Free field operators

Free field operators

Citation for published version (APA):

Eijndhoven, van, S. J. L., & Graaf, de, J. (1985). Free field operators. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8505). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1985

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Department of Mathematics and Computing Science

Memorandum 1985-0~

January 1985

FREE FIELD OPERATORS

by

S.J.L. van Eijndhoven

J. de Graaf

Eindhoven University of Technology

Department of Mathematics and Computing Science

PO Box 513, Eindhoven

Summary

by

S.J.L. van Eijndhoven and J. de Graaf

In the present paper we give a mathematical rigorization and interpretation

of the canonical (anti-) commutation relations which occur in the theory

of the quantized free field. In our set-up, with respect to the physics,

we feel inspired by the paper [Ro] of B. Robertson. His paper covers all

essential elements of the subject under consideration. The main part of

[Ro] will be mathematically justified and even generalized. In this

connec-tion we menconnec-tion also the monograph [Sh] of A.S. Shvarz. With respect to

the involved mathematics the papers [Co] of J.M. Cook and [KMP] of P.

Kristensen et al. have been a source of inspiration.

Our set-up is closely related to two of our earlier papers [EG 1-2] in which

we presented a mathematical interpretation of Dirac's formalism. This

inter-pretation is based on two new mathematical concepts: the concept of Dirac

basis and the concept of bracket. The bracket is no longer regarded as a

"number" but as an analytic function on the open right half of the complex

plane. As in [EG 1-2], we use the theory of generalized functions which has

Quantum theory and distribution theory seem to be dissolubly connected.

One might wonder whether the choice of the distribution theory plays an

essential role in a mathematical interpretation of Dirac's formalism and

related physical concepts. The answer is threefold: There are strong

indi-cations that similar interpretations can be derived in a very wide class of

distribution theories (which include Schwarz's theory of tempered

distri-butions). This class has been described in our papers [EK] and [EGK].

On the other hand, the distribution theory employed here, introduces

gene-ralized functions in a way very close to the physical intuitive view on

im-proper functions. Moreover, it seems natural to adapt the used distribution

theory to the concrete quantum mechanical system under consideration.

O. Introduction

The material of this paper is presented in the abstract setting of

functi-onal analysis. Thus we expect to be able to avoid notatifuncti-onal problems and,

more importantly, to clarify all mathematical concepts in the best possible

way.

We start with a separable Hilbert space X in which we choose a fixed

ortho-normal basis (Vj)jE~' Let X ®a X denote an algebraic tensor product of X.

Then the bilinear mapping ® sends each pair (vk,V£) to v

k ® v£' In a natural way the set {v

k ® v£

I

k,£ E ~} becomes an orthonormal basis for a Hilbert space X(2) which is a completion of X ®a X. X(2) may be called the two-fold

Hilbert tensor product of X. Proceeding inductively, the k-fold Hilbert

tensor product X(k) of X can be defined. The Fock space F is their countable direct sum,

(0,1) F

=

$ X(k)

k=O

where we take X(O) = t and X(l) = X.

The Hilbert space

F

is the middle component of the Gelfand triple

SF,H

c F c

TF,H'

Here

SF,H

is called the Fock analyticity space and

TF,H

the Fock trajectory space. Further, H denotes a well-known positive

self-adjoint unbounded operator in F,

The one-particle state is represented by the trajectory space Tx,A' where

A is the positive self-adjoint operator in X satisfying

(0.2) Av.=jv. ,

For each ~ E Tx A we define the annihilation operator

_,

a(~) and the crea-tion operator e(~). The operator a(~) is a continuous linear mapping from

Sf,H into Sf,H which 1s not extendible to Tf,H' 1n general. The operator

e(~) 1s a continuous linear mapping from Tf,H into Tf,H; its restriction to Sf,H does not map Sf,H into Sf,H'

Let (G~)tEM denote a Dirac basis in Tx,A' We present a mathematical inter-pretation of the following heuristic formulas

(0.3) a(~) =

)

<\1>

I

G

t

>

a(G~) d)l~ e(\1» =

_J

<G

t

I

~> e(G~) dtl~ • M

1 (d2 2)

,

We note that for X = L

2

(m),

A =

2'

- 2

_dx + x + 1 and G

t

=

0t' ~ €

m,

the Formulas (0.3) are the usual relations for the so-called free field

operators.

Next we consider the symmetric Fock space f(+) and the antisymmetric Fock

space f(-). Both f(+) and f(-) are closed subspaces of f. If p(+} and p(-)

denote the corresponding projections, then we define

(0.4)

a

(±) (~) = p(±} a(\p) pet) ,

We give a rigorous interpretation of the following heuristic (anti-)

commu-tat ion relations

(0.5)

[a(±) (\P) ,

e

(±) (IJI)]

=

0 etc.

+

We also devote attention to the concept of second quantization. For a

linear operator

V

from

Tx,A

into

Tx,A

we rigorize the following expression for the second quantization of

V

Contents

Q±(V)

=

If

dll~

_dll

n C (±) (Gl;) <Gt

I

V

I

Gn

>

ct (±) (Gn)

M2

1. The Gelfand triple Sy,B e Y e Ty,B'

2. Some results on continuous linear mappings.

3. The concepts: Dirac basis and bracket •

4. The Fock space •

5. Introducing the machinery.

6. Annihilation and creation operators.

7. The symmetric and anti-symmetric Fock space.

8. A mathematical interpretation of the field operator formalism in

quantum mechanics •

1. The Gelfand triple Sy B

.

e Y e Ty B

,

Let Y be an infinite dimensional separable Hilbert space and let (en)nEm

be an orthonormal basis in Y. Each element y E y is represented by the

co

R.₂-sequence «y,en»nE:fi' We extend Y by considering formal series

I

n=l

w e

n n

where the sequence (Wn)nE:fi does not necessarily belong to t2' Therefore

we introduce a sequence (An)nE:fi of positive numbers with the property

(1.1) lim e

-A

n t

=

0 , n-+<»

(1.2)

I

e n -A t

1 12

w

<"".

n=l n

Because of (1.1) the series

I

n=l the operator

8

by 00 (1.3) D(B) ={fEyl

I

n=l and 8f

=

I

n=l A (f,e)e , n n n w e

n n may be divergent in X. We introduce

f E D(8) .

Then

B

is a nonnegative self-adjoint operator in

Y.

Because of Condition -t8

(1.1) the semi group (e )t>O consists of compact operators on Y. We note

that the parameter t, we use in this paper, must not be regarded as the

time parameter.

To the sequence (wn)nE~ defined in (1.2) we link the mapping F from (0,"")

into Y defined by 00

-x

t e n w e n n t > 0 . F(t)

=

L

n=l

Observe that F(t + 1') = e -1'B F(t), t,1' > O. Conversely, for a mapping G from

(0,00) into Y which satisfies G(t + 1') = e -1'8 G(t) for all t, l' > 0, the

se-A t -quence (Yn)nE~' defined by Y

n

=

e n (en,G(t», has property (1.2). We have 00 G(t)

=

I

n=1 -Ant e y e . n n

(1.6) Definition.

The trajectory space

Ty,B

is the vector space of all mappings F with the property

"'t>O "'PO F(t + "C)

=

e-"CB F(t) . The elements of

Ty,B

are called trajectories.

Besides the trajectory space

Ty,B

we introduce the analyticity space

Sy,B'

(1.7) Definition.

The analyticity space

Sy,B

is defined by

u

t>O

-tB

I

=

{e y y E y, t > O} •

We mention the following important relations

(1.8)

'"

FETy

B

'"

t>O .

•

F(t)

E Sy,B

On the space

Ty B

_,

we impose the locally convex topology generated by the seminorms F 1+ IIF(t)lI, t > O.

Ty,B

is a Fr@chet space with this topology.

On

Sy,B

we impose the inductive limit topology brought about by the spaces

-tB

e (Y). We mention that both

Sy,B

and

Ty,B

are inductive and projective

limits of Hilbert spaces. The spaces

Sy,B

and

Ty,B

are bornological, barreled and Montel. If in addition the sequence (An)nEfi satisfies

(1.9)

I

e n -A t < <X>

n=l

(1.10) Definition

Let F E

Ty,B

and let g E

Sy,B'

Then the complex number <g,F> is defined by

.B

<g ,F>

=

(e g ,F(.»

where • > 0 has to be taken so small that e·B g E Y. The definition of <g,F> does not depend on the choice of • > O.

With the pairing <-,.>,

Sy,B

and

Ty,B

can be seen as each other's strong duals.

If Y

=

L2(M,~) where M is any measure space with a-finite measure ~, then for any nonnegative self-adjoint operator

B

in

y,

the space

Ty,B

consists of generalized functions on

M.

SO the theory of analyticity and trajectory spaces presents a functional analytic model for a particular type of

dis-tribution theories.

For more details and examples of this theory we refer to [G], [EGm] and

[EG 3-4].

2. Some results on continuous linear mappings

For the purposes of this paper some knowledge of operator theory for

ana-lyticity and trajectory spaces is required. We present here the necessary

material.

Let Z be a separable Hilbert space with orthonormal basis (gn)nEm' Further,

let the sequence (Un)nEm of nonnegative numbers satisfy Condition (1.1),

i.e. lim

e-~nt

= O. Accordingly we define the operator

C

in Z by

n-+<»

00

D (C)

=

{z E Z

I

I

u!

I

(z, gn)

12

< <X>}

and

QO

Cz =

I

_{l1n (z,gn) gn '}

n=l

z E D(C) •

Five types of linear mappings occur: the continuous linear mappings from

Sy,B

into Tz C' from Ty,B into

_,

Sz,C'

from Sy

_,

B

into

Sz

.

C and from Ty

_,

B

into Tz,C and, also, the extendible linear mappings, 1.e. the mappings from Sy

_,

B

into

_Sz,C

which can be extended to a continuous linear mapping from Ty,B into Tz,C' These five types admit the following characterizations.

(2.1) Theorem.

A linear mapping

V : Sy,B

+ Tz,C is continuous iff

-tC -tB

V t>O : e V e Y + Z is bounded .

A linear mapping

M :

Ty,B +

Sz.C

is continuous iff ,C

M .B

3 e e Y + Z is bounded .

• >0

A linear mapping L : Sy

_,

B +

Sz

_,

C is continuous iff

V 3

t>O .>0 e'C L e -tB y + Z is bounded .

A linear mapping R : Ty

_,

B

+ T_z

_,

C is continuous iff -tC R

.B

e e : Y + Z is bounded .

A linear mapping E :

Sy,B

+

Sz,C

is extendible iff E* maps

Sz,C

into

Sy,B'

Its extension t : Ty,B + Tz,C is defined by

<f,tG>

=

<E*f,G> , Both E and

E

are continuous.

""

co

I

-A t

I

e-llnt < "" for all t > O. Then both

Sy,B

Suppose that e n < co and

n=1 n=1

and Sz Care nuclear. In this case the above mentioned types of continuous

_,

linear mappings have characterizations based on infinite matrices. Now

Theorem (2.1> can be replaced by

(2.2> Theorem.

Let both Sy,B and

Sz,C

be nuclear.

A linear mapping

V :

Sy

B

+ T

z C is continuous iff

,

,

sup (e-tllm e-tAn I<g

,Ve

>1> < "" .

m n

n,mElN

A linear mapping M : Ty

_,

B

+ Sz C is continuous iff

_,

sup (eL)..lm e'rAn I (Me ,g > I> < "" •

n m

n ,mElN

A linear mapping L : S + S is continuous iff

y,B z,C

V 3

t>O T>O sup (e- tAn e Tllm I(Ve,g >1> n m < c o .

n,mElN

A linear mapping R : Ty

B

+ T_z C is continuous iff

J ,

V 3

t>O T>O sup (e -t II m e n <g, R e > > T A I m n 1

n,mElN

< co •

A linear mapping

E

is extendible iff

-tll 'rA -t>.. Tll

I

I

sup (e m e n + e n e m

(E

en,gm> > n,mElN

< co •

We denote the space of extendible linear mappings by E(Sy,B'SZ,C>' For

the other spaces of continuous linear mapping we employ the usual notations

L(Sy,B,TZ,C)' L(Ty,B'Sz,C)' L(Sy,B'SZ,C) and L(Ty,B,Tz,C)' In our

mono-graph [EGm] we have extensively studied the algebras L(Sy,B)' L(Ty,B)

and E(Sy

_,

B); and in particular their algebraic and topological structure.

3. The concepts: Dirac basis and bracket

In [EG1] the notion of Dirac basis has been introduced. Each orthonormal

basis is a Dirac basis. In general, a Dirac basis may consist of an

un-countable number of elements.

Let Ty B be a nuclear trajectory space. The definition of Dirac basis is

_,

the following.

(3.1) Definition.

Let M be a measure space with nonnegative a-finite measure ~. Let

G : a ~ G_a, a E

M,

be a mapping from M in Ty,B' Suppose the functions

a~ <ek,G

a> are ~-measurable and satisfy

I

<ek,Ga> _<et,Ga>

d~a

=

°kt • k,t Eli.

M

Then the quadruple (M,~,G,Ty,B) is called a Dirac basis. Instead of

(M,~,GJTy,B) we mostly write (Ga)aEM'

Remark.

It is appropriate to reserve the notion of Dirac bases for equivalence

classes. However, we will stick to a fixed "canonical" representant. Cf.

In [EG1] we have proved the following expansion theorem.

(3.2) Theorem.

Let f €

SY,B'

Then

TB

<e f,G > G (.) dJ.l

ex. ex. ex.

.B

where • > 0 has to be chosen so small that e f € Sy, B' The integral ex-pression does not depend on •• Further, the integral

H

I

I <e TB f,G ex. > I IIG ex. (T) II dJ.l ex. exists.

Next we give an outline of our paper [EG1].

For the ket space we take a nuclear trajectory space Ty

_,

B' The choice of Y and

B

may depend on the quantum mechanical system under consideration. In accordance with Diracfs formalism we denote the kets by IG>. SO any

element IG> can be written as

where the sequence (w

k) satisfies

and where Ie?, k € 1N, denotes the function

t > 0 •

con-tinuous linear fUnctional

y

on Y by y(x)

=

(x,y) , x

E

Y •

By the Riesz representation theoren, the mapping y» y is an antilinear

norm-preserving isomorphism from Y into

Y.

The norm in the Banach space

Y

originates from the inner product (.,-) defined by

So

Y

is a Hilbert space. In

Y

we define the nonnegative self-adjoint ope-rator

B

by

followed by the natural linear and self-adjoint extension. Then the

trajec-tory space

Ty,S

is well-defined. Each element of

Ty,S

is in 1-1 correspondence with an element of

_Ty

_,

_S' viz.

co co

Ty,S

3

a

L

-A

L

G

E

Ty,s

= _wkek ++ w k ek =

.

k=1 k=1

The correspondence G ++

a

is antilinear, i.e.

Now for the bra space we take the space

Ty

_,

S'

Its elements are denoted by

<GI.

It is clear that each bra

<GI

is in 1-1 correspondence with the ket

IG>.

Moreover, this correspondence is antilinear. We note that each bra

<GI

can be written as

where the sequence (wk)kE:tf satisfies

:;~

(I

wkl e -Akt) < "' _{and where <ek}

1

denotes the function

t > 0 .

To the elements f of Sy B

_,

there correspond the so-called test kets and test bras. The functions <fl and 1 f> from (0,"') into X can be extended to

func-tions from (_~,w) into X for some ~ > 0 dependent on the choice of f.

The bracket <-1-> is defined as follows.

(3.3) Definition.

The bracket of <FI and IG> denoted by <F I G> is the complex valued function

on (O,w) defined by

<F

I

G> t 1+ <F

I (I

G> (t» •

Definition (3.3) makes sense because <FI is a continuous linear functional

on Sy,B and

10>

(t) E Sy,B:

We observe that the function <F

I

G> can be extended to an analytic function on the open right half of the complex plane. Thus it can be seen as an

almost periodic distribution along the imaginary axis.

We mention the following relations

(3.4) <FI (t)

=

<F(t)

I

(0) , I G> (t)

=

1 G(t» (0)

<F I

0>

(t)

=

<F(t) 1 G> (0)

=

<F

I

G(t» (0) <FIG>=<GIF>

For each test bra <f 1 the function <f I G> can be extended to

Next we present a mathematical interpretation of Dirac's expansion

theo-rem for kets with respect to continuous basis.

(3.5) Theorem.

Let

(1a»aEM

be a Dirac basis in

Ty,B'

and let

IF>

be a keto Then we have

<a

I

F>

I a>

dll a

to be interpreted in the following sense

IF>

(t)

=)

<a

I

F>

('r)

Ia>

(t - T) dll ,

a t > 0 .

The integral expression does not depend on the choice of T, where 0 < T < t.

(3.6) Example.

Let

I j>, j

E IN, be an orthonormal bas is in X and let

I F>

be a ket. Then

IF>

=

L

<j

I

F> I j>

in the sense that

IF>

(t)

=

jElN

t >

o.

I

<j I F>

(t)

I

j>

(0) ,

jElN

With our concept of bracket we can interprete Dirac's orthogonality

rela-tions for continuous bases,

We only give a rough sketch of this interpretation here (cf. [EG1]). The

evaluation functional

Os :

f ~ f(a),

a

E

M,

belongs to the space

T

_L2(f~,l.l)

-

for a certain operator

B

which is unitarily equivalent to

B.

So 013 is a trajectory

and <a

I

13>

= 0B(a) has to be interpreted as <a

I

8>

(t) = 0e(a;t) , t

>

0 •

4. The Fock space

General vector space theory yields the existence of an algebraic tensor

product of Y and Z, we denote it by Y ® Z. The bilinear mapping ® sends a

the pair [en,gm]

N nations

I

a e n=1 n n to e ® g and more n M m and

L

13_m gm into m=l usual identifications,

generally the pair of linear combi-N 11

I I

a 8 (e ® g ) with the n=1 m=l n m n m

In Y ® Z we define a sesquilinear form through a

Then Y ® Z becomes a pre-Hilbert space with orthonormal basis

a

(e _n ® g) _{m n,}

mE-'

_{... ,} Its completion is denoted by Y ® Z; it is called the Hilbert tensor product of Y and Z, Naturally, Y ® Z consists of all series

with

Let X be a separable infinite dimensional Hilbert space with orthonormal

(4.1) with Ax

=

I

j (x, v j) Vj jEIi D(A)

=

{x E X 1

I

j2 1 (x,V J.)12 <

~}

jEIi

(E. g. we can take X

=

L2 (Il) and A

=

t (-

d 2

2 + x 2

+ 1).) Then the spaces dx

Sx

_,

A and Tx

_,

A are nuclear.

Next, we can inductively define the Hilbert spaces X(k)

=

X ® ••• ® X

(k times). X(k) is called the k-fold Hilbert tensor product of X. An

ortho-normal basis in X(k) is established by the vectors v. ® •••

J 1

j

=

(j1"'" jk) E lik. The space X(k) consists of all series

(4.2) where (4.3) where

I

Wj (v. jElNk J1

I

Iw.12 < ~ jElNk J ® ••• ® v . ) J_k

The inner product in

(1) (2)

I

(1) (2) (w ,w )X(k)

=

Wj Wj jElNk (i)

I

(i) (v. ® ••• ® w

₌

_Wj v. jElNk J1 Jk X(k) is given by ) i

₌

1,2

.

® v where jk Further, if w l ' ... , w

k belong to X, then the sequence (w j) jElNk with W

=

(w

1' v. ) ••• (wk,v. ), j E lN k

, uniquely determines an element of X(k).

j J₁ J

k

This element is denoted by w 1 ® tensor product of (w

1' ••• ,wk).

® w

k and it is called the k-fold

In X(k) we define the positive self-adjoint operator A(k) by

(4.4) ® ••• ® v. )

J_k

® ••• ® v. ) J_k

followed by linear and unique self-adjoint extension, i.e.

A (k)

=

A ® I ® ••• ® I + I ® A ® I ® ••• ® I + ••• + I ® I ® ••• ® I ® A •

So we take

Ijl

=

j1 + j2 + + jk'

Since

jJ~k

e-tljl < ro for all t > 0, the spaces SX(k),A(k) and TX(k),A(k)

are nuclear. We observe that

- t A(k) (

e v. ® ••• ®v.)

J 1 J k

It leads to the following results.

(4.5) Proposition. tt+- <v ,F > j1 1 - tj 1 -tjk = (e v.. ) ® ••• B (e v. ) J₁ J_k <v ,F > (V j jk k 1

is a member of TX(k),A(k). This mapping is denoted by Fl ® " . ® F_{k ,}

The space TX(k).A(k) is a k-fold topological tensor product of Tx,A'

Let

The

f1.···,f k € Tx,A' Then f1 ® ••• ® fk € X(k) belongs to SX(k),A(k). space SX(k),A(k) is a k-fold topological tensor product of Sx,A'

Proof: Cf. [EGmJ. Ch. III.

(4.6) Lemma.

Let Y1'Y2"" be Hilbert spaces, and let

B

1

,B

2, ••• be positive self-adjoint

operators in Y

1,Y2, ••• , respectively. Then we define the operator diag(Bk)

by

00

The operator diag(B

k) is positive and self-adjoint in

co

We put F

= •

X(k) and H

=

diag(A(k» with A(l)

=

A, X(l)

= x,

A(o)

=

0 k=O

and X(O)

=

t. The space

SF,H

is called the Fock analyticity space and TF

_,

H

the Fock trajectory space. The operator

H

has a complete orthonormal basis of eigenvectors with the set of natural number as its spectrum. The

multiplicity of 0 equals mO

=

_{1, and of N::: 1,2, • •• equals mN :::}

f

(:=~)

=

N-l

(N-1)

k-1

=

2 • (We note that k-l is the number of ways to write N as a sum of k natural numbers, e.g. 5

₌

1 + 1 + 3

=

1 + 3 + 1

=

3 + 1 + 1 = 2 + 2 + 1

=

2 + 1 + 2 = 1 + 2 + 2. ) It follows that the operators e

-tH

t > 0, are compact, but they are not Hilbert-Schmidt for o < t < log 2. So the spaces

SF,H

and

TF,H

are Montel but not nuclear.

5. Introducing the machinery

In the remaining of this paper we stick to Dirac's bracket notation

be-cause of its expressive nature. The technical results we will need at the

=

end of this paper, are contained in this section. We will rigorize a number

of heuristic formulas which are in use by theoretical physicists. We start

with some additional notations.

The trajectory space TX(k),A(k) represents the k-particles kets and bras;

<F ;

11

we shall also write

I

F> and

<Fl.

The "kets" and "bras" related to

T F , H are denoted by

{I

F ; k>}

~

and

{<F ;

k

I }

~

.

As observed in tbe previous section the set {v

j

1

an orthonormal basis in X(k) and each Vj ® ••• ® V. is an eigenvector of

1 J_k

A(k). These eigenvectors are denoted by

(5.1)

It follows that

(5.2) _{IF; k>}

₌

r

<j ; kiF ; k>

I

j ; k> .

jElNk

with the usual interpretation as given in Section 3,

The relation

(5.3)

leads to the following considerations: Associated to the k-tuple

From Theorem (4.5) it follows that

So it makes sense to write

and for the corresponding bra <G ; k I •

(5.5) (cf. [Di], p. 81) •

Similarly, for a k-tuple of test kets (lf

1> , ••• , Ifk» the k-particles ket

is a k-particles test ket, and <g ; k I ,

a k-particles test bra.

Now let I~> be a I-particle keto Identifying SX(k),A(k) with the space of all k-particles test kets. we define the operator d_k «<I>

I)

on SX(k). A(k) by

(5.5)

Remark.

For I fl> , ••• , I f_k> test kets we have

(5.6) Lemma.

Let Iw> be a normalizable ket, i.e. X - lim Iw> (t)

=

Iw> (0) exists. Then

HO

dk«wl) extends to a Hilbert-Schmidt operator from X(k) into X(k - 1).

Proof: We compute the matrix of dk«wl) with respect to the orthonormal

So

Thus the assertion has been proved.

o

(5.7) Lemma.

Let I~> be a ket, and let t > O. Then

k-l Proof: We observe that for all n,m

E

~ and n

1

E

~

=

<qJ

I n₁

>

(t) d =

<m;

k - l l a.k«~1 (t»

I

n1,n; k> (0) •

, nm

o

(5.8) Theorem.

The operator

ak«~I)

maps SX(k),A(k) continuously into SX(k-l).A(k-l)·

Proof: Combine Lemma (5.6) and Lemma (5.7), look at Theorem (2.2).

o

(5.9) Theorem.

The operator ak«~I) is extendible iff I~> is a test keto Proof: Following Theorem (2.5) we have to prove that V

Let t > O. If I~> is not a test ket, then for all T > 0 the expression

<*)

is infinite. If I~> is a test ket, then we can take 0 < T < t so small

that I ~> (-T) E X and hence the expression (*) is finite for such T. 0

We denote the dual mapping of ak«~I) by Ck(I~», Then Ck(I~» maps the (k - 1) -particles ket space T X(k-1) , A(k-1) cont inuously into the k-particles ket space TX(k),A(k), Easy computation shows that

(5.10)

and also that

We have the following corOllary,

(5.11) Corollary.

Let I~ be a test keto Then ~(I~) is a continuous linear mapping from SX(k-1),A(k-l) into SX(k),A(k) with Hilbert space adjoint

Ck(I~)*

=

= dk(<tp\).

Remark,

The mappings

and

\~ a test ket

Now let (1~»~€M be a Dirac basis. For each ket l~> we want to interprete the following heuristic expressions

and

a

_k

«411)

==

J

<41

I

s>

a

_k «l;

I )

dllE; M C k

(141))

==

J

<l;

I

41>

Ck

(Il;»

d 1.\ M

To this end, we first prove the following lemma.

(5.12) Lemma.

Let I~ be a test keto Then ak«~I) can be expressed by

co

I

«P

I JI.> (0) a

k «JI. I) with convergence in the following sense JI.=1

Proof: Let t > O. Then by Lemma (5.6) and (5.7)

-R,t

=

e Hence

co

I

I

«P

I

JI.> (0)

1

lie t A(k-1) a

k «9.

I)

e -t A(k) II :;; JI.=1

So for all t > 0 we have

co

=

I

«P

I

JI.> (0)

ak«Jl.1

(t» .

JI.=l

Employing the expansion theorem for Dirac bases, we derive ==

1:

<lP

I t> (0) d_k «R.I) == t==1 ==

~

(J

<lP

I

~>

(0)

<~

I t> (0)

dll~)

_{dk «t}

I)

=

R.-1

M

=

I

(J

<lP

I

~>

(-'1:)

<~

! R.> (T)

dll~)

dk«tl)

=

R,=1 M

<:)

J

<lP

I

~>

(-T)

(~ <~

I

t> (T) dk«t!»)

dll~

=

M

t-1 ==

J

<lP

I

~>

(-T)

dk«~1

(T»

dll~

• M

*) It is allowed to interchange summation and integration:

(~

f)

1<lP

Ii;>

(-T)<~

I D (T)llIdk«R.I)1I

dll~ ~

R,-1

M

~

CIl

.-2h)1

(MI

I«I!

1

,>

(-T)

12

d",f .

We come to the following theorem.

(5.13) Theorem.

Let I~> be a keto Then

dk

«~

I)

=

f

<~

_{I t;> d k}

«~

I )

dll~

M

in the sense that

V t>0 e t A(k-1) d

k

«4>

I)

e -t A(k)

=

J

<~

I

t;> (t - T) dk

«~

I

(T»

d1J~

•

Proof: Observe that

et A(k-l) ak«m 'i'

I)

e-t A(k)

=

a <

I

k ( ~ ( t

»

and that <q, I (t) is a test ket.

Similarly, by taking adjoints

(5.14) Corollary.

Let Iq,> be a keto Then

C

k ( I

~»

=

f

<t;

I

~>

Ck (

I

f;»

dllt;

M

in the sense that

Vt>o e -t A(k) C

k

(I

W»

e t A(k-l)

=

f

<i;

I

q,> (t - T) Ck

(I

t;> (T» dllt;

o

M 0

We finish this section with the presentation of the technicalities which

are involved in the concept of second quantization.

Let

V

E

L(T

_x

_,

A)' Then

V

satisfies -t

A

V T

A

e e : X + X is bounded •

So for all i; E M we obtain

We can compose the operators Ck(V I;» and ak«~I). The product Ck(V 1~»ak«~I) is a continuous linear mapping from SX(k),A(k) into TX(k),A(k)' Consider the following estimation

:ii lie -t A(k) C

k

(V

1

~»

e t A(k-l) II lie t A(k-l) ak

«~

I)

e -t A(k) II :ii

For all t > 0 and T > 0 sufficiently small we thus obtain

Mf

lie -t A(k)

c.

(V

I

~»

a

«~

I)

e -t A(k) II dj.l :ii

k k F,:

Hence the integral expression

M

J

e -t A(k)

c.

(V

I

~»

a

«F,:

I)

e -t A(k) dJJ

k k ~

denotes a bounded linear operator from X(k) into X(k). Now the expression

(5.14)

acts in the following way

(5.14')

(J

Ck(V IF;» ak«F;I) dlJF;) • Itp;

k>

M

(5.15) Proposition.

The linear mapping

defined in (5.14) and (5.14') belongs to L(SX(k),A(k),TX(k),A(k»' Further,

for all t > 0 and 0 < L < t, we have

=

f

<s'

I

VI

s>(t-L)Ck<ls'>('r»e-2tA(k-lak«sl

<t»dlJ~

® lJF;' .

MxM

Proof: Continuity of

MI

C

k (V Is» ak «I;

I)

dlJl; follows from the considerations which led to (5.14).

The other assertion in the proposition can be derived by the following

straightforward computation

=

Ck«V Is» (t»

=

J

<I;'

I V IF;> (t -L) ck<II;'> ('r» dlJk

M

6. Annihilation and creation operators

In Section 4 the Fock analyticity space

Sf.H

and the Fock trajectory space

T f • H have been introduced. We shall denote their elements by {I g ; IV} ~, { IF; IV} ~. etc. With the aid of the operators ~ «~ I). k E 1N J introduced

in the previous section, we define the operator a«~I) on

Sf.H

by

(6.1)

or, equivalently. by the operator matrix

(6.2)

o

_v'1

_a

1 «4>1 )

0

o

12

a2«~

I)

o

13

a

3 «4>1 )

.

o

• • (6.3) Theorem.

Let 14» be a keto Then the linear operator a«~I) maps

Sf.H

continuously into

Sf.H'

Proof: Let t > 0 and let 0 < , < t. Then by Lemma (5.6) and (5.7)

sup (ilik e' A(k-1) a

k«4> 1 ) e- t A(k)lI) ;:;i; k€1N

;:;i; sup

(Ik

e -k(t-1") lie' A(k-1) a

k «~ 1 ) e -, A(k) II) ;:;i;

k€lN

;:;i; 1l1<I»(,)11 sup (lke-k(t-,» < 0 0 .

k€lN

Hence the opera tor e'

H

a

«~

1 ) e -t

H :

f -+ f is bounded.

(6.4) Theorem.

Let I~ be a test keto Then a(~I) is extendible. Proof: We have to proof that

-t H I T H

"'t>O 3

T>0 : e a(~ ) e : F -+- F is bounded . Let t > 0 and 0 < T < t so small that I~ (-T) E X. Then

lie -t H

a(~I)

e't' Hii

~

sup

nlk

e -t A(k-l)

ak(~I)

e't' A(k) II

~

kE:N

::i e t - T II Iq:»(-"n Ii sup

(Ik

e-k(t-T»

kElN

We denote the dual mapping of a«~ I) by c. ( lIP». Then C. ( I ~» maps T F. H

con-o

tinuously into

TF H'

Further, if

I~

is a test ket, then

c.(I~)rs

belongs

,

F ,H

to

L(SF

.

H)' The linear mapping c.(I~) acts on

TF H

_,

as follows

where we put IF; -V = 0,

c.

o

(I ~»

=

O. It is represented by the operator matrix

o

o

11

c. 1 ( lIP» 0

12

c. 2 ( lIP» 0

13

c. 3 ( lIP» 0

o

• •

We mention that for any normalizable ket Iw>

Moreover, the operators (()(

I

w»

=

a«w

I )

+ c.(

I

w» and ({)

t (

I

w»

=

a«wl) - c.( Iw»

12

=

are self-adjoint in

f.

i12

From the previous section we obtain the following propositions which

eventually will lead to a rigorization of some heuristic formulas used

in the formalism of boson field operators and fermion field operators.

(6.5) Proposition.

Let I~> be a keto Then we write

with the following interpretation: Vt>O VO<t <t <t

2 1

e 1: 1 H

a«~

I )

e - t H =

=

J

<~

I

f;> (t

1 - t2)

a.«~

I

(1:2

»

e -(t-tl)H

djl~

M

where each of the integrals converges absolutely in

L(f).

Proof: The proof is a consequence of Theorem (5.13), and the estimation

Remark.

= sup

(Ik

lIa.k«~1

(t

2»lIlIe-(t-t 1 )A(k)lI)

~

kEm ~

1111;>

(t 2) II sup

(Ik

e -<t-t1)k) kEN

o

(6.6) Corollary,

Let I~> be a ket, Then we write

c.(

I

~»

=

I

<~

I

~>

c.(

I

~»

dllE;

M

with the following interpretation: V V

t>O 0<1"2<1"1 <t

Each of the integrals converges absolutely in L(F).

Further, we give an interpretation of the integral expression

) C(V

1<»

a

(<I;

I)

d"<

where V is a member of L(TX,A),

From the previous section for all E; E M the following estimation can be derived

~ 1I1E;>(t)III1I~>(1")lIl1e-tAVe1"AIi

sup (k e-t(k-l» ,

kElN

H ere L > 0 must be taken so small that the operator e -t A VeL A ].' s bounded,

So as in the previous section it follows that for all t > 0

J

e -t H C.(V

I

;»

a«~

I

)

e -t H dll~

M

converges absolutely and hence can be regarded as a bounded operator on

f. It follows that the integral expression

J

C.(V

I

;»

a«~

I )

dll~

M

denotes a continuous linear mapping from Sf,H into Tf,H' Its action on an

element

{I

f ; IV} E Sf ,H is given by

(6.7) t 1+

(f

e-tHc.(VI~»a«~I)e-1"Hd)lf;){lf ;IV(-T)}~

M

where

{I

f ; IV (_or)}~ belongs to f for 1" > 0 sufficiently small.

(6.8) Proposition.

The linear mapping

defined by (6.7) belongs to L(Sf,H,Tf,H)'

=

J

<~

t

I

v

I

~>

(t - r) c(

IE;

,> (1:» e -2t H a«s

I

(t» dlls dlJt;;' •

MxM

Each of the integral converges absolutely in

L(f).

Proof: Cf. Proposition (5.15). (6.9) Definition. The equality

f

c(ls'» <Sf

I

V

I

s>

a«E;

I

>

dlJ s dlJt;'

MxM

can be interpreted as: "t>o "O<'t<t :

e-tH (

f

C(V It;» a«sl) dlJt;)e-tH

=

M

=

f

<E;'

I

V

I

s> (t - T) c(

I

Sf> (T» e -2t H a(<l;;

I

(t» dllt; dlll;' .

MxM

7. The symmetric and anti-symmetric Fock space

The set of vectors

(7.1)

establishes an orthonormal basis in X(k). Let P

k denote the permutation

group of order k. For each (]

e:

P

k we define the linear operator

a :

X(k) ~ X(k) by (7.2)

...

The operator

a

is unitary on X(k). Next we define

(7.3) p(+) _k = -1

I

a

...

k!

aEP

k

Then (p(+»2

=

p(+)

=

(p(+»*,

k k k i.e.

P~+)

is an orthogonal projection. The

range of p!+) is called the k-fold symmetric Hilbert tensor product of X;

it is denoted by X(+)(k). For completeness we take

P~+)

=

1.

(7.4) Definition.

The orthogonal projection p(+) on F is defined by

Further we define F(+)

=

P(+)(F). The Hilbert space F(+) is called the Boson Fock space.

DP

We have F(+)

=

e X(+)(k). k=O

It is not hard to see that

P~+)

(A(k»

=

A(k) p!+), and We put A(+) (k) = p!+) A(k) p!+), and H(+)

=

p(+) Hp(+).

(7.5) Lemma. SF(+) H(+)

=

p(+)(S )

•

F,H and

T

_{F (+)}

_,

H(+)

=

p(+) (T _F,H ) (+) (+) hence P H = H P •

The eigenvalues of the self-adjoint operators H(+) are the numbers N

=

=

0,1,2, . . . . Because of the symmetrization the multiplicity

~+)

is strongly diminished in comparison with the multiplicity ~. In fact the

integer summands without regard to order". (E.g. 5

=

1 + 4

=

2 + 3

=

=

1 + 1 + 3

=

1 + 1 + 1 + 2

=

1 + 2 + 2

=

1 + 1 + 1 + 1 + 1, whence m!+)

=

7.) The asymptotics of

~+)

are given in [AS]. p. 825,

(7.6) ~ (+) ,..., - -I exp('Jf';~

12

vN) • r.: 4N13

(lO

\' (+) -t N

So the series I.. mN e converges for all t > O.

N=1

(7.7) Lemma.

The spaces SF(+).H(+) and TF(+),H(+) are nuclear. -t H(+)

Proof: Observe that the operators e • t > 0. are Hilbert-Schmidt.

0

Similarly. we introduce the anti-symmetric Fock space F(-). Let

e : P

k + {-1.1} denote the function which is one for even permutations and minus one for odd permutations. We introduce

(7.8) £(0)

cr

Then

p~-)

is an orthogonal projection from X(k) onto X(-)(k)

=

p~-)(X(k».

X(-)(k) may be called the k-fold anti-symmetric Hilbert tensor product of

X. For completeness we take

p~-)

=

1. (7.9) Definition.

The orthogonal projection p<-) on F is defined by

Moreover F(-)

=

P(-)(F). The Hilbert space F(-) is called the Fermion Fock space.

(-)

It is not hard to check that P

k A(k)

=

p(-)

H =

H

p(-). Put

H(-)

=

p(-)

H

p(-) .

(7.10) Lemma.

SF(-)

_,

H(-)

=

p(-)(S

_F

_,H )

and

A(k) p(-) and hence that

k

The eigenvalues of H(-) are the numbers N

=

0,1,2, . . . . The multiplicity

(-)

~ of N is equal to Uthe number of decompositions of N into distinct

integer summands without regard to order", (E,g. 5

=

1 + 4

=

2 + 3, whence m!-)

=

3.) The asymptotics of

~-)

for large N are given in [AS], p, 826,

~-)

"" YsN3

exp(1T

Ii

IN) .

(7.11) Lemma.

The spaces

SF(-),H(-)

and

TF(_),H(-)

are nuclear,

8. A mathematical interpretation of the field operator formalism in quantum

mechanics

The mappings a«~I) and c(I~» do not satisfy the wanted canonical (anti-) commutation relations. In order to obtain these CAR and CCR we modify the

(8.1) Definition.

Let I~> be a keto We introduce the following operators

From the properties of the linear mappings

a«~I), e(I~»,

and pet) we obtain

(8.2) Lemma.

a (±) «~

I)

= a«IP I )

P (±) ,

From Sections 6 and 7 we obtain the following results.

(8.3) Theorem.

Let

I'>

be a ket, and let I~ be a test keto The following statements are valid

- a(±)«'I) is a continuous linear mapping from SF(±),H(±) into

SF(±) H(±);

_,

a(±)(<qJI) 1s extendible.

- e(±)(I'» is a continuous linear mapping from TF(±),H(±) into

T _{F (±) ,H(±); \.;.} ~(±)(I,n') _'¥'

=

~±(~nl)*. _{... ""V}

Proof: Cf. Theorem (6.3) and (6.4).

The mappings

a

(±)

«~

I)

and

e

(±)

(I

~», ~

E M, can be regarded as "basic operators" because we have the following expansion results.

(8.4) Theorem.

Let I~> be a keto Then the formal expansion

a(±)

«~I)

=

J

<~

I s>

a(±)

«s I)

d ll

_s

M

can be interpreted as follows: V t>O V 0<'[2 <'[ 1 <t

H (±) (+) H(±)

eLl

a -

«~

I )

e -t

₌

=

Each of the integrals converges absolutely in

L(F(±».

Proof: Cf. Proposition (6.5) and Lemma (8.2).

(8.5) Corollary.

Let I~> be a keto Then the formal expansion

can be interpreted as follows: V V t>O 0<L

2<T1<t

Next we present a distributional interpretation of the so-called canonical

(anti-) commutation relations.

o

Let I~> be a ket and let I~ be a test keto Following Theorem (8.3) the (anti-) commutators

(8.6)

are well-defined continuous linear mappings on TF(±),H(±) and SF(±),H(±)'

respectively. Similarly, for all pairs of kets I~» I~> the (anti-) commu-tators

(8.7) [c. (±) (

I

~»

,

c.

(±) (

I

~»

I

+

are well-defined. We have the following result.

(8.8) Theorem.

Let I~> and I~> be kets, and let I~ be a test keto Then we have

[a.(±) «'VI ) •

c.(±)(1~)1

=

<'V I

~

(0)

1 •

+

[a.(±)

«~I) ,a.±«~1)1

= 0

+

Proof: We only prove the first commutator relation. The others can be dealt

with similarly.

Consider the following computations, where k E IN, j E lNk

k p(±)

c. (Iq:i»

a.

«'VI)

p(±)

I

j . k>

=

k k k k '

k

k = _{(k + 1) a} < I [ 1 1"''- p(±) I' ,-, + 1 \' (I' > k+1 ( '1' ) k + 1 'V'" k J ; IV' - k + 1 L J JI, • JI,=l JI, = <'1' I <P> (0)

p~±)

I j ; IV ±

,2:

<'1' I jJl,> (0)

p~±)

(IllY I j1"" ,jJl,-1 ' jJl,+l""

,j~)

J=l

Thus we obtain with (*) and (**)

From these relations (***) we get

+ +

[a-«'1'I) , c.-(I<p»l = <'1' I <p> (0) I .

o

+

For arbitrary kets I~> and 1'1'> we thus obtain

+ +

(8.9) [a-«'1'I) ,c.-(I~> (t»l = <'1' I ~> (t) I

+

This observation leads to the following definition.

(8.10) Definition.

denotes the operator valued function on (OJ~) defined by

+ +

t ~ [a-«lfl) ,c-(I~>

(t»l '

t > 0 .

+

(8.11) Corollary.

in the sense that for all t > 0

[a±«lfl)

,c(±)(I~>

(t»l

=

<If

I

~>

(t) I

+

For the members of the Dirac basis (I~»~EM we obtain

[a (±)

«~

I) ,

c

(±) (

1

n»

1

=

<~

I n>

I

=

+ 15 (~) 1 • n (Cf. Section 3.) Remark.

The canonical (anti-) commutation relations can also be interpreted with

the aid of operator valued distributions. Here we give some heuristic

ar-guments.

valued distributions. For fixed n E

M

we have

Hence

1

<.p) t+ [a (±)

«n

I) ,

c

(±) (

1<1»)]

is the operator valued distribution =+

In Proposition (6.8) and Definition (6.9) we presented an interpretation

of the formal integral expression

MxM

I

dill; dill;' c.(

It·»

<I;'

I

V

I

1;>

a

«I;

I) .

Multiplying both sides by p(±) should give a similar relation in the

Boson (Fermion) case.

(8.12) Definition.

Let V E

L(T

x,,4.)' Let

MJ

C.(V

I

E;»

a

«I;

I)

dllE; denote the operator in

L(Sf,H.Tf,H)

as defined in (6.7). Then the second quantization Q(±)(V)

of V is defined by Q (±) (V) = P (±) {

f

C.(V

11;»

a«E;

I)

dill;} P (±) • M (8.13) Theorem. Let V E

L(T

_x

,4.), Then

•

in the sense that for all t > 0:

-t H(±) (±) -t H(±)

e Q (V) e

=

Further,

o

(±) (V)

=

f

c.

(±)

(I

~

'»

<~'

I

V

I

~>

a. (±)

«~

I )

dll~ dll~,

MxM

in the sense that V t>O V O<T<t :

-t H(±) ± -t H(±)

e 0 (V) e

=

=

J

MxM

(±)

where the integral converge absolutely in L(F ).

Remark.

+

Appendix

In the spirit of this paper it can be shown that continuous operators

between Fock analyticity (c.q. trajectory) spaces can be represented in

their normal formal (cf. Sh)

c

=

I

m,n

in a rigorous way with respect to any Dirac basis.

The nature of the coefficients and the modus of convergence can be fully

interpreted within our theory.

It is very likely that most results in Shvarts' book [Sh] which are

semi-mathematical/semi-physical can be given a rigorous meaning with

References

[Co] Cook, J.M., The mathematics of second quantization. Trans. AMS,

74 (1953), pp. 222 - 245.

[Di] Dirac, P.A.M., The principles of Quantum Mechanics. Fourth edition,

Clarendon Press, Oxford, 1958.

[EG1] Eijndhoven, S.J.L. van, and J. de Graaf, A mathematical interpretation

of Dirac's formalism, Part a: Dirac bases in trajectory spaces.

[EG2]

[EG3]

[EG4]

[EGm]

To appear in Rep. Math. Phys ..

---, Part b: Generalized eigenfunctions in trajectory spaces. To

appear in Rep. Math. Phys ..

---, Analyticity spaces subjected to perturbations with applications

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Math. Anal..

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[EGK] Eijndhoven, S.J.L. van, J. de Graaf and P. Kruszynski, A dual system

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[G] Graaf, J. de, A theory of generalized functions based on

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[KMP]

Kristensen, P.,

L.

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