Durdevich, Micho
Institute of Mathematics, UNAM (Mexico City)
“Music of Quantum Circles”
Keywords : circle, symmetry, quantum geometry, diagrammatic
category.
Abstract : We illustrate the basic ideas and principles of
quantum geometry by considering mutually complementary quantum realizations of
circles. It is quite amazing that such a simple geometrical object as the
circle, provides a rich illustrative playground for an entire array of purely quantum phenomena.
On the other hand, the ancient Pythagorean musical scales naturally lead to a simple quantum circle. In this lecture
we explore different musical scales, their mathematical generalization
and formalization, and their possible quantum-geometric
foundations. In this conceptual framework, we outline a diagramatical-categorical formulation for a quantum theory
of symmetry, and further explore interesting musical connections and interpretations.
1. Introduction
Quantum geometry mirrors the ideas of quantum physics, into
the realm of geometrical spaces and their transformations. But quantum spaces, the analogies of atoms, mollecules and
quantum systems of physics in general, exhibit a nature essentially different from their classical counterparts. They are not
understandable in terms of points, parts, or local neighbourhoods. In general, these concepts do not apply at all to quantum
spaces. However the entire fabric of space is conceived as the one
indivisible whole.34
There is something profoundly quantum in all music. A
discrete space–the skeleton hosting any musical score, morphs into a true musical form, only after being symbiotically enveloped
by a geometry of sound. And this geometry is inherently quantum, as it connects the points of the discrete underlying structure,
invalidating the difference between now, then, here and there; thus creating an irreducible continuum for a piece of music:
continuous discreteness and discrete continuity.
All this inherently promotes simplicity in thinking, as we
are forced to look for some deeper structure, going far beyond the parts, points, local neighbourhoods, and fragmented
classical geometrical views. One such a way of thinking, transcending the nature of mathematical realms, is harmony: to look at
symmetries—the transformational modes of things—and understanding
the mathematical creatures in terms of them. Conceptual
roots of his thinking are found in the Erlangen Program by Felix Klein.
Circles are children of simplicity. A principal geometrical
realization of an infinite symmetry group. The idea of circle is observed in repetitions. Any continual change, movement,
transformation, in which there is something invariant before and after, naturally leads to the idea of circle. In music such
is the concept of octave. It leads to a circle representing the geometrical space of abstract tonalities. A more detailed geometrical
structure is given by a musical scale, interpretable as further ‘musical’
symmetries of the circle.
The aim of this lecture is to illustrate how these
symmetries of the circle lead to its own projected quantum realizations, and the complementary view of extending the circle into a
quantum counterpart. These examples are actually extremely rich in their internal structure. They reflect the spectrum of all
principal new phenomena of quantum geometry. In particular, the quantum circles are quantum groups in a proper sense. We
shall briefly talk about a general diagrammatical and categorical formulation of symmetry, which naturally includes our
quantum circles and their quantum siblings, as well as the variety of all classical structures.
2. Quantum Circles
The Pythagorean musical scale invites us to consider the
quotients of the classical circle over a free action of the infinite cyclic group of integers z, generated by a single irrational
rotation. The space of equivalence classes has a direct musical interpretation, as the space of abstract tonality classes
within a single octave. In the case of Pythagoreans, we have two principal frequency transformations: the octave itself,
given by doubling the frequency ω ↦ 2ω;
and the perfect fifth, given by the
shift ω ↦ 3ω.
If we consider the frequency range as covering all positive real numbers r+,
and pass to natural logarithms, then the multiplication becomes addition and frequency range
is the whole r. The octave space is given by r/z ln(2).
Within this space, the addition of ln(3) acts as a symmetry.
By transforming [r] ↦ exp(2irπ/ln(2)) we can identify the octave space with the circle of the unitary complex numbers. In
terms of this identification, the Pythagorean perfect fifth becomes a multiplication by exp(2iπ ln(3)/ln(2)) which represents an
irrational rotation, by the angle φ + 2π = 2π ln(3)/ln(2).
Another possibility is to consider rational rotations. In
terms of complex numbers, it corresponds to roots of unity, say primitive solutions of the equation z n = 1 for n ≥ 2. In
this case the action of z factorizes to the action of the cyclic group of order n on the circle. And the resulting factor space is
again a classical circle. So our tonality space is given by an n-fold covering of by [two circles].
Musical scales based on equal temperament provide a
realization of such a rational structure, and n is
the number of semitones. In terms of the original
frequencies, the simplest movement is given by ω ↦ 21/nω
However, in the irrational case, there exist infinitely many
connectable pitch values, dense in the octave space. In other words, every orbit of the action is dense in the circle. . The resulting orbit decomposition is ergodic in the
sense that there exist no no-trivial decomposition of the circle, into two
disjoint measurable sets consisting of whole orbits each. One of them always has measure 0 and hence another is of the normalized
measure one. To put it differently, there exist no no-trivial
measure theory on the orbit space Q . It exhibits a kind of
intrinsic wholeness. And if there is no measuring in Q, then there is simply no hope to build, in the spirit of classical
geometry, any meaningful higher-level theory.
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So Q is consisting of points, however the points are
behaving quite wildly, and there is no any effective and operational separability between them. Exactly the same kind of
phenomenon we encounter in studying certain aperiodic tilings of the Euclidian plane. A paradigmatic example is given by the
space of isomorphism classes of Penrose tilings. There exist (uncountably) infinitely many classes, however every two
tilings are indistinguishable by comparing their finite regions. Every finite region of one tiling is faithfully echoed, and
infinitely many times in any other tiling.
This space can also be described as the quotient space of
the full binary sequences space {–, +}N which is the same as the Cantor triadic set, by a relation of equivalence identifying
sequences which coincide on a complement of a finite subset of N.
One possibility to deal with such quantum points, is to
construct a noncommutative C*-algebra A, which captures the space
Q in terms of equivalence classes of its irreducible
representations. Such an approach is presented in detail in [1]. Another and inequivalent approach, is to apply the theory of quantum
principal bundles developed in [2], and consider non-trivial differential (necessarily quantum, as in music) structures
on discrete and extremely disconnected spaces and groups. We believe this is more in the spirit of the original Erlangen
Program.
So the rational rotations give us classical circle as the
tonality classes space. And irrational rotations produce quantum objects. It is interesting to observe that from a purely geometrical
perspective, the quantum behaviour is the generic one. Indeed, although the rational and irrational unitary complex numbers
are intertwined, both being everywhere dense in the circle, the roots of unity form a countable and therefore negligible,
subset. With probability one, will choose an infinite covering
mode, and cast a quantum shadow.
References: [1] Connes A. Noncommutative Geometry, Academic
Press (1994). [2] Prugovecki, E. Quantum Geometry, Kluwer (1990).
[3] Goldblatt, R. Topoi: The Categorical Analysis of Logic,
Elsevier Science (1984). [3] Đurđevich, M. “Geometry of Quantum
Principal Bundles I”, Communications in Mathematical Physics
175 (3) 457–521 (1996). [4] Đurđevich, M. “Geometry of Quantum
Principal Bundles II”, Reviews in Mathematical Physics, 9
(5) 531–607 (1997). [5] Đurđevich, M. “Diagrammatic Formulation of
Multi Braided Quantum Groups”, Contemp. Math. 318, 97–106
(2003). [6] Đurđevich, M. “Characteristic Classes of Quantum
Principal Bundles” (Extended Version), Algebras, Groups and
Geometries, Hadronic Press, Vol. 26 No. 3 241–341 (2009). [7]
Đurđevich, M. “Geometry of Quantum Principal Bundles III”,
Algebras, Groups and Geometries, Vol. 27, 247–336 (2010).
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