Noncommutative Spaces: a brief overview Francesco DAndrea - - PowerPoint PPT Presentation

Noncommutative Spaces: a brief overview

Francesco D’Andrea

Department of Mathematics and Applications, University of Naples Federico II P .le Tecchio 80, Naples, Italy

14/04/2011

University of Rome “La Sapienza” – April 14th, 2011

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Introduction to nc-geometry (` a la Connes)

Cartesian coordinates allow to translate geometric problems into algebraic ones. E.g. . . (x, y, z) ∈ R3 is a point of the unit sphere S2 iff (‡) x2 + y2 + z2 = 1 Coordinate functions generate the commutative algebra C(S2). One can forget the geometric object and work with the abstract commutative algebra generated by three self-adjoint elements x, y, z with relation (‡).

(from www.gps.oma.be) Alain Connes (Oberwolfach, 2004) ◮ In math. there are many interesting classes of spaces

that cannot be described using commutative algebras, but rather using noncommutative ones.

◮ Sometimes there is no space at all: just an algebra that

we want to treat like a “geometric object”.

◮ Noncommutative geometry (NCG) provides the tools to

study these “spaces”.

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Noncommutativity vs. quantization

Balmer series (hydrogen emission spectrum in the visible region) ◮ Quantum physics: C0(M) → B(H) ◮ Quantum = noncommutative:

◮ Noncommutativity there are physical quantities that cannot be simultaneously

measured with arbitrary precision (e.g. ∆x ∆p |[x, p]| /2 = h/2 ).

◮ Quantization (some) operators have a discrete spectrum, and the corresponding

physical observables are quantized (e.g. absorption and emission spectra of atoms).

◮ A physical quantity that takes discrete values is e.g. the angular momentum J2.

• Alt. point of view: let H = J2 be the Hamiltonian of a free particle on a 2-sphere S2.

(H + i)−1 ∈ K(H) and the energy is quantized. Spin geometry uses a square root of H (the Dirac operator) to study S2. This idea is generalized by NCG to nc-spaces.

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Spectral triples

In nc-geometry ` a la Connes, spaces (closed oriented Riemannian manifolds) are replaced by (unital) spectral triples.

Definition

A unital spectral triple is given by:

◮ a Hilbert space H; ◮ an algebra A of (bounded) operators on H; ◮ a (unbounded) selfadjoint operator D on H

that is of “order 1” in a suitable mathematical sense (i.e. [D, a] ∈ B(H) ∀ a ∈ A) and has a compact resolvent. Example: the unit 2-sphere S2

◮ H = L2(S2) ⊗ C2 ◮ A = C∞(S2) ◮ D = σ1J1 + σ2J2 + σ3J3 , where

σj’s are Pauli matrices and in cartesian coordinates: Jj = iǫjklxk ∂ ∂xl .

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Some examples from mathematics and physics

1 Famous applications: tilings, fractals, orbits, Moyal, foliations 2 Podle´

s quantum spheres and finite quantum field theories

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Some examples from mathematics and physics

1 Famous applications: tilings, fractals, orbits, Moyal, foliations 2 Podle´

s quantum spheres and finite quantum field theories

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Tilings

Roger Penrose, Texas A&M University Meredith College (North Carolina) (more @ www.eschertile.com/penrose.htm )

A Penrose tiling is a non-periodic tiling (partition) of the plane by ‘kites’ and ‘darts’. A classification (i.e. a parametrization up to equivalence defined by symmetries of the plane) of Penrose tilings requires the use of NCG. The set of (equivalence classes of) tilings is homeomorphic to the Cantor set.

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Fractals

Julia set Sierpinski gasket Fractal food (broccoli)

Fractal := metric space with Hausdorff (metric) dimension > the topological dimension. Fractals are not manifolds, but they are nc-manifolds! (studied by D. Guido, T. Isola et al.) An example related to the Riemann hypothesis is provided by p-adic numbers. There are also quantum spaces with metric dimension greater than the topological (Hochschild) dimension, e.g. SUq(2) and S2

• q. This is called dimension drop.

Spacetime has metric dimension 4 and KO-dimension 10! (Connes, JHEP11(2006))

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Orbit spaces

(image from en.wikipedia.org )

The algebra of the noncommutative torus C(T2

θ) is the universal C∗-algebra generated

by two unitary elements U, V with relation UV = e2πiθV U . If θ is irrational, C(T2

θ) ≃ C(S1) ⋊Rθ Z is an example of noncommutative C∗-algebra

associated to an ergodic group action (Rθ = rotation of an angle θ).

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The Moyal plane

Canonical and Lie algebra quantizations of R2: one replaces x = (x0, x1) ∈ R2 with ˆ x0, ˆ x1 generators of the Heisenberg algebra of 1D quantum mechanics and of sb(2, R): [ˆ x0, ˆ x1] = iθ [ˆ x0, ˆ x1] = iˆ x1/κ (Moyal plane) (κ-Minkowski) Bounded operator approach [Groenewold 1946, Moyal 1949]: Aθ := (S(R2), ∗θ) with (∗) (f ∗θ g)(x) := 1 (πθ)2

• f(x + y)g(x + z)e

2i θ y×zd2y d2z .

The canonical comm. rel. holds in the “Moyal multiplier algebra”: x0 ∗θ x1 − x1 ∗θ x0 = iθ . Huge literature about (∗). Wigner used it to formulate quantum mechanics in phase space. On the metric aspect see:

• E. Cagnache, FD, P

. Martinetti, J.-C. Wallet, The spectral distance on the Moyal plane, arxiv:0912.0906 [hep-th]

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Other examples of θ-deformations. . .

Marc A. Rieffel

M.A. Rieffel, Deformation quantization for actions of Rd, Memoirs of the AMS 506 (1993).

• A. Connes, G. Landi, Nc-manifolds, the instanton

algebra and isospectral deformations,

• Commun. Math. Phys. 221 (2001).

Giovanni Landi Michel Dubois-Violette

• A. Connes, M. Dubois-Violette,

Noncommutative finite-dimensional manifolds,

• Commun. Math. Phys. 230 (2002) and 281 (2008).

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Foliations

Hopf fibration S1 ֒ → S3 → S2 Reeb foliation S3 ≃ (D × S1) ⊔T2 (D × S1)

Index theory for foliations A. Connes, H. Moscovici, M.T. Benameur, P . Piazza, et al. An example of nc-Hopf fibration: S3

q ≃ SUq(2) S1

→ S2

q (other examples, relevant in the

construction of nc-instantons, have been studied by G. Landi and collaborators)

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Some examples from mathematics and physics

1 Famous applications: tilings, fractals, orbits, Moyal, foliations 2 Podle´

s quantum spheres and finite quantum field theories

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Compact quantum groups

A rich source of examples of nc-spaces is provided by quantum group theory, most notably the theory of compact quantum groups developed by S.L. Woronowicz.

Stanisław Lech Woronowicz (from www.fuw.edu.pl/~slworono/ )

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SUq(2)

. . . studied by Woronowicz in the ‘80s from the point of view of functional analysis. Algebraic approach: SUq(2) is the canonical quantization of SU(2) in the direction of its unique Poisson-Lie group structure, see e.g. [Semenov-Tian-Shansky, 1994]. An element of SU(2) is U = z1 z2 −z∗

2

z∗

1

• ,

with UU∗ = U∗U = 12 . ‘Coordinates’ on SUq(2) are given by ( 0 < q = e

h 1 ):

Uq =

• z1

z2 −qz∗

2

z∗

1

• ,

with UqU∗

q = U∗ qUq = 12 .

For example (UqU∗

q)12 = 0 gives:

z2z1 = qz1z2 (non-commutative if q = 1) SUq(2) is not only a quantum space, but also a quantum group.

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Podle´ s quantum spheres

For any fixed 0 s 1, x0 := s(z1z2 + z∗

2z∗ 1) + (1 − s2)z2z∗ 2 ,

x1 := s

• z2

1 − q(z∗ 2)2

+ (1 − s2)z∗

2z1 .

are ‘coordinates’ on a ‘quantum sphere’ (x0 = x∗

0). There are commutation relations, e.g.

x0x1 = q2x1x0 , and a relation fixing the ‘radius’. Symmetries are described by SUq(2). For s = 0, S1 →{point} and the sphere is a 1-point compactification of K (Moyal plane).

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Problems with infinities and possible cures. . .

◮ The great success of relativistic QFT: accepted for its excep-

tional agreement with experiments (e.g. the g-2 of e−). . . . nevertheless: even a successful theory like QED could be mathematically inconsistent ( “QED is not Q.E.D.” ).

◮ Perturbative QFT: one has to deal with infinities due to the

non-compact nature of the space (IR), and to the point-like nature of the particle (UV).

◮ Possible treatments: from point particles to extended objects

(e.g. strings) or to “delocalized” objects (quantum geometries, e.g. Loop Quantum Gravity and NcQFT).

◮ A toy model: on S2

q (s = 0) the only basic divergence of φ4 theory

in 2D, the tadpole diagram, becomes finite at q = 1 [Oeckl, 1999].

particles strings nc particles

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Higher dimensional generalizations

S2

q ≃ P1 q is a “quantum projective line”. The nc-geometry of Pn q has been studied in:

FD, G. Landi, L. Dabrowski, The Noncommutative Geometry of the Quantum Projective Plane,

• Rev. Math. Phys. 20 (2008), 979–1006.

FD, G. Landi, Geometry of the quantum projective plane, 5th ECM Satellite Conf. Proceedings, Royal Flemish Acad. (Brussels), 2008, pp. 85–102. FD, G. Landi, Anti-selfdual Connections on the Quantum Projective Plane: Monopoles,

• Commun. Math. Phys. 297 (2010), 841–893.

FD, G. Landi, Anti-selfdual Connections on the Quantum Projective Plane: Instantons, in preparation. FD, G. Landi, Bounded, unbounded Fredholm modules for quantum projective spaces, Journal of K-theory 6 (2010), 231–240. FD, L. Dabrowski, Dirac Operators on Quantum Projective Spaces,

• Commun. Math. Phys. 295 (2010), 731–790.

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