Hot Topics in General Relativity and Gravitation ( HTGRG-2) Lepton mass hierarchy in the light of time-space symmetry with microscopic curvatures Vo Van Thuan Vietnam Atomic Energy Institute (VINATOM) Email: vvthuan@vinatom.gov.vn ICISE, Quy Nhon-Vietnam, August 9-15, 2015.
Contents 1. Introduction 2. Geodesic equation in 6D time-space 3. Quantum equation and indeterminism 4. Charged lepton generations 5. Mass hierarchy of neutrinos 6. Conclusions
1- Introduction Objective : problem of consistency between QM and GR . Motivated by : Extra-dimension dynamics : Kaluza and Klein [1,2] Semi-classical approach to QM : de Broglie & Bohm [3,4]. (However: Violation of Bell inequalities in [5,6]). Technical tool: time-like EDs : Anti-de Sitter geometry: Maldacena [7]: AdS/CFT; Randall [8]: (hierarchy). Induced matter models: Wesson [9,10]; Koch [11,12] . o Our study based on space-time symmetry [13,14]: following induced matter models where quantum mechanical equations is identical to a micro gravitational geodesic description of curved time-like EDs. Present nt work : Ap Application tion of the e model to deal with lepton on gene nerati ation ons s and their r mass hiera rarchy rchy (all charge rged leptons ns and neutrino utrinos). 3 HTGRG-2, Quy Nhon, 9-15 August 2015 Vo Van Thuan 27th Rencontre de Blois June 3, 2015
2- Geodesic equation in 6D time-space (1) 𝑢 1 , 𝑢 2 , 𝑢 3 |𝑦 1 , 𝑦 2 , 𝑦 3 Constructing an ideal 6D flat time-space consisting of orthogonal sub-spaces 3D-time ( 3T ) and 3D-space ( 3X ): 𝟑 − 𝒆𝒚 𝒎 𝒆𝑻 𝟑 = 𝒆𝒖 𝒍 𝟑 ; summation: 𝑙, 𝑚 = 1 ÷ 3 . We are working further at its symmetrical “light - cone” : 2 = 𝑒𝑦 𝑚 𝒆𝒍 𝟑 = 𝒆𝒎 𝟑 (or 𝑒𝑢 𝑙 2 ; summation: 𝑙, 𝑚 = 1 ÷ 3 ) (1) Natural units ( ħ = 𝑑 = 1 ) used unless it needs an explicit quantum dimensions . For transformation from 6D time-space to 4D space-time let’s postulate A Conservation of Linear Translation principle (CLT) in transformation from higher dimensional geometries to 4D space-time for all linear translational elements of more general geometries . This means that the Eq. (1) of linear time & space intervals ( 𝒆𝒍 𝟑 = 𝒆𝒎 𝟑 ) is to be conserved not only for flat Euclidean/ Minkowski geometries, which bases on evidence of Lorentz invariance-homogeneity-isotropy of 4D space-time. 4 HTGRG-2, Quy Nhon, 9-15 August 2015 Vo Van Thuan 27th Rencontre de Blois June 3, 2015
2- Geodesic equation in 6D time-space (2) Introducing a 6D isotropic plane wave equation : 𝝐 𝟑 𝝎 0 (𝒖 𝒍 ,𝒚 𝒎 ) 𝝐 𝟑 𝝎 0 (𝒖 𝒍 ,𝒚 𝒎 ) = ; (2) 𝟑 𝟑 𝝐𝒖 𝒍 𝝐𝒚 𝒎 Where 𝝎 0 is a harmonic correlation of 𝑒𝑢 and 𝑒𝑦 , containing only linear variables. Assuming : Wave transmission (2) and “displacements” 𝑒𝑢 and 𝑒𝑦 serve primitive sources of formation of energy-momentum and vacuum potentials 𝑊 𝑈 or 𝑊 𝑌 (in terms of time-like or space-like cosmological constant Λ 𝑈 and Λ 𝑀 ): 𝑊 𝑈 & Λ 𝑈 ∈ 3𝑈 ; 𝑊 𝑌 & Λ 𝑀 ∈ 3𝑌 ; Potential 𝑾 𝑼 is able to generate quantum fluctuations with circular polarization about linear axis 𝒖 𝟒 , keeping evolution to the future , which is constrained by a time-like cylindrical condition and simultaneously leading to violation of space-time symmetry (in analogue to Higgs mechanism) . In according to CLT principle, during transformation from 6D- to 4D space-time: 𝝎 0 (6D) → 𝝎 0 ( 𝟓𝑬: 𝒖 𝒍 → 𝒖 𝟒 )= = 𝝎(𝟓𝑬)𝑓 𝑗𝝌(𝟓𝑬) ; It needs a suggestion equivalent to the Lorentz condition in 4D space-time (for 𝟑 𝟑 compensation of longitudinal fluctuations): 𝝐𝝌 𝝐𝝌 = ; (3) 𝝐𝒖 𝟒 𝝐𝒚 𝒎 5 HTGRG-2, Quy Nhon, 9-15 August 2015 Vo Van Thuan 27th Rencontre de Blois June 3, 2015
2- Geodesic equation in 6D time-space (3) We use for cylinder in 3D-time polar coordinates 𝛚(𝐮 𝟏 ), 𝛘(𝐮 𝟏 ), 𝐮 𝟒 : 𝟑 = 𝒆𝒕 𝟑 + 𝒆𝒖 𝟒 𝒆𝒖 𝟑 = 𝒆𝝎(𝒖 𝟏 ) 𝟑 + 𝝎(𝒖 𝟏 ) 𝟑 𝒆𝝌(𝒖 𝟏 ) 𝟑 + 𝒆𝒖 𝟒 𝟑 ; (4) linear time 𝒆𝒖 𝟒 in (4) is identical to 𝒆𝒍 in (1). 𝟑 + 𝒆𝒖 𝟒 𝟑 as definition of t. as 𝒆𝒖 𝟒 orthogonal to 𝒆𝒖 𝟏 : 𝜵𝒆𝒖 = 𝜵 𝟏 𝒆𝒖 𝟏 +𝜵 𝟒 𝒆𝒖 𝟒 𝒆𝒖 𝟑 = 𝒆𝒖 𝟏 And using in 3D-space spherical coordinates: 𝝎(𝒚 𝒐 ), 𝜾(𝒚 𝒐 ), 𝝌(𝒚 𝒐 ) : 𝒆𝝁 𝟑 = 𝒆𝝎(𝒚 𝒐 ) 𝟑 + 𝝎(𝒚 𝒐 ) 𝟑 [𝒆𝜾 𝟑 + 𝒕𝒋𝒐 𝟑 𝜾 𝒆𝝌(𝒚 𝒐 ) 𝟑 ] + 𝒆𝒚 𝒎 𝟑 𝟑 + 𝒆𝝉 𝑴 𝟑 + 𝒆𝒎 𝟑 ; = 𝒆𝝉 𝒇𝒘 (5) Where: 𝒆𝝉 𝒇𝒘 local interval characterizing P- even contribution of lepton spinning 𝒕 ; 𝒆𝝉 𝑴 P-odd contribution of intrinsic space-like curvature . 𝒕 𝑴 // 𝒚 𝒎 ( left-handed helicity ) local rotation in orthogonal plane 𝑸 𝒐 local proper 𝒚 𝒐 ∈ 𝑸 𝒐 serves an affine parameter to describe a weak curvature in 3D-space. EDs turn into the dynamical depending on other 4D space-time dimensions: 𝝎 = 𝝎(𝒖 𝟏 , 𝒖 𝟒 , 𝒚 𝒐 , 𝒚 𝒎 ) and 𝝌 = 𝜵𝒖 − 𝒍 𝒌 𝒚 𝒌 = 𝜵 𝟏 𝒖 𝟏 +𝜵 𝟒 𝒖 𝟒 − 𝒍 𝒐 𝒚 𝒐 − 𝒍 𝒎 𝒚 𝒎 . 6D time-space (1) generalized with curvature gets a new quadratic form: 𝟑 − 𝒆𝝉 𝑴𝟑 ; (6) 𝒆𝒖 𝟑 − 𝒆𝒕 𝟑 = 𝒆𝝁 𝟑 −𝒆𝝉 𝒇𝒘 Leading to generalized 4D Minkowski space-time with translation and rotation: 𝟑 − 𝒆𝝉 𝑴𝟑 = 𝒆𝒖 𝟑 − 𝒆𝝁 𝟑 ; 𝒆𝜯 𝟑 = 𝒆𝒕 𝟑 − 𝒆𝝉 𝒇𝒘 (7) 6 HTGRG-2, Quy Nhon, 9-15 August 2015 Vo Van Thuan 27th Rencontre de Blois June 3, 2015
2- Geodesic equation in 6D time-space (4) The derivation here is following [14]: Vo Van Thuan, arXiv:1507.00251[gr-qc], 2015 . Let’s assume that any local deviation from the linear translation in 3D-time should be compensated by a local deviation in 3D-space for conserving space-time 𝝐𝝎 symmetry (1): 𝑬𝒗 𝒖 𝟏 = 𝑬𝒗 𝒚 𝒐 ⧧𝟏 ; with velocity 𝒗 𝒕 = 𝝐𝒕 ; Their validity means a pumping of P- or T- violations, which are small. Then 3D local deviations are almost realized independently and exactly: 𝑬𝒗 𝒖 𝟏 = 𝑬𝒗 𝒚 𝒐 = 𝟏 ; (8) We derive a symmetrical equation of geodesic acceleration of the deviation 𝝎 : 𝝐 𝟑 𝝎 𝝐 𝟑 𝝎 𝝐𝒖 𝜸 𝝐𝒚 𝜹 𝝐𝒖 𝜷 𝝐𝒚 𝝉 𝝎 𝝎 𝝐𝒖 𝟏𝟑 + 𝜟 𝜷𝜸 = 𝟑 + 𝜟 𝜹 𝝉 𝝐𝒚 𝒐 ; (9) 𝝐𝒖 𝟏 𝝐𝒖 𝟏 𝝐𝒚 𝒐 𝝐𝒚 𝒐 Where: 𝒖 𝜷 , 𝒖 𝜸 ∈ 𝝎(𝒖 𝟏 ), 𝛘(𝒖 𝟏 ), 𝒖 𝟒 ; 𝒚 𝜹 , 𝒚 𝝉 ∈ 𝝎(𝒚 𝒐 ), 𝛘(𝒚 𝒐 ), 𝒚 𝒎 . There are two terms valid: = −𝝎. 𝒕𝒋𝒐 𝟑 𝜾 ; other terms with Γ αβ ψ = Γ 𝝎 𝝎 ψ 𝜟 𝝌(𝒖 𝟏 ) 𝝌(𝒖 𝟏 ) = −𝝎 and 𝜟 𝝌(𝒚 𝒐 ) 𝝌(𝒚 𝒐 ) = 0 . γ σ Applying a Lorentz-like condition (3) leads to the differential equation of linear elements 𝝐 𝟑 𝝎 𝝐 𝟑 𝝎 similar to (2) : 𝟑 = 𝟑 ; (10) 𝝐𝒖 𝟒 𝝐𝒚 𝒎 Adding (10) to (9) w e obtain equation including rotation and linear translation : 𝟑 𝟑 𝝐 𝟑 𝝎 𝝐 𝟑 𝝎 𝝐 𝟑 𝝎 𝝐 𝟑 𝝎 𝝐𝝌 𝝐𝒚 𝒐𝟑 − 𝝎 𝒕𝒋𝒐 𝟑 𝜾 𝝐𝝌 (11) 𝝐𝒖 𝟏𝟑 − 𝝎 + 𝝐𝒖 𝟒𝟑 = + 𝝐𝒚 𝒎𝟑 ; 𝝐𝒖 𝟏 𝝐𝒚 𝒐 7 HTGRG-2, Quy Nhon, 9-15 August 2015 Vo Van Thuan 27th Rencontre de Blois June 3, 2015
2- Geodesic equation in 6D time-space (5) Due to orthogonality of each pair of differentials ( 𝒆𝒖 𝟒 & 𝒆𝒖 𝟏 ) and ( 𝒆𝒚 𝒎 & 𝒆𝒚 𝒐 ) their second derivatives are combined together: 𝝐 𝟑 𝝎 +𝟑 + 𝝐 𝟑 𝝎 𝝐𝒖 𝟒𝟑 = 𝝐 𝟑 𝝎 𝝐𝒚 𝒐𝟑 + 𝝐 𝟑 𝝎 𝝐 𝟑 𝝎 𝝐𝒚 𝒎𝟑 = 𝝐 𝟑 𝝎 𝝐𝒚 𝒌𝟑 ; (13) 𝝐𝒖 𝟑 ; (12) and 𝝐𝒖 𝟏 Transformation from 6D-time-space to 4D-space-time is performed in the result of two operations: Defining 𝝎 as a deviation parameter ; The unification of time-like dimensions (12). Finally, from (11) we obtain the geodesic equation as follows: 𝝐 𝟑 𝝎 𝝐 𝟑 𝝎 𝟑 − 𝝐𝒖 𝟑 + 𝝐𝒚 𝒌𝟑 = − 𝜧 𝑼 − 𝐶 𝑓 𝒍 𝒐 . μ 𝒇 𝒇𝒘𝒇𝒐 − 𝜧 𝑴 𝝎 ; (14) Where : Effective potentials V 𝑼 of a time- like “cosmological constant” 𝚳 𝑼 and an odd 𝟑 𝟑 𝝐𝝌 𝝐𝝌 component 𝜧 𝑴 of the space-like 𝚳 : [𝚳 𝑼 −𝚳 𝑴 ]𝝎 = − 𝝎. + 𝑴 𝝐𝒖 𝟏 𝝐𝒚 𝒐 𝐶 𝑓 is a calibration scale factor and μ 𝒇 is magnetic dipole moment of charged lepton; its orientation is in correlation with spin vector 𝒕 and being P-even. 8 HTGRG-2, Quy Nhon, 9-15 August 2015 Vo Van Thuan 27th Rencontre de Blois June 3, 2015
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