**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Sat, 3 Apr 1999 07:40:09 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 109] RE: [time 104] Re: [time 102] Re: [time 99] Spacetime& consciousness"**Previous message:**Matti Pitkanen: "[time 107] Re: [time 106] Re: [time 92] Entanglement"**In reply to:**ca314159: "[time 106] Re: [time 92] Entanglement"**Next in thread:**Ben Goertzel: "[time 118] Re: [time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras"

On Wed, 31 Mar 1999, Ben Goertzel wrote:

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

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

increases.

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

field.

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.

MP

**Next message:**Hitoshi Kitada: "[time 109] RE: [time 104] Re: [time 102] Re: [time 99] Spacetime& consciousness"**Previous message:**Matti Pitkanen: "[time 107] Re: [time 106] Re: [time 92] Entanglement"**In reply to:**ca314159: "[time 106] Re: [time 92] Entanglement"**Next in thread:**Ben Goertzel: "[time 118] Re: [time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras"

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