Probably everybody who knows quantum mechanics can sense
the
same. It's a real pleasure by the way. Words emit other virtual words
just
like electron emits virtual photons.
Since
our brain behaves like dynamic system the words play
a
role of quasi-particles and memory of the brain plays a role of vacuum
which
determines interaction constants of words.
A formal language is usually defined as a set of sentences which can be
derived
using
a certain set of rules or generated by an automaton.
In QLT a language consists of sentences which can be generated by a memory
considered.
Generated
means that words of a sentence interact strongly in a memory exciting
long
lived states in it.
Process of recognition and generation of words is probably similar to
the
process of quantum mechanical scattering. It explains how we can look
through
all our memory so fast as we do, since quantum mechanical search
goes
in parallel.
A word really is a sort of "bare" word plus all its interactions with other
words
and
objects stored in a human memory. Word stimulation results in excitation
of
its links producing under suitable conditions the long lived quasi stationary
states
of brain which include many words.
A word as though sits in a "potential" created by other words.
Like
in quantum field theory one can define the mutual potential energy
of
two
words as the total propagation amplitude squared. The total propagation
amplitude
is the sum of propagation amplitudes taken over all possible links
between
these words. All possible means that all semantic networks, images
and
models
containing links between the words are taken into account.
In
a particular case only the links from selected models may participate.
Let’s consider a set of neurons nk(see the picture). Every neuron has many inputs and one output. Index k numbers neurons, index j numbers the inputs. In neuron nkevery input j has a connection weight wj(k) . Let’s consider how this set of neurons can recognize input vectors in classical and quantum mechanical case.
In process of education a set of input vectors forms the weights. Every vector comes to one (and only one) neuron. Components of the vector correspond to neuron inputs. In neuron k the weights wj(k) which are proportional to the vector vj(k) components are formed.
In process of recognition an arbitrary vector vj comes to all neurons in parallel. Suppose that output of a neuron is proportional to the scalar product s(k) of the input vector vj and vector of weights wj(k). Then the maximal output will be in the neuron which weights are similar to the input vector. The negative feedback between neurons helps to amplify the maximal signal and suppress others.
Consider simple mathematical model. Let the neuron outputs qk change according to the following equations.
(1) d/dt (qk) = (qk) * sum(j) vjwj(k) - sum(k) (qk)*(qk) ;
Here the first term gives linear amplification,
second term - negative feedback.
The whole is a process with mode competition. Modes
here are the outputs qk.
The output qk
with
max sum(j)
vjwj(k)
survives
resulting in the exitation of a certain neuron.
Now
about the difference between classical and quantum mechanics.
In
classical case in process of recognition all signals coming in parallel
to look
over
all neurons must be real signals (current pulses or just classical particles).
It
takes a lot of energy.
In
quantum case we just let a quantum particle pass through all possible paths.
First
we let the particle come to the neuron inputs with the probability amplitudes
proportional to the input vector components (vector we want to recognize).
For example, at every neuron input we emit particle with such amplitude.
Then the neuron internal weights are now the amplitudes for particle to
pass through specific input channel. Probability amplitude
for the particle to come in a neuron will be the scalar product of the
input vector and weight vector again.
Because
all is going in parallel the quantum mechanical case has an advantage of
saving energy.
In quantum case instead of equations (1) we get Shredinger equation
(2) ih d/dt (qk) = (qk) * sum(j) wj(k)vj ;
where
qk
- wavefunction. Now the scalar
product sum(j)
vjwj(k)
determines
the energy
level
value at individual neuron.
Suppose
that ions can transit between the neurons and some reservoir. Then
in case of underbarrier transition the transition probability exponentially
depends on energy levels considered.
This
exponential dependance gives us very sensitive method for maximum evaluation.
Let's consider a common case of a system which reacts on inputs and produces
outputs.
From
input side there are an input amplitude vj
and
a matrix
Wjk
. Together they
give the probability amplitudes Ak
=
sum(j)
wkjvj
to
excite a certain internal state k of
the system.
From
the output side there is another transition matrix Tlk
, which determines
the probability amplitude to excite a certain output l
from
a certain internal state k
.
Finally
the probaility to exite output l
is equal to sum(jk)
Tlk
wkjvj
.
From
such point of view a human memory is a huge number of internal models.
Many
of them are looked through in parallel (due to quantum mechanical
nature)
to
produce the result.
Let's
stress that classical machine cannot do it, or better
say
that can but with a lot of energy required.
That
explains why human intellect is more poweful than artificial one
till now.
Once more
example:
Let's
consider chains of objects (words, for example ) which can be linked.
Let’s
number words (objects) by index i.
Two words ai
and aj
can be connected by link Tij.
We
can record such chains on a lattice of cells (neurons). Link between words
will mean that signal (or particle in quantum case) can pass between cells.
Index k will number the different chains recorded (see the picture).
A
particle coming through input line i
will pass along any chain through the links existing in the chain. Total
amplitude for particle to come to line j
is the sum of amplitudes over all chains recorded.
Such
amplitude is large for those lines j
which have many links with line i
and
small
for other lines. Supposing that output signals are generated only for lines
which are excited significantly we get what we need. Entering such a system
an input word will excite or generate other words according to the chains
previously recorded in the memory.
If
particle (or particles) enters through several inputs (corresponding to
combination of input words) then the total amplitude will be the sum of
amplitudes for all inputs.
Again
in quantum mechanical case all is going in parallel.
Suppose
also that particle has a certain amplitude to tunnel between chains. Then
the amount of paths which particle can pass increases significantly
since a path now can be composed from parts of different chains.
In
classical case it takes a lot of signals and power to look over all these
combinations.
Now let's suppose that the particles are bosons in an active media.
Then
additional particles are produced due to induced radiation like in
a laser.
This
process leads to amplification of particle propagation between nodes
with
larger mutual propagation amplitude.
The process is similar to processes with mode competition taking place
in
biological
systems and lasers.
Let's consider interaction of words from path-integral
point of view.
Suppose
that every word is a quantized field.
Again
we have words ai interacting
through links Tij which
form a memory.
See
the network picture above.
Interaction
operator (part of Hamiltonian) is sum(k) { sum(i,j) ai
Tij(k)aj+
}, where
ai
, aj+ are
annihilation and creation operators of word quantum (semilon).
An
input word i comes at moment t0 and excites (creates) other
words through possible links Tij(k).
If
we do time slices t1, t2, t3 ... we shall see some word quanta excited
at moment t1,
others
at moment t2 and so on. This is the path integral picture usual for quantum
field theory.
Every
path has its probability amplitude. The total amplitude to
excite an output word is the sum of amplitudes over all possible paths.
When quanta considered are bosons in active media
they
amplify their production like photons in a laser. As a result the modes
with more branched
links
survive.
If
the number of semantic networks enumerated by index k is large
the
so called "classic" trajectories are possible which can be attributed to
long lived quasi
stationary
states of the memory considered.
It's
important that "classic" trajectories exist only if the quantum memory
is large enough.
This
may be the possible explanation of human intellectual superiority over
animals.
------------------------------------------------------------------------------------------
About generalization in quantum mechanical networks.
In classical neural networks the generalization is realized with the help
of next layer neurons which fire when certain subgroups of the lower
level neurons are exited.
This mechanism can exist in quantum mechanical networks too. But in addition
to this in quantum networks there is another mechanism due to ability of
quantum particle pass over all possible paths in parallel. So many quantum
chains may act cooperatively resulting in generalized answer on external
stimulus. Also generalized quantum states exist in such system.
------------------------------------------------------------------------------------------
some other remarks:
If
we have the word A, then A is really a sort of "bare word = [A]",
analogous
to a bare charge in QED, plus a series of other terms that are
dependent
upon the entire set of words.
A = [A] + z*sum_n([word_n]) + z^2*sum_{nm}([word_n]&[word_m]) ...
where
the & word means concatenation of two "bare words" into a bare
two
word combination, and z is some sort of weighting constant. By doing
this it might extend the notion of a word and a word combination into a
domain that might contain more information on semantics and the like.
Let's
try to imagine how the interaction of such words-particles can be organized.
First of all we need memory and a possibility to look
through
this memory in parallel to see all semantic nets written.
Input
set of word-particles comes, interacts with memory
and
forms output set of word-particles.
Suppose
one has words a(i) and set of semantic nets numbered
by
index k. In every semantic net k each bond T(i,j)
connecting
words a(i) and a(j) gives a term to interaction
operator
V.
V = g*sum(k) { sum(i,j) [ a(i)T(i,j,k)a(j)+ ] }
where
g is a constant, a(i) and a(j)+ are annihilation and
creation
operators respectively, T(i,j,k) is bond T(i,j) in net k.
When
input particles come, interaction goes through creation
of
intermediate virtual words-particles just like we want.
The
amplitude to create output word a(j) is the sum of amplitudes over
all
semantic nets. Note that number of combinations of bonds
from
different nets is huge when k grows, like situation for
NP-complete
problem. But quantum mechanical system sees them all.
This
is like path integral, but only the paths allowed by the memory
participate
in the game.
We
wrote a simple interaction operator for words without grammar .
To
include a grammar one must write interaction operator according to grammar
rules.
One
could similarly do this with a partition function. An interaction
term
is similar to the usual harmonic oscillator Hamiltonian. As such
a
quantum
interpretation would be that of a free field. It seems that this
theory
needs to be coupled to some other field, as in the case of the Dirac
field
where p ----> p - ieA and where the current in the Faraday equation
is
j = e(psi^*psi). For there we would get the sort of Feynman
diagrams
and
analogs of radiative corrections. It's not clear of what to
couple
this system to accomplish this.
Maybe
one could imagine that each symbol has a rather large space (Hilbert
>or
Banach space) assigned to it. Each term in the series is a projection
>along
one of the basis vectors in that space, something like a
>Gram-Schmidt
procedure. Then one could further imagine that the
>"concretizations"
of that symbol or a whole word comes about through a
>sort
of Landau-Ginzburg process that breaks the symmetry of the series and
>makes
the symbol or word have some sort of definite "meaning."
Memory T(i,j) itself can be a field. It plays a role of Higgs bosons, giving interaction constants for pairs of words-particles.
First, in process of education, one forms T(i,j,k) for each semantic net k . Diagram is
a(i) --->------\
|=====>== T(i,j)
a(j) --->------/
When
T(i,j) density will become significant enough, one can use the system
for
recognition or generation of sentences, sending a set of input words to
the system and looking at output words.
First order diagram is
a(i) - --->---\
\ ----->-- a(j)
//
T(i,j) ==>==//
In
absence of input signals a(i) the wave function
of
the system is given by a set of occupation numbers for fields T(i,j).
At
reading a(i) come, interact with the memory, output a(j) is produced.
So then the total Hamiltonian would be
H = a(i)^{dag}a(i) + T^{ij}T_{ij}+ a(i)T_{ij}a(j)
In
a sence T_{ij} would then coorespond to a machine stack. Presumably
this
stack is a function of some HO states, z_k, so that the equation of
motion
would contain terms
a(i)(&T_{ij}/&z_k)a(j) & = partial.
That might work. Maybe the Chomsky transformational grammars can be cast in this extended format.
The question is how to proceed with such complex systems, what to begin with.
What's interesting that we can imagine dynamic systems which implement
not
only
groups (as families of elementary particles do) but also grammars.
Obviously
grammar gives the rules for Feynman diagrams for such dynamic
system.
It means that together with gauge invariance and super symmetry one can
use
various grammars to build quantum field theory. Till now only very
simple
theories were used with only a few Higgs which can be considered
as
primitive memories.
QLT needs two types of fields. First type quanta are word-like,
let's
call them semilons.
Second type quanta are link-like, let's call them linkons.
Suppose that new more powerful language-like quantum field theory exists.
Than
we can understand why the Word was the beginning of the All.
The energetic word-particle of Nature language stroke another
word-particle
and created all our world variety which was packed before
in
a vacuum.
Example:
Quantum computations are executed by particles passing through networks.
Construction
of the networks is a separate complex problem - long-lived
memory
formation. Let's consider the simple examples supposing that the network
exist.
Suppose
there is a number of channels. One channel is open, others - closed.
Problem
to solve is to clear up which channel is open.
In
classical case one has to send particles to the input of all channels
and see
at
which output a particle will appear.
In
quantum case it's enough to send one particle propagating through all channels.
It
will appear at the output of the open channel.
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