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Geometry of Spin Particles

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Geometry of Spin Particles RESEARCH Revista Mexicana de F´ısica 61 (2015) 211–223 MAY-JUNE 2015 1 Geometry of spin 2 particles G. Sobczyk¤ Universidad de las Americas-Puebla,´ Departamento de F´ısico-Matematicas,´ 72820 Puebla, Pue., Mexico.´ Received 18 August 2014; accepted 18 March 2015 The geometric algebras of space and spacetime are derived by sucessively extending the real number system to include new mutually anticommuting square roots of §1. The quantum mechanics of spin 1/2 particles are then expressed in these geometric algebras. Classical 2 and 4 component spinors are represented by geometric numbers which have parity, providing new insight into the familiar bra-ket formalism of Dirac. The classical Dirac Equation is shown to be equivalent to the Dirac-Hestenes equation, so long as the issue of parity is not taken into consideration, the latter quantity being constructed in such a way that it is parity invarient. Keywords: Bra-ket formalism; geometric algebra; spacetime algebra; Schrodinger-Pauli¨ equation; Dirac equation; Dirac-Hestenes equation; spinor; spinor operator. PACS: 02.10.Xm; 03.65.Ta; 03.65.Ud 1. Geometric Concept of Number extended number system, and give the new quantity i := e12 the geometric interpretation of a directed plane segment or In his book “Number: The Language of Science” Tobias bivector. Note that Dantzig, in describing the invention of matrices has this to 2 2 2 say: i = e1e2e1e2 = ¡e1e2 = ¡1 (It is) a theory in which a whole array of elements is so that i has the same algebraic property as the imaginary unit regarded as a number-individual. These “filing cabi- of the complex numbers C. Indeed, every unit bivector of the nets” are added and multiplied, and a whole calculus of n-dimensional Euclidean space Rn shares this property and matrices has been established which may be regarded is the generator of rotations in the vector plane of that bivec- as a continuation of the algebra of complex numbers. tor. These abstract beings have lately found a remarkable The Euler identity interpretation in the quantum theory of the atom, and iθ in man’s other scientific fields, [2]. e = cos θ + i sin θ We now show how these “filing cabinets” take on the for θ 2 R and the unit bivector i = e12, and the hyperbolic interpretation of geometric numbers in a geometric number Euler identity system called geometric algebra. It was William Kingdon ee1Á = cosh θ + e sinh θ Clifford (1845-1879), himself, who first referred to his fa- 1 mous algebras as geometric algebras, but some authors still for Á 2 R and the unit vector e1, are easily established as call these algebras Clifford algebras [3]. David Hestenes and special cases of the general algebraic definition of the expo- other theoretical physicists over the last half-century have nential function shown this comprehensive geometric interpretation imbues X1 Xn the equations of physics with new meaning [4]. We shall re- eX ´ = cosh X + sinh X: n! strict our attention to developing geometric algebras of the n=0 plane, 3-dimensional space, and Minkowski spacetime and In [5, Chp. 2] and [6], the author shows that the hy- their application to quantum mechanics, but the ideas apply perbolic number plane, regarded as the extension of the real to higher dimensional spaces as well [5]. p number system to include a new sqrare root e1 := +1 of +1, has much in common with the more famous complex 1.1. Geometric algebra of the plane number plane. Since it took centuries for the complex num- We extend the real number system R to include two new an- bers to be recognised as bonified numbers, I suppose that it is not surprising that the sister hyperbolic numbers are dis- ticommuting square roots e1; e2 of +1, which we identify as unit vectors along the x and y axis of the coordinate plane criminated against even to this day. After all, it was Leopold R2. Thus, Kronecker (1823-1891) who said, “God made the integers, all the rest is the work of man”. 2 2 e1 = e2 = 1; and e12 := e1e2 = ¡e2e1: The standard basis over the real numbers of the geomet- 2 ric algebra G2 := G(R ) is We assume that the associative and distributive laws of mul- tiplication of real numbers remain valid in our geometrically G2 = spanf1; e1; e2; e12g: (1) 212 G. SOBCZYK With respect to this basis a geometric number g 2 G2 can be with respect to the spectral basis (3). From (4) it follows that µ ¶ written 1 g = (1 e )u [g] : (6) 1 + e g = ® + v = (a + bi) + (ce1 + de2) (2) 1 The interesting thing about equation (6) is that it can be where ® = a + bi = reiθ behaves formally like a complex turned around and solved directly for for the matrix of [g] of number and v = ce + de 2 R2 is a vector in the Euclidean 1 2 g. To accomplish this we define the e -conjugate of g 2 G plane of the bivector i = e . It can also be easily checked 1 2 12 by that v® = ®v, where ® = a ¡ ¯i is the (complex) conjugate ge1 := e ge : of ®. It is this property that gives ® the geometric significance 1 1 as the generators of rotations in the plane of e , [5, p.55]. Multiplying Eq. (6) on the left and right by 12 µ ¶ Another basis of G2, the spectral basis, gives the alge- 1 u+ bra of real 2 £ 2 matrices the interpretation of representing e1 geometric numbers in G2. We write µ ¶ and 1 (1 e1)u+; G2 = spanf u+(1 e1)g e1 respectively, gives µ ¶ µ ¶ u+ e1u¡ 1 = spanf g; (3) u+ g(1 e1)u+ e1u+ u¡ e1 µ ¶ µ ¶ or G2 = spanfu+; e1u+; e1u¡; u¡g, where the idempo- 1 e1 1 e1 = u+ u+[g] u+ = u+[g]: tents u§ := (1=2)(1 § e2). The relationship between the e1 1 e1 1 standard basis (1) and the spectral basis (3) is directly ex- By taking the e1-conjugate of the last equation, we get pressed by µ ¶ 0 1 0 1 0 1 1 e1 1 1 0 0 1 u+ u¡ g (1 e1)u¡ = u¡[g]: e1 B e C B 0 1 1 0 C Be u C B 1 C = B C B 1 +C @ e A @ 1 0 0 ¡1 A @e u A Adding the last two equations gives the desired result 2 1 ¡ µ ¶ µ ¶ e1 e12 0 1 ¡1 0 u¡ g g e1 g e1g [g] = u+ e1 u+ + u¡ u¡: (7) e1g g g e1 g Noting e1u+ = u¡e1, it follows that µ ¶ Let µ ¶ 1 1 (1 e1)u+ = u+ + e1u+e1 = u+ + u¡ = 1: gk = (1 e1)u+[gk] e1 e1 Using this last relationship, and the fact that where [gk] is the matrix of the geometric number gk 2 G2 for k = 1; 2 as given in (2), (4) and (6). The algebra of real 2 u+ = u+ = e2u+; and u+e1u+ = 0 = u+e12u+; 2 £ 2 matrices MatR(2) is algebraically isomorphic to the geometric algebra G2 because [g1 + g2] = [g1] + [g2] and with similar relationships for u¡gu+, u+gu¡ and u¡gu¡, [g1g2] = [g1][g2] for all 2 £ 2 matrices [g1]; [g2] 2 MatR(2) we calculate for the geometric number g given in (2), and their corresponding geometric numbers g ; g 2 G . For µ ¶ µ ¶ 1 2 2 1 1 example, using (6) and u+e1u+ = 0, g = (1 e1)u+ g(1 e1)u+ µ ¶ µ ¶ e1 e1 1 1 µ ¶ µ ¶ g1g2 = (1 e1)u+[g1] (1 e1)u+[g2] e1 e1 ® + v (® + v)e1 1 = (1 e1)u+ u+ µ ¶ µ ¶ e1(® + v) e1(® + v)e1 e1 u+ u+e1u+ 1 µ ¶ µ ¶ = (1 e1)[g1] [g2] a + d c ¡ b 1 u+e1u+ u+ e1 = (1 e1)u+ (4) µ ¶ µ ¶ c + b a ¡ d e1 1 1 = (1 e1)u+[g1][g2] = (1 e1)u+[g1g2] e1 e1 = (a + d)u+ which shows that [g g ] = [g ][g ]. + (c ¡ b)e u + (c + b)e u + (a ¡ d)u : (5) 1 2 1 2 1 ¡ 1 + ¡ Extending the real number system R to include the new We say that anticommuting square roots e1; e2 of +1 is well defined since µ ¶ the resulting geometric number system G2 is algebraically a + d c ¡ b [g] = isomorphic to the 2 £ 2 matrix algebra MatR(2). Indeed, c + b a ¡ d each geometric algebra Gp;q is either algebraically isomor- is the matrix of the geometric number phic to a real or complex square matrix algebra, or matrix subalgebra, of the appropriate dimension, [5, p.205]. Next, g = (a + be12) + (ce1 + de2) 2 G2 we consider the case of the geometric algebra G3 of space. Rev. Mex. Fis. 61 (2015) 211–223 1 GEOMETRY OF SPIN 2 PARTICLES 213 1.2. Geometric algebra of space Since the unit trivector i commutes with all the elements of G3, we can simply complexify the spectral basis (3) of G2 2 The geometric algebra G2 of the xy-plane in R can be ex- to obtain the spectral basis of G3, 3 tended to the geometric algebra G3 of the space R simply by µ ¶ extending G2 to include e3, a new square root of +1, which 1 G3 = spanf u+(1 e1)g anticommutes with both e1 and e2. Naturally, e3 has the ge- e1 ometric interpretation of a unit vector along the z-axis. The µ ¶ u+ e1u¡ standard basis of G3, with respect to the coordinate frame = spanf g; (13) 3 e1u+ u¡ fe1; e2; e3g of the Euclidean space R , is where in this case u := (1=2)(1§e ).
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