. (1) Definition 8.2 is a general definition of a QFT noting
only the basic relation (8.5) between quantum fields and the Hamiltonian.
It is independent of whether or not the theory considered is relativistic in
the sense of Einstein’s special theory of relativity.
As is well known, a classical scalar field is a function on the spacetime. Hence it would be natural to ask why a quantum field is defined
as an operator-valued distribution, not as an operator-valued function on
the space-time. One of the reasons comes from taking into account the
canonical quantization scheme which is a heuristic scheme to formulate a
QFT from a classical field theory. In this scheme, for a classical scalar field
ϕcl (x) and its canonical conjugate πcl (x) (x ∈ R1+d ), a quantum field ϕ(x)
and its canonical conjugate π(x) are supposed to exist as algebraic elements
satisfying the equal-time canonical commutation relations
[ϕ(t, x), π(t, y)] = i~δ(x − y),
[ϕ(t, x), ϕ(t, y)] = 0, [π(t, x), π(t, y)] = 0, (t, x), (t, y) ∈ R1+d ,
where δ(x − y) is the diagonal delta distribution on R2d (see Subsection
D.3.3 in Appendix D). But, since δ(x − y) is not a function, it is not
expected that the mapping:(t, x) 7→ ϕ(t, x) (resp. π(t, x)) is an operatorvalued function.5 Hence it may
R be legitimate to define a quantum field as
the smeared one: “ϕ(t, g) = Rd ϕ(t, x)g(x)dx” (g ∈ S (Rd )) or “ϕ(f ) =
R
ϕ(x)f (x)dx” (f ∈ S (R1+d )). Then the above equal-time canonical
R1+d
commutation relations are Zreplaced as follows:
g(x)h(x)dx,
[ϕ(t, g), π(t, h)] = i~
Rd
[ϕ(t, g), ϕ(t, h)] = 0,
[π(t, g), π(t, h)] = 0, t ∈ R, g, h ∈ S (Rd ).
5 Suppose that ϕ(·) and π(·) are operator-valued functions on R1+d . Then, by analogy
with classical fields, it would be natural to assume that there exists a subspace D such
that D ⊂ ∩x,y∈R1+d D(ϕ(x)π(y)) ∩ D(π(y)ϕ(x)) and, for all Ψ, Φ ∈ D, the function:
R1+d × R1+d ∋ (x, y) 7→ hΨ, ϕ(x)π(y)Φi (resp. hΨ, π(y)ϕ(x)Φi) is continuous. But, in
that case, the equation hΨ, [ϕ(t, x), π(t, y)]Φi = i~δ(x − y) hΨ, Φi is contradictory, since
the left hand side is continuous in (t, x, y).
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As we shall see later (Subsection 8.3.3 below), there are cases where the
object ϕ(t, g) is mathematically meaningful.
There is another reason to define a quantum field as an operator-valued
distribution. From a quantum mechanical point of view, the physical notion
of one point in the space-time may be meaningless.6 It may be natural to
suppose that the strength of a quantum field is measured in a finite region
in the space-time which may be variable, not at one point in the spacetime. The operator-value ϕ(f ) of the quantum field ϕ as an operatorvalued distribution corresponds to such a measurement in a space-time
region which includes the support of f .
In addition, there is an surprising theorem that, in a framework of
relativistic quantum field theory, a non-trivial (non-constant) quantum field
cannot be an operator-valued function on the space-time (Wightman’s
theorem7 ).
(2) A physical picture contained in Definition 8.2 is as follows. Since
f(t,0) physically means the time translation of f by t as mentioned above,
ϕr (f(t,0) ) is interpreted as the time translation of the quantum field ϕr by
t. Property (QF.2) shows that the time-translated quantum field ϕr (f(t,0) )
is unitarily implementable on D with the unitary operator e−itH/~ which
governs the time-development of states of the quantum system under consideration. This property is called the unitary covariance of the quantum
field (ϕr )sr=1 with respect to time or simply the time-translation covariance of (ϕr )sr=1 .
The operator eitH/~ ϕr (f )e−itH/~ is the Heisenberg operator of ϕr (f )
with respect to the Hamiltonian H (see Section 5.16), which describes the
quantum field at time t. Hence (8.5) means that the time-translated quantum field ϕr (f(t,0) ) coincides with the Heisenberg operator of ϕr (f ) on D.
This is a harmonious relation. Equation (8.5) is symbolically written in
terms of distribution kernels as follows:
ϕr (x0 + t, x) = eitH/~ ϕr (x0 , x)e−itH/~
on D.
(3) Let U (t) := e−itH/~ , t ∈ R. Then the mapping U : t 7→ U (t) is a
strongly continuous unitary representation of R (as a translation group) on
F (see Subsection 2.9.2), satisfying
ϕr (f(t,0) ) = U (t)∗ ϕr (f )U (t)
on D
(8.6)
6 Recall that the probability of finding a quantum particle at one point in the space is
theoretically zero.
7 See, e.g., [Bogoliubov et al. (1975), §2.4].
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for all t ∈ R, f ∈ S (R1+d ) and r = 1, . . . , s.
Conversely one can reformulate a time-translation covariant QFT as a
quartet (F , D, U, (ϕr )sr=1 ) with H replaced by a strongly continuous unitary representation U : R → GL(F ); R ∋ t 7→ U (t) ∈ GL(F ) satisfying
(8.6). Then, by Stone’s theorem, there exists a unique self-adjoint operator H on F such that U (t) = e−itH/~ , t ∈ R. Hence a time-translation
covariant QFT (F , D, H, (ϕr )sr=1 ) is obtained.
By these considerations, we see that the theory (F , D, H, (ϕr )sr=1 ) is
equivalent to the theory (F , D, U, (ϕr )sr=1 ).
(4) Definition 8.2 gives a general concept of a QFT which has a unitary covariance with respect to time. By adding covariances with resepct
to other transformations such as translations, rotations in space Rd and
Poincaré transformations in R1+d (see Subsection 8.8.5 below), one obtains
more specialized forms of the theory. In particular, a relativistic QFT is a
special form of time-translation covariant QFT. We shall see these aspects
below.
(5) The natural number s in Definition 8.2, which accounts the number
of the components of a multi-component quantum field, depends on the
kind of quantum field (scalar field, vector field, tensor field, spinor field
etc.). For more details on this point, see, e.g., [Streater and Wightman
(1964), p.99] and [Bogoliubov et al. (1975), Chapter 3].
(6) As is easily seen, if a self-adjoint operator H satisfies (8.5), then, for
all E ∈ R, (8.5) with H replaced by H + E holds. But we regard H + E as
an operator essentially same as H.
For (ϕr )sr=1 , a self-adjoint operator H satisfying (8.5) is not necessarily
unique up to addition of scalar operators. In Proposition 8.1 below, we
shall give a sufficient condition for H to be unique up to addition of scalar
operators.
8.3.2
Uniqueness of Hamiltonians
Proposition 8.1. Let (F , D, H, (ϕr )sr=1 ) be a time-translation covariant
QFT on R1+d such that, for all t ∈ R, eitH D ⊂ D and
{ϕr (f ) ↾ D|f ∈ S (R1+d ), r = 1, . . . , s}′ = CI,
where {· · · }′ denotes the commutant of {· · · } (see (5.40)). Suppose that
there exists a self-adjoint operator H ′ satisfying (8.5) with H replaced by
′
H ′ such that eitH D ⊂ D for all t ∈ R. Then there exists a constant E ∈ R
such that H ′ = H + E.
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Proof. By the assumption, we have
′
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eitH ϕr (f )e−itH Ψ = eitH ϕr (f )e−itH Ψ
for all Ψ ∈ D, t ∈ R, f ∈ S (R1+d ) and r = 1, . . . , s (we have replaced
′
′
t/~ by t). Hence ϕr (f )e−itH Ψ = e−itH eitH ϕr (f )e−itH Ψ. For each Φ ∈
′
′
D, Ψ := eitH Φ is in D. Hence, putting T (t) := e−itH eitH , we obtain
ϕr (f )T (t)Φ = T (t)ϕr (f )Φ. Hence
T (t) ∈ {ϕr (f ) ↾ D|f ∈ S (R1+d ), r = 1, . . . , s}′ .
Therefore T (t) = c(t)I with a constant c(t) depending on t. Hence
′
eitH = c(t)eitH . The unitarity of T (t) implies that |c(t)| = 1. The group
′
property of {eitH }t∈R and {eitH }t∈R yields c(t + t′ ) = c(t)c(t′ ), t, t′ ∈ R.
In particular, c(0) = 1. Since T (t) is strongly continuous in t, it follows
that c(t) is continuous in t. Hence c(t) = eiEt for some constant E ∈ R.
′
This means that eitH = eit(H+E) , ∀t ∈ R. Hence H ′ = H + E
8.3.3
Sharp-time quantum fields
There are classes of quantum fields in which each quantum field is welldefined even if it is not smeared in the time variable t. We next introduce
such a class of quantum fields.
Definition 8.3. A sharp-time QFT on R1+d is a quartet (F , D, H,
{ϕr (t, ·)}t∈R,r=1,...,s ) (s ∈ N) consisting of a Hilbert space F , a dense subspace D ⊂ F , a self-adjoint operator H on F and a set {ϕr (t, ·)}t∈R,r=1,...,s
of operator-valued distributions on Rd with values in L(F ) (ϕr (t, f ) ∈
L(F ), t ∈ R, f ∈ S (Rd )) satisfying the following (i) and (ii):
(i) For all t ∈ R, r = 1, . . . , s and f ∈ S (Rd ), D ⊂ D(ϕr (t, f )) ∩
D(ϕr (t, f )∗ ) and ϕr (t, f )D ⊂ D, ϕr (t, f )∗ D ⊂ D.
(ii) For all t ∈ R and f ∈ S (Rd ),
ϕr (t, f ) = eitH/~ ϕr (0, f )e−itH/~
on D.
(8.7)
The operator-valued distribution ϕr (t, ·) is called a quantum field at time
t and H is called the Hamiltonian.
A uniqueness theorem on Hamiltonians holds for a sharp-time QFT too:
Proposition 8.2. Let (F , D, H, {ϕr (t, ·)}t∈R,r=1,...,s ) be a sharp-time QFT
on R1+d such that, for all t ∈ R, eitH D ⊂ D and
{ϕr (0, f ) ↾ D|f ∈ S (Rd ), r = 1, . . . , s}′ = CI.
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Suppose that there exists a self-adjoint operator H ′ satisfying (8.7) with H
′
replaced by H ′ such that eitH D ⊂ D for all t ∈ R. Then there exists a
constant E ∈ R such that H ′ = H + E.
Proof. Similar to the proof of Proposition 8.1.
One can deduce a sharp-time QFT from a time-translation covariant QFT with an additional property in the following manner. Let
(F , D, H, (ϕr )sr=1 ) be a time-translation covariant QFT with the additional
condition that
e−itH/~ D ⊂ D,
t ∈ R.
(8.8)
Suppose that, for each t ∈ R and r = 1, . . . , s, there exists an operatorvalued distribution ϕr (t, ·) satisfying the following (a)–(c):
(a) For all t ∈ R, f ∈ S (Rd ) and r = 1, . . . , s, D ⊂ D(ϕr (t, f )) ∩
D(ϕr (t, f )∗ ).
(b) For all Ψ, Φ ∈ D and f ∈ S (Rd ), the function:R ∋ t 7→ hΦ, ϕr (t, f )Ψi
is continuousZ on R and
R
|u(t)| | hΦ, ϕr (t, f )Ψi |dt < ∞,
u ∈ S (R).
(8.9)
(c) For all Ψ, Φ ∈ D, u ∈ S (R) and f ∈ S (Rd ),
Z
u(t) hΦ, ϕr (t, f )Ψi dt.
hΦ, ϕr (u × f )Ψi =
R
If such operator-valued distributions {ϕr (t, ·)}t,r exist, then each ϕ(t, f ) ↾
D is unique. This is proved as follows.
Suppose that there exists another operator-valued distribution ηr (t, f )
satisfying the above (a)–(c) with ϕr (t, f ) replaced by ηr (t, f ). Then, it
follows from (c) that, for all u ∈ S (R),
Z
Z
u(t) hΦ, ϕr (t, f )Ψi dt =
u(t) hΦ, ηr (t, f )Ψi dt.
R
R
By (b), the functions hΦ, ϕr (t, f )Ψi, and hΦ, ηr (t, f )Ψi of t are locally
integrable. Hence, by du Bois-Reymond’s lemma, hΦ, ϕr (t, f )Ψi =
hΦ, ηr (t, f )Ψi a.e. t. Since the both functions are continuous in t, it follows that, for all t ∈ R, hΦ, ϕr (t, f )Ψi = hΦ, ηr (t, f )Ψi. This holds for
all Φ ∈ D and D is dense. Hence ϕr (t, f )Ψ = ηr (t, f )Ψ, implying that
ϕr (t, f ) ↾ D = ηr (t, f ) ↾ D.
Let us show that (8.7) holds. For each t ∈ R and a function u on R, we
define a function ut by
ut (x0 ) := u(x0 − t),
x0 ∈ R.
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The function ut is the time translation of u by t.
It is easy to see that, if u ∈ S (R), then ut ∈ S (R).
Let Ψ, Φ ∈ D, f ∈ S (Rd ), u ∈ S (R) and t ∈ R. Then, by (c), we have
Z
hΦ, ϕr (ut × f )Ψi =
ut (x0 ) Φ, ϕr (x0 , f )Ψ dx0
R
Z
u(x0 ) Φ, ϕr (x0 + t, f )Ψ dx0 .
=
R
On the other hand, by (8.5),
D
E
hΦ, ϕr (ut × f )Ψi = e−itH/~ Φ, ϕr (u × f )e−itH/~ Ψ
Z
D
E
=
u(x0 ) e−itH/~ Φ, ϕr (x0 , f )e−itH/~ Ψ dx0 ,
R
where we have used condition (8.8). Hence, by du Boi-Raymond’s lemma,
D
E
Φ, ϕr (x0 + t, f )Ψ = e−itH/~ Φ, ϕr (x0 , f )e−itH/~ Ψ
D
E
= Φ, eitH/~ ϕr (x0 , f )e−itH/~ Ψ .
Hence
ϕr (x0 + t, f )Ψ = eitH/~ ϕr (x0 , f )e−itH/~ Ψ.
Putting x0 = 0, we obtain ϕr (t, f )Ψ = eitH/~ ϕr (0, f )e−itH/~ Ψ. Thus (8.7)
holds.
Conversely, there exists a structure which makes it possible to construct
a time-translation covariant QFT from a sharp-time QFT. We outline it
briefly.
Let (F , D, H, {ϕr (t, ·)}t∈R,r=1,...,s ) be a sharp-time QFT such that, for
all Ψ, Φ ∈ D and f ∈ S (Rd ), the function:R ∋ t 7→ hΨ, ϕr (t, f )Φi is
continuous and (8.9) holds.
[Step 1] For each pair (Ψ, Φ) ∈ D × D, one defines a bilinear form LΨ,Φ
on S (R) × S (Rd ) by
Z
LΨ,Φ (u, f ) :=
u(t) hΨ, ϕr (t, f )Φi dt, u ∈ S (R), f ∈ S (Rd ).
R
[Step 2] Under additional conditions (if necessary), one shows that
LΨ,Φ is separately continuous (see Appendix D). Then, by Schwartz’s
nuclear theorem (Theorem D.5 in Appendix D), there exists a unique
e Ψ,Φ ∈ S ′ (R1+d ) such that
L
Z
e
LΨ,Φ (u × f ) =
u(t) hΨ, ϕr (t, f )Φi dt, u ∈ S (R), f ∈ S (Rd ).
R
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[Step 3] Under additional conditions (if necessary), one shows that,
for all Φ ∈ D and F ∈ S (R1+d ), the conjugate linear mapping: Ψ 7→
e Ψ,Φ (F ) ∈ C is a continuous linear functional.8 Then, by the density of D
L
and Riesz’s representation theorem, there exists a unique vector ΦF ∈ F
e Ψ,Φ (F ) = hΨ, ΦF i. So one defines a mapping
such that, for all Ψ ∈ D, L
ϕr (F ) : D → F by ϕr (F )Φ := ΦF , Φ ∈ D. The mapping ϕr (F ) is a linear
operator on F with D(ϕr (F )) = D and linear in F .
[Step 4] One shows that (F , D, H, (ϕr )sr=1 ) is a time-translation covariant QFT.
The procedures schematically described here are a method to
give
R a mathematically rigorous meaning to the heuristic expression
“ R u(t)ϕr (t, f )dt”, a sharp-time field smeared in the time variable, as an
operator-valued distribution on R1+d . We remark, however, that the above
procedures are not always possible. It may depend on each individual
sharp-time QFT.
Examples of time-translation covariant QFT and sharp-time QFT will
be discussed in later chapters.
8.4
Equations for Quantum Fields
Let us derive differential equations for a quantum field in a time-translation
covariant QFT. As in Appendix D, if a sequence {fn }n in S (R1+d ) conS
verges to f ∈ S (R1+d ) in the topology of S (R1+d ), we write fn −→ f
(n → ∞).
Lemma 8.1. For all f ∈ S (R1+d ),
f(t,0) − f S
−→ −∂0 f (t → 0),
t
where
∂
.
∂0 :=
∂x0
Proof. Let t ∈ R, t 6= 0 and
f(t,0) (x) − f (x)
gt (x) :=
+ ∂0 f (x), x ∈ R1+d .
t
Then, by elementary calculus, we have
Z ξ
Z
1 −t
gt (x) =
dτ ∂02 f (x0 + τ, x).
dξ
t 0
0
8 If
Cr (u, f, Φ) :=
Cr (u, f, Φ)kΨk.
R
R
(8.10)
e Ψ,Φ (F )| ≤
|u(t)| kϕr (t, f )Φkdt < ∞, then this holds, since |L
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For any multi-index α = (α0 , α1 , . . . , αd ), we have
∂ α gt (x) =
(∂ α f )(t,0) (x) − (∂ α f )(x)
+ ∂0 (∂ α f )(x),
t
Hence
∂ α gt (x) =
1
t
Z −t
0
dξ
Z ξ
x ∈ R1+d .
dτ ∂ β f (x0 + τ, x),
0
where β := (α0 + 2, α1 , . . . , αd ). By estimating the integral on the right
hand side, we obtain
|∂ α gt (x)| ≤ |t| sup |∂ β f (x0 + τ, x)|.
|τ |≤|t|
Let
xτ := (x0 + τ, x).
Then, for all x ∈ R1+d ,
|x| = |x − xτ + xτ | ≤ |x − xτ | + |xτ | = |τ | + |xτ |.
Let |τ | ≤ |t| ≤ 1. Then |x| ≤ 1 + |xτ |. Hence, for all m ∈ Z+ ,
(1 + |x|)m ≤ (2 + |xτ |)m ≤ 2m (1 + |xτ |)m .
Hence
(1 + |x|)m |∂ α gt (x)| ≤ 2m |t| kf km,β .
Therefore
kgt km,α ≤ 2m |t| kf km,β → 0
(t → 0).
Thus (8.10) holds.
For an operator-valued distribution ϕ on R1+d with values in L(F ),
we denote by Dµ ϕ ∈ S ′ (R1+d , L(F )) the partial derivative of ϕ in the
variable xµ (µ = 0, 1, . . . , d):
Dµ ϕ(f ) := −ϕ(∂µ f ),
ϕ ∈ S ′ (R1+d , L(F )), f ∈ S (R1+d ).
Theorem 8.1. Let (F , D, H, (ϕr )sr=1 ) be a time-translation covariant
QFT and suppose that the following conditions are satisfied:
(i) D ⊂ D(H).
(ii) For all f ∈ S (R1+d ) and r = 1, . . . , s, HD ⊂ D(ϕr (f )).
(iii) For all Ψ ∈ D, f ∈ S (R1+d ) and r = 1, . . . , s, the F -valued function
:R ∋ t → ϕr (f )e−itH/~ Ψ is strongly continuous.
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Then, for all Ψ ∈ D, f ∈ S (R1+d ), r = 1, . . . , s and n ∈ N,
D0n ϕr (f )Ψ =
i
[H, D0n−1 ϕr (f )]Ψ.
~
(8.11)
Proof. We first prove (8.11) with n = 1. Let Ψ, Φ ∈ D and f ∈ S (R1+d ).
Then
hΦ, D0 ϕr (f )Ψi = − hΦ, ϕr (∂0 f )Ψi .
By applying Lemma 8.1 to ∂0 f on the right hand side, we have
Φ, ϕr (f(t,0) )Ψ − hΦ, ϕr (f )Ψi
.
t→0
t
Using the time-translation covariance (8.5), we have
hΦ, D0 ϕr (f )Ψi = lim
Φ, eitH/~ ϕr (f )e−itH/~ Ψ − hΦ, ϕr (f )Ψi
.
t→0
t
To compute the right hand side, we note the following facts:
hΦ, D0 ϕr (f )Ψi = lim
Φ, eitH/~ ϕr (f )e−itH/~ Ψ − hΦ, ϕr (f )Ψi
t
−itH/~
e−itH/~ − 1
e
−1
Φ, ϕr (f )e−itH/~ Ψ + ϕr (f )∗ Φ,
Ψ
=
t
t
and, for all Ξ ∈ D(H),
i
e−itH/~ − 1
Ξ = − HΞ.
t→0
t
~
Condition (iii) implies that
lim
lim ϕr (f )e−itH/~ Ψ = ϕr (f )Ψ.
t→0
Using these facts, we obtain
hΦ, D0 ϕr (f )Ψi =
i
Φ, [H, ϕr (f )]Ψ .
~
Since D is dense, (8.11) with n = 1 follows.
To prove (8.11) with n arbitrary, we use the method by induction in n.
Suppose that (8.11) holds with n = k. Then
i
D0k+1 ϕr (f )Ψ = −D0k ϕr (D0 f )Ψ = − [H, D0k−1 ϕr (D0 f )]Ψ
~
i
= [H, D0k ϕr (f )]Ψ.
~
Hence (8.11) with n = k + 1 holds.
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In terms of operator-valued distribution kernels, (8.11) is symbolically
written
Z
Z
i
f (x) [H, D0n−1 ϕr (x)]Ψdx.
f (x)D0n ϕr (x)Ψdx =
~
1+d
1+d
R
R
Based on this picture, we say that
D0n ϕr (x) =
i
[H, D0n−1 ϕr (x)]
~
on D
in the sense of operator-valued distribution, meaning (8.11). In particular,
we have
i
(8.12)
D0 ϕr (x) = [H, ϕr (x)],
~
1
(8.13)
D02 ϕr (x) = − 2 [H, [H, ϕr (x)]] on D
~
in the sense of operator-valued distribution. Note that (8.12) has the form
of the Heisenberg equation of motion for ϕr (x) with respect to H.9
We remark that, in most of concrete models in QFT, (8.12) and (8.13)
are realized as dynamical equations of the quantum field (ϕr )sr=1 .
8.5
Vacuum Expectation Values and Wightman
Distributions
Let us consider a time-translation covariant QFT (F , D, H, (ϕr )sr=1 ) on
R1+d such that H is bounded from below. Then
E0 (H) := inf σ(H),
(8.14)
the infimum of the spectrum of H, is finite and called the lowest energy
of H.
If E0 (H) is an eigenvalue of H, then a non-zero vector Ψ in the
eigenspace ker(H − E0 (H)) 6= {0} is called a ground state or a vacuum
of H.
9 In general, an operator-valued function A : R ∋ t 7→ A(t) ∈ L(H ) (H is a Hilbert
space) is said to obey the strong Heisenberg equation of motion on a subspace
D ⊂ H with respect to a self-adjoint operator S on H if D ⊂ D([S, A(t)]), t ∈ R and,
for all ψ ∈ D, A(t)ψ is strongly differentiable in t with
i
dA(t)ψ
= [S, A(t)]ψ.
dt
~
This is heuristically written as dA(t)/dt = (i/~)[S, A(t)]. There is a weaker version of
Heisenberg equation of motion [Arai (2007)].
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The ground state (or vacuum) of H is said to be unique if dim ker(H −
E0 (H)) = 1 (i.e., the multiplicity of E0 (H) as an eigenvalue of H is one).
If dim ker(H − E0 (H)) ≥ 2, then the ground state (or vacuum) of H is said
to be degenerate.
In the literature, one sometimes notices the case where E0 (H) is called
the ground state energy even if H has no ground states. But this would not
be suitable in the case where H has no ground states. We want to emphasize
that, as is seen above, E0 (H) is defined independently of whether or not H
has a ground state.
By the way, it should be pointed out that it is non-trivial at all to prove
existence of a ground state in concrete models in QFT. Indeed, to prove
existence or absence of a ground state in QFT is one of the most important
and central issues in mathematical theory of QFT. We shall come back to
this aspect later.
We now consider the case where H has a ground state and suppose that
ker(H − E0 (H)) ⊂ D.
Let Ψ0 ∈ ker(H − E0 (H)) be a unit vector (kΨ0 k = 1). Then, for n ∈
(n)
N and (r1 , . . . , rn ) ∈ {1, . . . , s}n , one can define a functional Er1 ,...,rn on
S (R1+d )n by
(f1 , . . . , fn ) := hΨ0 , ϕr1 (f1 ) · · · ϕrn (fn )Ψ0 i
Er(n)
1 ,...,rn
for all (f1 , . . . , fn ) ∈ S (R1+d )n . These functionals are called the n-point
vacuum expectation values (VEV) of (ϕr )sr=1 with respect to the vacuum Ψ0 .
(n)
It is easy to see that Er1 ,...,rn is an n-linear form on S (R1+d ). Since the
correspondence S (R1+d ) ∋ f 7→ hΨ, ϕr (f )Φi (Ψ, Φ ∈ D) is continuous in
(n)
the topology of S (R1+d ), it follows that Er1 ,...,rn is separately continuous.
Hence, by Schwartz’s nuclear theorem (Theorem D.5 in Appendix D), there
(n)
exists a unique element Wr1 ,...,rn in S ′ ((R1+d )n ) such that
(f1 ×· · ·×fn ) = Er1 ,...,rn (f1 , . . . , fn ),
Wr(n)
1 ,...,rn
fj ∈ S (R1+d ), j = 1, . . . , n.
Therefore
(f1 × · · · × fn ) = hΨ0 , ϕr1 (f1 ) · · · ϕrn (fn )Ψ0 i .
Wr(n)
1 ,...,rn
(n)
The tempered distributions Wr1 ,...,rn are called the n-point Wightman
distributions with respect to the vacuum Ψ0 .
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In concrete models in QFT, it is heuristically shown that many physical
properties contained in QFT may be derived from the set of Wightman
distributions, although we do not discuss this aspect here.10
Remark 8.4. In a sharp-time QFT (F , D, H, (ϕr (t, ·))t∈R,r=1,...,s ), the
VEV’s are defined as follows:
hΨ0 , ϕr1 (t1 , f1 ) · · · ϕrn (tn , fn )Ψ0 i
for n ∈ N, rj ∈ {1, . . . , s}, tj ∈ R and fj ∈ S (Rd ) (j = 1, . . . , n).
8.6
Unitary Representations of Topological Groups
Before discussing a general theory of QFT further, we here make a mathematical interlude.
As is well known, a classical mechanical system may have various symmetries which are described by transformation groups on the space-time.
On the other hand, a symmetry in a quantum system is given in a form of
unitary representation of a group (see Subsection 2.9.2).
A basic fact on strongly continuous unitary representations of Rn as a
translation group is given in the following theorem:
Theorem 8.2. Let n ∈ N and H be a Hilbert space. Let U : Rn → U(H )
(the unitary group on H ) be a strongly continuous unitary representation
of Rn , i.e., for each a ∈ Rn , U (a) ∈ U(H ) and the mapping:a 7→ U (a) is
strongly continuous and satisfies
U (a + b) = U (a)U (b),
a, b ∈ Rn .
(8.15)
Then there exists a unique n-tuple (A1 , . . . , An ) of strongly commuting selfadjoint operators on H such that
1
n
U (a) = eia A1 · · · eia An ,
a = (a1 , . . . , an ) ∈ Rn ,
j
where the order of the exponential operators eia Aj (j = 1, . . . , n) in the
operator product on the right hand side can be arbitrary.
Proof. Let {ej }nj=1 be the standard basis of Rn so that, for all a =
Pn
j
(a1 , . . . , an ) ∈ Rn , a =
j=1 a ej . For each j = 1, . . . , n, we define
Uj (t) := U (tej ), t ∈ R. Then it is easy to see that {Uj (t)}t∈R is a strongly
10 For example, the so-called scattering-matrix (S-matrix), which describes probability
amplitudes for scattering phenomena of elementary particles, are represented in terms
of Wightman distributions.
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continuous one-parameter unitary group. Hence, by Stone’s theorem (Theorem 1.22), there exists a unique self-adjoint operator Aj on H such that
Uj (t) = eitAj , t ∈ R. Since U (a + b) = U (b + a), we have
a, b ∈ Rn ,
U (a)U (b) = U (b)(a),
i.e., the set {U (a)}a∈Rn is commutative. Hence
Uj (aj )Uk (ak ) = Uk (ak )Uj (aj ),
j
k
k
j, k = 1, . . . , n,
j
implying that eia Aj eia Ak = eia Ak eia Aj . Hence Aj and Ak are strongly
commuting. Thus (A1 , . . . , An ) is an n-tuple of strongly commuting selfadjoint operators on H .
By (8.15), we have for all permutations σ ∈ Sn
U (a) = U (aσ(1) eσ(1) + · · · + aσ(n) Aσ(n) )
= U (aσ(1) Aσ(1) ) · · · U (aσ(n) Aσ(n) )
= eia
σ(1)
Aσ(1)
· · · eia
σ(n)
Aσ(n)
.
This completes the proof.
The n-tuple (A1 , . . . , An ) in Theorem 8.2 is called the generator of the
strongly continuous unitary representation {U (a)|a ∈ Rn }.
Let X be a topological space and H be a Hilbert space. A mapping
f : X → H is said to be weakly continuous if, for all ψ ∈ H , the
function:X ∋ x 7→ hψ, f (x)i is continuous.
A mapping T : X → B(H ) is said to be weakly continuous if, for all
ψ ∈ H , the function:X ∋ x 7→ T (x)ψ is weakly continuous.
The following fact is useful in applications:
Lemma 8.2. Let U : X → U(H ); X ∋ x 7→ U (x) ∈ U(H ) be weakly
continuous. Then U is strongly continuous.
Proof. Let ψ ∈ H and a ∈ X be fixed arbitrarily. Put φ := U (a)ψ. Then,
by the weak continuity of U , for any ε > 0, there exists a neighborhood
Va of a such that, for all x ∈ Va , | hφ, U (x)ψi − hφ, U (a)ψi | < ε. Since
hφ, U (a)ψi = kψk2 , we have
| kψk2 − hU (a)ψ, U (x)ψi | < ε,
x ∈ Va .
Hence
kU (x)ψ − U (a)ψk2 = 2(kψk2 − Re hU (a)ψ, U (x)ψi) < 2ε,
Hence U (·)ψ is strongly continuous at x = a.
x ∈ Va .
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Lemma 8.2 can be brought into a more convenient form:
Lemma 8.3. Let X be a topological space, U : X → U(H ) and D be
a dense subspace in H . Suppose that, for all ψ, φ ∈ D, the function:
X ∋ x 7→ hφ, U (x)ψi is continuous. Then U is strongly continuous.
Proof. By Lemma 8.2, we need only to show the weak continuity of U .
For any ψ, φ ∈ H , we set F (x) := hψ, U (x)φi. By the density of D, there
exist sequences {ψn }n and {φn }n such that ψn → ψ, φn → φ (n → ∞).
Let Fn (x) := hψn , U (x)φn i. Then
|Fn (x) − F (x)| = | hψn − ψ, U (x)φn i + hψ, U (x)(φn − φ)i |
≤ kψn − ψkC + kψk kφn − φk,
where C := supn≥1 kφn k < ∞. For any ε > 0, there exist an n0 ∈ N such
that, for all n ≥ n0 , kψn − ψkC + kψkkφn − φk < ε Hence
sup |Fn (x) − F (x)| ≤ ε,
x∈X
n ≥ n0 .
Now we fix a ∈ X arbitrarily. For any x ∈ X, by the triangle inequality,
|F (x) − F (a)| ≤ |F (x) − Fn0 (x)| + |Fn0 (x) − Fn0 (a)| + |Fn0 (a) − F (a)|
≤ 2ε + |Fn0 (x) − F (a)|.
Since the function Fn0 is continuous on X, for any ε > 0, there exists a
neighborhood Va of a such that
Va ⊂ {x ∈ X| |Fn0 (x) − Fn0 (a)| < ε}.
Hence, for all x ∈ Va , |F (x) − F (a)| < 3ε. This means that F is continuous
at x = a.
Example 8.6. For each a ∈ R1+d , a mapping u(a) : L2 (R1+d ) → L2 (R1+d )
is defined as follows:
u(a)f = fa ,
f ∈ L2 (R1+d ),
where fa is the translation of f by a (see (8.4)). It is easy to see that u(a)
is unitary and
u(a + b) = u(a)u(b) = u(b)u(a),
a, b ∈ R1+d .
By applying Lemma 8.3 with (X, H , D) = (R1+d , L2 (R1+d ), C0∞ (R1+d )),
one can show that u(a) is strongly continuous in a. Hence, by Theorem 8.2,
there exists a unique (1 + d)-tuple (π0 , π1 , . . . , πd ) of strongly commuting
self-adjoint operators on L2 (R1+d ) such that
0
1
d
u(a) = eia π0 eia π1 · · · eia πd .
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By considering the strong derivative of u(0, . . . , 0, aµ , 0, . . . , 0)f (f ∈
C0∞ (R1+d )) at aµ = 0, one can show that
πµ = iDµ
(µ = 0, 1, . . . , d)
with Dµ being the generalized partial differential operator in the variable
xµ acting in L2 (R1+d ). As is seen, this example with 1 + d replaced by d
gives a proof of Theorem 1.32.
8.7
Translation Covariant QFT
Let F be a Hilbert space as before, D be a dense subspace in F and
(ϕr )sr=1 be an s-tuple of operator-valued distributions on R1+d with values
in L(F ) such that, for all f ∈ S (R1+d ) and r = 1, . . . , s, D ⊂ D(ϕr (f )) ∩
D(ϕr (f )∗ ) and ϕr (f )D ⊂ D, ϕr (f )∗ D ⊂ D. Suppose that there exists
a strongly continuous unitary representation U : R1+d → U(F ) of the
(1 + d)-dimensional translation group R1+d such that, for all f ∈ S (R1+d ),
r = 1, . . . , s and a ∈ R1+d ,
ϕr (fa ) = U (a)ϕr (f )U (a)−1
on D.
Then (F , D, U, (ϕr )sr=1 ) is called a translation covariant QFT on R1+d .
By Lemma 8.2, there exists a unique (1 + d)-tuple
P := (P0 , P1 , . . . , Pd )
of strongly commuting self-adjoint operators on F such that
0
1
d
U (a) = eia P0 /~ eia P1 /~ · · · eia Pd /~ ,
a ∈ R1+d .
The operator vector P is called the energy-momentum operator of the
quantum field (ϕr )sr=1 . The 0th component P0 and the d-tuple of the other
components
P := (P 1 , . . . , P d )
with P j := −Pj (j = 1, . . . , d) are respectively called the Hamiltonian
and the momentum operator of the translation covariant QFT under
consideration.11
By an application of Proposition 1.36, we have
eitP0 /~ Pµ e−itP0 /~ = Pµ ,
11 Sometimes (P
µ = 0, 1, . . . , d, t ∈ R.
1 , . . . , Pd ) also is called the momentum operator.
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447
Namely each Pµ is a conserved quantity.12 Hence, in particular, the energymomentum operator is a conserved quantity.
Since the energy-momentum operator P is strongly commuting, it
has the joint spectrum σJ (P ) (see Subsection 1.8.2). This set is called
the energy-momentum spectrum of the translation covariant QFT
(F , D, U, (ϕr )sr=1 ). This is an important quantity which characterizes the
QFT.
Since P is strongly commuting, ((P 1 )2 , . . . , (P d )2 ) also is strongly commuting (apply Theorem 1.28). Hence it follows from Corollary 1.6(ii) that
P 2 :=
d
X
(P j )2 =
j=1
d
X
Pj2
j=1
is a non-negative self-adjoint operator. Therefore one can define a nonnegative self-adjoint operator |P | by
1/2
|P | := P 2
.
We call |P | the modulus of the momentum operator P .
The dual space
(R1+d )∗ := {k = (k0 , k1 , . . . , kd )|kµ ∈ R, µ = 0, 1, . . . , d}
of the space-time R1+d is called the wave number vector space.13
By the de Broglie-Einstein-Planck relation, the momentum p of a quantum particle is related to the wave number vector k by
p = ~k.
Hence p ∈ ~(R
1+d ∗
) := {~k|k ∈ (R1+d )∗ }. Based on this relation, we write
R1+d
:= ~(R1+d )∗ .
p
: (x, p) 7→ R is given by
A natural bilinear form on R1+d × R1+d
p
X
xp :=
xµ pµ , x ∈ R1+d , p ∈ R1+d
.
p
µ=0
, the component p0 denotes the
For each p = (p0 , p1 , . . . , pd ) ∈ R1+d
p
energy coordinate and (p1 , . . . , pd ) the momentum coordinate. We
set
p := (p1 , . . . , pd ),
pj := −pj .
12 In a quantum system whose Hamiltonian is given by a self-adjoint operator H
on
a Hilbert space, independent of time t, a linear operator A (a quantum mechanical
quantity) on H is said to be a conserved quantity if its Heisenberg operator with
respect to H is independent of t: eitH/~ Ae−itH/~ = A, ∀t ∈ R. An n-tuple (A1 , . . . , An )
of linear operators is said to be a conserved quantity if each Aj is a conserved quantity.
13 Sometimes the symbol R1+d (resp. R1+d ) is used instead of R1+d (resp. (R1+d )∗ ).
x
k
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The above energy-momentum spectrum σJ (P ) is physically interpreted
as a subset of R1+d
.
p
An operator inequality for the components of P gives a condition on
σJ (P ):
Proposition 8.3. Let γ > 0 be a constant. Then
P0 ≥ γ|P |.
(8.16)
| p0 ≥ γ|p|}.
σJ (P ) ⊂ {p ∈ R1+d
p
(8.17)
if and only if
Proof. We denote by EP the joint spectral measure of P . Hence
supp EP = σJ (P ).
We first show by contraposition that (8.16) implies (8.17). Hence suppose that (8.17) does not hold. Then there exists a Borel set B ⊂ σJ (P )
such that, for all p ∈ B, p0 < γ|p| and EP (B) 6= 0. Let Kn := {p ∈
∞
R1+d | |p| ≤ n} (n ∈ N). Then R1+d = ∪∞
n=1 Kn . Hence B = ∪n=1 Bn with
Bn := Kn ∩ B. If EP (Bn ) = 0 for all n ∈ N, then EP (B) = 0. But this is a
contradiction. Hence, for some n0 , EP (Bn0 ) 6= 0. We set B0 := Bn0 . Then
B0 is bounded and EP (B0 ) 6= 0. Hence, for all non-zero ψ ∈ RanEP (B0 ),
ψ ∈ D(P0 ) ∩ D(|P |) and
Z
(p0 − γ|p|)dkEP (p)ψk2 < 0.
hψ, (P0 − γ|P |)ψi =
B0
Hence (8.16) does not hold. Thus (8.16) implies (8.17).
We next show that (8.17) implies (8.16). Hence suppose that (8.17)
holds. Let ψ ∈ D(P0 ). Then
Z
Z
γ 2 |p|2 dkEP (p)ψk2 .
p20 dkEP (p)ψk2 ≥
∞>
σJ (P )
σJ (P )
Hence ψ ∈ D(|P |). Therefore D(P0 ) ⊂ D(|P |). Moreover,
Z
hψ, P0 ψi =
p0 dkEP (p)ψk2
σJ (P )
Z
|p|dkEP (p)ψk2 = hψ, γ|P |ψi .
≥γ
σJ (P )
Hence (8.16) holds.
A condition for σJ (P ) such as (8.17) is called a spectral condition.
Proposition 8.3 gives a correspondence between a spectral condition and
an operator inequality for P0 and |P |.
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Proposition 8.3 will have an application to relativistic QFT discussed
in the next section.
In what follows, we sometimes write (F , D, P, (ϕr )sr=1 ) for the translation covariant QFT on R1+d .
In a translation covariant QFT, relations of eigenvalues of the Hamiltonian with those of the momentum operator also are important. To examine
this aspect, we note the following lemma:
Lemma 8.4. Let X be a Hilbert space and A, B be strongly commuting
self-adjoint operators on X . Suppose that σp (A) 6= ∅. Let λ ∈ σp (A) and
M := ker(A − λ) (the eigenspace of A with eigenvalue λ). Then:
(i) M reduces B.
(ii) If m := dim M < ∞, then there exists a CONS {ψj }m
j=1 of M such
that each ψj is an eigenvector of B.
Proof. (i) By Proposition 1.36(ii), AeisB ψ = eisB Aψ, ψ ∈ D(A). In
particular, for all ψ ∈ M , AeisB ψ = λeisB ψ. This means that eisB ψ ∈ M .
Hence, for all s ∈ R, eisB leaves M invariant. It follows that eisB M =
M for all s.14 Hence, for all s ∈ R, eisB can be regarded as a unitary
operator on M . Let PM be the orthogonal projection onto M . Then,
by the property of eisB just proved, PM eisB PM = eisB PM for all s ∈
R. Taking the adjoint of both sides and replacing −s with s, we have
PM eisB PM = PM eisB . Hence eisB PM = PM eisB . Therefore, for all
φ ∈ D(B), PM eisB φ = eisB PM φ. Considering the strong differentiation
of both sides at s = 0, we see that PM φ ∈ D(B) and BPM φ = PM Bφ.
Hence PM B ⊂ BPM . Thus M reduces B.
(ii) We denote by BM the reduced part of B to M . Since M is an mdimensional Hilbert space by the present assumption, the spectrum of BM
consists of only eigenvalues with finite multiplicity. By the self-adjointness
of BM , there exists a CONS {ψj }m
j=1 in which each ψj is an eigenvector of
BM .
A non-zero vector ψ in a Hilbert space X is called a simultaneous
eigenvector of some linear operators T1 , . . . , Tn on X if it is an eigenvector
of each Tj (j = 1, . . . , n).
Each vector ψj in Lemma 8.4(ii) is a simultaneous eigenvector of A and
B: Aψj = λψj , Bψj = µj ψj , where µj is an eigenvalue of BM .
14 In general, if W is a unitary operator on a Hilbert space H and D is a subspace in
H such that W D ⊂ D and W ∗ D ⊂ D, then W D = D (this is easy to prove).
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Thus Lemma 8.4 means that two strongly commuting self-adjoint operators have a simultaneous eigenvector if one of A and B has an eigenvalue
with finite multiplicity.
Theorem 8.3. Let (F , D, P, (ϕr )sr=1 ) be a translation covariant QFT.
Suppose that P0 is bounded from below and has a unique ground state Ψ0 .
Then Ψ0 is an eigenvector of each Pj (j = 1, . . . , d).
Proof. Let j = 1, . . . , d be fixed. Applying Lemma 8.4 with A = P0 and
B = Pj and using the assumption dim ker(P0 − E0 (P0 )) = 1, there exist
a unit vector Ψj ∈ ker(P0 − E0 (P0 )) and a real constant µj such that
Pj Ψj = µj Ψj . It is obvious that Ψ0 = cj Ψj for some constant cj 6= 0.
Hence Pj Ψ0 = µj Ψ0 .
Theorem 8.3 tells us that, in a translation covariant QFT, a ground
state of the Hamiltonian P0 is a simultaneous eigenvector of the energymomentum operator P if the ground state is unique.
Corollary 8.1. Let (F , D, P, (ϕr )sr=1 ) be a translation covariant QFT satisfying (8.16) (hence P0 ≥ 0). Suppose that E0 (P0 ) = 0 and P0 has a unique
ground state Ψ0 : P0 Ψ0 = 0. Then Pj Ψ0 = 0 (j = 1, . . . , d). In particular,
0 ∈ σJ (P ).
Proof. By Theorem 8.3, Pjq
Ψ0 = µj Ψ0 for some µj ∈ R (j = 1, . . . , d).
Pd
2
This implies that |P |Ψ0 =
j=1 µj Ψ0 . By (8.16), 0 = hΨ0 , P0 Ψ0 i ≥
qP
d
2
hΨ0 , |P |Ψ0 i ≥ 0. Hence hΨ0 , |P |Ψ0 i = 0. Hence
j=1 µj = 0, which
implies that µj = 0, j = 1, . . . , d. Thus Pj Ψ0 = 0.
By the result in the preceding paragraph, we have EPµ ({0})Ψ0 = Ψ0 .
Hence
EP ({0})Ψ0 = EP ({0} × · · · × {0})Ψ0
{z
}
|
(1+d) factors
= EP0 ({0})EP1 ({0}) · · · EPd ({0})Ψ0
= Ψ0 .
Hence 0 ∈ σJ (P ).
Remark 8.5. Let (F , D, H, (ϕr (t, ·))t∈R,r=1,...,s ) be a sharp-time QFT.
Suppose that there exists a d-tuple (P1 , . . . , Pd ) of strongly commuting selfadjoint operators on F such that each Pj (j = 1, . . . , d) strongly commutes
with H and, for all t ∈ R, a = (a1 , . . . , ad ) ∈ Rd and f ∈ S (Rd ),
1
d
1
d
ϕr (t, fa ) = eitH/~ eia P1 /~ · · · eia Pd /~ ϕr (f )e−itH/~ e−ia P1 /~ · · · e−ia Pd /~
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on D. Then we call (F , D, (H, P1 , . . . , Pd ), (ϕr (t, ·))t∈R,r=1,...,s ) a translation covariant sharp-time QFT.
8.8
Review of Some Aspects Related to the Theory of
Special Relativity
As is well known, there are two categories in classical field theory, namely,
the relativistic fields and the non-relativistic fields. This kind of classification is possible in QFT too. In the rest of this book, we will discuss both
of relativistic and non-relativistic quantum fields. But, before doing that,
we still need some mathematical preliminaries.
8.8.1
Minkowski space-time
We first recall a general concept of Minkowski space. Let V be a (1 + d)dimensional real vector space. A metric15 q : V × V → R on V is called a
Minkowski metric if there exists a basis (ǫµ )dµ=0 on V such that
q(ǫµ , ǫν ) = gµν ,
µ, ν = 0, . . . , d,
where the matrix (gµν ) is defined as follows:
g00 = 1,
gjj = −1,
j = 1, . . . , d,
(8.18)
µ 6= ν, µ, ν = 0, . . . , d.
(8.19)
Pd
P
d
µ
ν
It follows that, for all u =
µ=0 u ǫµ ∈ V and v =
ν=0 v ǫν ∈ V
µ ν
(u , v ∈ R),
gµν = 0,
q(u, v) =
d
X
gµν uµ v ν .
µ,ν=0
As is easily seen, q is an indefinite metric. The indefinite metric space (V, q)
is called a (1 + d)-dimensional Minkowski space.
Remark 8.6. There are many Minkowski metrics on V . Indeed, for each
bijective linear mapping T on V (i.e. T ∈ GL(V )), the mapping qT :
V × V → R defined by
qT (u, v) := q(T −1 u, T −1 v),
u, v ∈ V
(8.20)
is a Minkowski metric, since qT (T ǫµ , T ǫν ) = gµν and (T ǫµ )dµ=0 is a basis
of V . Conversely, it is not difficult to show that, for any Minkowski metric
15 See Footnote 3.
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q ′ on V , there exists a T ∈ GL(V ) such that q ′ = qT . Note that T as a
mapping from the Minkowski space (V, q) onto (V, qT ) preserves the metric:
qT (T u, T v) = q(u, v), u, v ∈ V . Hence (V, q) is isomorphic to (V, qT ). In
this sense, a Minkowski metric on V is essentially unique.
We now consider the case where V = R1+d and define a mapping gM :
R
× R1+d → R by
1+d
gM (x, y) := x0 y 0 −
d
X
j=1
xj y j ,
x = (xµ )dµ=0 , y = (y µ )µ=0 ∈ R1+d .
It is easy to see that gM is a metric on R1+d . Let {eµ }dµ=0 be the standard
basis of R1+d :
e0 = (1, 0, . . . , 0),
e1 = (0, 1, 0, . . . , 0),
..
.
ed = (0, 0, . . . , 0, 1).
Then
gM (eµ , eν ) = gµν ,
µ, ν = 0, . . . , d,
where (gµν ) is defined by (8.18) and (8.19). Hence gM is a Minkowski
metric on R1+d . In what follows, we write
xy := gM (x, y),
x, y ∈ M1+d ,
and call it the Minkowski inner product of x and y, although it is
indefinite.
In the special theory of relativity with space dimension d, the spacetime is taken to be the Minkowski space (R1+d , gM ) in which each element
in R1+d represents a space-time point. The Minkowski space (R1+d , gM )
in this sense is called the (1 + d)-dimensional Minkowski space-time.
We denote it by M1+d . In this context, x0 (resp. (x1 , . . . , xd )) in x =
(x0 , x1 , . . . , xd ) ∈ M1+d is called the time (resp. space) component.
The (1 + d) × (1 + d) matrix
g := (gµν )
is called the metric matrix. It follows that
g 2 = I1+d ,
(8.21)
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the (1 + d) × (1 + d) identity matrix. In terms of the metric matrix g, the
Minkowski inner product xy (x, y ∈ M1+d ) is written as
xy = x · gy = gx · y,
where x · y denotes the Euclidean inner product of R1+d :
x · y :=
1+d ∗
d
X
xµ y µ .
µ=0
We denote by (M ) the dual space of M1+d (i.e., the set of linear
functionals on M1+d ). For each µ = 0, 1 . . . , d, the mapping f µ : M1+d → R
defined by
f µ (x) := xµ ,
x ∈ M1+d ,
is an element of (M1+d )∗ . The {f µ }µ=0,...,d is a basis of (M1+d )∗ and called
the dual basis of {eµ }dµ=0 .
Each x ∈ M1+d is written as
x=
d
X
xµ eµ .
µ=0
Hence, for all k ∈ (M1+d )∗ ,
k(x) =
d
X
k(eµ )xµ =
µ=0
d
X
µ=0
kµ f µ
!
(x),
where kµ := k(eµ ), µ = 0, . . . , d. Hence
k=
d
X
kµ f µ .
µ=0
The real number kµ denotes the µth component of k with respect to the
dual basis {f µ }dµ=0 . We have
k(x) =
d
X
kµ xµ .
µ=0
We sometimes write this as kx or xk and call the mapping: (M1+d )∗ ×
M1+d ∋ (k, x) 7→ kx ∈ R the natural bilinear form on (M1+d )∗ × M1+d .
The dual space (M1+d )∗ also is a Minkowski space with metric
kl := k0 l0 −
d
X
j=1
kj lj ,
k, l ∈ (M1+d )∗ .
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For each x ∈ M1+d , the mapping x′ : M1+d → R defined by
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x′ (y) := xy,
y ∈ M1+d ,
is an element of (M1+d )∗ . Hence one can define a mapping ι : M1+d →
(M1+d )∗ by
ι(x) := x′ ,
x ∈ M1+d .
It is easy to see that ι is linear and bijective. Namely ι is an isomorphism
between M1+d and (M1+d )∗ . The linear functional ι(x) is called the dual
element of x. Note that this concept is independent of the choice of the
basis of M1+d .16
It is easy to see that
x′ = x0 f 0 −
d
X
xj f j .
j=1
′
Hence the coordinate representation (x µ )dµ=0 of x′ with respect to the basis
{f µ }dµ=0 is given as follows:
x′ 0 = x0 ,
x′ j = −xj , j = 1, . . . , d.
In terms of the metric matrix g, we have
x′ µ =
d
X
gµν xν .
ν=0
In what follows, we use the symbol xµ for x′ µ :
xµ :=
d
X
gµν xν .
(8.22)
ν=0
Then, as is seen from the above discussion, (xµ )dµ= is the coordinate representation of the dual element ι(x) of x ∈ M1+d with respect to the basis
{f µ }dµ=0 . This clarifies the meaning of the change from (xµ )µ to (xµ )µ .
The matrix g is invertible with g −1 = g. We denote by g µν the (µ, ν)component of g −1 :
g −1 = (g µν )µ,ν=0,...,d .
Then
d
X
ν=0
g
µν
gνρ = δρµ ,
d
X
gµν g νρ = δµρ ,
µ, ρ = 0, 1, . . . , d,
(8.23)
ν=0
16 This type of consideration is possible in any finite dimensional real vector space with
metric (not necessarily positive definite).
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where δµρ = δµρ is the Kronecker delta: δµρ = 1 for µ = ρ and δµρ = 0 for
µ 6= ρ.
It follows from (8.22) and (8.23) that
µ
x =
d
X
g µν xν .
ν=0
We have also
xy =
d
X
xµ yµ =
µ=o
d
X
xµ y µ ,
µ=o
2
We define the symbol x by
x2 := xx = (x0 )2 −
x, y ∈ M1+d .
d
X
(xj )2 .
j=1
In what follows, for notational simplicity, we use Einstein’s convention
ρλ···
for summation of object Tµν···
with an index which appears only once at
the upper index part and the lower one respectively. For example,
µλ···
Tµν···
:=
d
X
µλ···
Tµν···
.
µ=0
Following this convention, we have
xy = xµ yµ ,
xµ = gµν xν ,
xµ = g µν xν
etc.
The non-zero points in M
are classified into three categories:
1+d
A point x ∈ M
is said to be time-like (resp. space-like) if x2 > 0
2
(resp. x < 0).
A point x ∈ M1+d is said to be light-like if x2 = 0 and x 6= 0.
The set {x ∈ M1+d |x2 = 0} is called the light cone.
1+d
8.8.2
Lorentz group
A linear mapping Λ : M1+d → M1+d satisfying
(Λx)(Λy) = xy,
x, y ∈ M1+d
(8.24)
is called a Lorentz transformation on M1+d . Namely a Lorentz transformation is a linear mapping on M1+d which preserves the Minkowski inner
product. It follows that Λ is injective and hence bijective.
We denote by L (1+d) ) the set of Lorentz transformations on M1+d . It
is easy to see that L (1+d) is a group. The group L (1+d) is called the
(1 + d)-dimensional Lorentz group.
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In general, for a linear mapping T on M1+d , we denote by Tνµ (µ, ν =
0, 1, . . . , d) the (µ, ν)-component of the matrix representation of T with
respect to the standard basis {eµ }dµ=0 :
T eµ = Tµν eν ,
µ = 0, 1, . . . , d
where Einstein’s convention on the summation of ν is used. Hence
(T x)µ = Tνµ xν ,
x ∈ M1+d , µ = 0, 1, . . . , d.
In what follows, concerning Lorentz transformations Λ, we work with
matrix representations with respect to the standard basis {eµ }dµ=0 and identify Λ with its matrix representation (Λµν ). Then condition(8.24) is equivalent to the equation
t
ΛgΛ = g,
(8.25)
t
where Λ denotes the transposed matrix of Λ.
Multiplying (8.25) by Λg from the left and by Λ−1 g from the right and
using (8.21), we obtain
Λg t Λ = g.
(8.26)
This means that t Λ also is a Lorentz transformation, i.e., t Λ ∈ L (1+d) .
Taking the determinant of both sides on(8.25) and using det g 6= 0 and
det t Λ = det Λ, we obtain det Λ2 = 1. Hence
det Λ = ±1.
Also, computing the (0, 0)-component of both sides on (8.25) gives the
equation
d 2
X
2
(8.27)
Λj0 .
Λ00 = 1 +
j=1
Hence
Λ00
Therefore we obtain
2
≥ 1.
L (1+d) = L+↑ ∪ L+↓ ∪ L−↑ ∪ L−↓
with
L+↑ := {Λ ∈ L (1+d) | det Λ = 1, Λ00 ≥ 1},
L+↓ := {Λ ∈ L (1+d) | det Λ = 1, Λ00 ≤ −1},
L−↑ := {Λ ∈ L (1+d) | det Λ = −1, Λ00 ≥ 1},
L−↓ := {Λ ∈ L (1+d) | det Λ = −1, Λ00 ≤ −1}.
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457
Among these subsets, only L+↑ forms a subgroup of L (1+d) . The subgroup
L+↑ is called the proper or restricted Lorentz group.
An element of L+↑ is called a proper or restricted Lorentz transformation.
The following proposition shows that any proper Lorentz transformation
preserves the sign of the time component of a time-like vector:
Proposition 8.4. Let x ∈ M1+d be a time-like vector and Λ ∈ L+↑ .
(i) If x0 > 0, then (Λx)0 > 0.
(ii) If x0 < 0, then (Λx)0 < 0.
Proof. (i) We have
(Λx)0 = Λ00 x0 +
d
X
Λ0j xj
j=1
v
u d
uX
0 0
≥ Λ x − t (Λ0 )2 |x|
0
j
j=1
(by the Schwarz inequality).
v

u d
uX
|x| := t (xj )d 

j=1
2
Since x > 0 and x0 > 0, it follows that x0 > |x|. Hence
v


u d
X
u
(Λx)0 > Λ00 − t (Λ0j )2  |x|.
j=1
t
in L+↑ ,
Since Λ is also
qP
d
0 2
Λ00 −
j=1 (Λj )
(8.27) holds with Λ replaced by t Λ.
Hence
0
> 0. Thus (Λx) > 0.
(ii) In the present case, y := −x is a time-like vector with y 0 > 0.
Hence, by (i), (Λy)0 > 0. But Λy = −Λx. Hence (Λx)0 < 0.
Example 8.7. Let θ, a, a′ , b, b′ ∈ R satisfying
2
2
a2 = a′ = b2 = b′ = 1, ab = a′ b′ .
Then the matrix Λ = (Λµν ) defined below is a Lorentz transformation:


a cosh θ b sinh θ 0 0 · · · 0
 b′ sinh θ a′ cosh θ 0 0 · · · 0 




0
0
1 0 ··· 0


..
.
Λ=
.

0
.
0 1 0 .. 


..
.. 


.
.
0
···
··· ··· ··· 1
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It is easy to see that the following (i)–(iv) hold: (i) if a = a′ = 1, bb′ = 1,
then Λ ∈ L+↑ ; (ii) if a = a′ = −1, bb′ = 1, then Λ ∈ L+↓ ; (iii) if a = 1, a′ =
−1, bb′ = −1, then Λ ∈ L−↑ ; (iv) if a = −1, a′ = 1, bb′ = −1, then Λ ∈ L−↓ .
Therefore L±↑↓ are non-empty.
8.8.3
Dual operators on (M1+d )∗
For a linear operator T on M1+d , the linear operator T ′ : (M1+d )∗ →
(M1+d )∗ defined by
(T ′ k)(x) := k(T x),
k ∈ (M1+d )∗ , x ∈ M1+d
is called the dual operator of T . It is easy to see that
T ′ k = Tνµ kµ f ν ,
k = kµ f µ ∈ (M1+d )∗ .
Hence
(T ′ k)ν = Tνµ kµ .
It follows from (8.26) that, for all Λ ∈ L (1+d) ,
(Λ′ k)(Λ′ l) = kl,
k, l ∈ (M1+d )∗ ,
i.e., Λ′ is a Lorentz transformation on (M1+d )∗ .
8.8.4
↑
The Lie algebra of L+
It is easy to see that L+↑ is a linear Lie group (see Appendix F). Hence its
Lie algebra exists. We denote it by l↑+ :
l↑+ := {X ∈ M1+d (R)|etX ∈ L+↑ , ∀t ∈ R}.
An explicit form of this Lie algebra is given as follows:
Proposition 8.5.
l↑+ = {X ∈ M1+d (R)|t Xg + gX = 0}.
(8.28)
In particular,
dim l↑+ =
d(d + 1)
.
2
(8.29)
Proof. Let X ∈ l↑+ . Then, for all s ∈ R, t (esX )gesX = g, which is equivat
lent to es X gesX = g, ∀s ∈ R. Differentiating the both sides in s at s = 0,
we have
t
Xg + gX = 0.
(8.30)
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Conversely, suppose that X ∈ M1+d (R) satisfies (8.30). Then t X =
g(−X)g. Hence, using g 2 = I1+d , we have for all s ∈ R
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t
es X = esg(−X)g = ge−sX g.
t
t
This implies that es X gesX = g. Since es X = t (esX ), it follows that
esX ∈ L (1+d) . Hence (det esX )2 = 1. Since esX is continuous in s and
det e0X = 1, (e0X )00 = 1, it follows that det esX = 1 and (esX )00 ≥ 1, s ∈ R.
Hence esX ∈ L+↑ , s ∈ R. Thus (8.28) holds.
Let X ∈ l↑+ and Y := gX. Then, by (8.30), Y = −t Y . This means that
P
Y is anti-symmetric. Hence Y = 0≤µ<ν≤d Yνµ eνµ , where Yνµ is the (µ, ν)
component of Y and eνµ (µ < ν) is the anti-symmetric matrix satisfying
(eνµ )α
β = δµα δνβ − δµβ δνα (α, β = 0, . . . , d). Hence
X
X = gY =
Yνµ geνµ .
0≤µ<ν≤d
It is easy to see that geνµ ∈ l↑+ and {geνµ }0≤µ<ν≤d is a set of linearly in-
dependent matrices. The cardinal number of this set is d(d + 1)/2. Thus
(8.29) follows.
8.8.5
Poincaré group
The set
P (1+d) := {(a, Λ)|a ∈ R1+d , Λ ∈ L (1+d) }
is a group with the following operation:
(a2 , Λ2 )(a1 , Λ1 ) := (a2 + Λ2 a1 , Λ2 Λ1 ),
a1 , a2 ∈ R1+d , Λ1 , Λ2 ∈ L (1+d) .
The group P (1+d) is called the (1 + d)-dimensional Poincaré group.
Each element (a, Λ) ∈ P (1+d) can be regarded as a transformation on
1+d
M
defined by
(a, Λ)x := Λx + a,
x ∈ M1+d .
Under this identification, the product of two elements in P (1+d) is the
composition of those as transformations and P (1+d) is a transformation
group on M1+d .
The group P (1+d) is a topological group such that the convergence of
a sequence is defined as follows. A sequence {(an , Λn )}n in P (1+d) is said
to converge to (a, Λ) ∈ P (1+d) if limn→∞ |an − a| = and limn→∞ (Λn )µν =
Λµν , µ, ν = 0, 1, . . . , d. In this case we write limn→∞ (an , Λn ) = (a, Λ).
It is easy to see that the subset
↑
P+
:= {(a, Λ)|a ∈ R1+d , Λ ∈ L+↑ }
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of P (1+d) is a subgroup of P (1+d) . This subgroup is called the (1 + d)dimensional proper or restricted Poincaré group.
The Lorentz group L (1+d) and the translation group R1+d can be identified with the subgroup {(0, Λ)|Λ ∈ L (1+d) } and {(a, I)|a ∈ R1+d } of the
Poincaré group P (1+d) respectively.17
8.9
Axioms for Relativistic QFT
Schematically speaking, a translation covariant QFT discussed in Section
8.7 becomes a relativistic QFT (RQFT) if it gets a structure coming from
principles of special relativity in accordance with quantum mechanics. As
is well known, the special theory of relativity is based on the following two
principles:
(R.1) Invariance of light speed c (causality).
(R.2) Poincaré covariance (special relativity).
By incorporating these principles into the idea of QFT in a suitable manner,
one obtains a set of axioms for a RQFT. Four kinds of axioms for a RQFT
have been proposed so far: (i) the Gårding-Wightman axioms (1954); (ii)
the Wightman axioms (1956); (iii) the Osterwalder-Schrader axioms (1973);
(iv) the Nelson axioms (1973). The first two formulate minimum properties
that a RQFT on the four-dimensional Minkowski space-time M4 (more generally the (1 + d)-dimensional Minkowski space-time M1+d ) should satisfy,
while the last two are concerned with formulations of a Euclidean QFT,
a QFT on the Euclidean space R4 , which is heuristically obtained by an
analytic continuation of a RQFT on M4 to a region consisting of points
in C4 whose time component is purely imaginary. Below we outline these
axioms.
Remark 8.7. The axioms for relativistic quantum fields presented below
are of trial nature, because, in the case of 4-dimensional space-time, any
non-trivial model satisfying the axioms has not yet been shown to exist,
where a “non-trivial model” means that it is not equivalent to a free quantum field (for free quantum fields, see Chapters 10–12; it is shown that
relativistic free quantum fields obey the axioms). To show existence of a
non-trivial relativistic quantum field model on the 4-dimensional space-time
17 Strictly speaking, L (1+d) (resp. R1+d ) is isomorphic to {(0, Λ)|Λ ∈ L (1+d) } (resp.
{(a, I)|a ∈ R1+d }) under the mapping:Λ 7→ (0, Λ) (resp. a 7→ (a, I)).
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is left as a very difficult open problem which is closely related to one of the
millennium problems [Jaffe and Witten (2000)].
In what follows, we use the physical unit system where c = 1 unless
otherwise stated.
8.9.1
The Gårding-Wightman axioms
We first discuss the Gårding-Wightman axioms. There may be many kinds
of quantum fields corresponding to those of classical fields. For simplicity,
we restrict ourselves to the case of a neutral (Hermitian) quantum scalar
field on the (1 + d)-dimensional Minkowski space-time M1+d .18
A neutral (or Hermitian) quantum scalar field theory on M1+d is
defined to be a quintuple (H , U, φ, D, Ψ0 ) consisting of a separable infinite
dimensional Hilbert space H , a strongly continuous unitary representation
↑
U of P+
on H , an operator-valued distribution φ on R1+d with values in
L(H ), a common domain D of φ and a unit vector Ψ0 ∈ H , satisfying the
following (GW.1)–(GW.5):
(GW.1) (quantum field)
(a) For each f ∈ S (R1+d ), φ(f )D ⊂ D.
(b) (Hermiticity) If f ∈ S (R1+d ) is a real-valued, then φ(f ) ↾ D is
a symmetric operator.
The operator-valued distribution φ is called the neutral (Hermitian)
quantum scalar field.
(GW.2) (Poincaré covariance)
↑
(U.1) For all (a, Λ) ∈ P+
, U (a, Λ)D ⊂ D.
↑
1+d
(U.2) For all f ∈ S (R ) and (a, Λ) ∈ P+
,
−1
U (a, Λ)φ(f )U (a, Λ) = φ(f(a,Λ) ) on D,
where
f(a,Λ) (x) = f (Λ−1 (x − a)), x ∈ R1+d .
(U.3)
(8.31)
(8.32)
↑
For all (a, Λ) ∈ P+
,
U (a, Λ)Ψ0 = Ψ0 .
(8.33)
The vector Ψ0 is called the vacuum. Such a vector Ψ0 is
unique up to constant multiples, i.e., if a vector Ψ ∈ H satis↑
fies U (a, Λ)Ψ = Ψ for all (a, Λ) ∈ P+
, then Ψ = αΨ0 for some
α ∈ C.
18 For axioms for a more general quantum field, see, e.g., [Bogoliubov et al. (1975);
Streater and Wightman (1964)].
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It follows that R1+d ∋ a 7→ U (a, I) is a strongly continuous unitary
representation of the translation group R1+d . Hence, by Lemma 8.2, there
exists a (1 + d)-tuple P := (Pµ )dµ=0 of strongly commuting self-adjoint
operators on H such that
0
1
d
U (a, I) = eia P0 /~ eia P1 /~ · · · eia Pd /~ ,
a ∈ R1+d .
(8.34)
The operator vector P is called the energy-momentum operator of the
QFT under consideration.
Since the (1 + d)-tuple P is strongly commuting, it follows from a general theory of strongly commuting self-adjoint operators that Pµ and Pν
commute:
[Pµ , Pν ] = 0
(8.35)
on D(Pµ Pν ) ∩ D(Pν Pµ ).
The energy-momentum space in the special theory of relativity is given
by
M1+d
:= {~k|k ∈ (M1+d )∗ }.
p
(GW.3) (spectral condition) Let
V+ := {p = (p0 , p1 , . . . , pd ) ∈ M1+d
|p0 > 0, p2 > 0},
p
(8.36)
called the forward light cone. Then the joint spectrum σJ (P ) is a
subset of V + (the closure of V+ ):
|p0 ≥ 0, p2 ≥ 0}.
σJ (P ) ⊂ V + = {p ∈ M1+d
p
(8.37)
(GW.4) (locality or microscopic causality) If supp f (the support
of f ∈ S (R1+d )) is space-like separated from supp g (g ∈ S (R1+d )),
i.e., (x − y)2 < 0 for all x ∈ supp f and y ∈ supp g, then φ(f ) and φ(g)
commute on D:
[φ(f ), φ(g)]Ψ = 0,
Ψ ∈ D.
(GW.5) (cyclicity of the vacuum) The subspace D is algebraically spanned by Ψ0 and vectors φ(f1 ) · · · φ(fn )Ψ0 (n ∈ N, fj ∈
S (R1+d ), j = 1, . . . , n):
D = span {Ψ0 , φ(f1 ) · · · φ(fn )Ψ0 | f1 , . . . , fn ∈ S (R1+d ), n ≥ 1}.
The properties (GW.1)–(GW.5) are called the Gårding-Wightman
axioms.
Some comments on these axioms may be in order.
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Property (GW.1)-(a) is just a property that a quantum field should
posses (see Definition 8.2(QF.1)), while (GW.1)-(b) reflects the speciality
that the quantum field is neutral. A neutral (classical or quantum) field
means that it has no electric charge. A neutral classical scalar field on
M1+d is given by a real-valued function φcl onR M1+d . Hence, for all f ∈
SR (R1+d ) := {f ∈ S (R1+d )|f is real-valued}, R1+d φcl (x)f (x)dx is a real
number. Corresponding to this fact, the expectation value of a neutral
quantum scalar field φ(f ) in any physical state should be a real number
if f is real-valued. The subspace D in the Gårding-Wightman axioms is
interpreted to be included in the subspace of physical states. Hence, for all
Ψ ∈ D and f ∈ SR (R1+d ) , hΨ, φ(f )Ψi should be a real number. This is
equivalent to (GW.1)-(b).
A proper Poincaré transformation (a, Λ), which is a transformation on
the space-time M1+d , should accompany a transformation on the state
space. This transformation is U (a, Λ). The unitarity of U (a, Λ) is due
to the requirement that such a transformation on the state space should
preserve the transition probability between any two states and Wigner’s
↑
theorem.19 Since U is a representation of P+
, it obeys the following com↑
position law: for all (a1 , Λ1 ), (a2 , Λ2 ) ∈ P+
,
U (a2 , Λ2 )U (a1 , Λ1 ) = U (Λ2 a1 + a2 , Λ2 Λ1 ).
(8.38)
↑
In particular, for all (a, Λ) ∈ P+
,
U (a, Λ)−1 = U (−Λ−1 a, Λ−1 ).
(8.39)
Property (U.2) is a transformation law of the quantum field φ under
proper Poincaré transformations. The definition of the function f(a,Λ) given
by (8.32) is obviously extended to any function f ∈ L2 (R1+d ). It is easy to
↑
see that, for all (a, Λ) ∈ P+
and f ∈ L2 (R1+d ), f(a,Λ) is in L2 (R1+d ) and
kf(a,Λ) k = kf k.
Hence the mapping u(a, Λ) on L2 (R1+d ) defined by
u(a, Λ)f := f(a,Λ) ,
f ∈ L2 (R1+d )
(8.40)
19 Wigner’s theorem. Let H be a Hilbert space with dim H ≥ 2 and T be a mapping
on H such that, for all Φ, Ψ ∈ H , | hT Φ, T Ψi | = | hΦ, Ψi |. Then there exists a linear
isometry TL on H (i.e., TL is a linear operator on H satisfying kTL Ψk = kΨk, Ψ ∈
H ) or an anti-linear isometry TA (i.e., TA is an anti-linear operator on H satisfying
kTA Ψk = kΨk, Ψ ∈ H ) such that T Ψ = eiθL (Ψ) TL Ψ, Ψ ∈ H or T Ψ = eiθA (Ψ) TA Ψ, Ψ ∈
H , where θL (Ψ) and θA (Ψ) are real constants which may depend on Ψ. A detailed
proof of this theorem is given in [Arai and Ezawa (1999b)] (cf. also [Wigner (1959)]).
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↑
is a linear isometry. For all (a2 , Λ2 ), (a1 , Λ1 ) ∈ P+
and f ∈ L2 (R1+d ),
(u(a2 , Λ2 )u(a1 , Λ1 )f )(x) = (u(a1 , Λ1 )f )(Λ−1
2 (x − a2 ))
−1
= f (Λ−1
1 (Λ2 (x − a2 ) − a1 ))
= f ((Λ2 Λ1 )−1 (x − Λ2 a1 − a2 ))
= f(Λ2 a1 +a2 ,Λ2 Λ1 ) (x)
= (u((a2 , Λ2 )(a1 , Λ1 ))f )(x),
Hence
a.e.x ∈ R1+d .
u(a2 , Λ2 )u(a1 , Λ1 ) = u((a2 , Λ2 )(a1 , Λ1 )).
(8.41)
In particular,
u(a, Λ)u(−Λ−1 a, Λ−1 ) = u(0, I) = I.
Hence u(a, Λ) is surjective. Thus u(a, Λ) is unitary.
By using the Lebesgue dominated convergence, one can easily show that,
for all f, g ∈ C0∞ (R1+d ), the mapping: (a, Λ) 7→ hg, u(a, Λ)f i is continuous.
Hence, by Lemma 8.3, u(a, Λ) is strongly continuous in (a, Λ). Thus we
arrive at the following theorem.
↑
Theorem 8.4. The mapping u : P+
→ U(L2 (R1+d )) : (a, Λ) 7→ u(a, Λ) is
↑
a strongly continuous unitary representation of P+
.
S
S
We note that, if fn , f ∈ S (R1+d ), fn → f (n → ∞), then u(a, Λ)fn →
u(a, Λ)f (n → ∞). Hence the mapping u(a, Λ)′ φ : S (R1+d ) → L(H )
defined by
(u(a, Λ)′ φ)(f ) := φ(u(a, Λ)f ),
f ∈ S (R1+d )
is an operator-valued distribution. We call u(a, Λ)′ φ the Poincaré transform of the quantum field φ associated with (a, Λ).
Equation (8.31) is written
U (a, Λ)φU (a, Λ)−1 = u(a, Λ)′ φ
on D.
(8.42)
By this property, the quantum field φ is said to be unitarily imple↑
mentable under the action of the proper Poincaré group P+
.
R In terms of the operator-valued distribution kernel φ(x) of φ (φ(f ) =
φ(x)f (x)dx), (8.31) is symbolically written
R1+d
U (a, Λ)φ(x)U (a, Λ)−1 = φ(Λx + a).
In a physical picture, a vacuum is supposed to be a state in which there
exist no observed elementary particles or no visible matters in the spacetime. In such a state, every point in the space-time would be on an equal
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footing. Therefore a vacuum should be a state invariant under all proper
Poincaré transformations. This requirement is (U.3).
The joint spectrum σJ (P ) of P is interpreted as the set of measured
values of the energy-momentum P . In the special theory of relativity, an
energy-momentum vector of a “normal” motion (time-like motion) is timelike and the energy component is non-negative. Hence (8.37) is a natural
condition.
In the special theory of relativity, space-like separated points are
causally independent. Hence field measurements should be mutually independent if they are made at those points. On the other hand, independence of quantum mechanical observables is expressed as the commutativity
of them. This is a physical meaning of property (GW.4).
Physically a quantum field is supposed to have the function that creates
elementary particles from the vacuum or annihilates them into the vacuum.
Each state may be described by a configuration of elementary particles
whose number is finite and such a state may be given by a finite linear
combination of the vacuum Ψ0 and vectors of the form φ(f1 ) · · · φ(fn )Ψ0
(n ∈ N, fj ∈ S (R1+d ), j = 1, . . . , n). This is a physical picture behind
property (GW.5).
Let us see some consequences derived from (GW.1)–(GW.5).
Proposition 8.6. The vacuum Ψ0 is a simultaneous eigenvector of
P0 , . . . , Pd with eigenvalue 0:
Pµ Ψ0 = 0,
µ = 0, 1, . . . , d.
In particular, 0 ∈ σJ (P ).
Proof. By (8.33) and (8.34), we have for all t ∈ R and µ = 0, 1, . . . , d
eitPµ Ψ0 = Ψ0 .
By considering the strong differentiation of the both sides at t = 0, one sees
that Ψ0 ∈ D(Pµ ) and Pµ Ψ0 = 0.
Proposition 8.7. P0 ≥ |P |.
Proof. This follows from (GW.3) and an application of Proposition 8.3
with γ = 1.
Since each point of the joint spectrum σJ (P ) of P is considered as a point
in (M1+d )∗ , it is natural to assume that, for any Lorentz transformation Λ
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on M1+d , the operator vector P is transformed by the dual operator Λ′ as
Λ′ P :
Λ′ P = (Λµ0 Pµ , Λµ1 Pµ , . . . , Λµd Pµ ).
Since P is strongly commuting, (Λ′ P )µ = Λνµ Pν is essentially self-adjoint
for all µ = 0, 1, . . . , d. Moreover, the (1 + d)-tuple
Λ′ P := (Λµ0 Pµ , Λµ1 Pµ , . . . , Λµd Pµ )
of self-adjoint operators is strongly commuting, where Λµν Pµ denotes the
closure of Λµν Pµ .
The strong commutativity of P implies that, for all a ∈ R1+d ,
aP := a0 P0 +
d
X
aj Pj
j=1
is essentially self-adjoint and
0
1
d
eiaP /~ = eia P0 /~ eia P1 /~ · · · eia Pd /~ .
Hence
U (a, I) = eiaP /~ .
(8.43)
Proposition 8.8 (transformation law of the energy-momentum).
For all Λ ∈ L+↑ ,
U (0, Λ)Pµ U (0, Λ)−1 = (Λ′ P )µ ,
µ = 0, 1, . . . , d.
(8.44)
Proof. For all a ∈ R1+d , we have by (8.38) and (8.39)
U (0, Λ)U (a, I)U (0, Λ)−1 = U (Λa, I).
Hence
′
U (0, Λ)eiaP /~ U (0, Λ)−1 = ei(Λa)P /~ = eia(Λ P )/~ .
Taking a = teµ ∈ R1+d , t ∈ R, one has
′
U (0, Λ)eitPµ /~ U (0, Λ)−1 = eit(Λ P )µ /~ .
Hence (8.44) follows.
As a corollary to Proposition 8.8, we find the following natural fact:
Corollary 8.2. (Lorentz invariance of the energy-momentum spectrum)
For all Λ ∈ L+↑ ,
Λ′ σJ (P ) = σJ (P ).
(8.45)
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Proof. Let S = Λ′ σJ (P ). Then it is easy to see that S is a closed set. By
the spectral mapping theorem,
σJ (Λ′ P ) = S = S.
(8.46)
On the other hand, by (8.44),
σJ (Λ′ P ) = σJ (P ),
which, together with (8.46), gives (8.45).
One can derive the equations of motion for the quantum field φ:
Theorem 8.5. Suppose that the following conditions are satisfied:
(i) D ⊂ D(Pµ ), µ = 0, 1, . . . , d.
(ii) For all f ∈ S (R1+d ) and µ = 0, 1, . . . , d, Pµ D ⊂ D(φ(f )).
(iii) For all Ψ ∈ D and µ = 0, 1 . . . , d, f ∈ S (R1+d ), the H -valued
function :R ∋ t → φ(f )eitPµ Ψ is strongly continuous.
Then, for all Ψ ∈ D, f ∈ S (R1+d ) and n ∈ N,
Dµn φ(f )Ψ =
i
[Pµ , Dµn−1 φ(f )]Ψ
~
(µ = 0, 1, . . . , d).
(8.47)
Proof. Putting Λ = I in (8.31) and a = teµ (t ∈ R), we have
eitPµ /~ φ(f )e−itPµ /~ = φ(f(teµ ,I) ),
on D.
Hence one can apply Theorem 8.1 to obtain the desired result.
8.9.2
Angular momentum
In this subsection we show how the concept of angular momentum appears
in the framework of the Gårding-Wightman axioms. For each Λ ∈ L+↑ , we
set
U0 (Λ) := U (0, Λ).
Then U0 : L+↑ → U(H ) is a strongly continuous unitary representation of
L+↑ .
jk
1+d
by
For 1 ≤ j, k ≤ d, j 6= k, and θ ∈ R, a mapping R (θ) on M
(Rjk (θ)x)µ := xµ ,
jk
j
j
µ 6= j, k,
(R (θ)x) := x cos θ − xk sin θ,
(Rjk (θ)x)k := xj sin θ + xk cos θ,
x ∈ M1+d .
is defined
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It is easy to see that Rjk (θ) ∈ L+↑ . Geometrically Rjk (θ) means the Euclidean rotation of angle θ around the origin in the xj -xk plane.
Also we define R0j (θ) ∈ L+↑ (j = 1, . . . , d) by
(R0j (θ)x)µ := xµ ,
0j
0
0
µ 6= 0, j,
(R (θ)x) := x cosh θ + xj sinh θ,
(R0j (θ)x)j := x0 sinh θ + xj cosh θ.
Geometrically R0j (θ) means the Lorentz rotation of angle θ around the
origin in the x0 -xj plane. Also we introduce Rj0 (θ) ∈ L+↑ by
Rj0 (θ) := R0j (−θ),
θ ∈ R.
As is easily seen, for each (µ, ν) (µ, ν = 0, 1, . . . , d, µ 6= ν), {Rµν (θ)}θ∈R
is a subgroup of L+↑ and {U0 (Rµν (θ))}θ∈R is a strongly continuous oneparameter unitary group on H . Hence, by Stone’s theorem, there exists a
unique self-adjoint operator M µν on H such that
U0 (Rµν (θ)) = e−iθM
µν
/~
.
(8.48)
We set
M µµ := 0,
, µ = 0, 1, . . . , d.
Since Rµν (−θ) = Rνµ (θ), it follows that
M µν = −M νµ ,
µν
µ, ν = 0, 1, . . . , d, µ 6= ν.
The d(d + 1)/2-tuple (M )0≤µ<ν≤d is called the angular momentum
of the quantum field φ. In particular, we call (M jk )1≤j<k≤d the orbital
angular momentum.
We first note the following important fact:
Theorem 8.6 (conservation of the orbital angular momentum).
For all t ∈ R and j, k = 1, . . . , d,
eitP0 /~ M jk e−itP0 /~ = M jk .
Proof. Let θ ∈ R. Then
eitP0 /~ e−iθM
jk
/~ −itP0 /~
e
(8.49)
= U (te0 , I)U (0, Rjk (θ))U (−te0 , I)
= U (te0 − tRjk (θ)e0 , Rjk (θ)).
But Rjk (θ)e0 = e0 . Hence the last operator in the above equation is equal
jk
to U (0, Rjk (θ)) = e−iθM /~ . Therefore
eitP0 /~ e−iθM
jk
/~ −itP0 /~
e
= e−iθM
jk
/~
.
By this equation and the unitary covariance of functional calculus, we obtain (8.49).
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Let us calculate commutation relations among M µν ’s and P µ ’s (recall
that P µ := g µν Pν ).
Theorem 8.7. Suppose that D ⊂ D(M µν Pλ ) ∩ D(Pλ M µν ) for all µ, ν, λ =
0, . . . , d. Then, for all µ, ν, λ = 0, . . . , d,
[M µν , P λ ] = −i~(g µλ P ν − g νλ P µ )
(8.50)
on D.
Proof. Let Φ, Ψ ∈ D and j, k, ℓ = 1, . . . .d, µ = 0, 1, . . . , d. Then, by (8.44),
we have
D
E D
E
jk
jk
eiθM /~ Φ, Pµ Ψ = Rjk (θ)νµ Pν Φ, e−iθM /~ Ψ .
(8.51)
The following equations follow from the definition of Rµν (θ):
Rjk (θ)ν0 Pν Φ = P 0 Φ,
Rjk (θ)νℓ Pν Φ = δℓj [(cos θ)Pj Φ + sin θ)Pk Φ] + δℓk [(− sin θ)Pj Φ + (cos θ)Pk Φ].
By differentiating the both sides of (8.51) in θ at θ = 0, we see that
M jk Φ, P0 Ψ = P0 Φ, M jk Ψ ,
D
E i
i
M jk Φ, Pℓ Ψ = − (δℓj Pk − δℓk Pj )Φ, Ψ +
Pℓ Φ, M jk Ψ .
~
~
Hence it follows that
[M jk , P0 ]Ψ = 0,
[M jk , Pℓ ]Ψ = i~(δℓj Pk − δℓk Pj )Ψ.
Similarly one can show that
[M 0j , Pℓ ]Ψ = 0 (ℓ 6= j),
[M 0j , P0 ]Ψ = i~Pj Ψ,
[M 0j , Pj ]Ψ = i~P0 Ψ.
(8.52)
Rewriting these equations with P0 = P 0 and Pj = −P j (j = 1, . . . , d), we
obtain (8.50).
The (0, j) component M 0j of the angular momentum is not conserved,
but, the time-development of it obeys a simple beautiful formula:
Theorem 8.8. Suppose that the same assumption as in Theorem 8.7 holds
and that D ∩ E (P0 ) is dense in H , where E (P0 ) is the set of entire analytic
vectors of P0 (see Subsection 1.7). Then, for all t ∈ R and j = 1, . . . , d,
eitP0 /~ M 0j e−itP0 /~ = M 0j + tPj
on D ∩ E (P0 ).
(8.53)
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Proof. Let Φ, Ψ ∈∈ D ∩ E (P0 ). Then
∞ n
E
D
E X
1 D n
it
P0 Φ, M 0j e−itP0 /~ Ψ
Φ, eitP0 /~ M 0j e−itP0 /~ Ψ =
~
n!
n=0
D
E
= Φ, M 0j e−itP0 /~ Ψ
∞ n
X
it
1
+
Cn ,
(8.54)
~
n!
n=1
where Cn := P0n−1 Φ, P0 M 0j e−itP0 /~ Ψ . Using (8.52), we have
D
E
Cn = P0n−1 Φ, (M 0j P0 − i~Pj )e−itP0 /~ Ψ
D
E
D
E
= P0n−2 Φ, P0 M 0j P0 e−itP0 /~ Ψ − i~ P0n−1 Φ, e−itP0 /~ Pj Ψ ,
where we have used the strong commutativity of P0 and Pj . Repeating this
process, we obtain
D
E
D
E
Cn = M 0j Φ, P0n e−itP0 /~ Ψ − ni~ P0n−1 Φ, e−itP0 /~ Pj Ψ .
Putting this into (8.54) and using the fact that e−itP0 /~ Ψ ∈ E (P0 ) and
eitP0 /~ e−itP0 /~ = I, we obtain
D
E
Φ, eitP0 /~ M 0j e−itP0 /~ Ψ = M 0j Φ, Ψ +t hΦ, Pj Ψi = Φ, (M 0j + tPj )Ψ .
Since D ∩ E (P0 ) is dense by the present assumption, (8.53) follows.
With regard to commutation relations of M µν ’s, we give only a remark.
For r ∈ N, we denote by Mr (K) the set of r × r matrices whose components
are elements of K (K = R or C). For all µ, ν = 0, 1, . . . , d, we define
mµν ∈ M1+d (C) by
µα ν
(mµν )α
δβ − g να δβµ ),
β := i~(g
Then it is easy to see that, for all θ ∈ R,
Rµν (θ) = e−iθm
µν
µ, ν = 0, . . . , d.
/~
.
(8.55)
By direct computations, one can show that the following commutation relations hold:
[mµν , mρσ ] = −i~(g µρ mνσ − g µσ mνρ − g νρ mµσ + g νσ mµρ ).
By (8.55), −imµν ∈ l↑+ . It is easy to see that the set {−imµν |0 ≤ µ < ν ≤ d}
is linearly independent and the number of the elements in it is d(d + 1)/2.
Thus it is a basis of l↑+ .
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By (8.48) and (8.55), we have
µν
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U0 (e−iθm ) = e−iθM ,
µ, ν = 0, . . . , d, θ ∈ R.
Hence, by analogy with the theory of finite dimensional representations of
Lie algebras, the correspondence: imµν 7→ iM µν may give an infinite dimensional representation of the Lie algebra l↑+ . But it would need more
conditions for U0 (Λ) to justify this scheme in general, since each M µν may
be unbounded. If the scheme is justified, then one can expect that M µν ’s
satisfy the same commutation relation as those of mµν ’s on a suitable subspace:
[M µν , M ρσ ] = −i~(g µρ M νσ − g µσ M νρ − g νρ M µσ + g νσ M µρ )
(8.56)
on a subspace. For a structure related to this aspect, see Problem 3.
Commutations relations (8.35), (8.50) and (8.56) suggest that
{P µ , Mµν |µ, ν = 0, 1, . . . , d} may be a representation of a Lie algebra. This
Lie algebra is called the Poincaré algebra associated with M1+d .
8.9.3
Wightman axioms
Let (H , U, φ, D, Ψ0 ) be a neutral quantum scalar field theory on M1+d .
Then, by a general theory in Section 8.5, for each n ∈ N, there exists a
unique tempered distribution Wn on (R1+d )n such that, for all f1 , . . . , fn ∈
S (R1+d ),
Wn (f1 × · · · × fn ) = hΨ0 , φ(f1 ) · · · φ(fn )Ψ0 i .
(8.57)
The tempered distribution Wn is called the n-point Wightman distribution for the quantum field φ. The distribution kernel of Wn is denoted
Wn (x1 , . . . , xn ):
Z
Wn (f ) = Wn (x1 , . . . , xn )f (x1 , . . . , xn )dx1 · · · dxn .
In terms of the operator-valued distribution kernel φ(x) of the quantum
field φ, Wn (x1 , . . . , xn ) is written
Wn (x1 , . . . , xn ) = hΨ0 , φ(x1 ) · · · φ(xn )Ψ0 i
as a symbolical expression for (8.57).
One can characterize a QFT in terms of Wightman distributions instead
of using a Hilbert space and operators acting in it. This approach leads one
to the following axioms [Wightman (1956); Streater and Wightman (1964)].
A neutral quantum scalar field theory is a sequence {Wn ∈
′
S (R(1+d)n )}∞
n=0 (W0 := 1) of tempered distributions satisfying the following (W.1)–(W.6):
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(W.1) (Hermiticity) For all f ∈ S (R(1+d)n ),
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Wn (f )∗ = Wn (f † ),
n ≥ 1,
where f † is the function defined by
f † (x1 , . . . , xn ) := f (xn , xn−1 , . . . , x1 )∗
(8.58)
for all (x1 , . . . , xn ) ∈ (R1+d )n .
↑
(W.2) (relativistic invariance) For all (a, Λ) ∈ P+
and f ∈
(1+d)n
S (R
),
Wn (f ) = Wn (f(a,Λ) ),
n ≥ 1,
where
f(a,Λ) (x1 , . . . , xn ) := f (Λ−1 (x1 − a), . . . , Λ−1 (xn − a)).
(W.3) (positivity) For all N ∈ Z+ , f0 ∈ C, fn ∈ S (R(1+d)n ) (n =
1, . . . , N ),
N
X
n,m=0
Wn+m (fn† × fm ) ≥ 0.
(W.4) (spectral condition) For each n ≥ 2, there exists a tempered
(1+d)(n−1)
distribution Fn−1 ∈ S ′ (Rp
) such that
supp Fn−1 ⊂ (V + )n−1 := {(p1 , . . . , pn−1 )|pj ∈ V + , j = 1, . . . , n − 1}
and
Wn (x1 , . . . , xn )
Z
Pn−1
= Fn−1 (p1 , . . . , pn−1 )ei j=1 pj (xj+1 −xj )/~ dp1 . . . dpn−1 ,
(8.59)
where pj (xj+1 − xj ) is the natural bilinear form and equality (8.59) is
taken in the sense of tempered distribution, i.e., as the equality in the
form smeared by any f ∈ S (R(1+d)n ):
Wn (f ) = Fn−1 (f˜n−1 ),
where, for all (p1 , . . . , pn−1 ) ∈ (M1+d
)n−1 ,
p
Z
Pn−1
ei j=1 pj (xj+1 −xj )/~
f˜n−1 (p1 , . . . , pn−1 ) :=
(R1+d )n
× f (x1 , . . . , xn )dx1 . . . dxn .
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(W.5) (locality) For f ∈ S (R(1+d)n ) and j = 1, . . . , n − 1, let fj (x)
be f (x) with xj and xj+1 exchanged:
jth
⌣
fj (x) := f (x1 , . . . , xj+1 , xj , . . . , xn ), x = (x1 , . . . , xn ) ∈ R(1+d)n .
If supp f ⊂ {x ∈ R(1+d)n |(xj − xj+1 )2 < 0}, then Wn (f ) = Wn (fj ).
(W.6) (cluster decomposition proprety) For any space-like vector
a ∈ M1+d and f ∈ S (R(1+d)j ), g ∈ S (R(1+d)(n−j) , j = 1, . . . , n −
1, n ≥ 2,
lim Wn (f × g(λa,0) ) = Wj (f )Wn−j (g).
λ→∞
These axioms are called the Wightman axioms.
It is not so difficult to show that the Wightman distributions {Wn }∞
n=0
(W0 := 1) defined by (8.57) in a neutral quantum scalar field theory
(H , U, φ, D, Ψ0 ) satisfy the Wightman axioms.20
′
(1+d)n
Conversely, for each sequence {Wn }∞
) satisfyn=0 with Wn ∈ S (R
ing the Wightman axioms, there exists a quintuple (H , U, φ, D, Ψ0 ) obeying
the Gårding-Wightman axioms such that its Wightman distributions are
equal to {Wn }∞
n=0 , and it is unique up to unitary equivalents. This fact is
called Wightman’s reconstruction theorem.21
In this way, it is shown that the Gårding-Wightman axioms and the
Wightman axioms are equivalent (Fig. 8.1).
(GW.1)
(GW.2)
←→
←→
(W.1)
(W.2)
(GW.1)
positive definiteness of Hilbert space
(GW.2), (GW.3)
(GW.4)
(GW.2)–(GW.5)
Fig. 8.1
←→
←→
←→
←→ (W.3)
(W.4)
(W.5)
(W.6)
Rough correspondences
Remark 8.8. In general, a tempered distribution T ∈ S ′ (R(1+d)n ) is said
to be Poincaré (resp. Lorentz) invariant if, for all f ∈ S (R(1+d)n ) and
20 See, e.g., [Streater and Wightman (1964); Ezawa and Arai (1988)].
21 For details, see, e.g., [Streater and Wightman (1964); Ezawa and Arai (1988)].
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↑
(a, Λ) ∈ P+
(resp. Λ ∈ L+↑ ), T (f ) = T (f(a,Λ) ) (resp. T (f ) = T (f(0,Λ) )).
In terms of distribution kernel T (x1 , . . . , xn ) of T , the Poincaré invariance
of T is symbolically expressed as
T (Λx1 + a, . . . , Λxn + a) = T (x1 , . . . , xn ).
The case a = 0 in this equation gives the symbolical expression of the
Lorentz invariance of T . Axiom (W.2) is rephrased as follows: for all n ≥ 1,
Wn is Poincaré invariant.
8.9.4
The Osterwalder–Schrader axioms
The theory of functions of complex variables tells us that the essence of a
function may become apparent only after all possible analytic continuations
of it are found. This idea may be applicable to tempered distributions too.
From this point of view, one is naturally led to consider analytic continuations of Wightman distributions and may expect to obtain a set of axioms
for a QFT in terms of objects different from Wightman distributions. Below
we describe how this procedure is done and what kind of axioms appear.
One can show by using the relativistic invariance, the spectral condition
and the locality in the Wightman axioms that each Wightman distribution
Wn can be analytically continued to a function on the subset
{((it1 , x1 ), . . . , (itn , xn )) | tj ∈ R, xj ∈ Rd , (itj , xj ) 6= (itk , xk )
j 6= k, j, k = 1, . . . , n}
in C
:= C
× ··· × C
(the n direct product of C1+d ).22 This
subset is called the non-coincident Euclidean region. As is seen, the analytic continuation makes the time components of space-time points purely
imaginary ones. For this reason it is called the anaytic continuation to
purely imaginary times. On the level of the space-time, this analytic continuation changes the Minkoswki space-time metric to the Euclidean one.
In this sense the analytic continuation is called the Euclideanization of
the Wightman distributions. Let us denote the analytic continuation of Wn
by the same symbol Wn and introduce a sequence {Sn }∞
n=0 of functions as
follows: S0 := 1 and
(1+d)n
1+d
1+d
Sn ((t1 , x1 ), . . . (tn , xn )) := Wn ((it1 , x1 ), . . . , (itn , xn )).
It is shown that, for all n ∈ N, Sn is a real analytic function on the domain
En := {(y1 , . . . , yn ) ∈ R(1+d)n |yj 6= yk , j 6= k, j, k = 1, . . . , n}.
(8.60)
22 For the details, see, e.g., [Ezawa and Arai (1988), Chapter 2], [Osterwalder and
Schrader (1973, 1975)] and [Simon (1974)].
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General Theory of Quantum Fields
475
Ths function Sn is called the n-point Schwinger function or n-point
Euclidean Green’s function.
Properties of the Wightman distributions have their correspondents in
the Schwinger functions. To describe them, we need some notations.
Let
S6= (R(1+d)n ) := {f ∈ S (R(1+d)n )|for all multi-indices α, ∂ α f = 0 on
the hypersurface yj = yk (j 6= k, j, k = 1, . . . , n)}.
We consider the space S6= (R(1+d)n ) as a topological space with respect
to the relative topology in S (R(1+d)n ) and denote by S6=′ (R(1+d)n ) the
set of continuous linear functionals on S6= (R(1+d)n ). We also introduce a
subset of S6= (R(1+d)n ):
o
n
S+ (R(1+d)n ) := f ∈ S6= (R(1+d)n )|supp f ⊂ (R(1+d)n )+ ,
where
(R(1+d)n )+ := {((t1 , x1 ), . . . , (tn , xn )) ∈ R(1+d)n |0 < t1 < t2 < · · · < tn }.
One can prove that the sequence {Sn }∞
n=0 of the Schwinger functions
23
has the following properties (OS.1)–(OS.5) :
(OS.1) (temperedness and Hermiticity) For each n ≥ 1, Sn ∈
S6=′ (R(1+d)n ) and, for all f ∈ S6= (R(1+d)n )
Sn (f ) = Sn (θf † ),
where
Sn (f ) =
Z
Sn (y1 , . . . , yn )f (y1 , . . . , yn )dy1 · · · dyn ,
(θf )((t1 , x1 ), . . . , (tn , xn )) = f ((−t1 , x1 ), . . . , (−tn , xn )),
†
and f is defined by (8.58).
(OS.2) (Euclidean invariance) For all a ∈ R1+d and R ∈ SO(1 + d)
(the (1 + d)-dimensional Euclidean rotation group24 )
Sn (f ) = Sn (f(a,R) ),
f ∈ S6= (R(1+d)n ),
where
f(a,R) (y1 , . . . , yn ) := f (R−1 (y1 − a), . . . , R−1 (yn − a)),
y = (y1 , . . . , yn ) ∈ R(1+d)n .
23 For proofs, see, e.g., [Osterwalder and Schrader (1973, 1975)] and [Simon (1974)].
24 SO(1 + d) := {R ∈ M
1+d (R)|
t RR = I, det R = 1}.
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(OS.3) (reflection positivity) For all N ∈ Z+ and f0 ∈ C, f1 ∈
S+ (R1+d ), . . ., fN ∈ S+ (R(1+d)N ),
N
X
Sn+m (θfn† × fm ) ≥ 0, N ≥ 0.
n,m=0
(OS.4) (symmetry) For all n ∈ N and permutations σ ∈ Sn ,
Sn (yσ(1) , . . . , yσ(n) ) = Sn (y1 , . . . , yn ), y = (y1 , . . . , yn ) ∈ En .
(OS.5) (cluster property) Let m, n ∈ N. Then, for all f ∈
S6= (R(1+d)n ) ∩ C0∞ (R(1+d)n ), g ∈ S6= (R(1+d)m ) ∩ C0∞ (R(1+d)m ),
lim Sn+m (f × Tt g) = Sn (f )Sm (g).
t→∞
where, for each t ∈ R,
(Tt g)(y1 , . . . , ym ) := g(y1 − (t, 0), . . . , ym − (t, 0)),
y = (y1 , . . . , ym ) ∈ R(1+d)m .
(W.1)
(W.2)
(W.3)
(W.4)
(W.5)
(W.6)
−→
−→
−→
−→
−→
−→
Fig. 8.2
From the Wightman axioms to the Osterwalde-Schrader axioms
(OS.1)
(OS.2)
(OS.3)
analytic continuation to purely imagiary times
(OS.4)
(OS.5)
From an axiomatic point of view, forgetting about the Wightman axioms, one may define a neutral quantm scalar field theory as a sequence
{Sn }∞
In this scheme, properties (OS.1)–
n=0 satisfying (OS.1)–(OS.5).
(OS.5) are called the Osterwalder-Schrader axioms and, for any sequence {Sn }∞
n=0 satisfying (OS.1)–(OS.5), Sn ’s are called the Schwinger
functions. Then it is natural to ask if the Osterwalder-Schrader axioms
imply the Wightman axioms or the Gårding-Wightman axioms. In fact,
this involves some subtle problems technically. If the condition of the temperedness in (OS.1) is strengthened a little, then one can construct a sequence of tempered distributions satisfying the Wightman axioms from a
sequence of Schwinger functions obeying the Osterwalder-Schrader axioms
(Osterwalder-Schrader’s reconstruction theorem25 ).
25 For the details, see [Osterwalder and Schrader (1973, 1975)] and [Simon (1974)].
Zi-
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Gårding-Wightman axioms
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(Hilbert space, field operators, vacuum, etc.)
Wightman distributions ↓
↑ reconstruction theorem
Wightman axioms
(Wightman distributions)
analytic continuation to ↓
purely imaginary times
↑ reconstruction theorem
Osterwalder-Schrader axioms
(Schwinger functions))
Fig. 8.3
Relations among the three sets of axioms
The Osterwalder–Schrader axioms are not only interesting from purely
theoretical points of view but also useful for constructing relativistic quantum field models. This is suggested by the following considerations.
The equation of motion which a relativistic quantum field (φr )sr=1 on
the Minkowski space-time M1+d obeys is of the following form in many
cases:
mc 2 +
φr (x) = Fr (φ1 (x), . . . , φs (x), ∇φ1 (x), . . . , ∇φs (x)),
~
x ∈ M1+d , where c is the light speed (we have recovered it here),
:= Dµ Dµ = D02 −
d
X
Dj2
(8.61)
j=1
is the (1 + d)-dimensional d’Alembertian with distributional partial differential operators Dµ , µ = 0, . . . , d (the operator in this sense is called the
generalized (1 + d)-dimensional d’Alembertian),
∇ := (D0 , D1 , . . . , Dd ),
m ≥ 0 is a constant and Fr is a function on Rs × R(1+d)s . These equations
are of hyperbolic type.
noviev [Zinoviev (1995)] discovered a revised version of the Osterwalder-Schrader axioms
which is completely equivalent to the Wightman axioms.
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If one replaces the time component x0 in x ∈ M1+d by the purely
imaginary time ix0 , then one sees that the partial differential operator
Pd
2
+ (mc/~)2 changes to −∆ + (mc/~)2 , where ∆ =
µ=0 Dµ is the
generalized (1 + d)-dimensional Laplacian. Hence, schematically, by the
analytic continuation to the non-coincident Euclidean region, the quantum field φr on the Minkowski space-time may “metamorphose” into a
field φr,E on the Euclidean space R1+d , which is symbolically expressed as
φr,E (x) = φr (ix0 , x) (x = (x0 , x) ∈ R × Rd ), and the field equation takes
the form
mc 2 φr,E (x) = Gr (x),
−∆ +
~
where Gr is defined via the analytic continuation of Fr . These equations
are of elliptic type. As is well known, elliptic type equations may be more
tractable than hyperbolic ones in some respects. Thus the analytic continuation to purely imaginary times may make analysis easier equationtheoretically.
Moreover, the following aspect is also suggestive. For any two points
xE = (ix0 , x) and yE = (iy0 , y) which have a purely imaginary time component respectively, the Minkowski metric of them is −(x0 −y0 )2 −|x−y|2 < 0
and hence, in the symbolical sense, “xE and yE are space-like separated”.
Therefore, applying schematically (heuristically) the microscopic causality
to this case, one has that [φr (xE ), φr′ (yE )] = 0 (r, r′ = 1, . . . , s). Thus, if a
field (φr,E )sr=1 on the Euclidean space R1+d exists, then it is expected that
they forms a set of commuting operators. Needless to say, commutative
operators may be more tractable than non-commutative ones.
8.10
Euclidean Quantum Fields
As remarked in a heuristic way at the end of the last section, it is expected
that, in the analytic continuation of the Gårding-Wightman theory to the
non-coincident Euclidean region, a quantum field on the Minkowski spacetime may metamorphose into an operator-valued distribution whose values
are mutually commutative. One may call such a commutative operatorvalued distribution a Euclidean quantum field if it exists. In this connection,
the vacuum in the Gårding-Wightman theory should have a corresponding
non-zero vector in the Hilbert space in which the Euclidean quantum field
acts. If such a vector exists, we call it a Euclidean vacuum. In this scheme,
it is expected that the Schwinger functions are generated by the expectation
values of the Euclidean quantum fields with respect to a Euclidean vacuum.
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page 479
479
At least it is natural as a working hypothesis to suppose that such a class
of Schwinger functions may exist. These heuristic considerations leads one
to the following definition:
Definition 8.4. Let H be a Hilbert space. A Euclidean quantum field
(or a Euclidean field simply) on R1+d is an operator-valued distribution
ϕE on R1+d with values in L(H ) and common domain D, satisfying the
following (EF.1)–(EF.3):
(EF.1) (cyclicity) There exists a unit vector Φ0 in D such that, for all
n ∈ N and fj ∈ S (R1+d ) (j = 1, . . . , n), Φ0 ∈ D(ϕE (f1 ) · · · ϕE (fn ))
and the subspace
D0 := span Φ0 , ϕE (f1 ) · · · ϕE (fn )Φ0 |n ∈ N, fj ∈ S (R1+d ),
j = 1, . . . , n
is dense in H and D0 ⊂ D.
(EF.2) (commutativity) For all f, g ∈ S (R1+d ), [ϕE (f ), ϕE (g)] = 0 on
D0 .
(EF.3) (Euclidean invariance) Let n ∈ N and fj ∈ S (R1+d ) (j =
1, . . . , n). Then, for all (a, R) ∈ R1+d × SO(1 + d),
E
D
Φ0 , ϕE (f1 (a,R) ) · · · ϕE (fn (a,R) )Φ0 = hΦ0 , ϕE (f1 ) · · · ϕE (fn )Φ0 i ,
where fj (a,R) (x) := fj (R−1 (x − a)), x ∈ R1+d .
The vector Φ0 is called a Euclidean vacuum.
Let ϕE be a Euclidean field on R1+d as above. Then, by the Schwartz
nuclear theorem, for each n ∈ N, there exists a unique tempered distribution
Sn ∈ S ′ (R1+d ) such that
Sn (f1 ×· · ·×fn ) = hΦ0 , ϕE (f1 ) · · · ϕE (fn )Φ0 i ,
fj ∈ S (R1+d ), j = 1, . . . , n.
We call Sn ’s the Schwinger distributions of the Euclidean field ϕE .
Since a Euclidean field is commutative in the sense of (EF.2), there
may be the case where H = L2 (X, dµ) with a probability measure space
(X, Σ, µ), ϕE (f ) (f ∈ SR (R1+d )) is a random variable on (X, Σ, µ) and
Φ0 = 1. In this case ϕE is an example of a generalized random process
or a random distribution on R1+d . E. Nelson formulated a set of axioms
for such a Euclidean quantum field from which a RQFT on the Minkowski
space-time is reconstructed. The axioms is called the Nelson axioms. For
the details, see [Nelson (1973); Simon (1974)].
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8.11
PCT Theorem
Concerning space-time symmetries, we have so far considered only continuous symmetries, i.e., symmetries defined by continuous groups, such
as time-translation, space-translations, space-rotations and proper Lorentz
transformations. But, besides continuous symmetries, there are physically
important discrete symmetries which are described by finite groups consisting of transformations on the space-time. In this section we briefly describe
them and give a remark on an important theorem.
We define mappings I0 and Is on R1+d as follows:
I0 (x) := (−x0 , x),
0
Is (x) := (x , −x),
(8.62)
0
x = (x , x) ∈ R
1+d
.
(8.63)
The mappings I0 and Is are respectively called the time-reversal (or timeinversion) and the space-inversion. Note that
x ∈ R1+d .
(I0 Is )(x) = −x = (Is I0 )(x),
Hence
I0 Is = Is I0
and I0 Is gives the space-time inversion: R1+d ∋ x 7→ −x. It is easy to
see that I0 and Is are bijective linear mappings with
I02 = I,
Is2 = I.
I0−1 = I0 ,
Is−1 = Is .
In particular,
Hence {I, I0 }, {I, Is } and {I, I0 , Is , I0 Is } become discrete (finite) groups.
The mappings I0 and Is naturally induce mappings ut and us on
S (R1+d ) as follows:
(ut f )(x) := f (I0 x),
(us f )(x) := f (Is x),
f ∈ S (R1+d ), x ∈ R1+d .
It follows that u# (# = t, s) is bijective and
u2# = I,
ut us = us ut ,
u−1
# = u# .
The operators ut and us are respectively called the time-reversal operator and the parity operator on S (R1+d ). It is easy to see that the
eigenvalues of u# are ±1. The eigenvalues ±1 of us are called parities.
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Let ϕ = (ϕr )N
r=1 be the N -component quantum field acting in a Hilbert
space F and
ϕ(f ) =
N
X
ϕr (fr ),
r=1
We define
N
1+d
f = (fr )N
).
r=1 ∈ ⊕ S (R
ϕ(#) (f ) := ϕ(u# f ),
ϕ(st) (f ) := ϕ(us ut f ),
(t)
where u# f := (u# fr )N
and ϕ(s) are respectively called
r=1 . The field ϕ
the time-reversal of ϕ and the space-inversion of ϕ. Hence ϕ(st) may
be called the T P -transform of ϕ (T means “time-reversal” and P means
“parity” ).
(#)
(#)
In terms of operator-valued distribution kernels ϕr (x)) of ϕr , we
have
0
ϕ(t)
r (x) = ϕr (−x , x),
0
ϕ(s)
r (x) = ϕr (x , −x),
ϕ(st)
r (x) = ϕr (−x).
In an axiomatic framework of relativistic QFT, one can show that, under
a suitable assumption, there exist an involution C on ⊕N S (R1+d ) and an
anti-unitary operator Θ on F such that
Θϕ(f )Θ−1 = ϕ(Cus ut f )∗ ,
f ∈ ⊕N S (R1+d ),
(8.64)
where us ut f := (us ut fr )N
r=1 . This type of transformation law is called PCT
theorem (or CPT theorem), which tells us that the CPT transform
ϕ(Cus ut f )∗ of ϕ is anti-unitarily implementable.
Remark 8.9. In non-relativistic QFT, PCT theorem may take a slightly
different form from (8.64). See Subsection 9.4.3.
8.12
8.12.1
Scattering Theory and Spectral Analysis
Introduction
In an axiomatic QFT in which the basic concept is a quantum field (e.g., the
Gårding-Wightman axioms), no particle picture is given in advance. Hence
one needs to derive a particle picture from the axioms. Physically one
has a particle picture by observing free elementary particles. In scattering
phenomena of elementary particles, free elementary particles coming from
a long distance at time t ≈ −∞ are scattered by a target and then fly
away into the distance at t ≈ +∞, appearing as free elementary particles
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again.26 From these experimental facts, one infers that there exists an
asymptotic field φin (t, x) (resp. φout (t, x)) describing free particles which is
formally defined as an asymptotic limit (in a suitable sense) limt→−∞ φ(t, x)
(resp. limt→+∞ φ(t, x)) of an interacting quantum field φ(t, x), satisfying
φin/out (t, x) = eitH/~ φin/out (0, x)e−itH/~ , where H is the Hamiltonian of
the quantum system under consideration, and the free Klein-Gordon
equation
m2
+ 2 φin/out (t, x) = 0
~
in the sense of operator-valued distribution if φ(t, x) is a quantum scalar
field with m ≥ 0 being a constant.27 The quantum fields φin/out are called
free quantum Klein-Gordon fields and it is shown that they describe free
particles of mass m indeed (the details will be given in Chapter 10). But,
in the case of the asymptotic fields φin/out , m has to be identified with the
observed mass of a particle (not the “bare mass”28 ). With this particle
picture, one can ask the transition probability of an initial particle state
(in-state) at t = −∞ to a final particle state (out-state) at t = +∞, which
may be compared with experiments on scattering phenomena of elementary
particles. Such transition probabilities can be described in a unified way by
a unitary operator called a scattering operator or a scattering matrix.
A scattering theory in the framework of the Gårding-Wightman axioms
is established and called the Haag-Ruelle scattering theory. Concerning this theory, we refer the reader to textbooks on mathematical QFT (e.g.,
[Araki (1993), Chapter 4], [Bogoliubov et al. (1975), Chapter 4], [Glimm
and Jaffe (1987), Chapters 13–14], [Haag (1996), §II.4], [Reed and Simon
(1979), Chapter XI, §16]). In this section, we present another type of
scattering theory which can be applied to models in non-relativistic QFT
including those in relativistic QFT with cutoffs (see Chapter 14). From
mathematical point of view, it gives a method for spectral analysis of a
26 In the intermediate space-time where the interaction among the elementary particles
and the target is done, particle picture makes no sense.
27 We continue to use the physical unit system where c = 1, unless otherwise stated.
28 A “bare mass” means the mass of a particle before being in the interaction with
quantum fields (it may be somewhat a fictitious notion). In the conventional picture
in physical QFT, the bare mass of a particle may change through an interaction with
a quantum field to give an observed mass. It is known, however, that, in some models
in relativistic QFT, observed masses are found to be divergent in formal perturbation
theory. This is a reason why it is difficult to construct a model in relativistic QFT (see
Remark 8.7). In the physical QFT, one avoids such difficulties by an prescription called
“renormalization”, but this is not a mathematical solution.
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class of self-adjoint operators on Hilbert spaces of Fock space type.
8.12.2
Asymptotic annihilation and creation operators
For simplicity, we consider only a Bose field theory in an abstract framework. Let H be a Hilbert space and Fb (H ) be the boson Fock space over
H (see Chapter 5). As a free field Hamiltonian, we take
H0 := dΓb (T ),
the boson second quantization of a non-negative and injective self-adjoint
operator T on H (see Subsection 5.3.2). Let H be a self-adjoint operator on
Fb (H ), denoting the total Hamiltonian of a system of an interacting Bose
field. Then a symmetric operator H1 on Fb (H ) such that H − H0 ⊂ H1
(i.e. D(H) ∩ D(H0 ) ⊂ D(H1 ) and HΨ = (H0 + H1 )Ψ, Ψ ∈ D(H) ∩ D(H0 ))
describes an interaction of the quantum system under consideration. We
assume the following:
(H.1) (i) H is bounded from below and (ii) D(H) ⊂ D(H0 ).
Condition (H.1)(i) implies that the lowest energy
E0 := inf σ(H)
of H is finite and the energy-shifted Hamiltonian
b := H − E0 ,
H
b + 1)−1 is
is non-negative. By condition (H.1)(ii) and Lemma 1.8, H0 (H
bounded. Hence there exists a constant C1 > 0 such that
b + 1)Ψk, Ψ ∈ D(H).
kH0 Ψk ≤ C1 k(H
b 1/2 ) ⊂
It follows from this fact and an application of Theorem 1.26 that D(H
1/2
D(H0 ) and there exists a constant C2 > 0 such that
1/2
b + 1)1/2 Ψk,
kH0 Ψk ≤ C2 k(H
b 1/2 ),
Ψ ∈ D(H
where we have used the fact that, for all α > 0,
b + 1)α Ψk, Ψ ∈ D(H
b α ).
kΨk ≤ k(H
(8.65)
(8.66)
We denote by A(f ) the annihilation operator on Fb (H ) with test vector
f ∈ H (see Chapter 5).
b 1/2 ) ⊂ D(A(f )# ) and, for all
Lemma 8.5. For all f ∈ D(T −1/2 ), D(H
1/2
b ),
Ψ ∈ D(H
b + 1)1/2 Ψk,
kA(f )Ψk ≤ C2 kT −1/2 f k k(H
b + 1)1/2 Ψk.
kA(f )∗ Ψk ≤ (C2 kT −1/2 f k + kf k)k(H
(8.67)
(8.68)
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1/2
Proof. By Theorem 5.16, D(H0 ) ⊂ D(A(f )# ) and
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1/2
kA(f )Ψk ≤ kT −1/2 f k kH0 Ψk.
b 1/2 ) ⊂ D(A(f )# ) and (8.65) yields (8.67). Similarly we have
Hence D(H
1/2
kA(f )∗ Ψk ≤ kT −1/2 f k kH0 Ψk + kf k kΨk
b + 1)1/2 Ψk + kf k k(H
b + 1)1/2 Ψk,
≤ kT −1/2 f kC2 k(H
where we have used (8.66) with α = 1/2. Hence (8.68) holds.
In the rest of this section, for notatinal simplicity, we use the physical
unit system where ~ = 1 and c = 1.
For each (f, t) ∈ D(T −1/2 ) × R, we define
At (f )# := eitH e−itH0 A(f )# eitH0 e−itH ,
1/2
where # = ∅, ∗. Recall that D(H0 ) ⊂ D(A(f )# ) (Theorem 5.16). For any
self-adjoint operator Q on a Hilbert space and all α > 0, e−itQ D(|Q|α ) =
D(|Q|α ), ∀t ∈ R. Hence it follows that
b 1/2 ) ⊂ D(At (f )# ).
D(H
By Lemma 5.21, for all f ∈ H and t ∈ R, operator equality
e−itH0 A(f )# eitH0 = A(e−itT f )#
holds. Hence
At (f )# = eitH A(e−itT f )# e−itH .
(8.69)
b 1/2 ) ⊂ D(At (f )# ) and, for all
Lemma 8.6. For all f ∈ D(T −1/2 ), D(H
1/2
b
Ψ ∈ D(H ),
b + 1)1/2 Ψk,
kAt (f )Ψk ≤ C2 kT −1/2f k k(H
∗
kAt (f ) Ψk ≤ (C2 kT
−1/2
Proof. By Lemma 8.5 and (8.69),
b + 1)1/2 Ψk.
f k + kf k)k(H
(8.70)
(8.71)
kAt (f )Ψk = kA(e−itT f )e−itH Ψk
b + 1)1/2 e−itH Ψk.
≤ C2 kT −1/2 e−itT f k k(H
By the functional calculus, we have
kT −1/2 e−itT f k = kT −1/2 f k,
b + 1)1/2 e−itH Ψk = k(H
b + 1)1/2 Ψk.
k(H
Hence (8.70) follows. Similarly one can prove (8.71).
We assume the following condition too:
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b 1/2 ) and # = ∅, ∗, the strong limits
(H.2) For all Ψ ∈ D(H
#
Ψ#
± (f ) := lim At (f ) Ψ
t→±∞
exist.
Under assumptions (H.1) and (H.2), for each f ∈ D(T −1/2 ), one can
define linear operators A±,0 (f ) and A†±,0 (f ) as follows:
b 1/2 ),
D(A±,0 (f )) := D(H
A±,0 (f )Ψ := Ψ± (f ),
b 1/2 ),
D(A†±,0 (f )) := D(H
A†±,0 (f )Ψ := Ψ∗± (f ),
b 1/2 ).
Ψ ∈ D(H
It is obvious that A±,0 (f ) and A†±,0 (f ) are densely defined. It is easy to
see that
E
D
b 1/2 ).
hΦ, A±,0 (f )Ψi = A†±,0 (f )Φ, Ψ , Ψ, Φ ∈ D(H
b 1/2 ) ⊂ D(A±,0 (f )∗ ) and A±,0 (f )∗ Φ = A† (f )Φ, Φ ∈ D(H
b 1/2 ).
Hence D(H
±,0
∗
In particular, A±,0 (f ) are densely defined. Hence A±,0 (f ) are closable.
We denote the closures by A± (f ):
A± (f ) := A±,0 (f ).
b 1/2 ) ⊂ D(A± (f )) ∩ D(A± (f )∗ ) and
It follows that D(H
A± (f )∗ = A±,0 (f )∗ ⊃ A†±,0 (f ).
Hence
A± (f )# Ψ = lim At (f )# Ψ,
t→±∞
b 1/2 ).
Ψ ∈ D(H
We call A± (f ) and A± (f )∗ the asymptotic annihilation and creation operators associated with (H0 , H) respectively.
8.12.3
A vanishing theorem
The fact stated in the next lemma tells us an important property of the
annihilation operator A(·).
Lemma 8.7. Suppose that T is absolutely continuous (see Appendix C).
1/2
Then, for all f ∈ D(T −1/2 ) and Ψ ∈ D(H0 ),
lim A(e−itT f )Ψ = 0.
t→±∞
(8.72)
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Proof. We first consider the case where Ψ = A(f1 )∗ · · · A(fn )∗ ΩH (n ≥ 0,
fj ∈ D(T ), j = 1, . . . , n). Then we have
A(e−itT f )Ψ =
n
X
j=1
∗
\
∗
e−itT f, fj A(f1 )∗ · · · A(f
j ) · · · A(fn ) ΩH .
By the absolute continuity (see Theorem C.2 in Appendix C), we have
limt→±∞ e−itT f, fj = 0. Hence limt→±∞ A(e−itT f )Ψ = 0. Note
that the subspace algebraically spanned by all vectors of the form Ψ is
1/2
D0 := Fb,fin (D(T )). The subspace D0 is a core of H0 and hence of H0 .
1/2
Therefore, for any Ψ ∈ D(H0 ) and ε > 0, there exists a vector Φ ∈ D0
such that
kΦ − Ψk < ε,
1/2
1/2
kH0 Φ − H0 Ψk < ε.
By using the triangle inequality, we have
kA(e−itT f )Ψk ≤ kA(e−itT f )(Ψ − Φ)k + kA(e−itT f )Φk
1/2
1/2
≤ kT −1/2 f k kH0 Ψ − H0 Φk + kA(e−itT f )Φk
≤ εkT −1/2 f k + kA(e−itT f )Φk.
Since limt→±∞ kA(e−itT f )Φk = 0 as just shown above, it follows that
lim sup kA(e−itT f )Ψk ≤ εkT −1/2 f k.
t→±∞
Since ε > 0 is arbitrary, we obtain limt→±∞ kA(e−itT f )Ψk = 0. Thus
(8.72) follows.
Theorem 8.9 (vanishing theorem). Suppose that T is absolutely continuous. Let E be an eigenvalue of H and ΨE ∈ ker(H − E). Then, for all
f ∈ D(T −1/2 ),
A± (f )ΨE = 0.
(8.73)
1/2
Proof. Since ΨE ∈ D(H) ⊂ D(H0 ) ⊂ D(H0 ), it follows from Lemma
8.7 that limt→±∞ A(e−itT f )ΨE = 0. We have e−itH ΨE = e−itE ΨE , t ∈ R.
Hence
kA± (f )ΨE k = lim kA(e−itT f )e−itE ΨE k = lim kA(e−itT f )ΨE k = 0.
t→±∞
Thus (8.73) holds.
t→±∞
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8.12.4
487
Commutation relations and representations of CCR
Proposition 8.9. For all t ∈ R and f ∈ D(T −1/2 ),
eitH A± (f )# e−itH = A± (eitT f )
b 1/2 ).
on D(H
(8.74)
b 1/2 ) and s, t ∈ R. Then we have
Proof. Let Ψ ∈ D(H
eitH As (f )# e−itH Ψ = ei(t+s)H A(e−i(t+s)T eitT f )# e−i(t+s)H Ψ.
We have
lim eitH As (f )# e−itH Ψ = eitH A± (f )# e−itH Ψ
s→±∞
and
lim ei(t+s)H A(e−i(t+s)T eitT f )# e−i(t+s)H Ψ = A± (eitT f )# Ψ.
s→±∞
Thus (8.74) is derived.
Proposition 8.9 shows that (eitH , A± (f )) have the same commutation
property as that of (eitH0 , A(f )).
Since the strong convergence of vectors implies the norm convergence
of them, Lemma 8.6 immediately yields the following lemma:
b 1/2 ),
Lemma 8.8. For all f ∈ D(T −1/2 ) and Ψ ∈ D(H
b + 1)1/2 Ψk,
kA± (f )Ψk ≤ CkT −1/2 f k k(H
∗
kA± (f ) Ψk ≤ C(kT
−1/2
where C > 0 is a constant.
(8.75)
b + 1)
f k + kf k)k(H
1/2
Ψk,
(8.76)
b 1/2 ) and f ∈ D(T −1/2 ) ∩ D(T n ) (n ∈ N). Then
Lemma 8.9. Let Ψ ∈ D(H
the vector-valued functions : R ∋ t 7→ A± (eitT f )# Ψ are n times strongly
differentiable in t and the strong derivatives are given by
dn
A± (eitT f )# Ψ = A± ((iT )n eitT f )# Ψ.
dtn
b 1/2 -bounded by Lemma 8.8, one needs only to
Proof. Since A± (f )# are H
replace (S, dΓb (S), A(f )) in the proof of Lemma 5.22 by (T, H, A± (f )).
b 3/2 )
Theorem 8.10. For each f ∈ D(T −1/2 ) ∩ D(T ), A± (f )# map D(H
to D(H) and, for all t ∈ R,
eitH HA± (f ) = A± (eitT f )HeitH − A± (T eitT f )eitH ,
e
itH
∗
HA± (f ) = A± (e
itT
∗
f ) He
itH
+ A± (T e
itT
∗ itH
f) e
(8.77)
(8.78)
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b 3/2 ). In particular,
on D(H
HA± (f ) = A± (f )H − A± (T f ),
∗
∗
(8.79)
∗
HA± (f ) = A± (f ) H + A± (T f )
(8.80)
b 3/2 ).
on D(H
b 3/2 ) and Φ ∈ D(H). Then, by (8.74),
Proof. Let Ψ ∈ D(H
e−itH Φ, A+ (f )Ψ = R(t),
R(t) := A+ (eitT f )∗ Φ, eitH Ψ .
The left hand side is differentiable in t with
d −itH
e
Φ, A+ (f )Ψ = e−itH (−iH)Φ, A+ (f )Ψ .
dt
Hence R(t) also is differentiable in t. Using the preceding lemma, we have
dR(t)
= A+ (iT eitT f )∗ Φ, eitH Ψ + A+ (eitT f )∗ Φ, eitH iHΨ .
dt
Hence it follows that
HΦ, eitH A+ (f )Ψ = − Φ, A+ (T eitT f )eitH Ψ
+ Φ, A+ (eitT f )HeitH Ψ ,
b 1/2 ) and
where we have used the fact that eitH Ψ ∈ D(H 3/2 ) ⊂ D(H
itH
1/2
b
He Ψ ⊂ D(H ). Since this holds for all Φ ∈ D(H), it follows that
b 3/2 )
eitH A+ (f )Ψ ∈ D(H ∗ ) = D(H), which means that A+ (f ) maps D(H
to D(H), and (8.77) holds. Similarly one can prove (8.78). Putting t = 0
in (8.77) and (8.78), we obtain (8.79) and (8.80).
Theorem 8.10 shows that (H, A± (f )) obey the same commutation relations as those of (H0 , A(f )) (see Theorem 5.17).
A consequence of the mapping property of A± (f )# in Theorem 8.10 is
given in the following theorem.
Theorem 8.11 (CCR). Let f, g ∈ D(T −1/2 ) ∩ D(T ). Then the following
b 3/2 ):
commutation relations hold on D(H
[A± (f ), A± (g)∗ ] = hf, gi ,
[A± (f ), A± (g)] = 0,
(8.81)
∗
∗
[A± (f ) , A± (g) ] = 0.
(8.82)
b 1/2 ) and Ψ ∈ D(H
b 3/2 ). Then, by the preceding
Proof. Let Φ ∈ D(H
#
#
theorem, A± (f ) Ψ ∈ D(H) ⊂ D(A± (g) ). By Theorem 5.18, D(H) ⊂
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D(H0 ) ⊂ D(A(e−itT f )# A(e−itT g)# ) for all t ∈ R. Hence the following
calculations are mathematically meaningful:
hΦ, A± (f )A± (g)∗ Ψi = lim hAt (f )∗ Φ, At (g)∗ Ψi
t→±∞
= lim
t→±∞
A(e−itT f )∗ e−itH Φ, A(e−itT g)∗ e−itH Ψ
e−itH Φ, A(e−itT f )A(e−itT g)∗ e−itH Ψ
= lim e−itH Φ,
e−itT f, e−itT g
t→±∞
+ A(e−itT g)∗ A(e−itT f ) e−itH Ψ
= lim
t→±∞
= hf, gi hΦ, Ψi + hA± (g)Φ, A± (f )Ψi
= hΦ, (A± (g)∗ A± (f ) + hf, gi)Ψi .
Hence (8.81) follows. Similarly one can prove (8.82).
We set
DT := D(T −1/2 ) ∩ C ∞ (T ).
Lemma 8.10. For each f ∈ DT , A± (f )# leave C ∞ (H) invariant.
b 3/2 ). Hence, by Theorem 8.10,
Proof. Let Ψ ∈ C ∞ (H). Then Ψ ∈ D(H
A± (f )Ψ ∈ D(H) and
HA± (f )Ψ = A± (f )HΨ − A± (T f )Ψ.
b 3/2 ), the vector on the right hand side is in
Since HΨ ∈ C ∞ (H) ⊂ D(H
D(H) by Theorem 8.10 again. Hence HA± (f )Ψ ∈ D(H) (i.e. A± (f )Ψ ∈
D(H 2 )) and
H 2 A± (f )Ψ = HA± (f )HΨ − HA± (T f )Ψ.
Next one applies the same argument to each vector on the right hand side
to conclude that the vector on the right hand side is in D(H). Hence
H 2 A± (f )Ψ ∈ D(H), i.e., A± (f )Ψ ∈ D(H 3 ) and
H 3 A± (f )Ψ = H 2 A± (f )HΨ − H 2 A± (T f )Ψ.
Repeating this argument, one can show that, for all n ∈ N, A± (f )Ψ ∈
D(H n ).
Thus A± (f )Ψ ∈ C ∞ (H).
Similarly one can show that
∗ ∞
A± (f ) C (H) ⊂ C ∞ (H).
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The following theorem immediately follows from Theorem 8.11 and
Lemma 8.10:
Theorem 8.12. The triplets (Fb (H ), C ∞ (H), {A± (f )|f ∈ DT }) are representations of the CCR over DT .29
8.12.5
Scattering operator
We assume the following (H.3) and (H.4) in addition to (H.1) and (H.2):
(H.3) (existence of a ground state) H has a ground state Ψ0 :
HΨ0 = E0 Ψ0 ,
kΨ0 k = 1.
(H.4) The operator T is absolutely continuous.
Under these assumptions, we have by Theorem 8.9
f ∈ D(T −1/2 ).
A± (f )Ψ0 = 0,
Since Ψ0 ∈ C ∞ (H), it follows from Lemma 8.10 that, for all n ∈ N
and fj ∈ DT (j = 1, . . . , n), Ψ0 ∈ D(A± (f1 )# · · · A± (fn )# ). Hence we can
define vectors
Ψ± (f1 , . . . , fn ) := A± (f1 )∗ · · · A± (fn )∗ Ψ0 .
Physically the vector Ψ− (f1 , . . . , fn ) (resp. Ψ+ (f1 , . . . , fn )) is interpreted as
an asymptotically observed n particle state at t = −∞ (resp. +∞), called
an in-state (resp. out-state). In this way a particle picture emerges in
the interacting system under consideration.
Let
D0,± := span {Ψ0 , Ψ± (f1 , . . . , fn )|n ∈ N, fj ∈ DT , j = 1, . . . , n}
and
Fout := D0,± .
in
Then, by Theorem 8.12, (Fout , D0,± , {A± (f )|f ∈ DT }, Ψ0 ) are cyclic repin
resentations of the CCR over DT . Hence, by Theorem 5.42, there exists a
unitary operator U± : Fb (H ) → Fout such that U± ΩH = Ψ0 and, for all
in
−1
f ∈ DT , A± (f ) = U± A(f )U±
on D0,± . Hence the operator
−1
S := U+ U−
29 See Section 5.21.
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is a unitary operator from Fin onto Fout , satisfying
SΨ0 = Ψ0
(8.83)
A+ (f ) = SA− (f )S −1
(8.84)
and
on D0,+ . We call the operator S the scattering operator or the scattering matrix.
It follows (8.83) and (8.84) that, for all n ∈ N and fj ∈ DT (j =
1, . . . , n),
SΨ− (f1 , . . . , fn ) = Ψ+ (f1 , . . . , fn ).
Proposition 8.10. For all t ∈ R, eitH F in = F in and
out
e
−itH
S = Se
out
−itH
(8.85)
on Fin .
Proof. Let n ∈ N and fj ∈ DT (j = 1, . . . , n). Then, by Proposition 8.9,
e−itH Ψ± (f1 , . . . , fn ) = e−itE0 Ψ± (e−itT f1 , . . . , e−itT fn ),
where we have used the fact that e−itH Ψ0 = e−itE0 Ψ0 . Hence e−itH D0,± ⊂
D0,± . This implies that e−itH D0,± = D0,± . Since D0.± are dense in Fout ,
it follows that e−itH Fout = Fout . We have
in
e
−itH
in
in
SΨ− (f1 , . . . , fn ) = e−itE0 Ψ+ (e−itT f1 , . . . , e−itT fn )
= Se−itE0 Ψ− (e−itT f1 , . . . , e−itT fn )
= Se−itH Ψ− (f1 , . . . , fn ).
Hence e−itH S = Se−itH on D0,− . Since e−itH S and Se−itH are bounded
operators on Fin , (8.85) follows.
If Fin = Fout = Fb (H ), then the quantum system is said to be
asymptotically complete. In this case S is a unitary operator on
Fb (H ).
If F∞ := Fin = Fout , then the quantum system is said to have weak
asymptotic completeness. In this case S is a unitary operator on F∞ .
Since, for each in-state Ψ ∈ F∞ with kΨk = 1, SΨ is a state vector at
t = +∞ (an out-state) determined causally under no measurements in the
meantime, | hΦ, SΨi |2 is interpreted as the transition probability from Ψ at
t = −∞ to a state Φ ∈ F∞ with kΦk = 1 at t = +∞ (i.e. the probability
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that the state Φ is found under a measurement at t = +∞). In particular,
| hΨ, SΨi |2 gives the probability that the system is found to be in the same
state Ψ as the intitial one under a measurement at t = +∞. Based on this,
one says that there is no scattering with respect to the asymptotic intitial
state Ψ at t = −∞ if | hΨ, SΨi |2 = 1. We say that there is no scattering
if, for all unit vectors Ψ ∈ F∞ , | hΨ, SΨi |2 = 1. Hence, if | hΨ, SΨi |2 < 1
for some unit vector Ψ ∈ F∞ , then there is a scattering. “No scattering”
is characterized as follows:
Proposition 8.11. There is no scattering if and only if S = eiθ for some
θ ∈ R.
Proof. The “if part” is obvious. So we prove the “only if part”. Suppose that there is no scattering. Then, for all unit vectors Ψ ∈ F∞ ,
| hΨ, SΨi |2 = 1. Let Ψ ∈ F∞ with kΨk = 1 and P be the orthogonal
projection onto the one-dimensional space HΨ := {αΨ|α ∈ C}. Then we
have SΨ = hΨ, SΨi Ψ + (1 − P )SΨ. Hence
kSΨk2 = | hΨ, SΨi |2 + k(1 − P )SΨk2 = 1 + k(1 − P ))SΨk2 .
Since S is unitary, kSΨk2 = kΨk2 = 1. Hence k(1 − P )SΨk2 = 0. This
implies that SΨ ∈ HΨ . Hence SΨ = α(Ψ)Ψ with a constant α(Ψ) such that
|α(Ψ)| = 1. It follows that, for all non-zero vectors Φ ∈ F∞ , SΦ = β(Φ)Φ
with β := α(Φ/kΦk). Using the linearity of S, one can show that β(Φ) is
independen of Φ. Hence there exists a constant θ ∈ R such that β(Φ) = eiθ .
Thus S = eiθ .
8.12.6
Spectrum of H
We continue to assume (H.1)–(H.4). Then we have the orthogonal decomposition
⊥
.
Fb (H ) = Fin/out ⊕ Fin/out
Since e−itH Fin/out = Fin/out for all t ∈ R, it follows that H is reduced
⊥
by Fin/out and hence by Fin/out
too. We denote the reduced part of H to
⊥
⊥
Fin/out and Fin/out by Hin/out and Hin/out
respectively.
Lemma 8.11. The following operator equalities hold:
−1
U−
(Hin − E0 )U− = H0 ,
−1
U+
(Hout − E0 )U+ = H0 .
(8.86)
(8.87)
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Proof. Let fj ∈ DT (j = 1, . . . , n, n ∈ N) and t ∈ R. Then
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−1 it(Hin −E0 )
−1
U−
e
Ψ− (f1 , . . . , fn ) = U−
Ψ− (eitT f1 , . . . , eitT fn )
= A(eitT f1 )∗ · · · A(eitT fn )∗ ΩH
= eitH0 A(f1 )∗ · · · A(fn )∗ ΩH
−1
Ψ− (f1 , . . . , fn ).
= eitH0 U−
−1
−1 it(Hin −E0 )
are bounded,
Since D0,− is dense in Fin and U−
e
and eitH0 U−
−1 it(Hin −E0 )
itH0 −1
U− , i.e.,
it follows that U− e
=e
−1 it(Hin −E0 )
U−
e
U− = eitH0 .
By the unitary covariance of functional calculus, the left hand side is equal
−1
to exp(itU−
(Hin − E0 )U− ). Hence (8.86) follows. Similarly one can prove
(8.87).
by
For a subset X of R and a constant c ∈ R, we define a subset X + c ⊂ R
X + c := {x + c|x ∈ X}.
Theorem 8.13. Assume (H.1)–(H.4). Then
(i) σ(Hin/out ) = σ(H0 ) + E0 .
(ii) σ(H0 ) + E0 ⊂ σ(H).
(iii) If Fin = Fb (H ) or Fout = Fb (H ), then σ(H) = σ(H0 ) + E0 and
the ground state of H is unique.
Proof. (i) By the preceding lemma and the unitary invariance of spectrum,
we have σ(Hin/out − E0 ) = σ(H0 ). Hence the desired result follows.
(ii) Since H is a direct sum operator, we have
⊥
σ(H) = σ(Hin/out ) ∪ σ(Hin/out
).
Hence σ(H) ⊃ σ(Hin/out ) = σ(H0 ) + E0 .
⊥
(iii) Let Fin = Fb (H ). Then Fin
= {0} and hence Hin = H. Thus,
by (ii), σ(H) = σ(H0 ) + E0 . Suppose that a vector Φ0 6= Ψ0 were a ground
state of H. Without loss of generality, we can assume that Ψ0 ⊥ Φ0 . Then,
by A− (f )Φ0 = 0 for all f ∈ D(T −1/2 ), we have hΨ− (f1 , . . . , fn ), Φ0 i = 0
⊥
for all fj ∈ DT (j = 1, . . . , n, n ∈ N). Hence Φ0 ∈ D0,−
= {0}.Thus Φ0 = 0.
But this is a contradiction, Thus the ground state of H is unique. Similarly
one can prove the stated fact in the case Fout = Fb (H ) too.
Remark 8.10. Theorem 8.13(iii) shows that, if σ(H0 ) + E0 $ σ(H), then
Fin/out 6= Fb (H ). Physically an eigenvector of H with an eigenvalue in
σ(H) \ (σ(H0 ) + E0 ) is interpreted as a bound state of H.
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Let us consider a typical case of T appearing in quantum scalar field
models.
Corollary 8.3. Let H be separable and assume (H.1)–(H.4). Supppose
that T is absolutely continuous and
σ(T ) = σac (T ) = [m, ∞)
(8.88)
with a constant m ≥ 0. Then:
(i)
σ(Hin/out ) = {E0 } ∪ [E0 + m, ∞),
σp (Hin/out ) = {E0 }.
(8.89)
(ii) {E0 } ∪ [E0 + m, ∞) ⊂ σ(H).
(iii) If m = 0, then σ(H) = [E0 , ∞).
(iv) If Fin = Fb (H ) or Fout = Fb (H ), then σ(H) = {E0 }∪[E0 +m, ∞)
and the ground state of H is unique.
Proof. (i) By Theorem 5.3 and (8.88), we have σp (H0 ) = {0} and
σ(H0 ) = {0} ∪ [m, ∞).
By these facts and Theorem 8.13(i), we obtain (8.89).
Part (ii) follows from (i) and Theorem 8.13(ii).
(iii) Let m = 0. Then, by (ii), [E0 , ∞) ⊂ σ(H). But it is obvious that
σ(H) ⊂ [E0 , ∞).
Part (iv) follows from (i) and Theorem 8.13(iii).
Remark 8.11. The Hamiltonian H may have eigenvalues E other than E0
(E > E0 ). Let ΨE be a unit eigenvector of H with eigenvalue E. Then one
can construct the subspaces
span {ΨE , A± (f1 )∗ · · · A± (fn )∗ ΨE |n ∈ N, fj ∈ DT , j = 1, . . . , n}
and the same considerations as above can be applied to these subspaces.
8.12.7
Existence of asymptotic creation and annihilation
operators
In concluding this section, we give a sufficient condition for (H.2) to hold.
We assume (H.1). Then we have
H = H0 + H1 .
Concerning the perturbation operator H1 , we assume the following (H.1)’:
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(H.1)’ There exist a number p ≥ 3/2 and a core D of T −1/2 such that, (i)
b p ) ⊂ D(H1 A(f )# ) ∩
for all t ∈ R, e−itT D ⊂ D; (ii) for all f ∈ D, D(H
#
D(A(f ) )H1 ) and
b + 1)p Ψk,
k[H1 , A(f )# ]Ψk ≤ u# (f )k(H
b p ),
Ψ ∈ D(H
where u# (f ) is a non-negative constant depending on f such that
: R ∋ t 7→ u# (e−itT f ) is Borel measurable and
Rthe mapping
−itT
u (e
f )dt < ∞.
R #
b p ) and
Under assumptions (H.1) and (H.1)’, we have for all Ψ ∈ D(H
s∈R
b + 1)p Ψk.
keisH [H1 , A(e−isT f )# ]e−isH Ψk ≤ u# (e−isT f )k(H
(8.90)
Hence, for all t ∈ R ∪ {±∞}, the Bochner (strong) integrals
Z t
#
I(t) Ψ :=
eisH [H1 , A(e−isT f )# ]e−isH Ψds
0
exist (see Appendix E).
b p ), t ∈ R
Lemma 8.12. Assume (H.1) and (H.1)’. Then, for all Ψ ∈ D(H
and f ∈ D,
At (f )# Ψ = A(f )# Ψ + iI(t)# Ψ.
(8.91)
b p ). Then we have
Proof. Let Φ ∈ D(H) and Ψ ∈ D(H
hΦ, At (f )Ψi = e−itH Φ, Ψ(t) ,
where Ψ(t) := A(e−itT f )e−itH Ψ. We first show that Ψ(t) is strongly differentiable in t (this is non-trivial because of the unboundedness of A(e−itT f )).
For any ε ∈ R \ {0}, we have
1
b + 1)−1/2 Ψε + A(e−itT fε )e−itH Ψ,
(Ψ(t + ε) − Ψ(t)) = A(e−i(t+ε)T f )(H
ε
where
b + 1)1/2 e−itH
Ψε := (H
e−iεH − 1
Ψ,
ε
fε :=
(e−iεT − 1)f
.
ε
b + 1)1/2 e−itH (−iH)Ψ.
By the functional calculus, we have limε→0 Ψε = (H
−i(t+ε)T
−1/2
b
By (8.67), A(e
f )(H + 1)
is a bounded operator with
b + 1)−1/2 k ≤ C2 kT −1/2 f k
kA(e−i(t+ε)T f )(H
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uniformly in ε. Moreover,
b + 1)−1/2 − A(e−itT f )(H
b + 1)−1/2 k
kA(e−i(t+ε)T f )(H
≤ C2 k(e−iεT − 1)T −1/2 f k
→ 0 (ε → 0).
Hence
b + 1)−1/2 Ψε = A(e−itT f )e−itH (−iH)Ψ.
lim A(e−i(t+ε)T f )(H
ε→0
We have limε→0 fε = −iT f . Hence, in the same way as above, we obtain
lim A(e−itT fε )e−itH Ψ = A(e−itT (−iT )f )e−itH Ψ.
ε→0
Thus Ψ(t) is strongly differentiable in t and
d
Ψ′ (t) := Ψ(t) = A(e−itT f )e−itH (−iH)Ψ + A(e−itT (−iT )f )e−itH Ψ.
dt
Since e−itH Φ is strongly differentiable in t with the strong derivative
−itH
(e
Φ)′ = −iHe−itH Φ, it follows that hΦ, At (f )Ψi is differentiable in t
with
d
hΦ, At (f )Ψi = −iHe−itH Φ, Ψ(t) + e−itH Φ, Ψ′ (t)
dt
= Φ, ieitH [H1 , A(e−itT f )]e−itH Ψ ,
where we have used Theorem 5.17. Hence
Z t
Φ, ieisH [H1 , A(e−isT f )]e−isH Ψ ds.
hΦ, At (f )Ψi = hΦ, A(f )Ψi +
0
= hΦ, A(f )Ψ + iI(t)Ψi .
Thus (8.91) with At (f )# = At (f ) follows. Similarly one can prove (8.91)
with At (f )# = At (f )∗ .
b p ) and
Lemma 8.13. Assume (H.1) and (H.1)’. Then, for all Ψ ∈ D(H
#
#
f ∈ D, Ψ± (f ) := limt→±∞ At (f ) Ψ exist and are given by
#
Ψ#
± (f ) = A(f ) Ψ + iI(±∞)Ψ.
(8.92)
Proof. By (8.91), we have for t1 , t2 ∈ R
Z t1
eisH [H1 , A(e−isT f )# ]e−isH Ψds
At1 (f )# Ψ − At2 (f )# Ψ = i
t2
Hence
#
#
kAt1 (f ) Ψ − At2 (f ) Ψk ≤
Z t1
t2
u# (e
−isT
b + 1))p Ψk,
f )ds k(H
which tend to 0 as t1 , t2 → ±∞. Hence Ψ#
± (f ) exist and (8.92) holds.
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Theorem 8.14. Assume (H.1) and (H.1)’. Then, for all f ∈ D(T −1/2 )
#
b 1/2 ), Ψ#
and Ψ ∈ D(H
± (f ) := limt→±∞ At (f ) Ψ exist.
Proof. Step 1. We first extend the result of Lemma 8.13 to the case where
b p ). Since D is a core of T −1/2 , there exists a
f ∈ D(T −1/2 ) and Ψ ∈ D(H
sequence {fn }n in D such that fn → f and T −1/2 fn → T −1/2 f as n → ∞.
Let t1 , t2 ∈ R. Then, by the triangle inequality,
kAt1 (f )Ψ − At2 (f )Ψk
≤ kAt1 (f )Ψ − At1 (fn )Ψk + kAt1 (fn )Ψ − At2 (fn )Ψk
+ kAt2 (fn )Ψ − At2 (f )Ψk
b + 1)1/2 Ψk + kAt1 (fn )Ψ − At2 (fn )Ψk,
≤ 2C2 kT −1/2 (f − fn )k k(H
where we have used Lemma 8.6. By Lemma 8.13, we have
b + 1)1/2 Ψk.
lim sup kAt1 (f )Ψ − At2 (f )Ψk ≤ 2C2 kT −1/2(f − fn )k k(H
t1 ,t2 →±∞
Then, taking the limit n → ∞, we obtain limt1 ,t2 →±∞ kAt1 (f )Ψ −
At2 (f )Ψk = 0. Hence limt→±∞ At (f )Ψ exist. Similarly one cam show
the existence of the strong limits limt→±∞ At (f )∗ Ψ.
Step 2. We next extend the result in Step 1 to the case where f ∈
b 1/2 ). Since D(H
b p ) with p ≥ 3/2 is a core of
D(T −1/2 ) and Ψ ∈ D(H
1/2
b
b p ) such that Ψn → Ψ and
H , there exists a sequence {Ψn }n in D(H
1/2
1/2
b Ψn → H
b Ψ as n → ∞. Let t1 , t2 ∈ R. Then, by the triangle
H
inequality,
kAt1 (f )Ψ − At2 (f )Ψk
≤ kAt1 (f )(Ψ − Ψn )k + kAt1 (f )Ψn − At2 (f )Ψn k
+ kAt2 (f )(Ψn − Ψ)k
b + 1)1/2 (Ψ − Ψn )k + kAt1 (f )Ψn − At2 (f )Ψn k,
≤ 2C2 kT −1/2 f k k(H
where we have used Lemma 8.6. By Step 1, we have
b + 1)1/2 (Ψ − Ψn )k.
lim sup kAt1 (f )Ψ − At2 (f )Ψk ≤ 2C2 kT −1/2f k k(H
t1 ,t2 →±∞
Then, taking the limit n → ∞, we obtain limt1 ,t2 →±∞ kAt1 (f )Ψ −
At2 (f )Ψk = 0. Hence limt→±∞ At (f )Ψ exist. Similarly one can show
that limt→±∞ At (f )∗ Ψ exist.
b 1/2 ) \ D(H
b p ) or f ∈ D(T −1/2 ) \
Remark 8.12. In the case where Ψ ∈ D(H
D, (8.92) does not necessarily hold, i.e., the term iI(±∞)Ψ may be replaced
by an abstract one which is not explicitly written as a Bochner integral.
A simple model which satisfies (H.1) and (H.1)’ will be discussed in
Chapter 13.
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8.13
Problems
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(1) Let F be a Hilbert space and D be a dense subspace in F .
(i) Let ϕ1 , . . . , ϕn ∈ S ′ (Rd , F ) (n ∈ N) and Dj (j = 1, . . . , n) be
a common domain of ϕj . For αj ∈ C (j = 1, . . . , n), we define
ϕ : S (Rd ) → L(F ) by
ϕ(f ) :=
n
X
αj ϕj (f ).
j=1
Suppose that ∩nj=1 Dj is dense in F . Show that ϕ is an operatorvalued distribution on Rd .
(2) Let f ∈ S (R1+d ) and a ∈ R1+d . Show that fa ∈ S (R1+d ).
(3) Let
Lµν := i~(xµ ∂ ν − xν ∂ µ ), (µ, ν = 0, . . . , d)
acting in L2 (R1+d ) and u(a, Λ) be defined by (8.40).
(i) Let pµ := i~∂µ and pµ = g µν pν . Show that, for all µ, ν =
0, 1, . . . , d,
[pµ , pν ] = 0,
[Lµν , pλ ] = −i~(g µλ pν − g νλ pµ ).
on C0∞ (R1+d ).
(ii) Let Λ ∈ L+↑ . Show that u(0, Λ)C0∞ (R1+d ) ⊂ C0∞ (R1+d ) and,
µ = 0, 1, . . . , d,
u(0, Λ)xµ u(0, Λ)−1 = (Λ−1 x)µ ,
u(0, Λ)∂ µ u(0, Λ)−1 = (Λ−1 )µν ∂ ν
on C0∞ (R1+d ).
(iii) Show that, for all µ, ν = 0, . . . , d,
u(0, Λ)Lµν u(0, Λ)−1 = (Λ−1 )µα (Λ−1 )νβ Lαβ
on C0∞ (R1+d ).
(iv) Show that, for all µ, ν = 0, . . . , d,
[Lµν , Lρσ ] = −i~(g µρ Lνσ − g µσ Lνρ − g νρ Lµσ + g νσ Lµρ ).
on C0∞ (R1+d ).
(v) Show that there is a representation π : l↑+ → L(C0∞ (R1+d ))
of the Lie algebra l↑+ such that
π(imµν ) = iLµν ↾ C0∞ (R1+d ),
µ, ν = 0, . . . , d.
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(vi) Prove that each Lµν is essentially self-adjoint on C0∞ (R1+d )
and
u(0, Rµν (θ)) = e−iθL
µν /~
,
θ ∈ R.
◮ Hint. Apply Theorem 1.23.
Remark. This problem shows that {pµ , Lµν |µ, ν = 0, 1, . . . , d} generates a
representation of the Poincaré algebra associated with M1+d .
(4) Let H be a Hilbert space and H be a self-adjoint operator on H which
is bounded from below. Let E0 (H) be the infimum of the spectrum
of H (see (8.14)). We denote by P0 the orthogonal projection onto
H0 := ker(H − E0 (H)).
(i) Show that
s- lim e−β(H−E0 (H)) = P0 .
β→∞
◮ Hint. Use the functional calculus with respect to the spectral measure of H.
(ii) Suppose that dim H0 = 1 and let Ω ∈ H0 be a unit vector. Show
that, for all Ψ ∈ H satisfying hΩ, Ψi > 0,
e−βH Ψ
Ω = s- lim p
β→∞
hΨ, e−2βH Ψi
(iii) Let Ψ ∈ H be such that, for all sufficiently small ε > 0,
EH ([E0 (H), E0 (H) + ε))Ψ 6= 0. Show that
1
log Ψ, e−βH Ψ .
β→∞ β
E0 (H) = − lim
◮ Hints. Using the spectral representation
D
E Z
Ψ, e−βH Ψ =
e−βλ dkEH (λ)Ψk2 ,
[E0 (H),∞)
estimate the right hand side from above and below. In estimating from below, divide the interval [E0 (H), ∞) into two intervals
[E0 (H), E0 (H) + ε) and [E0 (H) + ε, ∞).
Remark. In applications to quantum physics, H may be the Hamiltonian
of a quantum system. The formula in (i) expresses the orthogonal projection onto the space of ground states of H in terms of the heat semi-group
{e−βH }β≥0 . Part (ii) is concerned with a formula for the ground state of H
in the case where H has a unique ground state. Part (iii) gives a method
to calculate the lowest energy of H from e−βH independently of whether or
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not H has a ground state. Generally speaking, it is important to investigate
ground states and the lowest energy of a quantum Hamiltonian. The above
formulae may be useful for this purpose.
(5) Let H and H be as in Problem 4. Suppose that there exists a non-zero
vector Ψ ∈ H such that
e−βH Ψ
ΩΨ = s- lim p
β→∞
hΨ, e−2βH Ψi
exists and ΩΨ 6= 0. Show that ΩΨ is an eigenvector of H.
◮ Hints. First, use the semi-group property of {e−tH }t≥0 to show that, for
each t ≥ 0, there exists a constant c(t) > 0 such that e−tH ΩΨ = c(t)ΩΨ .
Then, use the strong continuity and the semi-group property of {e−tH }t≥0
to derive the continuity of c(t) in t ≥ 0 and c(t + s) = c(t)c(s), t, s ∈ R.
(6) Prove the following: if there exists a vector Ψ ∈ H such that
D
E
lim Ψ, e−β(H−E0 (H)) Ψ 6= 0,
β→∞
then H has a ground state.
(7) Let (F , D, H, ϕ) be a time-translation covariant QFT with s = 1
such that H is bounded from below and has a ground state Ψ0 ∈ D.
Consider the two-point VEV
E(f, g) := hΨ0 , ϕ(f )ϕ(g)Ψ0 i ,
f, g ∈ S (R1+d ).
Show that there exists a bounded complex Borel measure µf,g on R
such that supp µf,g ⊂ [0, ∞) and
Z
eitλ dµf,g (λ), t ∈ R.
E(f, g(t,0) ) =
[0,∞)
◮ Hint. Use the functional calculus on H.
(8) Let (F , D, P, ϕ) be a translation covariant QFT with s = 1 such
that P0 is bounded from below and has a unique ground state Ψ0 .
Show that, for each pair (f, g) ∈ S (R1+d ) × S (R1+d ), there exists a
bounded complex Borel measure ρf,g on (R1+d )∗ such that supp ρf,g ⊂
[0, ∞) × (Rd )∗ and
Z
E(f, ga ) =
eiaλ dρf,g (λ), a ∈ R1+d ,
(R1+d )∗
where aλ is the natural bilinear form between a ∈ R1+d and λ ∈
(R1+d )∗ .
◮ Hint. Use the functional calculus on the joint spectral measure of P .
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(9) Let X be a non-empty set. For two mappings f, g : X → X, we define
f g : X → X by (f g)(x) := f (g(x)), x ∈ X. Let T (X) := {f : X →
X|f is bijective}.
(i) Show that T (X) is a group with operation (f, g) 7→ f g (f, g ∈
T (X)), where the unit element of T (X) is the identity mapping
I (I(x) := x, x ∈ X) and the inverse element of f ∈ T (X) is the
inverse mapping f −1 : X → X.30
(ii) Let G be a subgroup of T (X) (i.e., f, g ∈ G=⇒f g, f −1 ∈ G)31
and F(X) := {F : X → C} be the set of complex-valued functions
on X. Note that F(X) is a complex vector space with the usual
operation of addition and the scalar multiplication for functions.
For each g ∈ G, we define a mapping T (g) : F(X) → F(X) by
(T (g)F )(x) := F (g −1 x),
x ∈ X, F ∈ F(X).
Show that, for each g ∈ G, T (g) ∈ GL(F(X)) (the general linear
group on F(X)).32
(iii) Show that T : G ∋ g 7→ T (g) is a representation of G. 33
(10) (continued) Let X, G and T be as in Problem 9. Let Σ be a Borel field
consisting of subsets in X and µ be a measure on (X, Σ). Suppose
that µ is G-invariant, i.e., for all g ∈ G and B ∈ Σ,
µ(gB) = µ(B)
(gB := {g(x)|x ∈ B}).
Show that T : G ∋ g 7→ T (g) is a unitary representation of G on
L2 (X, dµ).
(11) Let d ≥ 2 and SO(d) be the d-dimensional rotation group: SO(d) :=
{R ∈ Md (R)| t RR = I, det R = 1}, where I is the d × d unit matrix.
For each R ∈ SO(d), define a mapping T (R) : L2 (Rd ) → L2 (Rd ) by
(T (R)f )(x) := f (R−1 x),
x ∈ Rd , f ∈ L2 (Rd ).
Show that T : SO(d) ∋ R 7→ T (R) is a strongly continuous unitary
representation of SO(d).
(12) Let (H , U, φ, D, Ψ0 ) be a neutral quantum scalar field theory such
that D ⊂ D(Pµ ) (µ = 0, . . . , d) and, for all f ∈ S (R1+d ), Pµ D ⊂
D(φ(f )), µ = 0, . . . , d. Suppose that, for all Ψ ∈ D and f ∈ S (R1+d ),
30 The group T (X) is called the general transformation group on X.
31 G is called a transformation group on X.
32 See Section 2.9.
33 See Section 2.9.
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the H -valued function : t 7→ φ(f )e−itPµ Ψ is strongly continuous.
Show that, for all Ψ ∈ D, f ∈ S (R1+d ),
i
Dµ φ(f )Ψ = [Pµ , φ(f )]Ψ, µ = 0, . . . , d,
~
(13) Let (H , U, φ, D, Ψ0 ) be a neutral quantum scalar field theory such
that D ⊂ D(M µν ) (µ, ν = 0, . . . , d) and, for all f ∈ S (R1+d ),
M µν D ⊂ D(φ(f )) (µ, ν = 0, . . . , d). Suppose that, for all Ψ ∈ D and
µν
f ∈ S (R1+d ), the H -valued function: t 7→ φ(f )e−itM Ψ is strongly
continuous. Show that, for all Ψ ∈ D and f ∈ S (R1+d ),
[M µν , φ(f )]Ψ = −i~(xµ Dν − xν Dµ )φ(f )Ψ, µ, ν = 0, . . . , d.
(14) Let I0 be the time-reversal on R1+d defined by (8.62). For each t ∈ R,
the time translation Tt : R1+d → R1+d by t is defined by
Tt x := (x0 − t, x),
x ∈ R1+d .
Let
G0 := {I, I0 , Tt , I0 Tt |t ∈ R},
where I is the identity mapping on R1+d . Show that G0 is a group.
(15) Let F be a Hilbert space and G0 be as in Problem 14. Suppose that
there exists a mapping U : G0 → B(F ) satisfying the following (a)
and (b):
(a) For all g ∈ G0 , U (g) is unitary or anti-unitary, satisfying
U (g1 )U (g2 ) = ω(g1 , g2 )U (g1 g2 ),
g1 , g2 ∈ G0 ,
where ω(g1 , g2 ) ∈ C with |ω(g1 , g2 )| = 1.
(b) There exists a non-negative self-adjoint operator H 6= 0 on F
such that, for all t ∈ R, U (Tt ) = e−itH/~ .
(i) Show that U (I0 )2 = eiθ for some θ ∈ R.
(ii) Assume that, for all t ∈ R, ω(T−t , I0 ) = ω(I0 , Tt ). Show that
U (I0 ) is anti-unitary and, for all t ∈ R, U (I0 )e−itH/~ U (I0 )−1 =
eitH/~ .
◮ Hints. Note that T−t I0 = I0 Tt , ∀t ∈ R and use the boundedness
from below of H.
Remark. A meaning of this problem in the context of quantum mechanics
is as follows. Let G be a group and H be a Hilbert space. A mapping U
from a group G to B(H ) is called a projective representation of G if,
for each g ∈ G, U (g) is unitary or anti-unitary and, for all g1 , g2 ∈ G,
U (g1 )U (g2 ) = ω(g1 , g2 )U (g1 g2 )
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with ω(g1 , g2 ) ∈ C satisfying |ω(g1 , g2 )| = 1. In quantum mechanics or
QFT, the symmetry defined by G is expressed most generally as a projective
representation of G on a Hilbert space of state vectors. This is a consequence
of Wigner’s theorem (for details, see, e.g., [Bogoliubov et al. (1975), §2.3]).
The conclusion of this problem is that, if the Hamiltonian H obeys the
assumption in Problem 15, the time-reversal eitH/~ of the time development
e−itH/~ is implemented by the anti-unitary operator U (I0 ).
(16) Let H be a Hilbert space and C be a conjugation on H (see Subsection 5.12).
(i) Show that, for each anti-unitary operator W on H , there exists
a unique unitary operator U on H such that W = CU .
(ii) Let H be a self-adjoint operator on H such that CH ⊂ HC.
Let U be a unitary operator on H such that U H ⊂ HU . Define
W := CU . Show that W is anti-unitary and, for all t ∈ R,
W eitH/~ W −1 = e−itH/~ .
(17) Let a > 0 be a constant and T := [−a, a] or R. Define TR : L2 (T ×
Rd ) → L2 (T × Rd ) by
(TR f )(t, x) := f (−t, x)∗ ,
a.e.(t, x) ∈ T × Rd .
(i) Show that TR is anti-unitary.
(ii) Let V : Rd → R be Borel measurable and a.e. finite. Suppose
that the Schrödinger operator
H := −
~2
∆+V
2m
is essentially self-adjoint as an operator on L2 (T × Rd), where ∆ is
the d-dimensional generalized Laplacian and m > 0 is a constant.
Show that
TR eitH/~ TR−1 = e−itH/~ ,
t ∈ R.
Remark. The anti-unitary operator TR is called the time-reversal
operator on L2 (T × Rd ).
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