Chiral asymmetry in the weak interaction Some applications of - PowerPoint PPT Presentation

Chiral asymmetry in the weak interaction Some applications of Clifford Algebras to the Standard Model Cristi Stoica September, 2019 Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering – Horia Hulubei, Bucharest, Romania. cristi.stoica@theory.nipne.ro, holotronix@gmail.com 10th MATHEMATICAL PHYSICS MEETING: School and Conference on Modern Mathematical Physics 9 - 14 September 2019, Belgrade, Serbia

Intro Discrete properties of leptons and quarks: They are due to certain symmetry groups and representations. Why these groups, and why these representations? 1

Intro This can be partially explained by grand unified theories (GUT) like SU( 5 ) (Georgi and Glashow 1974) and SO( 10 ) (actually Spin( 10 ) ) (Georgi 1975; Fritzsch and Minkowski 1975). All such models based on a simple gauge group predict still un- detected additional interactions and proton decay. But they contain important insights for future developments. 2

Intro But what if there is a simple mathematical structure which • has the symmetries of the Standard Model, • provides the right representations automatically, • and predicts no additional particles and forces? 3

Result A three-dimensional Hermitian space χ determines a Clifford al- � � χ † ⊕ χ gebra C ℓ , which is naturally split into left ideals. 4

Result In a basis adapted to the ideal decomposition, each column contains two 4-spinors associated to different flavors. 5

Result The Lie group SU( 3 ) c permutes the columns according to the representations 1 c , 3 c , 1 c , and 3 c . 6

Result Each ideal is indexed with an electric charge which is multiple of 1 3 partially representing the charge of the upper particle, 7

Result and its color is determined by the ideal to which belongs, having associated a particular representation of SU( 3 ) c . 8

Result The actions of the Dirac algebra and the Lorentz group are re- ducible, permute the rows of each ideal, and split it naturally into two 4-spinors, whose left chiral components are permuted by the representations 1 w and 2 w of SU( 2 ) L . 9

Exterior algebra patterns in the Standard Model

Exterior algebra patterns – Weak symmetry SU( 2 ) L The standard representation of SU( n ) is C n ∼ = C n . The fundamental and the trivial representations of SU( n ) are � k C n . The representations of the weak interaction group SU( 2 ) L Representation Particles Hypercharge � 0 C 2 ( ν e ) R (?) 0 � 1 C 2 ( ν e , e − ) L − 1 � 2 C 2 ( e − ) R − 2 ———————— (?) Does the right handed neutrino exist? 10

Exterior algebra patterns – Color symmetry SU( 3 ) c Combined internal charge and color spaces: Representation Particles Electric charge � 0 C 3 ν e 0 � 1 C 3 − 1 d r , d y , d b 3 � 2 C 3 u r , u y , u b − 2 3 � 3 C 3 e − − 1 11

Representations of complex Clifford algebras, dim 2 r i The classification of complex Clifford algebras says C ℓ 2 r ∼ = C ( 2 r ) . To see this, consider the orthonormal basis ( e 1 , . . . , e r , e r + 1 , . . . , e 2 r ) , where e 2 j = 1. Then, we can build the basis � a j := 1 2 ( e j + ie r + j ) (1) a † j := 1 2 ( e j − ie r + j ) Then, � { a j , a k } = { a † j , a † k } = 0 (2) { a j , a † k } = δ jk . This is a Witt basis . 12

Representations of complex Clifford algebras, dim 2 r ii The Witt decomposition of the vector space V is V := W ⊕ W † , where W is spanned by ( a j ) , and W is spanned by ( a † j ) . Let a ∈ � r W , a := a 1 ∧ . . . ∧ a r = a 1 . . . a r . Then, a is nilpotent ( a 2 = 0), so � • W † a is a left ideal. On the space � • W † a , a † and a act like creation and annihila- tion operators. Let φ ∈ � • W † . Then, � a † j φ a = a † j ∧ φ a (3) a j φ a = i a j φ a . Since dim W † = 2 r , this is our irreducible representation of C ℓ 2 r . 13

Representations of complex Clifford algebras, dim 2 r iii Algebraic spinors of C ℓ 2 r Since on W the inner product vanishes, the Clifford product on the subalgebra generated by W coincides with the exterior prod- uct. The same holds for W † . The algebra C ℓ 2 r is spanned by elements of the form a † j 1 . . . a † j p aa † a k 1 . . . a k q , (4) p , q ∈ { 0 , . . . , r } , 1 ≤ j 1 < . . . < j p ≤ r , 1 ≤ k 1 < . . . < k q ≤ r . j p span � • W † , C ℓ 2 r is the Since the elements of the form a † j 1 . . . a † direct sum of the minimal left ideals of the form � • W † aa † a k 1 . . . a k q . (5) On these ideals, C ℓ 2 r is represented just like on � • W † a (3). 14

Representations of complex Clifford algebras, dim 2 r iv From exterior algebra to Clifford algebra C ℓ 2 r Conversely, we can start with C r endowed with a Hermitian inner product h . Then, there is an isomorhism C r ∼ = C ∗ r , due to h . Let u , v ∈ � • C r . On � • ( C r ⊕ C r ) ∼ = � • C r ⊗ C � • C r , define an associative product by uv := u ∧ v , u † v † := u † ∧ v † , uv † = u ∧ v † + 2 v † ( u ) , u † v = u † ∧ v + 1 1 2 u † ( v ) . Then, if ( a j ) is an orthonormal basis of C r , � { a j , a k } = { a † j , a † k } = 0 (6) { a j , a † k } = δ jk . One obtains a Clifford algebra C ℓ ( C r ⊕ C r ) ∼ = C ℓ 2 r . Note that we don’t need h if we start from � • ( C ∗ r ⊕ C r ) . 15

Exterior algebra patterns – Chiral spinors Let γ µ in the Dirac algebra C ℓ 4 = C ℓ 1 , 3 ⊗ C . Define � e 1 = 1 2 ( γ 0 + γ 3 ) , e 2 = 1 2 ( − i γ 2 + γ 1 ) (7) f 1 = 1 2 ( γ 0 − γ 3 ) , f 2 = 1 2 ( − i γ 2 − γ 1 ) . Then, f 1 f 2 is nilpotent, and defines a minimal left ideal C ℓ 4 f 1 f 2 . In the basis ( e 1 f 1 f 2 , e 2 f 1 f 2 , 1 f 1 f 2 , e 1 e 2 f 1 f 2 ) of C ℓ 4 f 1 f 2 , the matrix form of γ µ is the Weyl representation. Let Σ be spanned by ( e 1 , e 2 ) . We see that the spinors from � − Σ f 1 f 2 are Weyl spinors of left chirality, and those from � + Σ f 1 f 2 are Weyl spinors of right chirality. 16

Exterior algebra patterns – Summary 1. Exterior algebras are the representations of the gauge groups in the Standard Model. The leptons and quarks in a generation correspond to these representations. 2. The classification of leptons and quarks and their antiparticles given by � • C 3 and � • C 3 suggests C ℓ 3 + 3 . 3. The weak force acts on the odd part � 1 C 2 = � − C 2 of � C 2 . This suggests C ℓ 2 + 2 . But since the same particles belong to representations of SU( 3 ) c , this suggests that C ℓ 2 + 2 is a subalgebra of the C ℓ 3 + 3 algebra above. The odd Dirac spinors are the left handed spinors. The weak force acts on left handed spinors. These suggest that the relation between chirality and the weak force is due to the inclusion of both the Dirac algebra and C ℓ 2 + 2 in the same algebra, which is C ℓ 3 + 3 . 17

SMA – Definition and main properties

SMA – Definition and main properties i Let us define χ := χ em ⊗ χ c . The space χ has complex dimension three and has the Hermitian inner product h = h em ⊗ h c . Orthonormal basis: � ( q 1 , q 2 , q 3 ) (8) † † † ( q 1 , q 2 , q 3 ) . The combined internal charge and color spaces for fermions and leptons are represented now in this table: 18

SMA – Definition and main properties ii On the space χ † ⊕ χ we define the inner product � � 3 + u 4 � := 1 � u † 1 + u 2 , u † u † 1 ( u 4 ) + u † 3 ( u 2 ) ∈ C , (9) 2 3 ∈ χ † and u 2 , u 4 ∈ χ (also see Gualtieri 2004). where u † 1 , u † We call Standard Model Algebra (SMA) the Clifford algebra de- fined by the inner product (9), A SM := C ℓ ( χ † ⊕ χ ) ∼ = C ℓ 6 , (10) together with the Witt decomposition χ † ⊕ χ of the base complex 6-dimensional space, and with the Hermitian inner product on χ and χ † . 19

SMA – Definition and main properties iii The elements of the bases defined in equation (8) satisfy the an- ticommutation relations { q j , q k } = 0 , † † (11) { q j , q k } = 0 , † { q j , q k } = δ jk for j , k ∈ { 1 , 2 , 3 } . 20

SMA – Definition and main properties iv We define an orthonormal basis of the vector space χ † ⊕ χ , ( e 1 , e 2 , e 3 , ˜ e 1 , ˜ e 2 , ˜ e 3 ) , by � † e j = q j + q j (12) � � † ˜ e j = i q j − q j , where j ∈ { 1 , 2 , 3 } . j = 1, e † e † Then, e 2 e 2 j = 1, ˜ j = e j , and ˜ j = ˜ e j . Also, � � � q j = 1 e j + i ˜ e j 2 (13) � � j = 1 † e j − i ˜ q e j . 2 21

Ideals and representation

SMA – Ideals The elements � q := q 1 q 2 q 3 , (14) † = † † † q q 3 q 2 q 1 , are nilpotent ( q 2 = 0 and q † 2 = 0). † and p ′ = q † q . We make the notation p := qq Then, p = 1 + i e 1 ˜ · 1 + i e 2 ˜ · 1 + i e 3 ˜ e 1 e 2 e 3 (15) . 2 2 2 The elements p and p ′ are idempotent, ( p ) 2 = p and ( p ′ ) 2 = p ′ . 22