From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Wavelets, squeezed states 7 Reproducing Kernels
Organization: Whole Earth 'Lectronic Link
Date: Sun, 10 Jan 1993 03:22:35 GMT


7. Reproducing Kernels
Nothing here is original it's all G Kaiser's stuff.

K(m,m') is called a "reproducing kernel". M is an arbitrary set. A set of
functions g(m) on M forms a Hilbert space F under some inner product <|>.
The exact form of the inner product does not matter.

Let K(m,m') be complex-valued on MxM such that

(42) for every m in M, Km(m') = K(m',m) in F.

(43) for every m in M and every g in F

g(m) = <K(m,m)|g> = <Km|g>    (p.30)

then F is a reproducing-kernel Hilbert space and K(m,m') is its reproducing
kernel.

K(m',m) = <K(m',m')|K(m,m)> = <Km'|Km>   (44)

K(m,m')* = K(m',m>     (45)

K(m,m) = |Km|^2 >= 0   all m  (46)

|K(m,m')| <= |Km||Km'|    (47)  Schwartz inequality

The kernel function K itself virtually generates the whole structure. All
the functions in F inherit the boundedness and growth properties of K. If K
has singularities, then some functions in F have those singularities.

*If M is a classical phase space, F is the corresponding Hilbert space of
the quantum system and |g(m)|^2 is the probability density in the quantum
state g of finding the system in the classical state m.

This assertion seems rather important!

*Basic idea of coherent and squeezed states:
The quantum state which maximizes the probability density of being in the
classical state mo is modulo over all phase gmo. That is, gmo is a quantum
wave packet which is optimally localized at the classical state (i.e.,
point) in phase space.


*Contact of this stuff to new quantum computer hardware? (half-baked
Sarfatti speculation or poetic precognition channeled from Q as in Star
Trek The Next Generation)

Sarfatti problem: How do we use this for fermions? Need anticommuting
Grassmann variables?  What does classical state mean for fermions? Is this
stuff only good for boson? What about fractional or para-statistics or
anyons as in 2-d solid state (quantum wells) fractional Hall effect? What
about quantum dots?

The reproducing kernel embodies the entire Hilbert space. Properties of K
are imaged in properties of g in F.

Given Hilbert space F whose elements are all functions on M, how do we know
whether F has a reproducing kernel K?  A necessary conditiion is

|f(m)| = |<Km|f>| <= |Km||f| , all f in F and all m in M (48)

For fixed m, there is an "evaluation map" Em of Hilbert space F to complex
plane C defined by

Em(f) = f(m)  (49)

is a bounded linear functional on F.

Riesz representation theorem implies

E(f) = <e|f>  (50)

for unique vector e in F. There exists a unique em such that

f(m) = <em|f>  (51)     (p.32)

But f(m) = <Km|f>, therefore

em = Km = K(m,m)  (52)

also

K(m,m') = <em|em'>  (53)

F has a reproducing Kernel if and only if all the evaluation maps are
bounded.

In applications M has more structure. M can be a Lie group G. M can be a
"homogeneous space" of Lie group G in which each g in G acts on homogeneous
space M as an invertible transformation preserving all other structure of M
like continuity or differentiability. These actions form the Lie group G.

Right action is

m -> mg  (54)

Tg:F -> F   (55)

(Tgf)(m) = f(mg)   (56)

is a "representation" of G on F, since

TgTg' = Tgg'     (57)

But f(m) = <Km|f> implies

Kmg = Tg*Km   (58)

therefore

K(m'g,mg) = <Km'g|Kmg> = <Km'|TgTg*|Km>   (59)

*Therefore, the reproducing kernel K is invariant under the action of G if
and only if all the operators Tg are unitary, i.e. the representation g->Tg
is unitary.

Sarfatti problem. In standard quantum mechanics (SQM) what is the group G
under which K is invariant in order that locally observable probabilities
(for the complete set of alternatives for part of an entangled nonlocal
whole) add up to 1? What if G is mutable? Suppose we go to a sub-group H?
Suppose G is a sub-group of some super-Group? Is this pointing not only to
something meaningful - but something profound about the relative nature of
probability in quantum mechanics?

to be continued