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New noise P'(u), u R d+

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n )N

n=1 is a given orthonormal sequenceP

N

(n )N

is

a

sequence

of

non-negative

numbers

with

n=1

n=1 n = 1, N  4.

Normal states which are represented by one wave function (N = 1) are

called pure states. So the space of signals can be identied with the set

of pure states on L2 (M (G)). Analogously, sometimes we identify normal

130

states \$ given by (2) with their density matrices (positive normalised tracePN

class operators) % = n=1 n h n , · n i.

We assume that G decomposes into disjoint regions G1 , . . . , Gn being

responsible for dierent tasks. So L2 (Gk , Gk ) represents the space of the

excited neurons in the region Gk . Hereby, B is the restriction of  to

the set B  G. The main reason to choose the bosonic Fock space as basic

space of signals is the possibility to identify the Fock space over L2 (G) with

the tensor product of the Fock spaces over L2 (G1 ), . . . , L2 (Gn ):

L2 (M (G1 ^ . . . ^ Gn )) 

= L2 (M (G1 ))
. . .
L2 (M (Gn )).

Now, let us be given functions f r 5 L2 (Gr ), r 5 {1, . . . , n}. For k  1

we dene functions fkr 5 L2 (M (Gr )) by

;s

Q r

A

f (x),

* 5 M (Gr ), |*| = k,

? k! ·

x5*

fkr (*) :=

.

A

=0

elsewhere.

Further, we set f0r (*) := 1Io (*). Observe that for each r the sequence

(fkr )k0 is an orthogonal system in L2 (M (Gr )) (being orthonormal if

kf r k = 1). An especially important class of functions in the Fock space are

the so-called exponential vectors exp{f r } dened by

exp{f r } =

4

X

1

s fkr .

k!

k=0

(3)

For f 5 L2 (G), we dene the exponential vector exp{f } by

¡

¢

exp{f }(*) := exp{f 1 }
. . .
exp{f n } (*1 , . . . , *n ) (* 5 M (G)) (4)

where f r := 1IGr · f is the restriction of f to Gr and *r = *(· _ Gr ) denotes

the restriction of the conguration * to points from Gr .

The states

2

\$ g := ekgk · hexp{g}, · exp{g}i

are called coherent states on L2 (M (G)) if g 5 L2 (G) resp. on L2 (M (Gr ))

if g 5 L2 (Gr ). Hereby, h·, ·i denotes the scalar product in the corresponding

Hilbert space. Roughly speaking, coherent states describe states of systems

of quantum particles where each particle is in the same one-particle state.

Furthermore, \$ 0 = hexp{0}, · exp{0}iis called the vacuum state.

131

Observe that hexp{f }, exp{g}i = e what implies k exp{af }k2 =

for a 5 C, f 5 L2 (G). We denote by N the number operator:

|a|2 ·kf k2

e

N (*) := |*| ·

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