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 1 / 18
Introduction to nc-geometry (` a la Connes) Cartesian coordinates allow to translate geometric problems into algebraic ones. E.g. . . ( x , y , z ) ∈ R 3 is a point of the unit sphere S 2 iff x 2 + y 2 + z 2 = 1 ( ‡ ) Coordinate functions generate the commutative algebra C ( S 2 ) . 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 ) ◮ 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”. Alain Connes (Oberwolfach, 2004) 2 / 18
Noncommutativity vs. quantization Balmer series (hydrogen emission spectrum in the visible region) ◮ Quantum physics: C 0 ( 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 J 2 . Alt. point of view: let H = J 2 be the Hamiltonian of a free particle on a 2 -sphere S 2 . ( H + i ) − 1 ∈ K ( H ) and the energy is quantized. Spin geometry uses a square root of H (the Dirac operator) to study S 2 . This idea is generalized by NCG to nc-spaces. 3 / 18
Spectral triples In nc-geometry ` a la Connes, spaces (closed oriented Riemannian manifolds) are replaced by (unital) spectral triples. Definition Example: the unit 2 -sphere S 2 A unital spectral triple is given by: ◮ H = L 2 ( S 2 ) ⊗ C 2 ◮ a Hilbert space H ; ◮ A = C ∞ ( S 2 ) ◮ an algebra A of (bounded) operators on H ; ◮ D = σ 1 J 1 + σ 2 J 2 + σ 3 J 3 , where ◮ a (unbounded) selfadjoint operator D on H σ j ’s are Pauli matrices and in that is of “order 1 ” in a suitable mathematical cartesian coordinates: sense (i.e. [ D , a ] ∈ B ( H ) ∀ a ∈ A ) and has a ∂ J j = i ǫ jkl x k . compact resolvent. ∂x l 4 / 18
Some examples from mathematics and physics 1 Famous applications: tilings, fractals, orbits, Moyal, foliations 2 Podle´ s quantum spheres and finite quantum field theories 5 / 18
Some examples from mathematics and physics 1 Famous applications: tilings, fractals, orbits, Moyal, foliations 2 Podle´ s quantum spheres and finite quantum field theories 6 / 18
Tilings Meredith College (North Carolina) (more @ www.eschertile.com/penrose.htm ) Roger Penrose, Texas A&M University 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. 7 / 18
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. SU q ( 2 ) and S 2 q . This is called dimension drop. Spacetime has metric dimension 4 and KO-dimension 10 ! (Connes, JHEP11(2006)) 8 / 18
Orbit spaces (image from en.wikipedia.org ) The algebra of the noncommutative torus C ( T 2 θ ) is the universal C ∗ -algebra generated by two unitary elements U , V with relation UV = e 2 πiθ V U . θ ) ≃ C ( S 1 ) ⋊ R θ Z is an example of noncommutative C ∗ -algebra If θ is irrational, C ( T 2 associated to an ergodic group action ( R θ = rotation of an angle θ ). 9 / 18
The Moyal plane Canonical and Lie algebra quantizations of R 2 : one replaces x = ( x 0 , x 1 ) ∈ R 2 with ˆ x 0 , ˆ x 1 generators of the Heisenberg algebra of 1D quantum mechanics and of sb( 2, R ) : [ ˆ x 0 , ˆ x 1 ] = iθ [ ˆ x 0 , ˆ x 1 ] = i ˆ x 1 /κ (Moyal plane) ( κ -Minkowski) Bounded operator approach [Groenewold 1946, Moyal 1949]: A θ := ( S ( R 2 ) , ∗ θ ) with � 1 2 i θ y × z d 2 y d 2 z . ( ∗ ) ( f ∗ θ g )( x ) := f ( x + y ) g ( x + z ) e ( πθ ) 2 The canonical comm. rel. holds in the “Moyal multiplier algebra”: x 0 ∗ θ x 1 − x 1 ∗ θ x 0 = 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] 10 / 18
Other examples of θ -deformations. . . M.A. Rieffel, Deformation quantization for actions of R d , Memoirs of the AMS 506 (1993). Marc A. Rieffel A. Connes, G. Landi, Nc-manifolds, the instanton algebra and isospectral deformations, Commun. Math. Phys. 221 (2001). Giovanni Landi A. Connes, M. Dubois-Violette, Noncommutative finite-dimensional manifolds, Commun. Math. Phys. 230 (2002) and 281 (2008). Michel Dubois-Violette 11 / 18
Foliations Reeb foliation S 3 ≃ ( D × S 1 ) ⊔ T 2 ( D × S 1 ) → S 3 → S 2 Hopf fibration S 1 ֒ Index theory for foliations � A. Connes, H. Moscovici, M.T. Benameur, P . Piazza, et al. S 1 An example of nc-Hopf fibration: S 3 → S 2 q ≃ SU q ( 2 ) q (other examples, relevant in the construction of nc-instantons, have been studied by G. Landi and collaborators) 12 / 18
Some examples from mathematics and physics 1 Famous applications: tilings, fractals, orbits, Moyal, foliations 2 Podle´ s quantum spheres and finite quantum field theories 13 / 18
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/ ) 14 / 18
SU q ( 2 ) . . . studied by Woronowicz in the ‘80s from the point of view of functional analysis. Algebraic approach: SU q ( 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 � z 1 � z 2 with UU ∗ = U ∗ U = 1 2 . U = , − z ∗ z ∗ 2 1 h � 1 ): ‘Coordinates’ on SU q ( 2 ) are given by ( 0 < q = e � � � z 1 z 2 with U q U ∗ q = U ∗ U q = , q U q = 1 2 . − qz ∗ z ∗ 2 1 For example ( U q U ∗ q ) 12 = 0 gives: z 2 z 1 = qz 1 z 2 (non-commutative if q � = 1 ) SU q ( 2 ) is not only a quantum space, but also a quantum group. 15 / 18
Podle´ s quantum spheres For any fixed 0 � s � 1 , x 0 := s ( z 1 z 2 + z ∗ 2 z ∗ 1 ) + ( 1 − s 2 ) z 2 z ∗ 2 , � z 2 1 − q ( z ∗ 2 ) 2 � + ( 1 − s 2 ) z ∗ x 1 := s 2 z 1 . are ‘coordinates’ on a ‘quantum sphere’ ( x 0 = x ∗ 0 ). There are commutation relations, e.g. x 0 x 1 = q 2 x 1 x 0 , and a relation fixing the ‘radius’. Symmetries are described by SU q ( 2 ) . For s = 0 , S 1 → { point } and the sphere is a 1-point compactification of K (Moyal plane). 16 / 18
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.” ). particles ◮ 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 strings (e.g. strings) or to “delocalized” objects (quantum geometries, e.g. Loop Quantum Gravity and NcQFT). q ( s = 0 ) the only basic divergence of φ 4 theory ◮ A toy model: on S 2 in 2D, the tadpole diagram, becomes finite at q � = 1 [Oeckl, 1999]. nc particles 17 / 18
Higher dimensional generalizations q is a “quantum projective line”. The nc-geometry of P n S 2 q ≃ P 1 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. 18 / 18
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