The Standard Model in Noncommutative Geometry and Morita equivalence

Transcription

1 The Standard Model in Noncommutative Geometry and Morita equivalence Francesco D Andrea e + u u W + ν g d Geometry, Symmetry and Quantum Field Theory Vietri, March 2015

2 The spectral action approach A.H. Chamseddine, A. Connes, The Spectral Action Principle, Commun. Math. Phys. 186 (1997), The noncommutative geometry approach to particle physics: algebraic reformulation of (quantum) field theory that works for spaces described by noncommutative algebras too. Two main goals: derive the Standard Model (the complicated Lagrangian) from simple geometric data; get some clues on unification with gravity. 1 / 20

3 The complicated Lagrangian... LSM = 1 2 νga µ νg a µ gsf abc µg a νg b µg c ν 1 4 g2 sf abc f ade g b µg c νg d µg e ν νw µ + νwµ M 2 W µ + Wµ 1 2 νz0 µ νz 0 µ 1 2c 2 M 2 Z 0 µz 0 µ 1 w 2 µaν µaν igcw( νz0 µ(w µ + Wν W ν + Wµ ) Z 0 ν(w + µ νw µ W µ νw + µ ) + Z 0 µ(w + ν νw µ W ν νw + µ )) igsw( νaµ(w + µ W ν W + ν W µ ) Aν(W + µ νw µ W µ νw + µ ) + Aµ(W + ν νw µ W ν νw + µ )) 1 2 g2 W + µ W µ W + ν W ν g2 W + µ W ν W + µ W ν + g 2 c 2 w(z 0 µw + µ Z 0 νw ν Z 0 µz 0 µw + ν W ν ) + g 2 s 2 w(aµw + µ AνW ν AµAµW + ν W ν ) + g 2 swcw(aµz 0 ν(w + µ W ν W + ν W µ ) 2AµZ 0 µw + ν W ν ) 1 2 µh µh 2M 2 αhh 2 µφ + µφ 1 ( 2M 2 µφ0 µφ 0 2 βh g 2 + 2M g H + 1 ) 2 (H2 + φ 0 φ 0 + 2φ + φ ) + 2M4 g 2 αh gαhm ( H 3 + Hφ 0 φ 0 + 2Hφ + φ ) 1 ( 8 g2 αh H 4 + (φ 0 ) 4 + 4(φ + φ ) 2 +4(φ 0 ) 2 φ + φ + 4H 2 φ + φ + 2(φ 0 ) 2 H 2) gmw + µ W µ H 1 2 g M c 2 w Z 0 µz 0 µh 1 2 ig ( W + µ (φ 0 µφ φ µφ 0 ) W µ (φ 0 µφ + φ + µφ 0 ) ) g ( W + µ (H µφ φ µh) +Wµ (H µφ + φ + µh) ) g 1 (Z 0 µ(h µφ 0 φ 0 µh) + M ( 1 Z 0 µ µφ 0 + W µ + µφ + Wµ µφ + ) ig s2 w MZ 0 µ(w µ + φ Wµ φ + ) + igswmaµ(w µ + φ Wµ φ + ) cw cw cw ig 1 2c2 w Z 0 µ(φ + µφ φ µφ + ) + igswaµ(φ + µφ φ µφ + ) 1 ( 2cw 4 g2 W µ + Wµ H 2 + (φ 0 ) 2 + 2φ + φ ) 1 1 ( 8 g2 c 2 Z 0 µz 0 µ H 2 + (φ 0 ) 2 + 2(2s 2 w 1) 2 φ + φ ) w 1 2 g2 s2 w Z 0 µφ 0 (W µ + φ + Wµ φ + ) 1 cw 2 ig2 s2 w Z 0 µh(w µ + φ Wµ φ + ) + 1 cw 2 g2 swaµφ 0 (W µ + φ + Wµ φ + ) ig2 swaµh(w µ + φ Wµ φ + ) g 2 sw (2c 2 w 1)Z 0 µaµφ + φ cw g 2 s 2 waµaµφ + φ + 1 ( 2 igs λa ij( q σ i γ µ q σ j )g a µ ē λ (γ + m λ e)e λ ν λ (γ + m λ ν)ν λ ū λ j (γ + m λ u)u λ j d λ j (γ + m λ d)d λ j + igswaµ (ē λ γ µ e λ ) (ūλ j γ µ u λ j ) 1 ) 3 ( d λ j γ µ d λ j ) + ig Z 0 µ{( ν λ γ µ (1 + γ 5 )ν λ ) + (ē λ γ µ (4s 2 w 1 γ 5 )e λ ) + ( d λ j γ µ ( 4 4cw 3 s2 w 1 γ 5 )d λ j ) + (ū λ j γ µ (1 8 3 s2 w + γ 5 )u λ j )} + ig 2 ( 2 W+ µ ( ν λ γ µ (1 + γ 5 )U lep λκe κ ) + (ū λ j γ µ (1 + γ 5 )Cλκd κ j ) ) + ig ( ) 2 2 W µ (ē κ U lep κλγ µ (1 + γ 5 )ν λ ) + ( d κ j C κλ γµ (1 + γ 5 )u λ j ) + ig 2M ( 2 φ+ m κ e( ν λ U lep λκ(1 γ 5 )e κ ) + m λ ν( ν λ U lep λκ(1 + γ 5 )e κ) + ig ( 2M 2 φ m λ e(ē λ U lep λκ(1 + γ 5 )ν κ ) ) m κ ν(ē λ U lep λκ(1 γ 5 )ν κ g m λ ν 2 M H( νλ ν λ ) g m λ e 2 M H(ēλ e λ ) + ig m λ ν 2 M φ0 ( ν λ γ 5 ν λ ) ig m λ e 2 M φ0 (ē λ γ 5 e λ ) 1 4 νλ MR λκ (1 γ5)ˆνκ 1 4 νλ MR λκ (1 γ5)ˆνκ + ig 2M ( ) 2 φ+ m κ d(ū λ j Cλκ(1 γ 5 )d κ j ) + m λ u(ū λ j Cλκ(1 + γ 5 )d κ ig ( ) j + 2M 2 φ m λ d( d λ j C λκ (1 + γ5 )u κ j ) m κ u( d λ j C λκ (1 γ5 )u κ j g m λ u 2 M H(ūλ j u λ j ) g m λ d 2 M H( d λ j d λ j ) + ig m λ u 2 M φ0 (ū λ j γ 5 u λ j ) ig m λ d 2 M φ0 ( d λ j γ 5 d λ j ) 1 Lagrangian of the Standard Model with neutrino mixing and Majorana mass terms (Minkowski space, Feynman gauge fixing). M. Veltman, Diagrammatica: the path to Feynman diagrams, Cambridge Univ. Press, / 20

4 The geometric data Properties of particles are encoded in the spectrum of a selfadjoint operator. L SM can be reconstructed from the following operator: 0 M ν ν L M ν 0 D F = 0 M e M e 0 0 M u M u 0 0 M d M d 0 ν R e L e R u L u R d L d R + M R : ν R ν R + charge conjugated where each M M n (C) mixes the n generations of particles (n = 3?) and contains the free parameters of the SM. (In the following, to simplify the discussion, I will work with n = 1 generation.) 3 / 20

5 Summary Talk based on: FD & L. Dabrowski, The Standard Model in Noncommutative Geometry and Morita equivalence, preprint arxiv: [math-ph]. Summary of the talk: 1 A Short Introduction to A. Connes Ideas. 2 (Some) Recent Advances. 4 / 20

6 PART I 5 / 20

7 The role of symmetries Symmetries often play a fundamental role in determining the dynamics of a theory. (e.g., in Dirac s equation: postulating local U(1) invariance forces us to replace derivatives with covariant derivatives, and introduce the electromagnetic field) [ In QFT ] Local gauge symmetry particle interactions. [ In GR ] Diffeomorphism invariance gravitational interaction. 1st step toward Standard Model + gravity find some generalized space whose symmetry group contains both local gauge transformations and diffeomorphisms. 6 / 20

8 Why noncommutative spaces? Smooth manifolds: by trading a (compact) space M for C (M), one recovers diffeomorphisms from automorphisms of the algebra: Diff(M) Aut(C (M)). A noncommutative algebra A has more symmetries, due to the presence of non-trivial inner automorphisms a uau, with u a unitary in A and a A. If A = C (M A I ) C (M, k) k A I ( dim A I < and k = R or C ) 1 Inn(A) Aut(A) Out(A) 1 C (M G) Diff(M) (with G := Inn(A I ) = U(A I )/Z I ) (if Out(A I ) = 1) Riemannian manifolds: metric informations are encoded in a Dirac-type operator D. For almost commutative spectral triples, the coupling constants are contained in D I. Symmetries are broken by interactions not every g Inn(A I ) commutes with D I. Nc manifold = algebra + a generalized Dirac operator. Let s see the precise definition. 7 / 20

9 Real spectral triples Definition A spectral triple is given by: Example: the Hodge-Dirac operator M = oriented Riemannian manifold. a complex separable Hilbert space H; A = C 0 (M) a -algebra A of bounded operators on H; a (unbounded) selfadjoint operator D on H s.t. [D, a] is bounded and a(d + i) 1 is a compact operator for all a A. It is called: unital if 1B(H) A ; even if γ = γ on H s.t. γ 2 = 1, γd = Dγ and [γ, a] = 0 a A; real if an antilinear J : H H with J 2 = ±1, JD = ±DJ, Jγ = ±γj and a, b A: [a, JbJ 1 ] = 0 [[D, a], JbJ 1 ] = 0 (reality) (1st order) H = Ω (M) = L 2 -diff. forms D = d + d γ = ( 1) degree J(ω) := ω It is unital M is compact. Example: finite nc spaces H = C n A = any -subalgebra of Mn (C) D = D arbitrary (even D = 0) 8 / 20

10 Gauge group and inner automorphisms Let (A, H, D, J) be a unital real spectral triple (with A real or complex) and U(A) the group of unitary elements of A. The reality axiom ensures that the map: U(A) B(H), u u JuJ 1, is a unitary representation (green and blue commute) called adjoint representation of U(A). The group of inner fluctuations of the spectral triple is: G(A, J) = { ujuj 1 : u U(A) } If the representation of A on H is faithful: G(A, J) U(A)/U(A J ) where A J := { a A : aj = Ja } is a -subalgebra of the center Z(A) of A, and Inn(A) = G(A, J), if A J Z(A). 9 / 20

11 Examples Einstein-Yang-Mills system: A I = M n (C) H I = M n (C) with Hilbert-Schmidt inner prod. (left regular representation) D I = 0 J I (a) = a G(A I, J I ) SU(n) ( means same Lie algebra) If A I = M n (R) or M n (H) one gets orthogonal or symplectic groups, respectively. Standard model coupled to gravity: A I = C H M 3 (C) =: A F H I C 32n, with n = number of generations G(A I, J I ) U(1) SU(2) SU(3) J I = charge conjugation D I = D F (see slide 3) 10 / 20

12 Spectral action The dynamics of (A, H, D, γ, J) is governed by an action [Chamseddine-Connes, 1997]: S[A, ψ] := S f [A, ψ] + S b [A] A = bosonic fields ψ = fermionic fields The fermionic part is where A = A Ω 1 D is a 1-form (def. later), S f [A, ψ] = Jψ, D A ψ D A := D + A + ɛ JAJ 1 }{{} ( ɛ = ±1 ) adjoint rep. of Ω 1 D and either ψ H or ψ H + := (1 γ)h (cf. fermion doubling ). The bosonic part is S b [A] = Tr H f(d A /Λ) p k=1 f kλ k Res z=k ζ DA (z) + f(0)ζ DA (0) + f(0) dim ker D A + O(Λ 1 ) where f is a cut-off function (Λ > 0), p the summability of D and f k := 0 f(t)tk 1 dt. 11 / 20

13 Inderlude: Where does A F come from? Some clues are in: D. Kastler, CPT internal report (1995). R. Coquereaux, Lett. Math. Phys. 42 (1997). L. Dabrowski, F. Nesti and P. Siniscalco, Int. J. Mod. Phys. A13 (1998). Consider SL(2, C) = Spin + (1, 3) = twofold covering of the Lorenz group. exact sequence: 1 Γ SL q (2) SL(2, C) 1 ( 2πi ) q = e 3 where Γ = finite quantum group, i.e. O(Γ) = finite dim. Hopf algebra. The dual is O(Γ) = S R with S = semisimple part = C M 2 (C) M 3 (C) A F R = radical ideal = irrep. π ker π 12 / 20

14 Stay tuned, it s a commercial break... Quantum group symmetries of the Standard Model: J. Bhowmick, FD & L. Dabrowski, Quantum Isometries of the Finite NcGeometry of the Standard Model, Commun. Math. Phys. 307 (2011), J. Bhowmick, FD, B. Das & L. Dabrowski, Quantum Gauge Symmetries in Noncommutative Geometry, J. Noncomm. Geom. 8 (2014), / 20

15 PART II Some Recent Advances 14 / 20

16 Back to the SM... The geometry is M F (continuous) (finite nc) with finite-dim. spectral triple (A F, H F, D F, γ F, J F ) given by: H F C 32 internal degrees of freedom of the elementary fermions. Total nr: = 32 (weak isospin) (lepton + quark (L,R chirality) (particle or in 3 colors) antiparticle) γ F = chirality operator A F = C H M 3 (C) J F = charge conjugation G(A F, J F ) U(1) SU(2) SU(3) D F encodes the free parameters of the theory. 15 / 20

17 Postdictions on D F Not every D F is allowed! Restrictions on the free parameters/on the interactions. Constraints of the 1st kind: 1 The parity (γ F D F = D F γ F ) and 1st order condition put constraints on D F : some matrix entries must be zero. For example, the 1st order cond. does not allow a vertex e? e Nothing forbids taking D F = 0 (all conditions are satisfied). Constraints of the 2nd kind: 2 The request that elements of H F are Dirac spinors on F ( property M, from Morita) imposes that some matrix entries cannot be zero. 16 / 20

18 Algebraic characterization of Dirac spinors From the example of Hodge-Dirac operator, we learn: Take M = R 4 : Real spectral triple oriented Riemannian manifold ψ Ω (M) has 16 components. A Dirac spinor ψ L 2 (M, S) has 4 components. Both carry a rep. of C 0 (M) and Cl 4,0 (R), but only the latter satisfies the following: If a bounded operator commutes with C 0 (M) and all γ µ s, then it is a function. This completely characterizes Dirac spinors. Theorem 1. A closed oriented Riem. manifold M admits a spin c structure iff a Morita equivalence C(M)-Cl(M) bimodule Σ, with Cl(M) the algebra of sections of the Clifford bundle. 2. Σ = C 0 sections of the spinor bundle S M (Dirac spinors in the conventional sense). Once we have S, we can canonically introduce the Dirac operator D of the spin c structure: 3. M is a spin manifold iff a real structure J on L 2 (M, S). 17 / 20

19 What is a noncommutative spin manifold? Definition (1-forms) If (A, H, D) is a spectral triple, we define Ω 1 D B(H) as: Ω 1 D := Span { a[d, b] : a, b A } The Dirac operator of M F is D = D M + γ 5 D F, and Ω 1 D = Ω1 D M + Ω 1 D F. Interpretation: Ω 1 D M = gauge fields Ω 1 D F = Higgs (or Higgs-like) Definition (Clifford algebra) We call Cl D (A) B(H) the algebra generated by A, Ω 1 D and possibly γ (in the even case). Let A := { JaJ 1 : a A }. The reality and 1st order cond. are equivalent to the statement Definition (Property M) A Cl D (A) := { b B(H) : [b, ξ] = 0 ξ Cl D (A) }. Elements of H are Dirac spinors iff ( ) is an equality: A = Cl D (A). ( ) 18 / 20

20 The 1st order condition Let (A F, H F, J F ) be as before. One can completely characterize D F s of 1st order: Theorem D F End C (H F ) satisfies the 1st order condition iff it is of the form D F = D 0 + D 1 with D 0 (A F ) and D 1 A F. D F is selfadjoint resp. odd iff both D 0 and D 1 are. J F D F = D F J F if and only if D 1 = J F D 0 J 1 F. } The only if part is not trivial. 16 free parameters or 25 with a non-standard γ F (for a toy model with 1 generation). In the Standard Model: 19 parameters, whose numerical values are established by experiment. One of these is the Higgs mass: m H 126 GeV. In Chamseddine-Connes original spectral triple, m H is not a free parameter. It was predicted m H 170 GeV, a value ruled out by Tevatron in / 20

21 Physical implications are under investigation (in collaboration with M. Kurkov and F. Lizzi). 20 / 20 On the Higgs mass Several modifications of the original model have been proposed. One can: 1. enlarge the Hilbert space thus introducing new fermions [Stephan, 2009]; 2. turn one element of D F into a field by hand, rather than getting it as a fluctuation of the metric [Chamseddine & Connes, 2012]; 3. break (relax) the 1st order condition, thus allowing more terms in the Dirac operator (or in the algebra) [Chamseddine, Connes & van Suijlekom, 2013]; 4. Grand Symmetry + twisted spectral triples [Devastato, Lizzi & Martinetti, 2014]. In 2,3,4: the Majorana mass term of the neutrino is replaced by a new scalar field Φ. Theorem In order to satisfy the property M, we must add two terms to Chamseddine-Connes D F. We get: a new scalar field close to the Φ above (but doesn t break the 1st order condition); a field coupling leptons with quarks.

22 A person who never made a mistake, never tried anything new. (A. Einstein) Thank you for your attention.