Mutual Unbias
Thanks to Carl Brannen for pointing us to a PI seminar by Thomas Durt, based on papers which study canonical bases for Hilbert spaces of dimension $p^{n}$ using, in particular, finite fields arising from subgroups of permutation groups on $p^{n}$ letters, where $p$ is prime.
Now it turns out that Durt has written a paper with John Corbett, who is one of few topos theory physicists, and who happens to be at Macquarie University in Sydney, the heartland of Category Theory. This makes it not unlikely that Durt et al are thinking very abstractly when they mention in passing their interest in foundational axioms for quantum mechanics.
Now it turns out that Durt has written a paper with John Corbett, who is one of few topos theory physicists, and who happens to be at Macquarie University in Sydney, the heartland of Category Theory. This makes it not unlikely that Durt et al are thinking very abstractly when they mention in passing their interest in foundational axioms for quantum mechanics.
posted by Kea | 2:01 PM
9 Comments:
So, Carl, I've downloaded the papers. He, he! I just love the 21st century. It seems we need four bases for the p=3 case. Still reading....
Thank you Kea!
This result could provide one additional reason why for p-adic physics. For instance, the discretization of one-dimensional line with length of p^n units would give rise to p^n-D Hilbert space of wave functions.
Article mentions that when one knows all transition amplitudes from a given state of to all states of all N unbiased basis, one can fully reconstruct the state. Thus a basis with p^n elements - which indeed emerge naturally in p-adic framework - would be optimal for quantum tomography. The explicit formulas for the unbiased basis would therefore have an immediate application in quantum information science.
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Article mentions that the problem of finding maximal number of unbiased basis is not solved when the dimension N of Hilbert space is arbitrary. Could anyone tell why the following idea does not work?
a) pxp-dimensional matrix algebras are prime tensor factors in the sense that NxN-D matrix algebra decomposes into a tensor product of prime matrix algebras and N-D Hilbert space decomposes to a tensor product of prime Hilbert spaces.
b) One can construct a maximal number of unbiased basis as a tensor product of these basis for prime tensor factors.
Is it here? Too good to be true?
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By the way, the notion of prime Hilbert space provides a new interpretation for infinite primes, which are in 1-1 correspondence with the states of a supersymmetric arithmetic QFT. The earlier interpretation was that the hierarchy of infinite primes corresponds to a hierarchy of quantum states. Infinite primes could also label a hierarchy of infinite-D prime Hilbert spaces with product and sum for infinite primes representing unfaitfully tensor product and direct sum.
a) At the lowest level of hierarchy one could interpret infinite primes as homomorphisms of Hilbert spaces to generalized integers (tensor product and direct sum mapped to product and sum) obtained as direct sum of infinite-D Hilbert space and finite-D Hilbert space. (In)finite-D Hilbert space is (in)finite tensor product of prime power factors.
b) The physical interpretation could be be as a decomposition of the universe to a subsystem plus environment or decomposition of degrees of freedom to those which are above and below measurement resolution.
c) The analog of the fermionic sea would be infinite-D Hilbert space which is tensor product of all prime Hilbert spaces H_p with given prime factor appearing only once in the tensor product.
d) The construction of these Hilbert spaces would reduce to that of infinite primes. One can "add n bosons" to this state by replacing of any tensor factor H_p with its n+1:th tensor power.
One can "add fermions" to this state by deleting some prime factors H_p from the tensor product and adding their tensor product as a finite-direct summand. One can also "add n bosons" to this factor.
e) At the next level of hierarchy one would form infinite tensor product of all infinite-D prime Hilbert spaces obtained in this manner and repeat the construction. This can be continued ad infinitum and the construction corresponds to abstraction hierarchy or statements about statements hierarchy or hierarchy of n:th order logics. Or hierarchy of space-time sheets of many-sheeted space-time. There are many interpretation.
f) Note that also infinite integers make sense since one can form tensor products and direct sums. Also infinite rationals exist. Zero energy ontology suggests that infinite integers correspond to positive energy states and their inverses to negative energy states. Zero energy states would be always infinite rationals which would correspond to infinite integers which would be units as real numbers giving for a given real number an infinite rich number theoretic anatomy so that single space-time point could represent quantum states of entire universe in its anatomy (number theoretical Brahman=Atman). Infinitesimals around given number would be replaced with infinite number of number-theoretically non-equivalent real units multiplying it.
He, he, Matti! Wonderful observations. I'm getting very confused trying to put the pieces together, but putting SUSY QFT at oo here totally makes sense to me. For others: so long as we're still just talking about matrices we're in the land of 2-cats, and even the mass matrices are a kind of projection into this 2-universe. By understanding how to generalise this construction to the heirarchy, we might better understand that damned number. That's really the only reason I care about polygons and Riemann zeta functions.
The analog of the fermionic sea would be infinite-D Hilbert space which is tensor product of all prime Hilbert spaces H_p with given prime factor appearing only once in the tensor product.
Perhaps integers and rationals could be seen as a reflection at the wall of Plato's cave of the "category" of Hilbert spaces generated by prime Hilbert spaces by tensor product and direct sum and also by tensor tensor products of positive and negative energy Hilbert spaces (rationals). The shadows would be generalized integers.
State function reduction would produce the shadow (in very concrete sense actually!). Recall that N-D Hilbert space basis is mapped to N-element set in quantum measurement projecting to N orthogonal rays defining this set.
This category (or whatever you might call it) would be closed with respect to tensor product and direct sum having direct quantum physical interpretation and mapped to ordinary sum and product at the wall of the cave.
For more detailed representation see my blog.
Okay, I've added a post writing the particle masses in terms of MUB theory.
Just for fun, I also wrote out a MUB set for the Hilbert space of the Dirac algebra. One ends up with 4+1 = 5 basis sets since the Dirac algebra has 2^4 = 16 degrees of freedom (i.e. bilinears).
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"Our main motivation is that most discussions of quantum mechanics use a background space-time that is the same as classical space-time, usually without any supportive arguments and even sometimes denying that quantum mechanics is a space-time theory. And yet many of the difficulties in understanding quantum phenomena derive from the use of classical space-time. We claim in the present paper that the space-time of quantum phenomena differs from that of classical phenomena in the nature of its continuum. According to our theory [10], the description of quantum phenomena requires a real number continuum that is not the classical continuum. It is not even a fixed element of the theory but varies with the quantum system in a way similar to the way the metric geometry of Einstein’s general relativity varies with the physical system [2]. This is not part of the usual paradigm of quantum theory but adopting it enables us to reformulate the paradoxes of the standard interpretation when each quantum system has its own real number continuum." - p2 of Quantum mechanics as a space-time theory, http://arxiv.org/abs/quant-ph/0512220
Thanks for this link, Kea. What I like about the paper (apart from the paragraph above) is the title, the abstract mentioning a comparison to Bohmian mechanics, and the fact that the paper contains a section called "physical interpretation" (a subject that is proudly missing from a lot of mathematical physics). I'm glad that Carl, you, and Matti, are investigating it.
If forces and thus accelerations are really due to quantum field interactions, then spacetime is not "curved" at the quantum scale or atomic size scale. A geodesic will only appear approximately curved when the number of graviton interactions (or whatever the field quanta are) which are causing interactions is large, e.g. motion on large scales. So any final theory has got to take account of how randomness emerges from field quanta interactions on small scales, and how these little zig-zag deflections add up on large scales (large numbers of field quanta being involved) to give something that is approximated by "curvature" i.e. differential geometry. I'm glad that this paper at least is trying to address this real physical problem. It's empirically known from GR that spacetime is classical on large distance scales, and it's well known from QM that spacetime is not classical on small distance scales - this is experimental, observational fact and not speculation like "problems" caused by spin-2 gravitons. However, looking at the paper carefully, it's just nowhere near radical enough. Still, it's a step towards tackling real problems. (Sorry, the first time I submitted this comment, there was a problem with the quotation, and I also typed "energes" instead of "emerges".)
Okay Kea, here's the blog post that derives the qutrit MUBs.
They come in groups of 4, with each basis set having three elements. Previously, we've associated the generations with the basis set elements via the Koide formula.
One might suppose that one basis set is the leptons, and the other three are the three colors of quarks.
But to get the real model, I believe you need to replace all the complex numbers in these 3x3 matrices with elements from the Pauli algebra. That makes the problem into a 6 dimensional Hilbert space; precisely the lowest dimension that is still an unsolved MUB problem.
Excellent, Carl! Great work. This really does put the mass matrices on a more solid theoretical footing.
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