The Standard Model in Noncommutative Geometry: fermions as internal forms

Transcription

1 1/35 The Standard Model in Noncommutative Geometry: fermions as internal forms Ludwik Dabrowski (based on JNCG 10 (2016) with F. D Andrea and arxiv: with F. D Andrea and A. Sitarz) Warsaw, 2 Nov. 2017

2 2/35 Goal Unveil the geometric nature of the multiplet of fundamental fermions in the Standard Model of fundamental particles Plan 1 few words on the SM and its nocommutative geometric formulation νsm 2 concept of quantum Dirac spinors and quantum de Rham forms as Morita equivalence bimodules of the algebra of sections of the quantum analogue of Clifford bundle 3 application to νsm. Proviso: quantum = noncommutative (NC)

3 (Unreasonably) successful Standard Model 3/35 & interactions u e + W + u ν g d governed by:

4 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 4/35

5 Conceptually/Geometrically: Φ U(1) SU(2) SU(3) gauge fields (bosons) minimally coupled to matter fields (fermions) & Higgs field (boson) + 2nd quantization with gauge fixing, spontaneous symmetry breaking, regularization & perturbative renormalization However unexplained: - (though constrained) contents of particles (especially 3 families) - several parameters, - the fourth known interaction not included: gravitation with its own fundamental symmetry: general relativity (diffeomorphisms) There have been various attempts to settle some of the above: GUT based on a simple group SU(5) or SO(10), modern variants of Kaluza-Klein model with compactified internal dimensions, and other more recent and fashionable... 5/35

6 νsm Another one, νsm, has been formulated in the framework of noncommutative geometry by A. Connes et.al. Rather than groups in the usual commutative geometry of SM: M U(1) SU(2) SU(3) connection ( multiplet of vectors) on (a multiplet of) spinors; & a doublet of scalars νsm is primarily based on algebras. It adds to the 75 years-old Gelfand-Naimark (anti)equivalence: topological spaces commutative C algebras and to the Serre-Swan equivalence: vector bundles modules other data to encode smoothness, calculus and metric on M. 6/35

7 Intro 7/35 The first datum is a Hilbert space H that carries a unitary representation of a (possibly noncommutative) -algebra A (thus also of its norm closure C -algebra). The second one is an analogue of Dirac operator D = D on H. Together with the algebra A they satisfy some analytic conditions, [D, A] B(H), (D z) 1 K(H), z / spec(d) so that they form a spectral triple (S.T.) (A, H, D). A S.T. is called even if a Z 2 -grading χ of H, χ 2 = χ = χ, s.t. [χ, A] = 0, {χ, D} = 0.

8 Intro 2 A S.T. is called real if a real structure, i.e. antiunitary J on H, s.t. denoting B the commutant of B B(H), JAJ 1 A, (order 0 condition). (1) In addition, we call and JAJ 1 [D/, A], (order 1 condition) (2) J[D/, A]J 1 [D/, A], (order 2 condition). (3) Names: A-bimodule spanned by [D, A] - 1-forms A-algebra generated by [D, A] - forms (with derivative [D, ]) or, quite as in [Lord et.al.] - Clifford algebra denoted Cl D (A). This permits right actions b := Jb J 1 on H, so that the order 0, 1, 2 conditions mean that H is a A A, A Cl D (A) and Cl D (A) Cl D (A) bimodule, respectively. 8/35

9 Intro 3 9/35 Further A. Connes formulated few other important properties (which permit to reconstruct the geometrical data); one of them is the request that J 2 = ɛ id H, JD = ɛ DJ, Jχ = ɛ χj (in the even case) (4) for some ɛ, ɛ, ɛ {±1}, that specify the KO-dimension mod. 8.

10 Canonical S.T. The prototype example is the canonical S.T. on a spin manifold M (C (M), L 2 (S), D/ ), where C (M) is the algebra of smooth complex functions on M, S is the rank C = 2 [n/2] bundle of Dirac spinors on M, whose sections carry a faithful irrep γ of the algebra of sections of the (simple part of) Clifford bundle Cl(M) γ : Γ(Cl(M)) End C (M)Γ(S) Γ(S) C (M) Γ(S) (5) and D/ is the usual Dirac operator on M: n D/ = γ = γ j j (locally). (6) (!) Note that (5) means that (after norm completion) j Γ(S) is a Morita equivalence Γ(Cl(M)) C (M)bimodule (7) and this exactly characterizes Spin c manifolds M [Plymen]. 10/35

11 Canonical S.T. 2 11/35 Note also that since [D/, a] = γ(da) indeed we have Cl D/ (C (M)) Γ(Cl(M)). Next if dimm is even, a chiral Z 2 -grading χ S of L 2 (S). Furthermore a real structure J S (given by charge conjugation ), that satisfies order 0 and 1 condition, and obviously not the order 2 condition since it implements the Morita equivalence (12). Such J S, together with (12) provides precisely the algebraic characterization of spin manifolds. We also mention that the signs ɛ, ɛ, ɛ {±1} associated to D/, χ S, J S correspond to the KO-dimension equal n mod. 8. Finally, the canonical S.T. fully encodes the geometric data on M, that can be indeed reconstructed [Connes].

12 12/35 de Rham-Hodge S.T. But it is not the only natural S.T.; on an oriented Riemannian manifold there is also (C (M), L 2 (Ω(M)), d + d ), where Ω(M) is the space of de Rham differential forms on M with the hermitian form induced by the metric g on M, d is the exterior derivative and d its adjoint. The operator d + d is Dirac-type: where the representation d + d = λ, (8) λ : Γ(Cl(M)) End C (M)Ω(M), λ(v) = v v, v T M (9) is equivalent to the left regular self-representation of Γ(Cl(M)). Clearly [d + d, a] = λ(da) so we again have Cl d+d (C (M)) Γ(Cl(M)).

13 de Rham-Hodge S.T. 2 13/35 There is also an anti-representation ρ : Γ(Cl(M)) End C (M)Ω(M), ρ(v) = (v +v )χ Ω, v T M, (10) where χ Ω = ±1 (11) on even forms Ω(M) even, respectively odd forms Ω(M) odd, that is equivalent to the right regular self anti-representation of Γ(Cl(M)). Furthermore, since λ v and ρ v commute, Ω(M) is a Γ(Cl(M))-Γ(Cl(M)) bimodule, which is equivalent to Γ(Cl(M)). Thus (after norm completion) Ω(M) is a Morita equivalence Γ(Cl(M)) Γ(Cl(M))bimodule (12) which characterizes Ω(M) up to with a complex line bundle.

14 de Rham-Hodge S.T. 3 14/35 Besides the grading χ Ω by parity which on any M, if dimm = n = 2m is even another grading given by the normalized Hodge operator χ Ω := i k(n k)+m : Ω k (M) Ω n k (M) (13) On any M also a real structure on Ω(M) J Ω := c.c, which satisfies the order 0 and 1 conditions but not order 2, and so can not implement the Γ(Cl(M))-Γ(Cl(M)) self-morita equivalence.

15 de Rham-Hodge S.T. 4 15/35 For that we need another J Ω on Ω(M) that interchanges the actions λ and ρ. It turns out that there is one: J Ω(e j1 e jk ) = e jk e j1, 0 k n, (14) which corresponds to the main anti-involution on Γ(Cl(M)) and can be simply written on Ω k (M) as J Ω = ( ) k(k 1)/2 c.c. (15) It satisfies all the order 0, 1 and 2 conditions and does implement the Γ(Cl(M)-Γ(Cl(M)) self-morita equivalence (!).

16 de Rham-Hodge S.T. 5 16/35 We mention that for d + d, and respectively: χ Ω, J Ω one has ɛ = 1, ɛ = 1, ɛ = 1 and so KO-dim=0; χ Ω, J Ω one has ɛ = 1, ɛ = 1, ɛ = ( 1) m and so KO-dim=0 if n=0 mod 4, and 6 if n=2 mod 4 [Rubin, M. Thesis]; χ Ω, J Ω one has ɛ = 1, ɛ = 1, ɛ = 1 and so KO-dim=0; χ Ω, J Ω one has ɛ = 1, ɛ = 1, but on k-forms J Ωχ Ω = ( ) k χ ΩJ Ω (16) so ɛ is not just a sign but a grading (given by χ Ω ), which requires a generalization of the notion of KO-dimension. Actually it is not known if, and with which additional conditions, the de Rham-Hodge S.T. equipped with any choice of χ s and J s as above may faithfully encode the geometric data on M, that can be then reconstructed.

17 17/35 νsm: A F The underlying arena of νsm [Connes, Chammseddine,...] is ordinary (spin) manifold M a finite quantum space F, described by the algebra C (M) A F, where A F = C H M 3 (C). The Hilbert space is L 2 (S) H F, where H F = C 96 =: H f C 3, with C 3 corresponding to 3 generations, and

18 18/35 νsm: H f H f = C 32 M 8 4 (C) with basis labelled by particles and antiparticles, we arrange 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 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).

19 19/35 νsm: π F The representation of A F, diagonal in generations, on H f is: λ 0 0 λ 0 0 π F (λ, q, m) = q 0 4 λ m Note that π F (A F ) is a real -algebra of operators, and to get its complexification just replace λ by an independent λ C, and take q M 2 (C). (17)

20 20/35 νsm: γ F & J F The grading (the chirality operator) is γ M γ F, where γ F (diagonal in generations) on H f reads: 2 γ F = [ ] [ ]. (18) The real conjugation is J M J F, where J F on H f is [ ] v1 J F = v 2 [ ] v 2 v 1 (19) that satisfies the order 0 and 1 conditions (as in the classical case).

21 21/35 νsm: D F Finally, the Dirac operator is D = D/ M id + γ M D F, where Chamseddine-Connes : Υ ν D F = Υ e Υ R e 55 Ῡ ν Ῡ e Ῡ R 0 0 Ῡ ν Ῡ e Υ ν Υ e 0 0 Υ u e 11 + Υ d Ῡ u Ῡd +(e 66 +e 77 +e 88 ) e Ῡ u Ῡ d Υ u Υ d 0 0 where Υ s are in Mat(3, C), or just in C for one generation.,

22 νsm With all that: G := {U = ujuj 1 u A, det U = 1} U(1) SU(2) SU(3) (S.M. gauge group) all the fundamental fermions in H have the correct S.M. charges w.r.t. G (broken to U(1) em SU(3)) the 1-forms a[d, b], a, b A yield the S.M. gauge fields A µ, W ±, Z, G µ (from the part D M of D), plus the Higgs complex scalar (weak doublet) Higgs field (from the part D F of D). MERITS: gauge & Higgs field as a connection, explains why only the fundamental reps of G, a simple spectral action Trf(D/Λ) reproduces the bosonic part of L SM as the lowest terms of asymptotic expansion in Λ, & and < φ, Dφ > the (Wick-rotated) fermionic part couples to gravity on M Connes&Chamseddine claim to predict a new relation among the parameters of S.M. 22/35

23 Geometry of νsm 23/35 The above almost commutative geometry is described by a S.T. (C (M), L 2 (S), D/ ) (A F, H f, D F ), that is mathematically a product of the external canonical S.T. on spin manifold M with the internal finite S.T. What is its geometric interpretation of (A F, H f, D F )? Does it also correspond to a (noncommutative) spin manifold? Are the elements of H f spinors in some sense? In particular Dirac spinors? Or does it correspond rather to de-rham forms? Or else?

24 Dirac spinors: quantum 24/35 To answer this question, basing on a deeper understanding of the classical case, to accomplish the noncommutative case we define Def (c.f. FD A, LD) An even spectral triple (A, H, D, χ) is called spin c if H is a Morita equivalence Cl D (A)-A bimodule (i.e. after norm-completion the algebras Cl D (A) & A are maximal one w.r.t. the other), and it is called spin if the right action of a A is Ja J 1 (implemented by a real structure J satisfying the 1st O.C.). Furthermore the elements of H are called quantum Dirac spinors (sometimes named charged or neutral, respectively).

25 νsm: 1st result 25/35 Is the internal S.T. of νs.m. spin? (like the external one = the canonical S.T. on M) Bulding on and extending the classifications of [Krajewski] and [Paschke,Sitarz] the answer [FD A, LD] is: NO In fact, after a tedious and sistematic search we constructed X = e 55 (1 e 11 ), s.t. X (A) but X / JAJ. (Can be evaded with a different grading and two extra 0 matrix elements in D F - the status of which is however under study since though desirable for the correct renormalized Higgs mass, they would have unobserved couplings to fermions). But then, without such additions, may be the internal S.T. of νsm is rather an analogue of the other natural classical spectral triple, namely de-rham forms?

26 de Rham forms: quantum 26/35 To answer this question, basing on a deeper understanding of the classical case of Ω(M) with χ Ω and J Ω, we define Def (c.f. FD A, LD, AS) An even spectral triple (A, H, D, χ) is called complex Hodge if H is a Morita selfequivalence Cl D (A)-Cl D (A) bimodule, and Hodge if the right Cl D (A)-action is implemented by J satisfying the 2nd O.C.). Furthermore we then say that H consists of quantum complex or real de-rham forms, respectively. Theorem (LD, FD A, AS) For the internal spectral triple of the νs.m. with one generation the Hodge property holds whenever Υ x 0, x {ν, e, u, d} and Υ ν Υ u or Υ e Υ d.

27 27/35 About the proof: A F Lemma (1 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, (20) 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.

28 28/35 About the proof: J F A F J F J F A F J F End C (H F ) consists of elements of the form: [ ] λ [ ] λ m λ q Note that A and A C have the same commutant in End C (H F ). The map a J F āj F gives two isomorphisms A F J F A F J F and (A F ) C (J F A F J F ) C, and also the map x J F xj F is an isomorphism between A F and (J F A F J F ).

29 About the proof: (J F A F J F ) 29/35 From this, Lemma The commutant (J F A F J F ) of J F A F J F has elements [ b a e 11 + c ] [ b e 22 + with a M 8 (C), b, c, d M 4 (C). d ] (e 33 + e 44 ) (21) Lemma A F (J F A F J F ) C 10 M 2 (C). It follows that dim C (A F + (J F A F J F ) ) = 210 (= ). The (real) subspace of hermitian matrices has dim R = 210.

30 About the proof: Cl D (A), Lemma A 30/35 Any unital complex -subalgebra of End C (H), dimh <, is a finite direct sum of matrix algebras: B s i=1 M m i (C). Denote P i the unit of M mi (C), then P 1,..., P s are orthogonal projections and H decomposes as H s i=1 H i, with H i = P i H C m i C k i, (22) where k i is multiplicity of the (unique) irrep C m i of M mi (C) in H i, and M mi (C) acts on the 1st factor of C m i C k i by matrix product. Lemma (A) The commutant of B in End C (H) is B s i=1 M k i (C) and the action of B on H i C m i C k i is given by matrix multiplication by M ki (C) of the second factor in the tensor product.

31 About the proof: Cl D (A), Lemma B Lemma (B) Let (A, H, D, J) be a finite-dimensional real spectral triple and B End C (H) a unital complex -algebra satisfying: Cl D (A) B and B = JBJ. The following are equivalent: (a) Cl D (A) = JCl D (A)J (the Hodge property) (b) Cl D (A) JBJ (c) Cl D (A) = B. Proof (a) (b) the hypothesis Cl D (A) B implies J F Cl D (A)J F JBJ; and thus if from (a), i.e. Cl D (A) = J F Cl D (A)J F, follows (b) (b) (c) Cl D (A) J F BJ F = B implies B Cl D (A) and, since holds by hypothesis, Cl D (A) = B. (c) (a) If (c), then B = J F BJ F translates to 31/35

32 Now, in our case we take About the proof: Cl D (A) B := C M 3 (C) M 4 (C) M 4 (C) (23) with (λ, m, a, b) B represented on H F as [ ] λ 0 e 0 m a e 11 e 11 + b e 11 (1 e 11 ), (24) check that Cl DF (A F ) B, i.e. 1st assumption of Lemma B check that B and JBJ commute, and so JBJ B. (24) is equivalent to the rep of B on (the 1st factors of): (C C 4 ) (C 3 C 4 ) (C 4 C) (C 4 C 3 ) given by matrix multiplication on the first factor. by Lemma A B M 4 (C) M 4 (C) C M 3 (C) B and we have JBJ = B, i.e. 2nd assumption of Lemma B find that Cl DF (A F ) JBJ, so get (b) and thus (a) of Lemma B which ends the proof. 32/35

33 Conclusions The Connes-Chamseddine νsm interprets the geometry of the SM as gravity on the product of a (Riemannian) manifold M with a finite noncommutative internal space F. The multiplet of fundamental fermions (each a Dirac spinor on M) are fields on F that constitute H F. We show that the geometric nature of the latter is not a noncommutative analogue of Dirac spinors on F (unless >2 new parameters are introduced in the matrix D F, so fields on M with physical status under scrutiny), but rather of de-rham forms on F (for one generation). What happens for 3 generations (96 96 matrices)? Can be seen (not easily) that also then NO spin property, and that Cl D (A) JCl D (A)J (order 2 condition). Whether the Hodge property is satisfied is under investigation. 33/35

34 THANK YOU! 34/35

35 Refs 35/35 [CC12] A.H. Chamseddine and A. Connes, Resilience of the Spectral Standard Model, JHEP 1209 (2012) 104. [C13] A. Connes, On the spectral characterization of manifolds, J. Noncommut. Geom. 7 (2013) [DD14] F. D Andrea and L. Dabrowski, The Standard Model in Noncommutative Geometry and Morita equivalence, J. Noncommut. Geom. 10 (2016) [DDS17] L. Dabrowski, F. D Andrea and A Sitarz, The Standard Model in noncommutative geometry: fundamental fermions as internal forms, arxiv: [Kra97] T. Krajewski, Classification of Finite Spectral Triples, J. Geom. Phys. 28 (1998) [LRV12] S. Lord, A. Rennie and J.C. Várilly, Riemannian manifolds in noncommutative geometry, J. Geom. Phys. 62 (2012) [PSS97] M. Paschke, F. Scheck and A. Sitarz, Can (noncommutative) geometry accommodate leptoquarks?, Phys. Rev. D59 (1999)