Quanta of Geometry and Unification

Transcription

1 Quanta of Geometry and Unification Memorial Meeting for Abdus Salam 90 th Birthday Ali Chamseddine American University of Beirut (AUB) and Institut des Hautes Etudes Scientifique (IHES) January 25-28, 2016 Work in Collaboration with Alain Connes and Slava Mukhanov A

2 We opened for thee a manifest victory A

3 Degree of the map At very small distances, such as the Planck scale, we assume structure of space-time changes. Is there is a fundamental unit volume in terms of which volume of space-time is quantized? Introduce a mapping Y from a four-manifold M 4 to S 4 and equate the volume form ω on M 4 with degree of the map Y ω = ɛ ABCDE Y A dy B dy C dy D dy E, Y A Y A = 1, A = 1, 2,, 5. The volume of M 4 is the degree of the map Y (winding number), an integer multiple of the unit four sphere ω = d 4 x g = deg (Y ) Z A

4 Heisenberg relation We rewrite this equation by Feynman slashing the coordinates Y A on S 4 with the Clifford algebra generated by the five gamma matrices Γ A where { Γ A, Γ B } = 2δ AB Y = Y A Γ A, Y = Y We also use the Dirac operator D on M 4 which we assume is a spin-manifold D = γ a e a µ ( ) µ + ω µ Quantization condition is then Y [D, Y ] 4 = γ where γ = γ 5 is the chirality operator and is tracing over Γ A. A

5 A picture of bubbles The pullback of the volume form is a four-form that does not vanish anywhere. Thus, the Jacobian of the map Y does not vanish anywhere, and is a covering of the sphere S 4. Since Sphere is simply connected, manifold has to be covering of sphere. Therefore, manifold must be disconnected, each piece being a sphere. This gives a bubble picture of space, every bubble is of Planckian volume A

6 The need for Noncommutative Geometry There is an ambiguity in defining the Feynman slash of the mappings Y { Γ A, Γ B } = 2κδ AB, κ = ±1 There are two possible maps Y = Y A Γ (+) A and Y = Y A Γ ( ) A. Quantization condition becomes Y [D, Y ] 4 = κγ Noncommutative geometry is characterized by a spectral triple (A, H, D) where A is an associative algebra with involution and unit element.

7 The need for Noncommutative Geometry There is an ambiguity in defining the Feynman slash of the mappings Y { Γ A, Γ B } = 2κδ AB, κ = ±1 There are two possible maps Y = Y A Γ (+) A and Y = Y A Γ ( ) A. Quantization condition becomes Y [D, Y ] 4 = κγ Noncommutative geometry is characterized by a spectral triple (A, H, D) where A is an associative algebra with involution and unit element. H a Hilbert space

8 The need for Noncommutative Geometry There is an ambiguity in defining the Feynman slash of the mappings Y { Γ A, Γ B } = 2κδ AB, κ = ±1 There are two possible maps Y = Y A Γ (+) A and Y = Y A Γ ( ) A. Quantization condition becomes Y [D, Y ] 4 = κγ Noncommutative geometry is characterized by a spectral triple (A, H, D) where A is an associative algebra with involution and unit element. H a Hilbert space D is a self-adjoint operator on H with bounded spectrum for ( D ) 1.

9 The need for Noncommutative Geometry There is an ambiguity in defining the Feynman slash of the mappings Y { Γ A, Γ B } = 2κδ AB, κ = ±1 There are two possible maps Y = Y A Γ (+) A and Y = Y A Γ ( ) A. Quantization condition becomes Y [D, Y ] 4 = κγ Noncommutative geometry is characterized by a spectral triple (A, H, D) where A is an associative algebra with involution and unit element. H a Hilbert space D is a self-adjoint operator on H with bounded spectrum for ( D ) 1. A chirality operator γ and reality operator J

10 Properties J 2 = ɛ, JD = ɛ DJ, Jγ = ɛ γj, ɛ, ɛ, ɛ { 1, 1} which defines KO dimension Z (mod 8).

11 Properties J 2 = ɛ, JD = ɛ DJ, Jγ = ɛ γj, ɛ, ɛ, ɛ { 1, 1} which defines KO dimension Z (mod 8). For Riemannian manifold A = C (M), H = L 2 (S), D = Dirac operator, γ =chirality, J =Charge Conjugation.

12 Properties J 2 = ɛ, JD = ɛ DJ, Jγ = ɛ γj, ɛ, ɛ, ɛ { 1, 1} which defines KO dimension Z (mod 8). For Riemannian manifold A = C (M), H = L 2 (S), D = Dirac operator, γ =chirality, J =Charge Conjugation. J sends the algebra into its commutant [ a, Jb J 1] = 0, a, b A

13 Properties J 2 = ɛ, JD = ɛ DJ, Jγ = ɛ γj, ɛ, ɛ, ɛ { 1, 1} which defines KO dimension Z (mod 8). For Riemannian manifold A = C (M), H = L 2 (S), D = Dirac operator, γ =chirality, J =Charge Conjugation. J sends the algebra into its commutant [ a, Jb J 1] = 0, a, b A J sends Y to Y where Y = JY J 1 and [Y, Y ] = 0.

14 Properties J 2 = ɛ, JD = ɛ DJ, Jγ = ɛ γj, ɛ, ɛ, ɛ { 1, 1} which defines KO dimension Z (mod 8). For Riemannian manifold A = C (M), H = L 2 (S), D = Dirac operator, γ =chirality, J =Charge Conjugation. J sends the algebra into its commutant [ a, Jb J 1] = 0, a, b A J sends Y to Y where Y = JY J 1 and [Y, Y ] = 0. From Y define idempotent element e = 2Y 1, with e 2 = e. Similarly e = 2Y 1 then

15 Properties J 2 = ɛ, JD = ɛ DJ, Jγ = ɛ γj, ɛ, ɛ, ɛ { 1, 1} which defines KO dimension Z (mod 8). For Riemannian manifold A = C (M), H = L 2 (S), D = Dirac operator, γ =chirality, J =Charge Conjugation. J sends the algebra into its commutant [ a, Jb J 1] = 0, a, b A J sends Y to Y where Y = JY J 1 and [Y, Y ] = 0. From Y define idempotent element e = 2Y 1, with e 2 = e. Similarly e = 2Y 1 then Let Z = 2ee 1, Z 2 = 1 then quantization condition becomes the two sided equation Z [D, Z ] 4 = γ

16 Clifford Algebras Y = Y A Γ A, Y = Y A Γ A where { Γ A, Γ B } = 2δ AB generate Clifford Algebra { C + = M 2 (H) M 2 (H) where H is algebra of quaternions and Γ A, Γ B } = 2δ AB generate the Clifford Algebra C = M 4 (C). This defines the algebra of a finite space A F = M 2 (H) M 4 (C) We can define a finite noncommutative space (A F, H F, D F ) with J F and γ F. The dimensions of H F is 32 dimensional ( c ), D F is a Hermitian matrix. We form a NC space A = C (M) A F, H = L 2 (S) H F, J = C J F, γ = γ 5 γ F and the non-trivial mixing D = D M 1 + γ 5 D F A

17 Mathematical Problem Let M be an oriented four-manifold then a solution of the two sided equation exists and equivalent to existence of two maps Y and Y : M 4 S 4 such that the sum of the two pullbacks Y (ω) + Y (ω) does not vanish anywhere and vol (M) = ( deg Y + deg Y ) vol ( S 4) The two sided relation in four dimensions gives the sum of two terms γ Z [D, Z ] 4 = Y [D, Y ] 4 + Y [ D, Y ] 4 Proof very diffi cult because in four dimensions because the kernel of the map Y is of co-dimension 2. There are no obstacles for spin-manifolds and if the volume of M is more than five units. A

18 Quanta of Geometry Four dimensional manifold emerges as the inverse maps of the product of two spheres of Planckian size. M 4 is folded many times in the product, unfolds to macroscopic size. The two different spheres, associated with the two Clifford algebras can be considered as quanta of geometry, which are the building blocks to generate an arbitrary oriented four dimensional spin manifold. The manifold M 4, the two spheres with their associated Clifford algebras define a noncommutative space which we will now show is the unified geometrical space that yields unification of all interactions. A

19 Emergence of fields and interactions The algebra of the finite space is M 2 (H) M 4 (C) and the fundamental element of Hilbert space Ψ H =L 2 (S) H F satisfies JΨ = Ψ, γψ = Ψ where J = C J F, J F (x, y) = (y, x ) with pair x, y elements of the two algebras. A

20 Emergence of fields and interactions The algebra of the finite space is M 2 (H) M 4 (C) and the fundamental element of Hilbert space Ψ H =L 2 (S) H F satisfies JΨ = Ψ, γψ = Ψ where J = C J F, J F (x, y) = (y, x ) with pair x, y elements of the two algebras. These simultaneous conditions are possible because the KO dimensions of the finite algebra is 6. A

21 Emergence of fields and interactions The algebra of the finite space is M 2 (H) M 4 (C) and the fundamental element of Hilbert space Ψ H =L 2 (S) H F satisfies JΨ = Ψ, γψ = Ψ where J = C J F, J F (x, y) = (y, x ) with pair x, y elements of the two algebras. These simultaneous conditions are possible because the KO dimensions of the finite algebra is 6. Ψ = ( ψ ψ c ), ψ ααi, α = 1, 4, α = 1, 4, I = 1, 4, where α is a space-time spinor index, α is M 2 (H) index, and I is M 4 (C) index. This gives 16 spinors. A

22 Spinors The γ F = diag(1 2, 1 2 ) grading acts on M 2 (H) and breaks it into H R H L. ψ αi (ψ ȧi ) R + (ψ ai ) L or (2 R, 1, 4) + (1, 2 L, 4). For I = 1, a = 1 is neutrino, a = 2 is electron. (lepton color) For I = i, i =, 2, 3, 4 (quark colors), a = 1 is up quarks, a = 2 down quarks ψ αi = ( ψ 11., ψ 21., ψ ( a1) + ψ. 1i, ψ 2i., ψ ) ai = (ν R, e R, l a ) + ( ur i, d R i, ) qi L ( ) ( ) ν u i where l a =, q e i = d i. L L The 16 space-time spinors arise uniquely in correct representations of Pati-Salam symmetry of SU(2) R SU (2) L SU (4) A

23 Gauge fields Under automorphisms of the algebra A A o we have where [u, û] = 0. Ψ UΨ, U = u û, û = Ju J 1 Define the Dirac operator D A = D + A such that D A Ψ UD A Ψ where A is given by [ ] A = aâ D, b b We get A = A (1) + JA (1) J 1 + A (2) A (1) = a [D, b], where JA (2) J 1 = A (2). A (2) = â [ ] A (1), b A

24 Which models? [ [ ]] If the condition a, D, b = 0 then we get a family of Pati-Salam models with fixed Higgs structures depending on the initial form of the [ finite [ space ]] Dirac operator all with SU(2) R SU (2) L SU (4) If a, D, b = 0 (order one condition) then we get Standard Model with H R H L M 4 (C) C H L M 3 (C) yielding the gauge symmetry U (1) SU (2) L SU (3) and one Higgs doublet. Compute the connection A starting with elements of the algebra and a given D F.

25 Which models? [ [ ]] If the condition a, D, b = 0 then we get a family of Pati-Salam models with fixed Higgs structures depending on the initial form of the [ finite [ space ]] Dirac operator all with SU(2) R SU (2) L SU (4) If a, D, b = 0 (order one condition) then we get Standard Model with H R H L M 4 (C) C H L M 3 (C) yielding the gauge symmetry U (1) SU (2) L SU (3) and one Higgs doublet. Compute the connection A starting with elements of the algebra and a given D F. e.g. DaI bj = γ (D µ µ δa b δi J i 2 g LWµL α (σα ) b a δj I i ) 2 gδb a VµI J D bj. ai = γ 5 Σ bj. ai (2 R, 2 L, ) Higgs

26 Which models? [ [ ]] If the condition a, D, b = 0 then we get a family of Pati-Salam models with fixed Higgs structures depending on the initial form of the [ finite [ space ]] Dirac operator all with SU(2) R SU (2) L SU (4) If a, D, b = 0 (order one condition) then we get Standard Model with H R H L M 4 (C) C H L M 3 (C) yielding the gauge symmetry U (1) SU (2) L SU (3) and one Higgs doublet. Compute the connection A starting with elements of the algebra and a given D F. e.g. DaI bj = γ (D µ µ δa b δi J i 2 g LWµL α (σα ) b a δj I i ) 2 gδb a VµI J D bj. ai = γ 5 Σ bj. ai (2 R, 2 L, ) Higgs Higgs fields are components of the connection along discrete directions.

27 Pati-Salam with Higgs fields There are three classes of Pati-Salam models depending on the form of D F. A

28 Pati-Salam with Higgs fields There are three classes of Pati-Salam models depending on the form of D F.. 1 ΣbJ ai = (2 R, 2 L, 1) + (1, 1, ), H aibj = (1, 1, 6) + (1, 3, 10), H ȧi ḃj = (1, 1, 6) + (3, 1, 10) A

29 Pati-Salam with Higgs fields There are three classes of Pati-Salam models depending on the form of D F.. 1 ΣbJ H ȧi ai = (2 R, 2 L, 1) + (1, 1, ), H aibj = (1, 1, 6) + (1, 3, 10), ḃj = (1, 1, 6) + (3, 1, 10) 2 H aibj = 0. A

30 Pati-Salam with Higgs fields There are three classes of Pati-Salam models depending on the form of D F.. 1 ΣbJ H ȧi ai = (2 R, 2 L, 1) + (1, 1, ), H aibj = (1, 1, 6) + (1, 3, 10), ḃj = (1, 1, 6) + (3, 1, 10) 2 H aibj = 0.. ḃj = ȧj ḃi, ΣḃJ ai φb a Σ J I where ȧj = (2, 1, 4) and 3 H ȧi Σ J I = (1, 1, ). A

31 Pati-Salam with Higgs fields There are three classes of Pati-Salam models depending on the form of D F.. 1 ΣbJ H ȧi ai = (2 R, 2 L, 1) + (1, 1, ), H aibj = (1, 1, 6) + (1, 3, 10), ḃj = (1, 1, 6) + (3, 1, 10) 2 H aibj = H ȧi ḃj = ȧj ḃi, ΣḃJ ai φb a Σ J I where ȧj = (2, 1, 4) and Σ J I = (1, 1, ). 4 All these models admit unification of coupling constants at energies of order Gev.

32 Standard Model as a special case [ ] If order one condition [D, a], b = 0 is satisfied (equivalent to linearity of connection, or no mixing between algebra and its opposite, except for a singlet) then Pati-Salam models give uniquely and unambiguously the Standard Model W ± µr = 0, g R WµR 3 = g 1 B µ, H =. ȧi σδ ḃj. 1ȧ δ 1ḃ δ 1 I δ1 J, A

33 Standard Model as a special case [ ] If order one condition [D, a], b = 0 is satisfied (equivalent to linearity of connection, or no mixing between algebra and its opposite, except for a singlet) then Pati-Salam models give uniquely and unambiguously the Standard Model W ± µr = 0, g R WµR 3 = g 1 B µ, H =. ȧi σδ ḃj. 1ȧ δ 1ḃ δ 1 I δ1 J, This is the SM plus a singlet σ whose vev gives Majorana masses to the right handed neutrinos. It stabilizes the λ Higgs coupling and make it consistent with a low mass. Predicts top quark mass 170 Gev.

34 Action Fermionic action always given by (JΨ, D A Ψ)

35 Action Fermionic action always given by (JΨ, D A Ψ) Bosonic action is given by the Spectral action principle which states that the spectrum of the Dirac operator specified by its eigenvalues are geometric invariants: Trf (D A /Λ) which amounts to counting spectrum below scale Λ.

36 Action Fermionic action always given by (JΨ, D A Ψ) Bosonic action is given by the Spectral action principle which states that the spectrum of the Dirac operator specified by its eigenvalues are geometric invariants: Trf (D A /Λ) which amounts to counting spectrum below scale Λ. Function f is a positive function of real variables, the form of f is not important.

37 Standard Model as a special case [ ] If order one condition [D, a], b = 0 is satisfied (equivalent to linearity of connection, or no mixing between algebra and its opposite, except for a singlet) then Pati-Salam models give uniquely and unambiguously the Standard Model W ± µr = 0, g R WµR 3 = g 1 B µ, H =. ȧi σδ ḃj. 1ȧ δ 1ḃ δ 1 I δ1 J, A

38 Standard Model as a special case [ ] If order one condition [D, a], b = 0 is satisfied (equivalent to linearity of connection, or no mixing between algebra and its opposite, except for a singlet) then Pati-Salam models give uniquely and unambiguously the Standard Model W ± µr = 0, g R WµR 3 = g 1 B µ, H =. ȧi σδ ḃj. 1ȧ δ 1ḃ δ 1 I δ1 J, This is the SM plus a singlet σ whose vev gives Majorana masses to the right handed neutrinos. It stabilizes the λ Higgs coupling and make it consistent with a low mass. Predicts top quark mass 170 Gev.

39 Action Fermionic action always given by (JΨ, D A Ψ)

40 Action Fermionic action always given by (JΨ, D A Ψ) Bosonic action is given by the Spectral action principle which states that the spectrum of the Dirac operator specified by its eigenvalues are geometric invariants: Trf (D A /Λ) which amounts to counting spectrum below scale Λ.

41 Action Fermionic action always given by (JΨ, D A Ψ) Bosonic action is given by the Spectral action principle which states that the spectrum of the Dirac operator specified by its eigenvalues are geometric invariants: Trf (D A /Λ) which amounts to counting spectrum below scale Λ. Function f is a positive function of real variables, the form of f is not important.

42 Action S b = 48 π 2 f 4Λ 4 d 4 x g 4 π 2 f 2Λ 2 d 4 x g (R + 12 ahh + 14 ) cσ π 2 f 0 d 4 x [ 1 g ( 18Cµνρσ R R ) ( ) 2 ( ) 2 3 g 1 2 Bµν 2 + g2 2 Wµν α + g 2 3 Vµν m arhh + b ( HH ) 2 + a µ H a 2 +2eHH σ d σ crσ c ( µ σ ) ] 2 A

43 Summary and Conclusions Quantization condition is a two sided relation Z [D, Z ] 4 = γ where Z = 1 2 (1 + Y ) (1 + iy ) 1 = Z 2 = 1. D Momenta, Z are coordinates, both Feynman Slashed with gamma matrices. The maps (Y, Y ) : M 4 S 4 S 4 satisfies vol (M) = ( deg (Y ) + deg ( Y )) vol ( S 4) If M is a oriented spin manifold then a solution of the two sided relation exists if vol (M) 5 unit volume. A noncommutative space is generated which is a product of a continuous four dimensional manifold times a discrete space of the algebra M 2 (H) M 4 (C) and give rise to a unified model of all interactions with the Pati-Salam symmetries. Standard model is a special case, and is unique, including a singlet responsible for Majorana masses for right-handed neutrinos. A