The Standard Model in Noncommutative Geometry: fermions as internal Dirac spinors

Transcription

1 1/21 The Standard Model in Noncommutative Geometry: fermions as internal Dirac spinors Ludwik Dabrowski SISSA, Trieste (I) (based on JNCG in print with F. D Andrea) Corfu, 22 September 2015

2 Goal Establish if in the noncommutative geometry approach to the Standard Model (νsm) the fundamental fermions can be regarded as quantum Dirac spinors of the internal space. 2/21

3 2/21 Goal Establish if in the noncommutative geometry approach to the Standard Model (νsm) the fundamental fermions can be regarded as quantum Dirac spinors of the internal space. Plan 1 Introduction 2 Formulate the concept of quantum spin space and of Dirac spinors in terms of Morita equivalence involving the underlying algebra A and certain quantum analogue of Clifford bundle algebra 3 See what happens in νsm.

4 2/21 Goal Establish if in the noncommutative geometry approach to the Standard Model (νsm) the fundamental fermions can be regarded as quantum Dirac spinors of the internal space. Plan 1 Introduction 2 Formulate the concept of quantum spin space and of Dirac spinors in terms of Morita equivalence involving the underlying algebra A and certain quantum analogue of Clifford bundle algebra 3 See what happens in νsm. Proviso: quantum = noncommutative (NC)

5 Intro 3/21 Framework: NC or spectral geometry à la Connes et. al.

6 Intro 3/21 Framework: NC or spectral geometry à la Connes et. al. The arena of νsm [Connes, Chammseddine,...] is ordinary (spin) manifold a finite quantum space, described by the algebra C (M) A F, where A F = C H M 3 (C).

7 Intro 3/21 Framework: NC or spectral geometry à la Connes et. al. The arena of νsm [Connes, Chammseddine,...] is ordinary (spin) manifold a finite quantum space, described by the algebra C (M) A F, where A F = C H M 3 (C). The matter fields are Dirac spinors Γ(S) H F, where H F = C 96 (its basis labels the fundamental fermions).

8 Intro 3/21 Framework: NC or spectral geometry à la Connes et. al. The arena of νsm [Connes, Chammseddine,...] is ordinary (spin) manifold a finite quantum space, described by the algebra C (M) A F, where The matter fields are A F = C H M 3 (C). Dirac spinors Γ(S) H F, where H F = C 96 (its basis labels the fundamental fermions). The gauge fields are encoded by the usual Dirac operator D/ on M plus certain Hermitian matrix D operating on H F.

9 Intro 2 4/21 This almost commutative geometry is mathematically a product of two spectral triples (S.T.): The first one is the canonical S.T. on M ( C (M), L 2 (Σ), D/ ), (1) a prototype of commutative S.T. Under some assumptions one can reconstruct the data M, g, Σ & D/.

10 Intro 2 4/21 This almost commutative geometry is mathematically a product of two spectral triples (S.T.): The first one is the canonical S.T. on M ( C (M), L 2 (Σ), D/ ), (1) a prototype of commutative S.T. Under some assumptions one can reconstruct the data M, g, Σ & D/. The scond one is finite-dimensional internal S.T. (A, H, D).

11 Intro 2 4/21 This almost commutative geometry is mathematically a product of two spectral triples (S.T.): The first one is the canonical S.T. on M ( C (M), L 2 (Σ), D/ ), (1) a prototype of commutative S.T. Under some assumptions one can reconstruct the data M, g, Σ & D/. The scond one is finite-dimensional internal S.T. (A, H, D). Does it also correspond to a (noncommutative) spin manifold? Are the elements of H spinors in some sense? In particular Dirac spinors?

12 Intro 2 4/21 This almost commutative geometry is mathematically a product of two spectral triples (S.T.): The first one is the canonical S.T. on M ( C (M), L 2 (Σ), D/ ), (1) a prototype of commutative S.T. Under some assumptions one can reconstruct the data M, g, Σ & D/. The scond one is finite-dimensional internal S.T. (A, H, D). Does it also correspond to a (noncommutative) spin manifold? Are the elements of H spinors in some sense? In particular Dirac spinors? Answer: Yes, if two extra fields are added (& a different grading).

13 Other modifications 5/21 In fact to get the correct experimental value of the Higgs mass, various modifications of the CC model have been proposed: - enlarge H thus introducing new fermions [Ste09] - turn one of the elements in D into a field by hand [CC12] rather than getting it as a fluctuation of the metric; - relax the 1st order condition [CCvS13] & allow new terms in D; - enlarge A [DLM13] and use the twisted spectral triple [DM13]

14 Other modifications 5/21 In fact to get the correct experimental value of the Higgs mass, various modifications of the CC model have been proposed: - enlarge H thus introducing new fermions [Ste09] - turn one of the elements in D into a field by hand [CC12] rather than getting it as a fluctuation of the metric; - relax the 1st order condition [CCvS13] & allow new terms in D; - enlarge A [DLM13] and use the twisted spectral triple [DM13] Actually, much before νsm a GUT was proposed, with the group Spin(10) and fundamental fermions in representation 16.

15 Other modifications 5/21 In fact to get the correct experimental value of the Higgs mass, various modifications of the CC model have been proposed: - enlarge H thus introducing new fermions [Ste09] - turn one of the elements in D into a field by hand [CC12] rather than getting it as a fluctuation of the metric; - relax the 1st order condition [CCvS13] & allow new terms in D; - enlarge A [DLM13] and use the twisted spectral triple [DM13] Actually, much before νsm a GUT was proposed, with the group Spin(10) and fundamental fermions in representation 16. NCG gives a possibility to employ also quantum groups [BDDD13], and actually just algebras instead of groups.

16 Dirac spinors: classical 6/21 An oriented Riemannian manifold M is spin c iff... SO(n) frame bundle lifts to Spin c (n) w 2 (M) is a Z 2 -reduction of a class in H 2 (M, Z) there is a Morita equivalence Cl(M) C(M) bimodule Σ (Cl(M) = Λ(M) C with Clifford product).

17 Dirac spinors: classical 6/21 An oriented Riemannian manifold M is spin c iff... SO(n) frame bundle lifts to Spin c (n) w 2 (M) is a Z 2 -reduction of a class in H 2 (M, Z) there is a Morita equivalence Cl(M) C(M) bimodule Σ (Cl(M) = Λ(M) C with Clifford product). Automatically Σ = Γ(S), where S is the complex vector bundle of Dirac spinors on M in conventional diff. geom., on which C(M) acts by pointwise multiplication, Cl(M) by Clifford multiplication, and one constructs canonical D/. Now, for f C (M), i[d/, f] = df and such operators generate Cl(M). If dimm is even the S.T. (1) is Z 2 -graded; a grading γ Cl(M).

18 Dirac spinors: classical 6/21 An oriented Riemannian manifold M is spin c iff... SO(n) frame bundle lifts to Spin c (n) w 2 (M) is a Z 2 -reduction of a class in H 2 (M, Z) there is a Morita equivalence Cl(M) C(M) bimodule Σ (Cl(M) = Λ(M) C with Clifford product). Automatically Σ = Γ(S), where S is the complex vector bundle of Dirac spinors on M in conventional diff. geom., on which C(M) acts by pointwise multiplication, Cl(M) by Clifford multiplication, and one constructs canonical D/. Now, for f C (M), i[d/, f] = df and such operators generate Cl(M). If dimm is even the S.T. (1) is Z 2 -graded; a grading γ Cl(M). There is an algebraic characterization for spin manifolds as well: a spin c manifold is spin iff a real structure (charge conjugation).

19 Another class of N.C. S.T. Def A finite-dimensional spectral triple (A, H, D) consists of: - (real or complex) subalgebra A of matrices acting on - fin. dim. Hilbert space H, and - Hermitian matrix D on H. (A, H, D) is even if H is Z 2 -graded, A is even and D is odd; we denote by γ the grading operator. (A, H, D) is real if an antilinear isometry J on H, s.t. J 2 = ɛ id H, JD = ɛ DJ, Jγ = ɛ γj for some ɛ, ɛ, ɛ {±1}, plus the 0th order condition: and the 1st order condition: [a, JbJ 1 ] = 0 a, b A, [[D, a], JbJ 1 ] = 0 a, b A. (2) 7/21

20 Dirac spinors: quantum 8/21 Quite as in [Lord,Rennie,Varilly12]: Def We call Clifford algebra the complex -algebra Cl(A) generated by A, [D, A] and by γ (if any).

21 Dirac spinors: quantum 8/21 Quite as in [Lord,Rennie,Varilly12]: Def We call Clifford algebra the complex -algebra Cl(A) generated by A, [D, A] and by γ (if any). Now, can t have a Cl(A)-A bimodule (for noncommutative A), but the 0th and 1st order H is a Cl(A)-A bimodule, where A := JAJ.

22 Dirac spinors: quantum 8/21 Quite as in [Lord,Rennie,Varilly12]: Def We call Clifford algebra the complex -algebra Cl(A) generated by A, [D, A] and by γ (if any). Now, can t have a Cl(A)-A bimodule (for noncommutative A), but the 0th and 1st order H is a Cl(A)-A bimodule, where A := JAJ. Def (FD A, LD) A real spectral triple (A, H, D, J) is spin (and H are quantum Dirac spinors) if H is a Morita equivalence Cl(A)-A bimodule (i.e. Cl(A) & A are maximal one w.r.t. the other).

23 Dirac spinors: quantum Quite as in [Lord,Rennie,Varilly12]: Def We call Clifford algebra the complex -algebra Cl(A) generated by A, [D, A] and by γ (if any). Now, can t have a Cl(A)-A bimodule (for noncommutative A), but the 0th and 1st order H is a Cl(A)-A bimodule, where A := JAJ. Def (FD A, LD) A real spectral triple (A, H, D, J) is spin (and H are quantum Dirac spinors) if H is a Morita equivalence Cl(A)-A bimodule (i.e. Cl(A) & A are maximal one w.r.t. the other). Examples: - Classical case - H = A, J(a) = a and D = 0. 8/21

24 9/21 H F We identify (for any of 3 generations) H F = C 32 M 8 4 (C) with basis labelled by particles and antiparticles arranged as ν R u 1 R u2 R u3 R e R d 1 R d2 R d3 R ν L u 1 L u2 L u3 L v 1 = e L d 1 L d 2 L d 3 L ν R ē R ν L ē L ū 1 R d R 1 ū1 L d L 1 ū 2 R d 2 R ū2 L d 2 L ū 3 R d 3 R ū3 L d 3 L (1,2,3=colors).

25 10/21 A F A general linear operators on H F is a finite sum L = i a i b i, with a i M 8 (C) acting on the left and b i M 4 (C) on the right. In particular A F = C H M 3 (C) (λ, q, m) is represented by a = λ 0 0 λ q 0 4 λ m 1 4. (3)

26 11/21 J F & A F The real conjugation J F is the operator [ ] [ ] v1 v J F = 2 v 2 v 1. (4) A F = J F A F J F End C (H F ) consists of elements of the form: [ ] λ [ ] λ 0 a 14 0 = m λ q

27 11/21 J F & A F The real conjugation J F is the operator [ ] [ ] v1 v J F = 2 v 2 v 1. (4) A F = J F A F J F End C (H F ) consists of elements of the form: [ ] λ [ ] λ 0 a 14 0 = m λ q Note that if A End C (H F ) is a real -subalgebra, its complex linear span A C has the same commutant in End C (H F ). The map a a = J F āj F (here ā = (a ) t ) gives two isomorphisms A F A F and (A F ) C (A F ) C.

28 12/21 A F Lemma (By direct computation) The commutant of A F in M 8 (C) is the algebra C F with elements q 11 q 12 α β1 2 q 21 q, (5) 22 δ1 3 where α, β, δ C, q = (q ij ) M 2 (C). The commutant of A F in End C (H) is A F = C F M 4 (C) M 4 (C) 3 M 8 (C), of dim C = 112.

29 13/21 (A F ) The map x J F xj F is an isomorphism between A F and (A F ). From this, Lemma The commutant (A F ) of A F has elements [ ] [ b b a e 11 + e c 22 + d with a M 8 (C), b, c, d M 4 (C). ] (e 33 + e 44 ) (6)

30 13/21 (A F ) The map x J F xj F is an isomorphism between A F and (A F ). From this, Lemma The commutant (A F ) of A F has elements [ ] [ b b a e 11 + e c 22 + d with a M 8 (C), b, c, d M 4 (C). Lemma A F (A F ) C 10 M 2 (C). ] (e 33 + e 44 ) (6) It follows that dim C (A F + (A F ) ) = 210 (= ). The (real) subspace of hermitian matrices has dim R = 210.

31 D F : the 1st order condition 14/21 Prop (Krajewski) D F End C (H F ) satisfies the 1st order condition (2) iff where D 0 (A F ) and D 1 A F. Furthermore [D F, J F ] = 0 iff D F = D 0 + D 1 D 1 = J F D 0 J F. If D F = DF, we have 112 free real parameters.

32 D F : reformulation of the 1st order 15/21 Cl(A F ) and the property spin constrains only D 0, and it is useful to reformulate the Props above as follows. Let D R = (Υ R e 51 + ῩRe 15 ) e 11, (7) with Υ R C. Note that D R A F (A F ) and J F D R = D R J F.

33 D F : reformulation of the 1st order 15/21 Cl(A F ) and the property spin constrains only D 0, and it is useful to reformulate the Props above as follows. Let D R = (Υ R e 51 + ῩRe 15 ) e 11, (7) with Υ R C. Note that D R A F (A F ) and J F D R = D R J F. Prop Most general D F = D F satisfying the 1st order condition is D F = D 0 + D 1 + D R (8) where D 0 = D0 (A F ) and D 1 = D1 A F have null coefficeint of e 15 e 11 and e 51 e 11, and D 1 = J F D 0 J F if D F and J F commute.

34 Grading 16/21 Consider now a grading operator γ F anticommuting with J F, which means KO-dimension 6 (or every except 0 and 4 if J F J F γ F, D F D F γ F, and/or forgetting γ F ).

35 Grading 16/21 Consider now a grading operator γ F anticommuting with J F, which means KO-dimension 6 (or every except 0 and 4 if J F J F γ F, D F D F γ F, and/or forgetting γ F ). Lemma Any γ F -odd Dirac operator satisfying the 1st order condition can be written in the form D F = D 0 + D 1 + ɛd R as in (8), with both D 0 and D 1 γ F -odd operators and ɛ = 0 or 1 depending on the parity of D R. If moreover γ F either commutes or anticommutes with J F, then D 1 = J F D 0 J F. Some natural choices of γ F, and the corresponding form of odd D 0 (D 1 is spurious) are:

36 The standard grading γ F = +1 on p R, p L, and 1 on p L, p R (9) (KO-dim =6). Any γ F -odd D 0 has the form: e 11 + (1 e 11 ), 17/21

37 A non-standard grading 18/21 Let γ F = γ F (L B). (10) (opposite parity of chiral leptons w.r.t. quarks, still KO-dim =6). Any γ F -odd D 0 has the form: e 11 + e 11 where can differ for e 22 and (e 33 + e 44 ).

38 Special case: (Modified) Chamseddine-Connes s D 0 Ῡ ν Ῡ e Ω Υ ν Υ e Ω e 11 + Ῡ u Ῡd Υ u Υ d e 11, where Υ s, Ω C, and R (term mixing leptons and quarks). 19/21

39 Special case: 19/21 (Modified) Chamseddine-Connes s D 0 Υ ν Υ e Ω Ῡ ν Ῡ e Ω e 11 + Υ u Υ d Ῡ u Ῡd e 11, where Υ s, Ω C, and R (term mixing leptons and quarks). The Chamseddine-Connes is a further special case with Ω = = 0 (which is odd w.r.t. the both gradings γ F and γ F ).

40 The spin property 20/21 By straightforward (though meticulous and tedious) considerations: Prop A spectral triple with γ F -odd operator D F is not spin. For a γ F -odd Modified Chamseddine-Connes s operator D F with all coefficients different from zero, if at least one of 1. Υ ν ±Υ u, 2. Υ e ±Υ d, holds, then: i) the odd spectral triple (A F, H F, D F, J F ) is not spin; ii) the even spectral triple (A F, H F, D F, γ F, J F ) is spin;

41 Conclusions 21/21 Corollary i) the spectral triple with D F of Connes-Chamseddine ( = 0 or Ω = 0), is not spin. ii) our D F is a minimal modifiation of Connes-Chamseddine D F, for which the even spectral triple is spin.

42 Conclusions Corollary i) the spectral triple with D F of Connes-Chamseddine ( = 0 or Ω = 0), is not spin. ii) our D F is a minimal modifiation of Connes-Chamseddine D F, for which the even spectral triple is spin. Final remarks i) γ F does not belong to the algebra generated by A F and [D, A F ], so the modified even spectral triple is not orientable. Thus it is pin rather than spin, and the elements of H are quantum Dirac pinors. ii) It is irreducible in the sense that {0} and H are the only subspaces stable under A, D F, J and γ F. iii) So far Math. Phys. implications, e.g. small Ω, ; lepto-quark interactions (cf. [PSS97]); how the Higgs mass is modified...: under scrutiny. 21/21