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THE LEPTON SECTOR AS AN AXIOMATIC-LIKE CONSTRUCTION O. Kosmachev INTRODUCTION Necessity of conversion to axiomatics. What means axiomatic-like construction? Unique approach and unification of mathematical formalism. STABLE LEPTONS

  1. THE LEPTON SECTOR AS AN AXIOMATIC-LIKE CONSTRUCTION O. Kosmachev INTRODUCTION Necessity of conversion to axiomatics. What means axiomatic-like construction? Unique approach and unification of mathematical formalism. STABLE LEPTONS Suppositions, necessary and sufficient conditions. Massive leptons Massless leptons Comparison with well known results. UNSTABLE LEPTONS Setting of the problem Groups ∆ 1 , ∆ 2 , ∆ 3 . Discussion of the results. PERSPECTIVES AND REPRESENTING PROBLEMS

  2. Our suppositions are: 1 the equations must be invariant and covariant under homogeneous Lorentz transformations taken into account all four connected components; 2 the equations must be formulated on the base of irreducible representations of the groups determining every lepton equation; 3 conservation of four-vector of probability current must be fulfilled and fourth component of the current must be positively defined; 4 the lepton spin is supposed equal to 1/2; 5 every lepton equation must be reduced to Klein-Gordon equation; We see here two kinds of symmetries: with respect to the homogeneous Lorentz group and relativistic quantum wave equations. These requirements are necessary and sufficient for obtaining of lepton equations.

  3. The unified form of four connected components We will use contracted form for notation of connected components. It looks for d γ -group as: { b i , b k } = 2 δ ik , ( i, k = 1 , 2 , 3) . (1) Lie algebra of d γ -group is: [ a 1 , a 2 ] = 2 a 3 , [ a 2 , a 3 ] = 2 a 1 , [ a 3 , a 1 ] = 2 a 2 , [ b 1 , b 2 ] = − 2 a 3 , [ b 2 , b 3 ] = − 2 a 1 , [ b 3 , b 1 ] = − 2 a 2 , [ a 1 , b 1 ] = 0 , [ a 2 , b 2 ] = 0 , [ a 3 , b 3 ] = 0 , [ a 1 , b 2 ] = 2 b 3 , [ a 1 , b 3 ] = − 2 b 2 , [ a 2 , b 3 ] = 2 b 1 , [ a 2 , b 1 ] = − 2 b 3 , [ a 3 , b 1 ] = 2 b 2 , [ a 3 , b 2 ] = − 2 b 1 . The obtained commutation relations coincide with commutation relations of the infinitesimal matrices of the proper homogeneous Lorentz group. Due to construction of commutation relation, all six operators a 1 , a 2 , a 3 and b 1 , b 2 , b 3 have a definite physical meaning. O. Kosmachev THE LEPTON SECTOR AS AN AXIOMATIC-LIKE CONSTRUCTIO

  4. P-conjugate representation and duality of d γ -group The duality means, that d γ contains apart from Q 2 [ a 1 , a 2 ] one more group of the eighth order q 2 [ a 1 , a ′ 2 ] . Here a ′ 2 = a 2 · c, c = σ x σ y σ z . Lie algebra is [ a 1 , a ′ 2 ] = 2 a ′ [ a ′ 2 , a ′ 3 ] = − 2 a 1 , [ a ′ 3 , a 1 ] = 2 a ′ (2) 3 , 2 , where a ′ 3 ≡ a 1 a ′ 2 . Let us coll this group quaternion group of the second kind q 2 [ a 1 , a 2 ] . As corollary we have another Lie algebra. We will denote it as f γ [ a 1 , a ′ 2 ] = 2 a ′ 3 , [ a ′ 2 , a ′ 3 ] = − 2 a 1 , [ a ′ 3 , a 1 ] = 2 a ′ 2 , [ b ′ 1 , b ′ 2 ] = − 2 a ′ [ b ′ 2 , b ′ 3 ] = 2 a 1 , [ b ′ 3 , b ′ 1 ] = − 2 a ′ 3 , 2 , [ a 1 , b ′ 1 ] = 0 , [ a ′ 2 , b ′ 2 ] = 0 , [ a ′ 3 , b ′ 3 ] = 0 , [ a 1 , b ′ 2 ] = 2 b ′ [ a 1 , b ′ 3 ] = − 2 b ′ 3 , 2 , [ a ′ 2 , b ′ 3 ] = − 2 b ′ 1 , [ a ′ 2 , b ′ 1 ] = − 2 b ′ 3 , [ a ′ 3 , b ′ 1 ] = 2 b ′ [ a ′ 3 , b ′ 2 ] = 2 b ′ 2 , 1 . The contracted defining relations for f γ -group take the form { b 1 , b k } p = 2 δ 1 k , ( k = 1 , 2 , 3) , (3) { b i , b k } p = − 2 δ ik , ( i, k = 2 , 3) . O. Kosmachev THE LEPTON SECTOR AS AN AXIOMATIC-LIKE CONSTRUCTIO

  5. T-conjugate representation The contracted defining relations for b γ -group take the form { b ′ i , b ′ k } = − 2 δ ik , ( i, k = 1 , 2 , 3) . (4) Lie algebra of b γ -group is: [ a 1 , a 2 ] = 2 a 3 , [ a 2 , a 3 ] = 2 a 1 , [ a 3 , a 1 ] = 2 a 2 , [ b ′ 1 , b ′ 2 ] = 2 a 3 , [ b ′ 2 , b ′ 3 ] = 2 a 1 , [ b ′ 3 , b ′ 1 ] = 2 a 2 , [ a 1 , b ′ [ a 2 , b ′ [ a 3 , b ′ 1 ] = 0 , 2 ] = 0 , 3 ] = 0 , [ a 1 , b ′ 2 ] = 2 b ′ [ a 1 , b ′ 3 ] = − 2 b ′ 2 , 3 [ a 2 , b ′ 3 ] = 2 b ′ [ a 2 , b ′ 1 ] = − 2 b ′ 1 , 3 , [ a 3 , b ′ 1 ] = 2 b ′ 2 , [ a 3 , b ′ 2 ] = − 2 b ′ 1 , O. Kosmachev THE LEPTON SECTOR AS AN AXIOMATIC-LIKE CONSTRUCTIO

  6. (PT)-conjugate representation The contracted defining relations for c γ -group take the form { b ∗ 1 , b ∗ k } pt = − 2 δ 1 k , ( k = 1 , 2 , 3) , (5) { b ∗ i , b ∗ k } pt = 2 δ ik , ( i, k = 2 , 3) . [ a 1 , a ′ 2 ] = 2 a ′ 3 , [ a ′ 2 , a ′ 3 ] = − 2 a 1 , [ a ′ 3 , a 1 ] = 2 a ′ 2 , [ b ∗ 1 , b ∗ 2 ] = 2 a ′ 3 , [ b ∗ 2 , b ∗ 3 ] = − 2 a 1 , [ b ∗ 3 , b ∗ 1 ] = 2 a ′ 2 , [ a 1 , b ∗ [ a ′ 2 , b ∗ [ a ′ 3 , b ∗ 1 ] = 0 , 2 ] = 0 , 3 ] = 0 , [ a 1 , b ∗ 2 ] = 2 b ∗ [ a 1 , b ∗ 3 ] = − 2 b ∗ 2 , 3 [ a ′ 2 , b ∗ 3 ] = − 2 b ∗ [ a ′ 2 , b ∗ 1 ] = − 2 b ∗ 1 , 3 , [ a ′ 3 , b ∗ 1 ] = 2 b ∗ 2 , [ a ′ 3 , b ∗ 2 ] = 2 b ∗ 1 . Here (PT)=(P)(T)=(T)(P) means sequential action (P)- and (T)-conjugation. Now we have complete system of constituents for constructing of lepton wave equations. O. Kosmachev THE LEPTON SECTOR AS AN AXIOMATIC-LIKE CONSTRUCTIO

  7. This set of the groups was become closed with respect to discrete transformations. It is expressed by following equalities: � T � d γ = b γ , � P � d γ = f γ , � PT � d γ = c γ , � T − 1 � b γ = d γ , � T − 1 P � b γ = f γ , � P � b γ = c γ , � T − 1 � c γ = f γ , � P − 1 � c γ = b γ , � T − 1 P − 1 � c γ = d γ , � P − 1 � f γ = d γ , � P − 1 T � f γ = b γ . � T � f γ = c γ , Here: � T � means transition b k → b ′ k = ib k ( k = 1 , 2 , 3) , � P � means transition a 2 → a ′ 2 = ia 2 . It is so-called analytic continuation by group parameters.

  8. Structure of the stable lepton groups. 1 The Dirac equation — D γ ( II ) : d γ , b γ , f γ , structural invariant In [ D γ ( II )] = − 1 . 2 The equation for a doublet of massive neutrinos — D γ ( I ) : d γ , c γ , f γ , structural invariant In [ D γ ( I )] = 1 . 3 The equation for a quartet of massless neutrinos — D γ ( III ) : d γ , b γ , c γ , f γ , structural invariant In [ D γ ( III )] = 0 . 4 The equation for a massless T -singlet — D γ ( IV ) : b γ , structural invariant In [ D γ ( IV )] = − 1 . 5 The equation for a massless P -singlet — D γ ( V ) : c γ , structural invariant In [ D γ ( V )] = 1 .

  9. Structure of the unstable lepton groups. Group ∆ 1 has the following defining relations: Γ µ Γ ν + Γ ν Γ µ = 2 δ µν , ( µ, ν = 1 , 2 , 3 , 4 , 5) (6) As a result we obtain the following composition: ∆ 1 { D γ ( II ) , D γ ( III ) , D γ ( IV ) } , In [∆ 1 ] = − 1 . (7) Group ∆ 3 has the following defining relations: Γ s Γ t + Γ t Γ s = 2 δ st , ( s, t = 1 , 2 , 3 , 4) , Γ s Γ 5 + Γ 5 Γ s = 0 , ( s = 1 , 2 , 3 , 4) , Γ 2 5 = − I. It follows from here: ∆ 3 { D γ ( II ) , D γ ( I ) , D γ ( III ) } , In [∆ 3 ] = 0 . (8)

  10. Structure of the unstable lepton groups. Group ∆ 2 has the following defining relations: Γ s Γ t + Γ t Γ s = 2 δ st , ( s, t = 1 , 2 , 3) , Γ s Γ 4 + Γ 4 Γ s = 0 , ( s = 1 , 2 , 3) , Γ 2 4 = − I. Γ u Γ 5 + Γ 5 Γ u = 0 , ( u = 1 , 2 , 3 , 4) , Γ 2 5 = − I. We obtain in this case: ∆ 2 { D γ ( I ) , D γ ( III ) , D γ ( V ) } , In [∆ 2 ] = 1 . (9) All three groups have its own structures.

  11. conclusion Nearest representing problems are determined by perspectives, which was opened on the base of obtained results. Obviously that immediate task among the perspective problems is extension of obtained results on hadron sector. It should be noted that this goal raises a set of problems. Some of them are well known, but some of them appear for the first time.

  12. APPENDICES

  13. 8 < X : A new (for physical applications) and effective tool for analysis and constructing lepton equations was found, i.e. numerical characteristic of irreducible matrix group. Theorem. If D = { γ 1 , ..., γ ρ } is an irreducible matrix group, then 1 ρ In [ D ] = 1 χ ( γ 2 i ) = − 1 (10) ρ 0 i =1 Here ρ - is order of the group, χ ( γ 2 i ) - is a trace of i-matrix squared. In [ D ] — will be called structural invariant of D -group.

  14. The defining relations for the groups of stable leptons Dirac D γ ( II ) : d γ , b γ , f γ . γ µ γ ν + γ ν γ µ = 2 δ µν , (11) µ, ν = 1 , 2 , 3 , 4 . Majorana D γ ( I ) : d γ , c γ , f γ . γ s γ t + γ t γ s = 2 δ st , (12) γ 4 γ s + γ s γ 4 = 0 γ 2 4 = − 1 , s, t = 1 , 2 , 3 Pauli D γ ( III ) : d γ , b γ , c γ , f γ . γ s γ t + γ t γ s = 2 δ st , (13) γ 4 γ s − γ s γ 4 = 0 γ 2 4 = 1 , s, t = 1 , 2 , 3

  15. The defining relations for the groups of stable leptons T-singlet D γ ( IV ) : b γ . γ s γ t + γ t γ s = − 2 δ st , s, t = 1 , 2 , 3 (14) γ 4 γ s − γ s γ 4 = 0 , s = 1 , 2 , 3 , γ 2 4 = 1 P-singlet D γ ( V ) : c γ . γ s γ t + γ t γ s = 0 , s � = t, s, t = 1 , 2 , 3 (15) γ 2 1 = γ 2 2 = 1 , γ 2 3 = − 1 , γ 4 γ s − γ s γ 4 = 0 , γ 2 4 = 1 , s = 1 , 2 , 3 .

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