Motive Madness III
A quick note: check out John Baez's TWF 259 for more details on the absolute point! Clearly I'm not the only crazy one who sees ghosts in the machine. To quote the summary:
In short, a mathematical phantom is gradually taking solid form before our very eyes! In the process, a grand generalization of algebraic geometry is emerging ...
posted by Kea | 9:38 AM
4 Comments:
I looked at Baez's page. Fascinating. I hope I had time to look at it.
q-integers become ordinary integers and n-D vector spaces reduce to n-element sets: each dimension becomes point just as S^1becomes effectively point in previous example. For quantum logic this would mean that 2^N-D spinor space becomes 2^N-element set. N qubits are replaced with N bits. Brings in mind what happens in the transition from quantum theory to classical theory.
The quantum phase q=exp(i2pi/n) at the limit n-->infty would correspond to q=1 limit. This also correspond to Jones inclusions at the limit when the discrete group Z_n or or its extension-both subgroups of SO(3)- to contain reflection has infinite elements. Therefore this limit where field with one element appears might have concrete physical meaning. Does the system at this limit behave very classically?
In TGD framework this limit can correspond to either infinite or vanishing Planck constant depending on whether one consider orbifolds or coverings. For the vanishing Planck constant one should have classicality: at least naively! In perturbative gauge theory higher order corrections come as powers of g^2/4*pi*hbar so that also these corrections vanish and one has same predictions as given by classical field theory....
In TGD framework this limit can correspond to either infinite or vanishing Planck constant...
Great comment, Matti. Yes, it really seems these difficult mathematics questions are very closely related to the physics. See, for example, my thesis (link on sidebar) on a comparison between Set and Vect as toposes (not that I'm recommending it as good reading, by the way).
Thank you for link! And congratulations: it seems that I have missed something;-).
Non-commutativity might also relate to this quite closely via q-measurement theory which differs from quantum measurement theory.
Since the components of quantum spinor do not commute, one cannot perform state function reduction but only measure the modulus squared of spinor components which indeed commute and have interpretation as probabilities for spin up or down. They have a universal spectrum of eigen values. The interpretation would be in terms of fuzzy probabilities and finite measurement resolution but may be in different sense as in case of HFF:s.
At q--> 1 limit quantum measurement becomes possible in the standard sense of the word and one obtains spin down or up. This in turn means that the projective ray representing quantum state is replaced with one of n possible projective rays defining the points of n-element set. Thus the set of orthogonal coordinate axis replaces the state space in quantum measurement (we do this all the time when at blackboard!). For HFF:s of type it would be N-rays which would become points, N the included algebra.
All this should have space-time correlates by quantum classical correspondence. A TGD inspired geometro-topological interpretation for the projection postulate might be that quantum measurement at q-->1 limit corresponds to a leakage of 3-surface to a sector of imbedding space with q-->1 (Planck constant near to 0 or infty depending on whether one has covering of CP_2 or division of M^4 by subgroup of SU(2) becoming infinite cyclic, roughly) and Hilbert space is indeed effectively replaced with n rays. For q not equal to 1 one would have only probabilities for different outcomes since things would be fuzzy.
Classical physics and classical logic would be the physical counterparts for the shadow world of mathematics and would indeed result only as an asymptotic notion.
Still a comment, one element field(s actually) could correspond to Galois fields associated with infinite primes represented as phases. See also the posting at my blog.
One would have infinite hierarchy of Galois fields with infinite number of elements and the map to real numbers would reduce the elements of these fields to 0 or 1, that is Z_2.
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