One element field, quantum measurement theory and its q-variant, and the Galois fields associated with infinite primes
Kea mentioned John Baez's This Week's Finds 259, where John talked about one-element field - a notion inspired by the q=exp(i2π/n)→1 limit for quantum groups. This limit suggests that the notion of one-element field for which 0=1 - a kind of mathematical phantom for which multiplication and sum should be identical operations - could make sense. Physicist might not be attracted by this kind of identification.
In the following I want to articulate some comments from the point of view of quantum measurement theory and its generalization to q-measurement theory which I proposed for some years ago (see this).
I also consider and alternative interpretation in terms of Galois fields assignable to infinite primes which form an infinite hierarchy. These Galois fields have infinite number of elements but the map to the real world effectively reduces the number of elements to 2: 0 and 1 remain different.
1. q→ 1 limit as transition from quantum physics to effectively classical physics?
The q→limit of quantum groups at q-integers become ordinary integers and n-D vector spaces reduce to n-element sets. For quantum logic the reduction would mean that 2N-D spinor space becomes 2N-element set. N qubits are replaced with N bits. This brings in mind what happens in the transition from wave mechanism to classical mechanics. This might relate in interesting manner to quantum measurement theory.
Strictly speaking, q→1 limit corresponds to the limit q=exp(i2π/n), n→∞ since only roots of unity are considered. This also correspond to Jones inclusions at the limit when the discrete group Zn 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 g2/4πhbar so that also these corrections vanish and one has same predictions as given by classical field theory.
2. Q-measurement theory and q→ 1 limit.
Q-measurement theory differs from quantum measurement theory in that the coordinates of the state space, say spinor space, are non-commuting. Consider in the sequel q-spinors for simplicity.
Since the components of quantum spinor do not commute, one cannot perform state function reduction. One can however measure the modulus squared of both spinor components which indeed commute as operators 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. Probability would become the observable instead of spin for q not equal to 1.
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 states is replaced with one of n possible projective rays defining the points of n-element set. For HFF:s of type II1 it would be N-rays which would become points, N the included algebra. One might also say that state function reduction is forced by this mapping to single object at q→ 1 limit.
On might say that the set of orthogonal coordinate axis replaces the state space in quantum measurement. We do this replacement of space with coordinate axis all the time when at blackboard. Quantum consciousness theorist inside me adds that this means a creation of symbolic representations and that the function of quantum classical correspondences is to build symbolic representations for quantum reality at space-time level.
q→ 1 limit 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 dark sector of imbedding space with q→ 1 (Planck constant near to 0 or ∞ depending on whether one has n→∞ covering or division of M4 or CP2 by a subgroup of SU(2) becoming infinite cyclic - very 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.
In this picture classical physics and classical logic would be the physical counterpart for the shadow world of mathematics and would result only as an asymptotic notion.
3. Could 1-element fields actually correspond to Galois fields associated with infinite primes?
Finite field Gp corresponds to integers modulo p and product and sum are taken only modulo p. An alternative representation is in terms of phases exp(ik2π/p), k=0,...,p-1 with sum and product performed in the exponent. The question is whether could one define these fields also for infinite primes (see this) by identifying the elements of this field as phases exp(ik2π/Π) with k taken to be finite integer and Π an infinite prime (recall that they form infinite hierarchy). Formally this makes sense. 1-element field would be replaced with infinite hierarchy of Galois fields with infinite number of elements!
The probabilities defined by components of quantum spinor make sense only as real numbers and one can indeed map them to real numbers by interpreting q as an ordinary complex number. This would give same results as q→ 1 limit and one would have effectively 1-element field but actually a Galois field with infinite number of elements.
If one allows k to be also infinite integer but not larger than than Π in real sense, the phases exp(k2π/Π) would be well defined as real numbers and could differ from 1. All real numbers in the range [-1,1] would be obtained as values of cos(k2π/Π) so that this limit would effectively give real numbers.
This relates also interestingly to the question whether the notion of p-adic field makes sense for infinite primes. The p-adic norm of any infinite-p p-adic number would be power of π either infinite, zero, or 1. Excluding infinite normed numbers one would have effectively only p-adic integers in the range 1,...Π-1 and thus only the Galois field GΠ representable also as quantum phases.
I conclude with a nice string of text from John'z page:
What's a mathematical phantom? According to Wraith, it's an object that doesn't exist within a given mathematical framework, but nonetheless "obtrudes its effects so convincingly that one is forced to concede a broader notion of existence".
and unashamedely propose that perhaps Galois fields associated with infinite primes might provide this broader notion of existence! In equally unashamed tone I ask whether there exists also hierarchy of conscious entities at q=1 levels in real sense and whether we might identify ourselves as this kind of entities? Note that if cognition corresponds to p-adic space-time sheets, our cognitive bodies have literally infinite geometric size in real sense.
For details see the chapter Was von Neumann Right After All?.