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De-Sheng Li. Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold

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De-Sheng Li. Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold Physical Science & Biophysics Journal MEDWIN PUBLISHERS ISSN: 2641-9165 Committed to Create Value for Researchers Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold De-Sheng Li1,2,3* Review Article 1College of Physics, Mechanical and Electrical Engineering, Jishou University, China Volume 5 Issue 2 2Interdisciplinary Center for Quantum Information, National University of Defense Technology, Received Date: July 07, 2021 China Published Date: August 11, 2021 3 Institute of High Energy Physics, Chinese Academy of Sciences, China DOI: 10.23880/psbj-16000186 *Corresponding author: De-Sheng Li, College of Physics, Mechanical and Electrical Engineering, Jishou University, Jishou 416000, P. R. China, Email: [email protected] Abstract There is a UU(4') × (4) -bundle on four-dimensional square root Lorentz manifold. Then a Pati-Salam model in curved space-time (Lagrangian) and a gravity theory (Lagrangian) are constructed on square root Lorentz manifold based on self-parallel transportation principle. An explicit formulation of Sheaf quantization on this square root Lorentz manifold is shown. Sheaf quantization is based on superposition principle and construct a linear Sheaf space in curved space-time. The in Standard Model (SM) of particle physics and Einstein gravity are found in square root metric and the connections of bundle. transition amplitude in path integral quantization is given which is consistent with Sheaf quantization. All particles and fields ±c Thegravity interactions theory is betweenEinstein-Cartan particles/fields kind with are torsion. described There by are Lagrangian new particles, explicitly. right There handed are neutrinos, few new physics dark photon, in this Fiona,model. TheX 01212 and YYYYY,,,,**. Keywords: Lorentz manifold; Pati-Salam model; Curved space-time; Gravity theory; Yang-Mills theory; Sheaf quantization Introduction of extra dimensions. Four-dimensional pseudo-Riemann geometry with signature (−+++,,,) , Lorentz manifold, is the geometry Mills theory takes gauge invariance as its basic principle and background of the general relativity, space-time is described to beLater, the theoretical the Yang-Mills framework theory of electromagnetic,[1] was confirmed. weak Yang- and strong interaction in Standard Model (SM) of particle physics the curve of space-time. In general relativity, the geodesic [2-7]. The Yang-Mills theory is the theoretical framework by the metric, and the gravitational field is described as equation describes the trajectories of free particles, and the of SM and has a good correspondence with the complex Einstein equation determines how matter curves space-time. At the last life time of Einstein, he attempted to establish a new geometry unifying electromagnetic interaction and theorystructure also, group except G fiber that bundle the G theorystructure [8]. Generalgroup of relativity general gravity. This idea was developed by Weil into the early idea can actually be rewritten in the framework of fiber bundle of gauge invariance and by the Kaluza and Klein into the idea the tangent and cotangent bundles. Another way to build relativity is real, and the corresponding fiber bundles are Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold Phys Sci & Biophys J 2 Physical Science & Biophysics Journal extraunified dimension field theory were is made.introducing Is it possible extra dimensions to fuse the tangent to give (cotangent)all fields their bundle geometric of general positions. relativity And lotswith of the attempts complex in orthonormal frames describe µgravitational∂ field (Figure 1). θθaa( xx) = ( ) µ . structure group G-bundle of Yang-Mills theory? ∂x spinors through making square root of the Klein-Gordon equation,Inspired we byresearched the Dirac’s the way four-dimensional of finding his equation square rootand Lorentz manifold, which similar with the papers in Clifford algebra or Clifford bundle [9-52], spin-gauge theory in Riemann-Cartan space-time [53,54], sedenion [55] and Einstein-Cartan theory [56-58] etc. Four-dimensional square root Lorentz manifold has extra UU(4') × (4) principal bundle than Lorentz manifold. Two Lagrangians based on four-dimension square root Lorentz manifold are constructed which describe a U(4') × UU(4) × ( 4) Pati- LR Figure 1: Picture visualization of geometry background of Salam model in curved space-time and a gravity theory, (a) general relativity (b) Yang-Mills theory and (c) square respectively. In the Pati-Salam model [59], the ' is SU (4 ) root metric. (a) Geometry background of general relativity color group with “lepton number as the fourth color”, and the is pseudo-Riemann geometry, which is a smooth curved SU44× SU ( )LR( ) manifold. (b) Geometry background of Yang-Mills theory Einstein kind with torsion. We realize an explicit formulation of Sheaf quantizationis chiral [60-76] flavor scheme group. which The gravity consistent theory with is gauge group of Yang-Mills theory. (c) Square root metric path integral quantization. The particles spectrum on this geometryis complex in G-bundle four-dimension with flat has base everything manifold, of Gpseudo- is the model is analyzed. Riemann geometry and extra UU(4') × (4) -bundle. Geometry and Lagrangian The notations are introduced here. abcd,,, represent The definition of gamma matrices is γγa b+= γγ b a2. η ab I (4) frame indices and abcd,,, are equal to 0,1,2,3 . µ,,v ρσ , 44× represent coordinates indices and µ,,v ρσ , are equal to 0,1,2,3 . α represent group indices with α equals to The Hermiticity conditions for gamma matrices are 0,1,....,15. i, jklm , ,, are equal to 1,2,3,4. C is quarks color a b†† b a ab and equals to RGB, ,( 1, 2, 3) . k is Sheaf space index. Repeated γγ+= γ γ 2,II44× (5) indices are summed by default. where I ab is diag (1,1,1,1) The pseudo-Riemann manifold is described by a metric 0†a l= iγγik( x) kj. (We xe) define j⊗ e j θ a ( x), (6) g( x) = -, g( x) dxµν dx (1) µν a 0† l= iγγik( x) kj( xe) j⊗ e i θ a ( x), (7) where the metric is symmetric where ei are the orthogonal bases expanding four-dimension 4 4 gµν ( x) = gxνµ ( ), (2) complex space . The orthogonal bases of satisfy µ ††⊗= =δ (8) and{x|, x=( x) = ( tx)} is a four-dimensional topological tr( ej e i) e i e j ij. space. Here we discuss the four-dimensional pseudo- One simple choice of e is Riemann manifold with signature (−+++,,,) , Lorentz i manifold. And it can be described by orthonormal frame θθ = ii12= (9) (vierbein) formalism as ee12( ,0,0,0) , e( 0, e ,0,0) , − θ θ gx1 = −ηθab x θ x, e= 0,0, eei 3 ,0 ,= 0,0,0, ei 4 . (10) ( ) ab( ) ( ) (3) 34( ) ( ) where After using γaa†= γγγ 00 η ab =diag (1, −−− 1, 1, 1) . , we find that De-Sheng Li. Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold. Copyright© De-Sheng Li. Phys Sci & Biophys J 2021, 5(2): 000186. 3 Physical Science & Biophysics Journal 1 UU(44' )× ( ) -bundle are Hermitian -1 éù . (11) g( x)= trëûêú l( x) l( x) 4 †* The gauge fieldsVxVxWxWxµ (on) the= µµ( ),.i j ( ) = µji ( ) Then lx( ) and lx ( ) are the square root of metrics in some a sense. The representation freedom of γ ij( x) can be shown from restriction (4), (5) and (8). The equation as follow can as beThe derived uniqueness from (16)of definition of gauge fields is originated 0''aa†0 a ∇=γγa0 γγaa00 − γγ γik( xx) γ kj ( ) = ψik( x) γ kl γ lm ψ mj ( x) = ψi ( x) γψj ( x), µµ( ) iVx( ) Vx µ( ) , (18) '' γa( xx) γ0( ) = ψ †0( x) γγ aa ψ( x) = ψ( x) γ† ψ( x), ik kj ik kl lm mj i j where we write where ψ 00 i Vx ( ) = γγ V. index i equals to 1,2,3,4. And µ are the Dirac fermions field with flavor related V (x) and W (x) can be decomposed by the ψ( x) = ψ†0( x) γψ,4( xU)∈ ( ) µ µij ii generators of the U(4) group is 44× The gauge field = α α follow Vxµ ( ) Vµ ( x) , (19) matrix. So, the squarea root† metrics are defined as lx( ) = iψi( x) γψ j( xe) j⊗ e ia θ ( x), (12) αα Wµµij( xWx) = ( )ij, (20) a †† l( x) = iψi( x) γψj( xe) j⊗ e ia θ( x). (13) where α is equals to 0,1,2,....15. There are Hermitian gauge The square root Lorentz manifold is described by square root ᆆ αα α bosons fieldsVµ( xVxWxWx) = µµ( ),.( ) = µ( ) (12) and (13) satisfy (11) and metric (12), (13). Direct calculations show that the definition The α are the generators of U (4) and an expilict one ††=−=− lxlxlxlx( ) ( ),.( ) ( ) can be seen in appendix. An equation is constructed which satisfying the UU(44′)× ( ) gauge invariant, locally Lorentz invariant and generally covariant principles The coefficients of the affine' × connections on coordinates, UU(44) ( ) tr∇=lx( ) 0. (21) followscoefficients of spin connections on orthonormal frame [77] and gauge fields on the -bundle are defined as ρ This equation originated from generalized self-parallel ∇∂ =Γ( x) ∂, (14) µµvv p transportation principle. Eliminating index x, the explicit b formula of equation (21) is ∇=Γµµθθa( x) ab( xx) ( ), (15) 0a 00aa ∇=µµ(γγ) iVx( ) γγ − γγ Vx µ( ) , (16) ∇=e iW( x) e†. (17) µµi ij j A relation between coefficients of the affine connections We define a Lagrangian orthonormal frame can be easily found a µi b aµ on coordinates and coefficients of spin connections on =ψγi(iV ∂+ µ ψ i µ ψ i − ψ j W µji) θ a + ψγψ i i Γ bµ θ a . (22) bvρρ ρ 2 Γaµµ( xx)θ b( ) =∂+Γ θθ a( x) av( x) µ( x). The last term in Lagrangian (22) is Yukawa coupling term ψ φψ If the covariant derivative is compatible with metric, ii ∇=gx( ) 0, and the scalar (Higgs) field is gamma matrix valued and originated from gravitationali field.
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