Lecture Notes in Mathematics
Editors:
A. Dold, Heidelberg
B. Eckmann, Zfirich
F. Takens, Groningen

1472


Torben T. Nielsen

Bose Algebras:
The Complex and Real
Wave Representations

Springer-Verlag
Berlin Heidelberg NewYork
London Paris Tokyo
Hong Kong Barcelona
Budapest


Author
Torben T. Nielsen
Mathematical Institute, ,~rhus University
and DIAX Telecommunications A/S
F~elledvej 17, 7600 Struer, Denmark

Mathematics Subject Classification (1980): 81 C99, 81D05, 47B47

ISBN 3-540-54041-5 Springer-Verlag Berlin Heidelberg New York

ISBN 0-387-54041-4 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1991
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Contents

I

0.

Introduction

1.

The Bose

2.

Lifting

3.

The c o h e r e n t

4.

The W i c k

5.

Some

6.

The

7.

The real w a v e

representation

72

8.

Bose

algebras

of o p e r a t o r s

79

9.

Wave

representations

algebra
operators

special
complex

to

vectors

ordering

4

F0}(,<,>

23

F]{
in

33

F}[

and the W e y l

relations

53

operators
wave

representation

10. A p p e n d i x

I: Halmos'

11. A p p e n d i x

2: G a u s s i a n

of

45

F(][+][*)

len~a
measures

66

89
94
96

12.

References

130

13.

Subject

132

index


Introduction
The

aim

consequences

Though

having

theoretical
further.

communication

[15,16,17]

well

Hilbert

by

the

a

eleme n t s
with

that

algebra

an

main

is

that

theory,

has

is

a

process

Bose--Fock
by

much

filtering

the

spaces

Irving

and

have been
an

fixed

E.

of
were

Segal

accompanying

combined

is

algebra.

to form

algebra.

Given

algebra

multiplicative

H

extended

operators

scalar

the

commutative

from

the

in the

(in

of

The

product

algebra

literature

treating

the

unit

over

F0H

known
the

a

F0H

F0K

multiplication

is

physical

subject.

fermions

their

Weyl

It

having

e,
in
by

provided

as

the

base

as

important
become

which
facts

easier

complex

to

are

Bose

space

There

are

inner

product

extra

structure
known

in

those

be

directly

relations

the

is

the

closely

to

expose

understand

also

an
a

and

can be b e t t e r
purely

whereas

in

work

free

without

mathematical

the

with.

product,

Bose

algebra

knowledge

an

algebra

for

Weyl

the

fermion

relations

and

to

areas

Moreover,

a

number

traditional
particular

of
of

manner
the

of
for

belong

over

which

reasons

associative

and

introducing
objects

the
prior

in

physics,

now

role

of

understood.

mathematical

manner

In

this

quantum

no b a c k g r o u n d

to

the

algebra,
related.

free commutative,
inner

no

compared

Clifford

so

point,

with

in

of

in q u a n t u m

anti-commutator

operators,

not

Introducing

space with

starting

spaces
study

with

situation

for

difficult

conjugation

algebras.

to the
its

within

the

to m a t h e m a t i c i a n s

as

also

its

by

clearer

creation

counterpart

mathematics

relations

Bose--Fock

originated

In c o n t r a s t

can

and

of

available

intuitively

annihilation

well

of

namely

free

and

to

mathematics,

physics.

the

formalism

linear

K

the

advantage
the

becomes

mathematical

the

extends

[26],

in

"the one-boson--space").
The

which

formalism

product

adjoints

H,<,>

spaces

space

object,
the

space

derivations

base

[22].

are

space.

application

processing

Bose--Fock

scalar

of

of

description

which

Bose--Fock

in the sixties.

the

operator

extension

with

manner
field

the

area

Bose-Fock

society

consider

The

results,
of

mathematicians

mathematical

linear

in H are

such

called

the

of

signal

of

both

some

concept

principal

human

(cf.[3])

~,<,>,

the

formalized

the

known

vacuum.

way

and

attention

space

generated

such

new

paper

present

of

digital

and a n n i h i l a t i o n

single

called

a

within

this

to

usefulness

in

and others

In
creation

the

in

the

is

origin

appear

and

to

paper

algebraization

physics,

[8]

brought

this

an

their

They

theory

a

of

of

at the

the
new
are

same

obtained
relations.
actually

studying

Bose

multiplication

for

on a

time

extending

algebra,
There
Bose

the

provide
are

an

several

algebras

in a


more

or

less

scalar

polynomials
3)

one

natural

also

attach

[6]

which

the
is

the

taking

n-fold

very

direct

in

original

to

find

polynomials

generated

a

is

by

rather

natural

with

Hermite
obscure),

manner

to

also

admits

LZ--Fock--space

a way

ours,

exponential

based

Hilbert

n-particle

of

products
of

the

constructed

correspondence

on

the
a

is

in

one

the

Bose--Fock

of

symmetric

of

has,

space

the

the

algebra

for

each

obtained

base

by

space

and

from each other

product.

n-particle

Bose

the

concept

Here

can be o b t a i n e d

these

the

the

elements

elements

to

introducing

which

which

from

of

spaces.

space,

products

order

is

complex

multiplication

algebra

the

will

tensor

the

space

of

of

algebra

structure.

tensor

those

permuting

Hilbert

4)

reader

or

the

Bose

[24],

so-called

identifying

a

algebra

2)

operation

similar

spaces

] ) the

[1 ],

the

Bose a l g e b r a

Hilbert
n6~,

in

calculus

In
space

way:

as

(here

can

stochastic

by

obvious

product

The

symmetric

spaces,

generated

and

by

is

the

in

base

space.
One of the best k n o w n
Segal--Bargmann
the

elements

construction
of

complex

are

elements

the

of

the

conjugate--entire.
e lem e n t s

of

vectors,

which

[5].
has

The

view;

here we
real

a

wave

representation,

the

exponentials

as

for

laser

is

the

from

into the

one.

the

which

to

the

representation

complete
point

In this

following
Hilbert

transforms

the

case

optics

L2-space

abstract

(complex)

Contrary

the real w a v e

in

paper

space,

the

constitutes

of

way:

complex

of

the

coherent

appropriate

an

become

quantum

representation.

constructed

constructed

in

paper

of

so-called

and m a t h e m a t i c a l l y

representation

it the real wave

functions

beams

within

contained

a self

the

in w h i c h

exponential

that

known

is the

this

well

gave

real

taking

In

mentioning

surjective.

is

[16],

plane.

the

representation

the

and

space

thus

is i n t r o d u c e d

into

[I]

involves

of not being

call

functor

representation

cf.

complex

[5,20],

states

representation

conjugation

this

are

Schr~dinger

shall

wave

the

cf.

worth

the

wave

Segal

the

space

is

on

functions

space

provide

[15]

of

complex
from

It
base

complex

In

the

the

base

of the Bose--Fock

representation,

functions

complex

the d i s a d v a n t a g e

account

wave

entire

of

realizations

a

and
wave

complex

a unitary

map.
The
the real

functer

one

that

by a so--called s q u e e z e d
a

subject

experiments
aspects

of

special
with

attract
It

transforms

is an o p e r a t o r

turns

out

adjoint

state with

interest

squeezed
growing

the c o m p l e x

whose

attention
that

infinite

in q u a n t u m

light

the

are

wave

representation

corresponds

q ui t e

energy.

optics.
recent

Squeezed
The
and

into

to m u l t i p l i c a t i o n

first
the

states

are

successful
theoretical

[13].
so-called

normal-product-algebra

of


creation
which
adjoint
using

and annihilation

we

analyse

f unc t i o n

[21].

of o p e r a t o r s
It

The

[11],

important
measures
principal

to

since

the

be

the

a short

excellent

I

Graversen
for m a k i n g

would
and

Bodil

numerous

the

consisting

of

determined

one

that

real

who

Hilbert

may

the

conjugation

to

the

elements
by the

the

the

method

and
This

Wigner
of

the

conjugation

book
we

of

Louisell

apply

without

fits

making

not

Steengaard
suggestions

which

are p r o v i d e d

space.

These

algebras.

already

be

spaces,

measures

are
With

acquainted
can

find

in the a v a i l a b l e

in

express

representations

of Bose

linear

on g a u s s i a n

to

reads

wave

interpretations,

expositions

like

taking

representation.

analogous

the m a t h e m a t i c s

information

appendix

elementary

complex

wave

of operators,
of

the

it too

physicist.
and

dimensional

necessary

the

real

if

clear

for the t h e o r y

reader,

operation

operators

that

dimensional

importance

in i n f i n i t e

seek

added

complex

infinite

as

a

space

rigorizing

probabilistic
on

of

a Bose a l g e b r a
The

above.

immediately

the

9.

conjugation

mentioning

for a t h e o r e t i c a l
Both

theory

kernels

Bose--Fock

of a p p l i c a t i o n ,

yields

and

construct

a complex

worth

it will

obscure

and

has

described

is

we

produces

representation

8

can now be taken

conjugation

representation

operators

chapter

of an o p e r a t o r

this

spirit

in

have

by g a u s s i a n
therefore

of

this

in mind

with

measure

it v e r y t i r e s o m e

literature,

in H i l b e r t

very

spaces,

we

have

b a s e d on

[]8].

my

thanks

for r e a d i n g

to

parts

and corrections.

David
of

Adams,

Krista

the m a n u s c r i p t

and



Chapter

A:

]:

The Bose alqebra

The free c o m m u t a t i v e
Let

linear

H,<,>

in

the

commutative

alqebra

F0~

be a separable
second

algebra

vacuum)

and the Hilbert

space).

We

denote

by

Hilbert

variable.

generated

Let

space with

then

X

the

(called

set

of

the inner product

F0H

by a m u l t i p l i c a t i v e

space
~

FOH,<,>

denote

the

free

o

(called

the

unit

the base or the one-particle

positive

integers.

For

n6~

we

fulfilling

the

define

{

n
H0 = span
where

ala2...a n

additional

ala2...a n

denotes

linearity

the

}

al,a2,...,aneH

free commutative

,

product,

relation

(t-a + b)a2a3...a n : t.a.a2a3...a n + b.a2a3...a n
with

a,a2,a3,...,an,b6~
We

linear

consequently

and

commutative

and

tEC

identify

.
elements

operations,

can

which,

be

by

reduced

repeating

to

the

same

where almost all

fn

0

these
form.

M o r e o v e r we set
0 = span
~0

{e} = C.o

and
co
co

F0 H = n=0
@ }{0
n = {

~ fn

fn 6 ~0n '

n=O
FOK an a l g e b r a by d e f i n i n g

We make

the addition

:

~ fn +
n=0

the m u l t i p l i c a t i o n

:

~ fn
n:0

for every

f n , g n 6 Hn0

~ gn =
n=0

with

~ gn =
n=0

nE~ 0 = ~ U {0}

,

~ (fn + gn )
n=0
~

~

n=0

f j'gk

j+k=n

and defining

e-f=f-e=f
for

f6FoH

.

It is an easy exercise
are

associative

and

to show that addition

co~utative,

thus

making

and m u l t i p l i c a t i o n

F0H

a

commutative


algebra

with
We

multiplicative

shall

use

the

unit

following

r =
i.e.

rk6~ 0
Irl

for

.

notation:
n
e ~0

(r I , r 2 , . . . , r n )

k=1 , 2 , . . . , n

= rI + r2 +

e

...

,

'

we define

+ rn

= r] !-r2!- . . .-r n.t

r!

r
rl
r2
rn
Irl
e-- = e I -e 2 - . . . . e n
e ~0
0
e
= e ,
where

{e I , e 2 , . . . , e n }

Proposition
an

orthonormal

is a n o r t h o n o r m a l

I .IA:

system

a I ,a2,...,an6}{

every

r 6 ~k

to

It

is

sufficient

.

We

define

dim

an

orthonormal

K < n

Then

basis

it

}{

with

find

and

that

=

dimensional

}

k6~

a

= n }

f

{e] , e 2 , . . . , e k }
to

correspond

,r,

{ a l , a 2 ..... a n

is p o s s i b l e

){ .

such

consider

a finite

in

there

in

{ er

~ = span
Choose

fe){~

{e] , e 2 , . . . , e k }

f 6 span

Proof:

To

system

a la 2. ..a n

space

K

,

where

,

•

in t h e

complex

space

K

numbers

with

k =

{t]}i, j

such

that
k
=

ai

tl..e

.

3

]

for

i=],2,...,n

,

j=1
and

we

get
k

al-a2.....a

k

k
1

~

n =

J] =]

j2=I
s r •e ~

for

some

s

r

In t h e
choose

the

6 C

,

which

case

space

2

n
e

• ''

of
~

f

t]]'t]2

"'tin

. . . .

e31

e

]2

3n

Jn=1
with

_r £ ~

evidently
being

sufficiently

and

Irl
_

= n

is a s u m

of

the

a sum

of

large.

generators

desired
for

type.

n
}[0 '

we

just


8

The a b o v e

argument

also

Proposition
{e],e2,...,ek}

verifies

1.2A:

Let

the

X

an o r t h o n o r m a l

denote

basis

{e~l
spans

the w h o l e

B: T h e B o s e

use

alqebra

e+(x)

We

shall

demanding
+
(x) ,
i.e.

for
to

fulfilling

the

with

H

inner

the

operator

defined

on

the

product

operators

the

e(x)

whole

lemma

for

to

the

Then

the

in p h y s i c s

,

by

x6H

the

whole

dual

and

~+(x)

for e v e r y

determined

set

to

of
be

V0H

the
a

we w i l l

by

operator

derivation,

f,geFoK

.

by defining

: ]

and e(x)

shall

be c a l l e d

the

creation

and

respectively.

Leibniz

rule,

we o b t a i n

the

recursive

formula

> = <x2x3..Xm,~(x])(ylY2..Yn)>
I + y]~(Xl)(Y2...yn

)>

,

relation

1.1B b e l o w

we o b t a i n

= <xl,Yk>e

for

n~m

and

H

,

F0K

+ f-e(x)g

~(Xl)(Yk)
from

employed

in

= the

.

and

0 }

product

uniquely

operators


using

K

space

rule

= g-~(x)f

inner

the a n n i h i l a t i o n

and

space

of m u l t i p l i c a t i o n

the

the Leibniz

Applying

dimensional

k

the n o t a t i o n

<o,~>
The

in the

~c~

the o p e r a t o r

extend

e(x)(f-g)
We m a k e

a finite

FOK,<,>

for

x 6
be

proposition.

FOX

To be in a g r e e m e n t
often

following

> = 0

n=m
n


> : ~ <xl,Yk>'k:]
=

where

~

runs

~ <xl'Y ~ 1 >'<x2,Y~2>'...'
through

the

set

of

all

n'

>

Y~n >

permutations

of

the

numbers


{I,2,...,n}

Lemma

I.|B:

For

1)

~(x)(~)

=

2)

e(x)(y)

= <x,y>o

Further,

every

we

have

o

for

n,m6~

we

have

3)

~ ( x ) ( y n)

= n - < x , y > - y n-I

4)

~(x)m

=

yn

x,yEK

n!m ) !
(n-

"<x'y>m'yn-m

for

m
Proof : Since
o 2 = ~.~
by applying

the

¢ ( x ) ( e 2)
and

operator

=

~(x)

(e(x)e)o

+ o(~(x)o)

sides

we

: 20-e(x)~

get

= 2e(x)e

= ~(x)e

,

hence

every

n>1

and

=

c I ,c2, . . . , C n e H

< ~ ( x ) y , c l c 2. • .c n > = < y , x - c l c 2. . . C n >

Thus

,

¢~

on both

,~(x)~

For

=

the

0

.

we get
= <~,~(y)(x-clc2...Cn)>

=
> +

= <~,(~(y)x)clc2...Cn

> +
= < ~ ( c I ) e , c 2 c 3 . . . c n- ( e ( y ) x ) ) > + 0 = 0
n
a(x)y
is o r t h o g o n a l
to
K0
for every

element

n6~

)>
)>

,

and we

get
~(x)y
As

t = <~,t.e>
Moreover

= <o,~(x)y>
we

have

by

e ( x ) y n = e ( x ) ( y . y n-1 ) =
= <x,y>yn-1+
The

last

We

= t-~

for

= <x,y>

easily

often

identify

we

t6G

have

.
proved

identity

2

induction

( ~ ( x ) y ) y n-I

y. ( n - 1 ) < x , y > y n - 2

identity

,

some

follows

an

by

+ y- ( ~ ( x ) y n-1 )
= n - < x , y > y n-1
induction.

element

and

the

operator

consisting

of


multiplication

b y the

the

for

symbol

x

Given
write

x

linear

for

the

the

operator.

be

theory,

To
a

the

a

presentation,

element

itself,

operator

e+(x)

operator

x

operator

to

mathematician

,

to

the
is

intuitions

will

x6H

in m a t h e m a t i c s

operator

use

for

we

shall

one

will

often

use

.

adjoint

annihilation
able

i.e.

gladly

the

and

use

operator
adjoint

x
to

techniques

this

as

parallel

a rule
In

this

the

creation

from

operator

notation

when

w

computing.
operator

Hence
e(x)

we

,

shall

alternatively

write

x

for the

annihilation

i.e.
+

for e v e r y

fEVoH

(x)f

= x-f

~(x)f

= x f

.

Proposition

I.2B:

For

arbitrary

al , a 2 , . . . , a m , b 6 H

we

have

the

identity
<ale2. • .am,bn>

Proof:

The

= ~ 0
[ n'.-<a] , b > < a 2 , b > . . . < a n , b >

proposition

follows

by

induction

for
for

n~m
n=m

and

the

following

calculation.
<ala2...am,bn>

= <a2a3...am,~(a])(bn)>
=

will

<a2a3...am,al (bn)> = n.<al,b>.<a2a3...am,bn-l>

We

are

now

able

turn

out

to be v e r y

Proposition

to

Hilbert

space

Assuming
H

,

and

prove

a result,

which

later

on

useful.

] .3B: To e v e r y
~

Proof:

formulate

= span

that

we w i l l

~

n£~
{ an

we have
a6H

is a f i n i t e

prove

that

}

dimensional

subspace

of the


K~ = span { an
Notice

K n0

that
Take

prove

f6K~

that

n

and

dimensional

assume

that

subspace

f 6 { an

as well.
a6K

we

}I

have

to

f = 0 .

Choose
with

is a finite

J a 6K }

an

k6~

orthonormal

Since

we e x p a n d

the

set

basis

{e ~

in a finite

J ~6~

with

a6K

we then

0 = <f,an>

= ~ ~r.<e[,an>

is a basis

I~I = n }

in

.

rI

= ~ ~r-n!-<e],a>

>.<e2,a>r2>...

>rk>

r
rl

= ~ ~r'n''al
r

r2
"a 2

where

a i = <ei,a>

whole

K

,

space

have

r
,

the

sum
f = ~ tr'e[
r

For e v e r y

in

{el,e2,...,ek}

the

rk
...a k

for

> ,
i=1,2,...,k

variables

As

a

al,a2,...,a k

is r u n n i n g

range

the

through

the

C

and

whole

,

consequently
t

= 0

r

for e v e r y

r .

i

As e v e r y

element

n
}{0

in

is

finitely

generated,

the p r o p o s i t i o n

holds.

The

above

result

polarization

should

not

be

surprising,

since

the

general

identity
n

n! • XlX 2 . . .x n =

~ (-I) n-k
k=1

for c o m m u t i n g

variables

Theorem

1.4B:

as usual

[A,B]

operator.

For e v e r y

(The

denote

~

(xil + x i 2 + . . + X i k )n
il<..
Xl,X2,...,x n

canonical
the

x,y6H

is well

known.

commutation

commutator
the i d e n t i t y

AB -- BA

relation,
and

I

cf.

[2])

Let

the

identity


10

[e(x),e+(y)]
holds

on the w h o l e

Proof:
derivation.

F0H

This

For

.

is an easy

f6FoH

W

(y'f)

Definition
generated
(called
a(x)

=

= x*

,

is d e f i n e d

(called

shall

dual

to

be

the

on the w h o l e

algebra

the vacuum)

called

operator

algebra,

a

for e v e r y

mentioned

operators,

and

By a p p l y i n g
n
H0

in

we

:

n-particle
(apart

state

n
~0

in

we

inner p r o d u c t

algebra

if

<,>

space
the

of m u l t i p l i c a t i o n

,

H,<,>

operator
by

x6K

,

0

= <x,y>'I

a+(x)

annihilation
operator

(n+1)-particle

and by a p p l y i n g

with

Bose

operators

are called

an

+ y" (x f)

.

the

the c r e a t i o n

get

normalization),

x,y£H

earlier

a(x)

is a

and if the i d e n t i t i e s

[a(x),a+(y)]

As

a(x)

that

and a H i l b e r t

a+(x)

a(x)~

are f u l f i l l e d

fact

W

+ y. (x f) = <x,y>f

A commutative

~

base)

of the

W

(x y)f

1.5B:

by a unit
the

consequence

we get

W

x

= <x,y>'I

the

an

creation

to an n--particle
.n+]
~0

in

annihilation

get

called

operators.

a+(x)
state

are

(apart

operator

(n--1)-particle

state

a(x)

state

from
to an

in

H~ -]

from n o r m a l i z a t i o n ) .
Every

commutation
bosons
photons,
physics

Bose
relation

(particles
phonons,
all

force

carriers

of

are

carriers

the

algebra

the

CCR

then

for the c r e a t i o n

with

integer

mesons,

and

exchanging

the

the

strong

Bosons

helium

force,

axiomatizes

and a n n i h i l a t i o n

spin).

particles

electromagnetic
of

automatically

are

are

nucleus
bosons.

the

interaction

and

operators

particles
~He.

The

~-mesons

the

photons

and

the

In

the

W +,

W-

of

like

particle
are

the

K--mesons
and

Z°


11

particles

are

the

carriers

Proposition
Bose

algebra

1.6B:

Consider

is a u t o m a t i c a l l y

Proof:

We must

x*(f-g)
It is

of t h e w e a k

sufficient

=

prove

interaction.

x,y6K

.

~(x)

The dual

: x

in t h e

a derivation.

that

(x*f).g

+ f-(x*g)

to c o n s i d e r

for every

f = an

for

some

f,g6FoH
a6H

•

and

n6~ 0

The

identity
x*(an.g)

is an e a s y

consequence

in t h e v a r i a b l e

= an.x*g

of t h e

n6~

x a

x*o

easily

commutation

It r e m a i n s
•

which

+ n.<x,a>.an-l.g

follows

by

n

by induction

g = ~

n-1
in

the

above

formula

and

using

~+(y)

and

a(x)

= 0

The
satisfy

requirement

the

that

for

x,yeH

:

[~(X),e+(y)]
thus

demanding

-- in
the

Lepta
be o f t e n
The

2)

Letting

the

1.7B:

We

in t h e

subspaces

n
H0

e(x)

shall

operators

the

= <x,y>I
a

Bose

,
algebra

-- b e

replaced

by

to be a d e r i v a t i o n .

prove

are pairwise

the

following

orthogonal

b e an o r t h o n o r m a l

rI r2
rn
= e] e 2 ...e n
inner

of

0

assertions,

which

will

sequel.

el , e 2 , . . . , e n
eZ

we have

definition

operator

needed

I)

the

relations
~(x)~

can

proved

to s h o w t h a t

= n-<x,a>-a

setting

relation,

I~I
6 HO

<e~,e~>

= r!-6
--

system

for

product
r,s

for different
in

K

n
[ 6 N0

,

indices.

and defining


12
n
r,s 6 ~0

for e v e r y

Proof:

])

I .3B. R e g a r d i n g

is

"

a consequence

2) we get

of

proposition

f r o m the c o m m u t a t i o n

I .2B a n d

proposition

that

[ek,a+(ei ) ] = 6i,k.I
By i n t r o d u c i n g

the n o t a t i o n
- 1k = ( r ] , r 2 , . . . , r k _ ] , r k - ] , r k + ] , . . . , r n )

we c o n c l u d e

,

that
e k ( e [ ) = rk.e([-] k)

Assuming

[ = s

<e[,e~>

= <el,el>

we get by i n d u c t i o n
r. r 2

r
= <el]e2...enn,e[>

rl.<e(~-11 ),e(~-11)>

=

If

,

instead

we

fulfilling

have

[ # s

rk~s k

,

Without

:

= <e([-l]),e~e[>

rl !.r2!....

then

there

loss

of

*

rk
Because

indices,

we get

the o p e r a t o r s

ek

.

r n.t

=

exists

_r !

a positive

integer

k

g e n e r a l i t y we w i l l a s s u m e that
+
and
(el)
c o m m u t e for d i f f e r e n t

rk
(r-r k ) s k
(~-s k )
< e Z , e ~> = >
<~(e~k
=

rk
)e k

We w r i t e

(~-r k)
.e

FH

(~-Sk) >
,e

for

the

:

Hilbert

0

space

which

is

the

completion

of

FOK,<,>
The
norm

following

proposition

by multiplication,

which

gives

is not

an

estimate

a bounded

of

the

operator.

We

growth
shall

in
use

the n o t a t i o n
Ifl = <f,f>

Pr__o p o s i t i o n

1.8B:

with

being

the

feFK

.

m
f6K
and
g6K 0
!
n+m 2
~ ( n ) " If!" Igl '

To e v e r y

If'gl
n
(m)

for

binomial

coefficient

we h a v e

n!
ml(n-m)!

for

positive


13

integers

n,m6~

Before

notation.

.

stating

For

the

i,k 6 ~

with

p6~

we

shall

need

a

lemma

and

a

concise

we define

i < k
-

proof

if

i.< k.

--

3-

for

j=1,2 .... ,p

]

.

Lemma

inequality

Consider

1.9B:

m ! n .

positive

To every

k6~

m,n6~

integers
with

length

fulfilling

I~l = n

the

we have

k!
il!l=m
Notice

that the length

of the vector

Proof:

the binomial

Consider

(s

+

t)

~ =

does not vary.

series

k.
l

k.

i

ki

L

of

(s + t)

k.
l

J. ki_J

(j).s

t

j=0
By multiplication

of the series

(s + t) n = (s + t)
kI

k1+k2+. "+k

k2

kp

for

i=I ,2,...,p

we get

P

k]

k2

lj)lj2

..(kp)

p'S

j1+J2+..+jp.

J1=0 j2=0 jp=0
k I+k2+. .+kp-(j1+J2 +" "+Jp)
t
k!
=

k!.

i]+i2+..+i

(k_--_~) ! . s

n-(i I+i2+. .+ip)
P

- t

i<_k_Notice

that the length

of the vector

Take the binomial

series

for

i

varies.

(s + t) n ,

n

(S + t) n =

~ (n)'sm-tn-m
m=1

By collecting
lil : m

the terms

in the previous

sum,

involving

indices

and comparing

with

i

of constant

the above

length

result,

we


14

get

(n)

k!

i_
Ill

Proof

of

=m

proposition

common

orthonormal

Fourier

coefficients)

] .8B:

system

From

proposition

{e],e2,...,ep}

in

1.2A
H

we

(and

find

matching

fulfilling

f = ~ a r-e ~

for

~6~

with

length

Irl_ = n

for

s6~ p

with

length

I~_1 =

r

and

g

b s" e s

= ~

m

s

with

both

sums

finite.

I

f g:

s

= I
k
indices

_k

i+j:k

1

i

~
ai'bk-i'ek

i
fulfilling
lil = n ,

Moreover

get by m u l t i p l i c a t i o n

= I

r

with

We then

lJl = m

and

Ikl = n+m

.

we get
IfI 2 = <f,f>

Igl 2 =

~ a n - a n , - < e -r, e - r' > = ~ larl 2._r!
r'
r

= ~
r

~ Ibs 12-s:_
S

and
2
If'gl 2 = ~

Ifl2-tgl

2 = [

~ ai-bk_ i

~ larl2"lbs

-k!

with

Iil=

n , I~l : n+m

t2"r~'s!

I

r

S

~ lail2"lbk_il2"i!(~-i)
i

k

i
!

,

lil=

n,

a

I~1 = n*m


15
In what

follows

l!l : n

2
k!-

~

and

lhl = n+m

I

ai-bk_ i

=

.

By lemma

1.9B we get

2

~ h ! ai" bk_ i
i

k

:

]i

kl

2. (_!i (k - i ) ! ) .ai'bk_ i

2

i~k
k!
~ !!'(~-!1!

~ !:'(~-!)!'[a!12"Ibk_il2

= (~)- ~ !l" (~-!)!" fail2" Ibk_i 12
i~k
The d e s i r e d

result

We n o t i c e
aeK

and

m,n6~

follows

that

the constant

Proposition
in

over

k .

[.n+m.
n )

cannot

(n+m)!

.[aj

n+m

k6~

and

a

be improved,

as for

we have
lan'aml

{an}n6 ~

by summation

] .]0B:

=

[an+m

I

=

Consider

convergent

sequence

H ,

an

Then the sequence

Proof:

{a k'n}n6~

We start by m a k i n g

by using p r o p o s i t i o n
la k - bkl

:

n ~ a 6 K .
will converge to
ak
, a k e FK
n
n

an estimate.

1.8B several

k
(a - b). ~ ak-J.b j-!

in

ak

FH ,

To e v e r y

a,b6K

times,
i

_< k2-[a

k
- b • ~

j=1

a k - J ' b j-1

j=1

k
_I
< k ~. In - b I- ~ (k-I)2.
j-1

ak-J I. IbJ-11

j=1
k
i

<- k ~ " la

-

I
k-1

b I"

j=l

,( "~j- I ,- ~

•

(k-j)I ~"

lal k-j- (j-1)!~" Ibl j-1

we have,


16

k

la k _ bk[

=

k~.la - bl.(k_~):~.

~

lalk-J'lbl

j-1

j=l
k

< k,~.l a _ b I. ~

(max(lal,lbl))

k-1

9=I

<_ k! ½" la - b I-k- (laj
N o w we e a s i l y

get the d e s i r e d

la k _ ak[
n

The
therefore

Theorem
Consider

inequality

To e v e r y

k6~

will

and

bkl ~ k~'k'(lal

-

] .12B:
the

la I) k-l. In
n
- an.l

be

0 .
n~

frequently

used,

and

it as a lemma.

1.11B:

la k

+

calculated

we p l a c e

Ib[ )k-1

result

< k!~-k- (la[
-

above

Lema

+

(Wiener)

Let

a,b6H

we h a v e

Ibl)k-l"la

÷

-

bl

be an o r t h o n o r m a l

{en}n6 ~

basis

in

set of i n d i c e s ,
00

i

u .[ rclN~

=

}

n=l

T h e n the v e c t o r s
1

reI
form

an

orthenermal

Proof:

basis

in

Fg

It has a l r e a d y

.

been proved
r,

that

•e r

r6 I
forms
that
p6~

an o r t h o n o r m a l
this

orthonormal

system

in

system

T h e n we c a l c u l a t e

F}{ .
is

total

We
in

using proposition

n
a p = (lira
n

~ <ek,a>ek)P
k= I

)rove c o m p l e t e n e s s
F0H

Consider

1.10B,
n

= lim
n

~ <ek,a>e k
k= I

)P

by

showing
aeH

and


17
Since
n

_~
)P 6 span { r!

( ~ <ek'a>ek
k=]
and

{ ap

a6H

and

p£N 0 }

spans

-e r }

the whole

r6I

F0H

the p r o p o s i t i o n

,

holds.

Since the set of indices

I

is countable,

the Hilbert

space

FH

is separable.

Definition
of the space

|.13B:

The spaces

Lemma
of

F~

onto

We define

Kn

1.14B:
Kn

and

Denote

Hm

feFK

,

~n

are orthogonal

by

P

n ,

as the closure

subspaces

for

the orthogonal

ner o ,

n~m

Pn(f)

=

we have

f =

~ fln

FK

in

.

n=O

Proof:

To a r b i t r a r y

6>0
If

As

geFoH

,

it is possible

choose
-

gl

<

to find

geFoH

c;2

fulfilling

.

N6~

and

gn6H~

N

g =
Defining

gn = 0

for

n>N

n

f-

,

~ gn
n=0

<- I f - g l

"

we have for
n

flk
k=O

in FH

+

flk
k=O

n>N

that

.

projection

Defining
fln

for

the space

n
~0 "

such that


18

n

n

< e/2

+

~ (gk
k=0

k=0

- flk )

'

N

and

because

Pk g = P k ( ~ g n ) = gk
n=0

n

f -

'

n

~ flk

_< 6 / 2

+

~

k=0

(pkg

- pk f)

k=0
n
: 6/2

+

( ~ pk)(g

- f)

k=0
_< 6 / 2

As
sum

of

a

+

g

-

consequence,

orthogonal

Corollary

fl

we

closed

1 .15B:

< 6/2

have

+ 6/2

expressed

subspaces

F?~ =

{

= 6

FK

as

space

FH

This

and

From

is

lemma

~ fn

operation

whole

,

FK

easy

fn

n

] .SB

in
thus

F0K

we

becoming

multiplication

to

F0H

For

fixed

n,m6~ 0

numbers
n

m
K0 × H0

continuous

extended

according

uniquely

to

9
to

9

that

of

the

an

completeness

the

multiplication

cannot

algebra.
class

of

the

Still
of

be

is

extended

it

elements

is

not

a

to

the

possible

to

than

those

of

can

be

mapping

(f,g)

~ f.g

proposition

(f,g)

the

therefore

a broader

a commutative

Hn ì Km

know

and

the

is

"

n=0

consequence

extend
ã

direct

] .]4B.

proposition

continuous
FK

an

infinite

Hn

n=0

Proof:

an

~n+m
6 ~0

] .8B,

and

multiplication
, f-g

e K n+m

thus

it


19
Definition
it is p o s s i b l e

1.16B:

to w r i t e

Consider
f,g

f,geFH

According

to

lemma

1.14B

in the f o r m
00

fn

with

fn eK n

~ gn

with

gn6~ n

f =
n=0
00

g =

n=0
We d e f i n e

the p r o d u c t

f-g

by the

series

00

n=0

j+k=n

provided
co

2

This
in

F0K

new

,

and

product
it

n=0

j+k=n

is

clearly

is

still

both

an

extension

of

commutative

and

the

multiplication

associative

when

defined.

Lemma
Hilbert
] .12B,

1.17B:

space

Let

{en}n6 ~

By

expanding

denote
f,geFoK

an

orthonormal

basis

in

from

the

base

in

the

theorem

i.e.
f = ~ a r - e ~ e r0K
r

g =

~ b s . e £ 6 FOK ,
s

we get

f'g = ~ ( ~
aj_.bk_ )
n_ j+k=n
Proof:
have t h a t

Notice

that

the

indices

•

en

run

6 FO~{

through

countable

sets.

We


20

r!-a
--

= <er, f>
r

s!'b
--

= <eS,g>

n

= <f * e--,g>
= ~ <f * e -n , b s e S >

<en, f-g>

.

S

= ~ b s-<en,es.f>

S

S

n a r.e-r> = ~ ~ ar'bs'<e-n, e-r+-s>
= ~ ~ b s ' < ( -e)s *e -,
s r

s r
ar bs•n

=

'

r+s=n
Since

f.geFoH

,

by the theorem

of W i e n e r

f.g

where

we

have

shown

(theorem

= ~ Cn'e~
n

] .12B)

we get

,

that
cn

ar bs
r+s:n

Example

consider

I . | 8B:

the

conjugate

Let

n6Z

linear

function

a
defined

b y its v a l u e s

be

fixed.

: Cn

on v e c t o r s

the

Hilbert

the

inner

,

of t h e

form
...

£ = (zl,z2,...,Zn)

as

+ anZ n

space

~ : { a[-]
with

a],a2,...,an6C

~ C

_a[z] = a ] z I + a 2 z 2 +
We d e f i n e

For

a = (a],a 2 ..... a n ) 6 C n

}

product
n
< a[- ],b[- ] > =

By

treating

easily

the

prove
The

integration

the

Bose

following
algebra

z] , z 2 , . . . , ~ n 6 C

,

have

product

the

inner

and

over

as

.

integration

over

~2

we

can

statement:
F0K

the

~

~ ak-bk
k=]

consists

vacuum

e

of
as

polynomials

the

constant

in t h e
function

variables
]

We