[time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sat, 3 Apr 1999 07:40:09 +0300 (EET DST)

On Wed, 31 Mar 1999, Ben Goertzel wrote:

> > Could "entanglement entropy" be similar to "mutual entropy"?
> I don't think they are the same
> However, it is tempting to think of the "wholeness" of a system as having
> to do with
> the mutual entropy generated by placing the parts of the system together as
> a whole
> Mutual entropy is a special case of what I call "emergent pattern" -- the
> emergent
> pattern in a set {A,B} being roughly
> | Patterns({A,B}) - Patterns(A) - Patterns(B) |
> where | | is a norm operation on the space of patterns in question and
> Patterns(X)
> refers to the set of patterns in X
> Mutual entropy results from this definition if one restricts consideration
> to "Markovian
> patterns" in dynamical trajectories, i.e. to statistical analysis of
> transition probabilities
> between cells in state space.
> > To put in my 2 cents :), we hope that a discussion with a wide variety
> >of people with differing backgrounds and specializations but with the
> >common goal of a good model of quantum gravity will accomplish more that
> >individuals working independently. ;)
> I have been through this process before, I note.
> I was involved in a FOUR YEAR LONG e-mail dialogue involving the radical
> physicist
> Tony Smith, the philosopher Kent Palmer and a Norwegian physics/math student
> Onar Aam (who now works with me at Intelligenesis). We made a lot of
> conceptual
> progress in a very abstract way, beginning from the shared intuition that
> octonionic
> algebra and Clifford algebras are essential to the structure of the
> universe. However,
> we didn't solve the crucial problems we set out to solve -- not yet,
> anyway; and the discussion
> sort of petered out a year ago, although we're all still good friends. I felt
> that Tony clung too tightly to all the details of his theory; and Onar and
> Kent didn't
> have the math background to really get into the nitty-gritty details

Also we had very interesting and fruitful discussions with Tony
Smith and I have studied Tony's homepage. One one results on my side was
the realization that 8-dimensional
imbedding space H of TGD allows octonionic structures as tangent space
structure: one could say that H is locally a number field. Future
lightcone of Minkowski space allows very natural quaternion structure.
This was a surprise for me since I had used to think quaternions are
 as inherently Euclidian space: one can however define also Minkowksi
metric: inner product is defined as the real part of xy instead of
real part of x^*y.

Actually future lightcone can be regarded as an especially natural
quaternionic manifold since Lorentz invariant lightcone proper time
corresponds to real quaternion unit. Also CP_2 is very natural quaternion
manifold: even more, its isometry group SU(3) is by its 8-dimensionality
the only simple Lie group allowing octonionic tangent spacet structure.

Induction of octonion structure of H=M4^_+xCP_2 to spacetime surface makes
sense and gives algebraic structure in tangent space of spacetime surface:
this algebra is not necessarily isomorphic to quaternion algebra.
Thus H is extremely natural from octonionic view point.

I learned from Tony also that the length squared of octonionic and
quaterionic primes are orinary primes: R^2= p and this is nothing but
p-adic length scale hypothesis which is basic corner stone of p-adic TGD.
The construction of infinite primes is like repeated quantization of
a QFT with states labelled by primes and one cannot avoid the idea that
entire quantum TGD could at some level reduce to a theory of infinite
octonionic primes(;-). Physics as number theory!

> I don't mean to be negative in any way -- this kind of conversation is
> great fun, and if
> it never leads to anything but fun, it's worth more than most things in
> human life!!!!
> But if we really want to solve the puzzle of modern physics, we need to be
> resolute
> about not clinging too closely to our pet ideas -- taking what is best from
> them and paring
> away the inessentials, and moving always toward the essence.
> There are after all literally hundreds of radical physics theories out
> there, and probably
> at least 10-20% of them have a big element of truth to them. But they are
> all too
> complicated, and they lack the basic conceptual simplicity that to me has
> the "ring
> of truth" about it.
> Hitoshi's theory does have that "ring of truth" in its articulation of a
> very simple principles:
> wholes may have different laws than parts

Yes. I agree with this. I do not believe in reductionism. Macroscopic
physics is not just phenemonological models forced by our inability to
calculate but involves completely new laws. I formulate this idea in terms
of the hierarchy of p-adic physics. The larger the p the richer the
structures since p-adic topology becomes more and more refined when p

> Tony Smith's theory also had that ring of truth to me, in the way that it
> derived ALL
> structures from the same finite algebra, the octonions. Space was an
> 8-dimensional
> discrete lattice, formed from integral octonions. At each corner of the
> lattice was
> an octonion element -- first generation particles are single octonions,
> then second generation
> particles are pairs and third generation particles are triples. (the fact
> that there are only
> 3 generations, and 2^3=8 is probably important). Particle interactions are
> explained by
> octonion multiplication.
> Gravity is explained in a way that I don't like -- the MacDowell-Mansouri
> mechanism is used
> to explain gravity as a spin-2 field.... This loses the conceptual
> intuition of General Relativity,
> which feels wrong to me.
> (I was going to point you to the URL for Tony's website, but it seems to
> have moved.
> He does have some papers at xxx.lanl.gov, but they don't describe the
> discrete physics
> framework that we worked out together.)
> I would like to express the whole/part distinction algebraically. An
> algebra for parts, an
> algebra for wholes, and an algebraic mapping (homomorphism?) from the part
> algebra
> into the whole algebra. This is very tricky; the standard model is
> described nicely by
> clifford algebras and lie algebras, but general relativity's algebras are
> different. I haven't
> studied this kind of math in many years so I am a bit rusty here.

Your idea is interesting also from my point of view. I am pondering
analogous problem: spacetime sheets decomposes in QFT limit to regions
with different p-adic primes: the physics in different regions are
described by different p-adic number fields. How to relate these
p-adic physics to each other?: this is the basic problem. I believe that
this is possible.

For instance, anticommutation relations for fermions involve only 0 and
1 at right hand side. Rather remarkably, canonical identification between
reals and p-adics (SUM x_np^n --> SUM x_np^(-n)) maps 0 to 0 and 1 to 1.
Hence one can say binary numbers 0 and 1 are common to reals and all
p-adic number fields and that anticommutation relations for fermions are
number field independent: same oscillator operators can be regarded as
operators in Hilbert spaces with arbitrary number fields as coefficient

The generalization of the unitarity concept at configuration space level
makes also sense: super S-matrix decomposes into sub-S-matrices belonging
to various p-adic number fields. Also this generalization is possible only
because the right hand side of unitarity relations is either 0 or 1: the
elements common to all number fields! Unitarity relations involve some
fascinating and rather surreal features of p-adic probabilities.

This physical picture might give rise to very elegant algebraic structures
constructed from p-adic number fields and reals.


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