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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 identied with the set
of pure states on L2 (M (G)). Analogously, sometimes we identify normal
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states $ given by (2) with their density matrices (positive normalised tracePN
class operators) % = n=1 n hn , ·n i.
We assume that G decomposes into disjoint regions G1 , . . . , Gn being
responsible for dierent 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 dene 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 )k0 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 } dened by
exp{f r } =
4
X
1
s fkr .
k!
k=0
(3)
For f 5 L2 (G), we dene 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 conguration * to points from Gr .
The states
2
$ g := ekgk · 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.
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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 (*) := |*| ·