Total derivative for gandgg. Sab is==1 d two dgfor = to obtain the tensor kind.is thetransformed from (44).The following derivation only make use of the property Sab = -Sba . For , we’ve d = 1abcd f a Sbc g,= -1dabc f d f a Sbc g.(54)4. The Classical Approximation of Dirac Equation Within this section, we derive the classical mechanics for any charged spinor moving in gravity, and disclose the physical which means of connections and . By covariance principle, the Dirac Equation (18) is valid and covariant in any standard coordinate technique; nonetheless, as a way to SC-19220 Formula acquire the energy eigenstates of a spinor we need to have to solve the Hamiltonian program of quantum mechanics, and so as to derive its classical mechanics we will need to calculate the spatial integrals of its Noether charges for example coordinates, energy and momentum. These computations can’t be realized in an arbitrary coordinate system, but have to be performed within a coordinate program with realistic worldwide simultaneity; that is definitely, we need to have the Gu’s organic coordinate system (NCS) [12,32] ds2 = gtt dt2 – gkl dx k dx l , d =gtt dt = f t 0 dt,dV =gd3 x.(55)in which ds could be the suitable time element, d the Newton’s absolute cosmic time element and dV the absolute volume element on the space at time t. NCS usually exists along with the global simultaneity is special. Only in NCS we are able to clearly establish the Hamiltonian formalism and calculate the integrals of Noether charges. In NCS, we’ve ft 0 =gtt ,1 f t0 = , gttt =gtt 0 ,1 t = 0 . gtt(56)Then by (20) we get =1 t lng, f k a j f a k lnjg ,t = gtt t ,k = – gkl l .(57)In NCS, to lift and reduce the index of a vector implies t = gtt t , k = – gkl l . Extra commonly, we look at the Dirac equation with electromagnetic possible eA and nonlinear prospective N = 1 w2 , exactly where = 0 . Then (18) may be rewritten in two Hamiltonian formalism itt= H,^ H = -k pk et At S (m – N )0 ,(58)where H may be the Hamiltonian or energy in the spinor, t = f t0 0 = ( gtt )-1 and = 0 dt would be the realistic time with the universe, only it . Because d = f t t = i will be the trueSymmetry 2021, 13,10 ofenergy operator to get a spinor. gtt represents the gravity, and it can’t be generally merged into d as performed inside a semi-geodesic coordinate technique. In traditional quantum theory, we simultaneously take coordinate, speed, momentum and wave function of a particle as original ideas. This scenario will be the origin of logical confusion. As a matter of truth, only wave function is independent concept and dynamical Equation (58) is fundamental in logic. Other ideas of the particle really should be defined by and (58). Similarly towards the case in flat space-time [33], we define some classical ideas for the spinor. SB 271046 GPCR/G Protein Definition 2. The coordinate X and speed v on the spinor is defined as X k (t) x k | |two gd3 x =RRx k qt gd3 x,vkd k d X = f t0 X k , d dt(59)exactly where R3 stands for the total simultaneous hypersurface, q= = is definitely the current.By definition (59) and existing conservation law q;= ( g)-1 (qg) = 0, we havevj= =f t0 f tR3 Rx j t (qt g)d3 x = – f tqjR3x j k ( q k g ) d3 x(60)gd xRqjgd x.RSince a spinor has only a really tiny structure, with each other with normalizing situation qt gd3 x = 1, we obtain the classical point-particle model for the spinor as [33] q u1 – v2 three ( x – X ), v2 = gkl vk vl , u= dX v= , ds 1 – v2 (61)exactly where the Dirac- meansR3 ( x – X ) gd3 x = 1.R^ Theorem 6. For any Hermitian operator P, P following generalized Ehrenfest theorem, dP = dt where^ g Pd3 x is true for any . We’ve got theR^ ^ ^ g t t P -.