Morphic towards the matrix algebra, we need to have not distinguish tetrad a and

Morphic towards the matrix algebra, we need to have not distinguish tetrad a and matrix a in algebraic calculation. There are many definitions for Clifford algebra [27,28]. Clifford algebra can also be called geometric algebra. If the definition is directly related to geometric ideas, it is going to bring terrific convenience for the study and analysis of geometry [12,29]. Definition 1. Assume the element of an n = p q dimensional space-time M p,q over R is given by (4). The space-time is endowed with distance ds = |dx| and oriented volumes dVk calculated by dx2 dVk= =1 ( )dx dx = gdx dx = ab X a X b , two dx1 dx2 dxk = dx1 dx2 dxk , (1 k n),(11) (12)Symmetry 2021, 13,4 ofin which the Minkowski metric (ab ) = diag( I p , – Iq ), and Grassmann basis = k M p,q . Then the following quantity with basis C = c0 I c c c12 12 , (ck R) (13)with each other with multiplication rule of basis given in (11) and associativity define the 2n -dimensional true universal Clifford algebra C p,q . The geometrical meanings of elements dx, dy, dx dy are shown in Figure 1.Figure 1. Geometric meaning of vectors dx, dy and dx dy.Figure 1 shows that the exterior solution is oriented volume of the parallel polyhedron from the line element vectors, as well as the Grassmann basis ab is just the orthonormal basis of k-dimensional volume. Because the length of a line element and the volumes of every single grade constitute the basic contents of geometry, the Grassmann basis set becomes units to represent numerous geometric and physical quantities, that are particular kinds of tensors. By simple calculation we have [5,12,29] Theorem 1. For C I,1,three ,we’ve got the following helpful relations ab =i abcd cd five , abc = i abcd d 5 , 0123 = -i5 . 2 g , = g – g .a , (14) (15)=The above theorem offers a number of generally utilized relations amongst the Clifford items as well as the Grassmann merchandise. Because the calculations of geometric and physical quantities are mostly within the type of Clifford merchandise, but only by expressing these forms as Grassmann solutions, their geometric and physical GYY4137 Cancer significance is clear. Hence the above transformation relations turn into fundamental and critical. For Dirac equation in curved space-time devoid of torsion, we’ve got [1,30], (i- eA) = m,= ,(16)in which the spinor connection is offered by 1 1 1 ;= ;= ( – ). 4 4(17)Symmetry 2021, 13,5 ofThe total spinor connection 1 3 . Clearly, is actually a Clifford solution, and its geometric and physical significance is unclear. Only by projecting it onto the Grassmann basis a and abc , its geometric and physical meanings become clear [12]. Theorem 2. Dirac equation (16) is often rewritten in the following Hermitian form ^ (p- S) = m0 , ^ in that is present operator, pmomentum and Sspin operator, = diag(, ), ^ p= i – eA, S= 1 diag(, -), 2 (19) (18)where is Keller connection and Gu ester prospective, they are respectively defined as1 1 (ln g) – f a f a , f a ( f – f a ) = two two 1 1 f f a f b f e abcd ce = ab f a (f b – f ). 2 d 4 g(20) (21)Proof. By (14) and (15), we’ve got the following Clifford AZD4625 Cancer calculus = = = = = =1 1 ( – ) = ( g )( – ) 4 4 1 1 1 (; ) = ( ln( g)) f a f f c ab c b four four 4 1 a 1 [ f a ( f f a )] f a f f d ab c cd a b 4 four 1 1 f a (-f a f ) f a f f d ( bc a – ac b abc )cd b four 4 1 1 f a ( f – f a ) f a f b f e abc ce two four i 5 .(22)Substituting it into (16) and multiplying the equation by 0 , we prove the theorem. The following discussion shows that and have unique physical.