Thursday, March 08, 2012

Quantum mathematics

The comment of Pesla to previous posting contained something relating to the self-referentiality of consciousness and inspired a comment which to my opinion deserves a status of posting. The comment summarizes the recent work to which I have associated the phrase "quantum adeles" but to which I would now prefer to assign the phrase "quantum mathematics".

To my view the self referentiality of consciousness is the real "hard problem". The "hard problem" as it is usually understood is only a problem of dualistic approach. My hunch is that the understanding of self-referentiality requires completely new mathematics with explicitly built-in self-referentiality. During last weeks I have been writing and rewriting chapter about quantum adeles and end up to propose what this new mathematics might be. The latest draft is here.

1. Replace of numbers with Hilbert spaces and + and × with direct sum and tensor product

The idea is to start from arithemetics : + and × for natural numbers and generalize it .

  1. The key observation is that + and x have direct sum and tensor product for Hilbert spaces as complete analogs and natural number n has interpretation as Hilbert space dimension and can be mapped to n-dimensional Hilbert space.

    So: replace natural numbers n with n-D Hilbert spaces at the first abstraction step. n+m and n×m go to direct sum n⊕m and tensor product n⊗m of Hilbert spaces. You calculate with Hilbert spaces rather than numbers. This induces calculation for Hilbert space states and sum and product are like 3-particle vertices.

  2. At second step construct integers (also negative) as pairs of Hilbert spaces (m,n) identifying (m⊕r,n⊕r) and (m,n). This gives what might be called negative dimensional Hilbert spaces! Then take these pairs and define rationals as Hilbert space pairs (m,n) of this kind with (m,n) equivalent to (k⊗m,k⊗n). This gives rise to what might be called m/n-dimensional Hilbert spaces!

  3. At the third step construct Hilbert space variants of algebraic extensions of rationals. Hilbert space with dimension sqrt(2) say: this is a really nice trick. After that you can continued with p-adic number fields and even reals: one can indeed understand even what π-dimensional Hilbert space could be!

    The essential element in this is that the direct sum decompositions and tensor products would have genuine meaning: infinite-D Hilbert spaces associated with transcendentals would have different decompositions and would not be equivalent. Also in quantum physics decompositions to tensor products and direct sums (say representations of symmetry group) have phyiscal meaning: abstract Hilbert space of infinite dimension is too rough a concept.

  4. Do the same for complex numbers, quaternions, and octonions, imbedding space M4×CP2, etc.. The objection is that the construction is not general coordinate invariant. In coordinates in which point corresponds to integer valued coordinate one has finite-D Hilbert space and in coordinates in which coordinates of point correspond to transcendentals one has infinite-D Hilbert space. This makes sense only if one interprets the situation in terms of cognitive representations for points. π is very difficult to represent cognitively since it has infinite number of digits for which one cannot give a formula. "2" in turn is very simple to represent. This suggests interpretation in terms of self-referentiality. The two worlds with different coordinatizations are not equivalent since they correspond to different cognitive contents.

Replace also the coordinates of points of Hilbert spaces with Hilbert spaces again and again!

The second key observation is that one can do all this again but at new level. Replace the numbers defining vectors of the Hilbert spaces (number sequences) assigned to numbers with Hilbert spaces! Continue ad infinitum by replacing points with Hilbert spaces again and again.

You get sequence of abstractions, which would be analogous to a hierarchy of n:th order logics. At lowest levels would be just predicate calculus: statements like 4=22. At second level abstractions like y=x2. At next level collections of algebraic equations, etc....

Connection with infinite primes and endless second quantization

This construction is structurally very similar to - if not equivalent with - the construction of infinite primes which corresponds to repeated second quantization in quantum physics. There is also a close relationship to - maybe equivalence with - what I have called algebraic holography or number theoretic Brahman=Atman identity. Numbers have infinitely complex anatomy not visible for physicist but necessary for understanding the self referentiality of consciousness and allowing mathematical objects to be holograms coding for mathematics. Hilbert spaces would be the DNA of mathematics from which all mathematical structures would be built!

Generalized Feynman diagrams as mathematical formulas?

I did not mention that one can assign to direct sum and tensor product their co-operations and sequences of mathematical operations are very much like generalized Feynman diagrams. Co-product for instance would assign to integer m all its factorizations to a product of two integers with some amplitude for each factorization. Same for co-sum. Operation and co-operation would together give meaning to 3-particle vertex. The amplitudes for the different factorizations must satisfy consistency conditions: associativity and distributivity might give constraints to the couplings to different channels- as particle physicist might express it.

The proposal is that quantum TGD is indeed quantum arithmetics with product and sum and their co-operations. Perhaps even something more general since also quantum logics and quantum set theory could be included! Generalized Feynman diagrams would correspond to formulas and sequences of mathematical operations with stringy 3-vertex as fusion of 3 -surfaces corresponding to ⊕ and Feynmannian 3-vertex as gluing of 3-surfaces along their ends, which is partonic 2-surface, corresponding to ⊗! One implication is that all generalized Feynman diagrams would reduce to a canonical form without loops and incoming/outgoing legs could be permuted. This is actually a generalization of old fashioned string model duality symmetry that I proposed years ago but gave it up as too "romantic": see this.


At 4:48 AM, Blogger hamed said...

The posting was very interesting. For me understanding of it(Quantum mathematics) was simpler than your previous postings about quantum adeles. :)
Best wishes for you.

At 5:38 AM, Anonymous said...

Dear Hamed,

you are right. I was on wrong track in the definition of quantum p-adics. It seems that one can simply replace the coefficients a_n<p by their quantum counterparts when one performs the map to reals by canonical identification.

Quantum map means mapping of prime factors of a_n to quantum primes defined by q=exp(i*pi/p). Similar map should make sense also for Hilbert spaces and in case of Jones inclusions would give rise to quantum 2-spinors identifiable as quantum Hilbert spaces 2_q. Also p_q, p=3,5,7, predicted and should correspond to analogs of Jones inclusions. The realization was that this quantum map can be separated from the challenge of defining quantum p-adics completely.

As the number of pages in "Quantum Adeles" exceeded 50 I was sure that something IS wrong;-). The replacement of integer n with n-D Hilbert space and + and x with direct sum and tensor product is so simple that it could be fundamental, and gives beautiful connection with basic vision about generalized Feynman diagrams and might answer to the question about origin of self-referentiality of consciousness.

One must however define also co-sum and co-product and this is a fascinating challenge for both mathematician and physicist.

At 8:15 AM, Blogger Stephen said...

Nice post!!!.. the sqrt(2) features prominently in one of my articles at "Mellin and Laplace Integral Transforms Related to the Harmonic Sawtooth Map and a Diversion Into the Theory of Fractal Strings" and digit sequences can be understood in terms of "deterministic chaos" .. I even gave a shout-out to the "adelic product" at the end of my article. Actually, if everyone can stop obsessing about the LHC and get back to reality (number theory) then some of this stuff might start to jive in better ways. :) The sqrt(2) also shows up in reference to Clifford(geometric) algebras quite naturally see p.16 of and again the transcendental equation aspect leading to some very curious "integer sequences"

I'm not trying just advertise my own stuff but I'm hoping someone will stumble across the work and perhaps notice something that I have not.. there appears to be structure and ground for tilling.. but it is not known the "value" of these structures if that makes sense.. in other words, if this stuff wasn't fun and enjoyable then it would all be for naught...

My hope is that eventually these disparate threads can be woven into a more coherent whole. It'll require some fancy symbolic manipulation inspired by numerical evidence though ;)


At 8:35 AM, Anonymous said...

Dear Stephen,

nice to learn that we have similar perversions;-). Speaking seriously, it would be high time for physicists to take number theory seriously. Thank you for links. I try to find time.

The idea about Hilbert spaces with algebraic number valued dimensions is so beautiful and crazy that I cannot resist the temptation in the case of sqrt(2).

sqr(2) defines algebraic extension of rationals as pairs (m,n)== m+sqrt(2)*n. Define the product as
(m,n)*(r,s)= (mr+2ns, ms+nr). Replace m and n with Hilbert spaes with dimesions m and n and here it is. There is no need to bring in sqrt(2) explicitly! It is just the product!

For p-adics for which sqrt(2) exists one can even define Hilbert space as infinite direct sum a_n\otimes p^n where a_n is finite-D Hilbert space: also p^n denotes Hilbert space.

You mention complexity and chaos. Here it emerges also: pi-dimensional Hilbert space would correspond to infinite-D Hilbert space defined by pinary expansion of pi. No formula for the dimensions of coefficients of a_n. For rationals one can the dimensions appear periodically from some digit. For algebraic numbers one has similar situation since one can construct them as n-tuples of rational Hilbert spaces for n-D extension.

At 2:17 AM, Blogger Santeri Satama said...

Wow. Few days ago I was thinking of Ramanujan saying that thinking math was thinking God's ideas - and wondering about the mathematical self-referentiality! And Platonia as divine world of possible words is at least philosophically and scientifically consistent theology, and quantum math of Hilbert spaces emerges as divine gnothi seauton.

And transcendentals included, as it seems, as kinds of infinite-D singularities but each unique! But how to get to these actualities of our kind of observer participation, does this landscaping generalisation predict and necessitate any way the experience of 3D and 4D spaces we inhabit? As infinite-D n=pi/3,14 is near and between finite-D n=3 and n=4 (euclidean and non-euclidean), does quantum math succeed somehow in explaining or necessitating directly unifying our collective spatio-temporal experience?

At 4:07 AM, Anonymous said...

I dare feel that this is certainly one of the "big" ideas;-). Utterly simple but extremely non-trivial and indeed says something about the basic mystery of consciousness which to me is self-referentiability- which is also the essence of mystics. Someone has said that fractals are in some very strange sense very familiar. I think I understand why;-). What else the endless zooming up is but replacing point with Hilbert space!

Transcendentals would give rise to infinite-D Hilbert spaces for which the representation as infinite direct sum would correspond to expression as a series of digits, say powers of some prime.

For rationals the direct sum would have periodic pattern corresponding to the periodicity of digits for rationals. Second representation for rationals would be as pairs of integer-dimensional Hilbert spaces with finite dimension. For algebraics one would have n-tuples of finite-D Hilbert spaces.

For transcendentals infinite amount of information would be needed. The states in these Hilbert spaces would be that would matter in physical realization, and one might perhaps interpret finite approximation as use of states which correspond to 0 after some digit. Decimal cutoff realized quantally.

The interesting question is whether various representations for say pi are cognitively equivalent. Is representation in decimal series equivalent with binary series which involves direct sum of Hilbert spaces whose dimensions are 2^n. Bit 1 gives this kind of space, bit 0 gives nothing.

It is quite possible that they are not and that cognition is sensitive to this representation. Physics would not care about it. Even more: general coordinate invariance would mean that the representation of space-time point by integer valued coordinates in some coordinate system is equivalent with a representation in terms of transcendentals in some other coordinate system. GCI is blind to computational difficulties. If you force you Turing machine to perform the coordinate change from integer values to transcendental values of coordinates, it fails miserably: never-ending mission!

I think that all could be about cognition. Cognition would provide space-time points with the number theoretic anatomy.

At 2:46 PM, Blogger Ulla said...

Apparently spiders have their brains in their body cause their brains are so large they overfill the body cavity and legs. It seems the smaller the animal, the more it has to invest in its brain,

How was it? Consciousness can only be diminished?

At 9:25 PM, Blogger Santeri Satama said...

Another noob question: Lee Smolin said somewhere that the problem of combining quantum and relativity arose from different notions of time, quantum still carrying on the newtonian notions into Hilbert spaces. This leads to question about the relation of Minkowski spaces and relativity to this Grand New Quantum Math of Hilbert space(s inside...), especially as Minkowski space does not have inner product or just indefinite inner product. In other words, does the TOE of Hilbert with inner product have also GUTs? ;)

Consciousness of self-reflection may be more funda-mental as purely Hilbert-mathematical self-reflection, but life and biomatter as we know us this still seems to require a Minkowski wrapped around us. And how does the no-cloning theorem relate to quantum states and maths of purely Hilbert self-reflection?

At 11:27 PM, Anonymous said...


Thank you for a good question. This topics is new to me;-): do not take me too seriously;-)

a) This is *not* about replacing Minkowski space itself with Hilbert space but numbers with Hilbert spaces whose points are numbers and can be replaced with Hilbert spaces whose.... Minkowski space, M^4xCP_2, and basic TGD would be still there. These Hilbert spaces relate to the new, quantal view about numbers themselves. + and x have interpretation in terms of direct sum and tensor product for Hilbert spaces.

What is new would represent cognition and self-reference via number theoretical, fractal anatomy of points. The construction could be equivalent with the construction of infinite primes (and integers, and rationals). This does *not* mean GUTs.

b) M^4 itself is *not* replaced with Hilbert space, only the values of its 4 coordinates at each point of M^4 so that no problems with indefinite metric appear. Hilbert space replacing the values of 8 coordinates of a given point of - say -imbedding space - or any mathematical object is *not* analogous to "internal space" - say CP_2 replacing points of M^4 in TGD.

c) Only now I realized what it could mean that the dimension of Hilbert space (having rational, algebraic, and even transcendental values in certain sense) varies from point to point for each coordinate value.

*For a given imbedding space point there is 8-ple of Hilbert spaces whose dimensions code for the values of the coordinates. Should one interpret this 8-ple as Cartesian product of Hilbert spaces? If so then the dimension D of this tensor product would be sum of the dimensions coding for the values of these 8 coordinates. Forcing D to be same for
allowed sets of points of partonic 2-surface would give constant dimension for the Cartesian product but would be very weird condition.

*Quantum states* as points of these Hilbert spaces are what actually matter: quantum states in say infinite-D space can be restricted to finite-D sub-space and this is probably very relevant and could allow effectively constant value for Hilbert space dimension for a region of 4-surface.

d) The hierarchy of Planck constants and braids are also something completely new from the standard physics point of view. A connection is highly suggestive even at the level of details. The hierarchy of Planck constants associated with dark matter means in TGD Universe local covering spaces of imbedding space and defining finite-D Hilbert spaces assigned with space-time points. Dark matter would correspond to this Hilbert hierarchy.

At 4:37 AM, Blogger Santeri Satama said...

Please bear with me, as I fail to comprehend even what "inner product" means. Under-standing is a reductionistic movement, comprehension holistic and embodied state of wrapping yourself around universe, platonia etc. ;)

I imagine walking the vector of real line, starting from two (to make it line). At three I'm first genuine prime and euclidean space, at 3,14... I fall into the pit of the best known transcendental infinity that makes lines curves and vibrating, at four I remember being the first square of multiplying myself with myself (2*2) according to no-cloning-theorem, and the number of dimensions that is the characteristic of Minkowski carrying the curves and vibrations of pi within. What algebraic function is this inclusion of pi into square of two? And is there a relation between Minkowski having no inner product and second law of thermodynamics (which according to you is the square of quantum math?!); I sense that what we are reaching for is inflation cosmology in the most platonic sense. :D

You have given much attention to 2-adics, but how about 3-adics? Could they reveal something more about secrets of pi?

At 12:03 AM, Anonymous said...

To Santeri:

I answer those questions that I understand;-).

Inner product codes for angles and distances. For two vectors with components x_i and y_i the inner product is simply sum_i x_i*y_i. Complex case can be generalized in obvious manner.

You have interesting motorization of the basic idea. A serious warning: Hilbert spaces are always in question. The analog of Hilbert space with metric replaced with Minkowski metric allows zero norm states (light-like vectors) and Hilbert does not want his name to be associated with it. Hilbert space inner product is necessarily Euclidian.

I repeat myself: what is done is the replacement of number with Hilbert spaces or rather -states of Hilbert spaces. The dream would be quantum physicalization of entire mathematical cognition. Every sequence of mathematical manipulations would correspond to zero energy state.

Another new thing is the notion of co-operation: co-sum and co-product are inverses of sum and product. For instance, co-product of n=12 gives product quantum superposition of two-particle states 2*6, 3*4, and three-particle state 2*2*3 with space coefficients and arithmetic QFT gives these coefficients. One must have unitary S-matrix!

These ideas are fascinating but as always deep mathematical ideas do they go over the human head;-).

As a matter fact, the physically most interesting p-adic numbers correspond to very large primes p.
p=M_127= 2^127-1 corresponds to electron. This number is about 10^38 and represents what physicist would have called equivalent of infinity.

The view about practically infinite changed when M-theorists gave us the stringy landscape and about 10^500 solutions to stringy equations. Even more: this number has been increasing steadily since then while the optimism of M-theorists has been steadily decreasing. Even Lubos Motl has ceased to report about the victories of M-theory.

At 12:40 AM, Blogger Ulla said...

"the notion of co-operation: co-sum and co-product are inverses of sum and product."

Like undoing the borders (squares)? This is undoing the selves? Unentanglements?

I have been thinking of the hidden networks, like the genome networks in a population - could these be an expression of p-adics?

Networks are descreate things said Endre Szemeredi as is the quantum jump (perpendicular motion as in spin?)

"The analog of Hilbert space with metric replaced with Minkowski metric allows zero norm states (light-like vectors) and Hilbert does not want his name to be associated with it. Hilbert space inner product is necessarily Euclidian."


But Euclidean space is instantanous, no curvature, that is 3-D space? You said the lightlike space was 4-D earlier when I asked? Dimensions are emergent?


Post a Comment

<< Home