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 ∈ 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 ee1φ = 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 φ ∈ 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 X∞ 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]. √ 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 = span{1, e1, e2, e12}. (1) 212 G. SOBCZYK

With respect to this basis a geometric number g ∈ 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 = α + v = (a + bi) + (ce1 + de2) (2) 1 The interesting thing about equation (6) is that it can be where α = 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 ∈ R2 is a vector in the Euclidean 1 2 g. To accomplish this we define the e -conjugate of g ∈ G plane of the bivector i = e . It can also be easily checked 1 2 12 by that vα = αv, where α = a − βi is the (complex) conjugate ge1 := e ge . of α. It is this property that gives α 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 = span{ u+(1 e1)} e1 respectively, gives µ ¶ µ ¶ u+ e1u− 1 = span{ }, (3) u+ g(1 e1)u+ e1u+ u− e1 µ ¶ µ ¶ or G2 = span{u+, e1u+, e1u−, u−}, 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 ¶       1 e1 1 1 0 0 1 u+ u− g (1 e1)u− = u−[g]. e1  e   0 1 1 0  e u   1  =    1 +  e   1 0 0 −1  e u  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 ∈ 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] ∈ MatR(2) we calculate for the geometric number g given in (2), and their corresponding geometric numbers g , g ∈ 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 α + v (α + v)e1 1 = (1 e1)u+ u+ µ ¶ µ ¶ e1(α + v) e1(α + 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) ∈ 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 = span{ u+(1 e1)} 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 = span{ }, (13) 3 e1u+ u− {e1, e2, e3} of the Euclidean space R , is where in this case u := (1/2)(1±e ). We can then directly G : =G(R3)=span{1, e , e , e , e , e , e , e }. ± 3 3 1 2 3 12 13 23 123 (8) apply the analogous formulas (4), (5), (6), and (7) to elements g ∈ G merely by allowing the values of a, b, c, d to be of the The elements e , e , e are unit bivectors in the xy-, xz-, 3 12 13 23 form s + it where s, t ∈ R and i = e . The famous Pauli and yz-planes and generate rotations in those planes, and 123 matrices of the coordinate frame {e , e , e } are simply ob- i := e := e e e is the unit trivector or oriented vol- 1 2 3 123 1 2 3 tained using the spectral basis (13), with u = (1/2)(1±e ), ume element of R3. The unit trivector i also has the property ± 3 getting that i 2 = −1, as follows from µ ¶ µ ¶ 0 1 0 −i 2 2 2 [e ] = , [e ] = i = e1(e2e3)e1(e2e3) = e1(e2e3) = −1. 1 1 0 2 i 0 µ ¶ Any geometric number g ∈ G can be written in the form 1 0 3 [e ] = −i[e ][e ] = . (14) 3 1 2 0 −1 X3 g = αkek (9) In this representation, the imaginary unit i acquires the k=0 geometric interpretation of the oriented volume element in G , as follows from where e := 1 and α = a + ib for a , b ∈ R and where 3 0 k k k k k µ ¶ 0 ≤ k ≤ 3. The conjugation known as the reverse g† of the 1 0 [e ] = [e ][e ][e ] = i . (15) geometric number g, is defined by reversing the orders of all 123 1 2 3 0 1 the products of the vectors that make up g, giving In terms of the geometric product, all other products in X3 the geometric algebra are defined. For example, given vec- † g = αkek. (10) 1 3 tors a, b ∈ G3 ≡ R , k=0 ab = a · b + a ∧ b ∈ G0+2, (16) In particular, writing g = s + v + B + T, the sum of a real 3 number s ∈ G0, a vector v ∈ G1, a bivector B ∈ G2 and 3 3 3 where a · b := (1/2)(ab + ba) ∈ R is the symmetric inner a trivector T ∈ G3, g† = s + v − B − T as can be easily 3 product and a ∧ b := (1/2)(ab − ba) is the antisymmetric checked. outer product of the vectors a and b, respectively. The outer Another conjugation widely used in geometric algebra is product satisfies a∧b = i(a×b), expressing the duality rela- the grade inversion, obtained by replacing each vector in a tionship between the standard Gibbs-Heaviside cross product product by its negative. It corresponds to an inversion in the a × b and the outer product a ∧ b ∈ G2. In what follows the origin, otherwise known as a parity inversion. For the geo- 3 outer product a ∧ b ∧ c of three vectors is also used. Similar metric number g given in (9), the grade inversion is to (16), we write X3 − a(b ∧ c) = a · (b ∧ c) + a ∧ (b ∧ c), (17) g := α0 − αkek. (11) k=1 where When g is written in the form g = s + v + B + T, the grade 1¡ ¢ inversion takes the form g− = s − v + B − T. a · (b ∧ c) : = a(b ∧ c) − (b ∧ c)a 2 Alternatively, we can obtain the geometric algebra G3 by 1 extending or complexifying G2 to include a commuting imag- = −a × (b × c) ∈ G3, inary i, which we subsequently identify as the unit trivector. We write and 1¡ ¢ a ∧ (b ∧ c) : = a(b ∧ c) + (b ∧ c)a G3=G2(i):=span{1, e1, e2, e21i, e12, −e2i, e1i, i}, (12) 2 = a · (b × c)i ∈ G3. and then define e3 := e21i, e13 = −e2i and e23 = e1i. 3

Rev. Mex. Fis. 61 (2015) 211–223 214 G. SOBCZYK

A much more detailed treatment of G3 is given in [5,Chp.3], where h i0 denotes real scalar part of the enclosed expres- and in Ref. 7 I explore the close relationship that exists be- sion, and using a·b = habi0 that we found in (16). A closely tween geometric algebras and their matrix counterparts. Ge- related calculation gives almost the same answer ometric algebra is developed in Ref. 5 as the geometric com- ˆ † ˆ ˆ ˆ ˆ ˆ pletion of the real number system, providing a new founda- (aˆ+b+) (aˆ+b+) = (b+aˆ+)(aˆ+b+) = b+aˆ+b+ tion for much of mathematics and physics. bˆ bˆ = + (1 + aˆ)(1 + bˆ) = + (1 + aˆ + bˆ + aˆbˆ) 4 4 1.3. Properties of idempotents and nilpotents ˆ b+ ˆ ˆ ˆ 1 ˆ ˆ 2 = (1 + aˆ + b − baˆ + 2aˆ · b) = (1 + aˆ · b)b+, An idempotent p ∈ G3 has the deﬁning property p = p. 4 2 Of course +1, 0 are idempotents, and so is p = (1/2)(1 + √ ˆ but multiplied by the idempotent b+. In summary, 2e1 + i e2). For a general geometric number g = ω + m + in ∈ G3 to be an idempotent, we must have ˆ † ˆ ˆ ˆ ˆ ˆ (aˆ+b+) (aˆ+b+) = b+aˆ+b+ = P rob(aˆ+|b+)b+, (22) g2 = ω2 + m2 − n2 + 2i m · n ˆ ˆ † † † where P rob(aˆ+|b+) := (1/2)(1+aˆ·b) and (g1g2) = g2g1 g , g ∈ G + 2ω(m + i n) = ω + m + i n, (18) for all 1 2 3 is the operation of reverse deﬁned in (10). Another important property that easily follows is or equivalently, 1 bˆ mbˆ = bˆ m(1 + bˆ) + + 2 + ω2 + m2 − n2 + 2i m · n = ω 1 ˆ ˆ ˆ ˆ ˆ = b+(m − bm + 2b · aˆ) = (b · m)b+. (23) and 2 (2ω − 1)(m + i n) = 0. A geometric number g is a nilpotent if g2 = 0. A similar analysis to (18) for a general idempotent shows that a nilpo- If m + i n 6= 0, it follows that ω = (1/2), m · n = 0, and tent N has the canonical form m2 − n2 = (1/4). Equivalently, a geometric number A, not 1 1 equal to 0 or 1, is an idempotent iff N = (r + i s) = r (1 + i r−1s) = rnˆ = nˆ r, (24) 2 2 + − 1 −1 1 A = (1 + m + i n), where where nˆ := i r s ∈ G3 for the orthogonal vectors r · s = 0 2 and r2 = s2. Thus, every nilpotent N has associated with it 2 2 m − n = 1, and m · n = 0, (19) a corresponding idempotent nˆ+.

1 for m, n ∈ G3. It follows immediately that any idempotent A has the important cannonical form 2. Quantum mechanics of spin 1 1 The observable corresponding to the spin of an electron in A = (1 + m + i n) = m (1 + m−1 + i m−1n) 2 2 a coordinate frame {e1, e2, e3} is a unit vector mˆ = m1e1 +m e +m e in the Euclidean space R3 of that frame. The 1 2 2 3 3 = m (1 + aˆ) = maˆ+, (20) Hermitian matrix of the observable mˆ is 2 µ ¶ µ ¶ m3 m1 − im2 m3 m− −1 −1 1 [mˆ ] = = , where aˆ = m + i m n ∈ G3, aˆ+ = (1/2)(1 + aˆ) and m1 + im2 −m3 m+ −m3 aˆ · m = 1. We use the notation aˆ± = (1/2)(1 ± aˆ) for the mutually annihiliating idempotents aˆ+ and aˆ−. as given in (13) and (14), with u± = (1/2)(1 ± e3) and For the idempotents aˆ±, note that m± = m1 ± im2, where µ ¶ 1 aˆaˆ+ = aˆ+ and aˆaˆ− = −aˆ−. mˆ = (1 e1)u+[mˆ ] e1

Because of these properties, we would like to identify aˆ± as = m3u+ + m−e1u− + m+e1u+ − m3u−. (25) the two pure eigenprojector spin states of an electron. Unfor- tunately, as is discussed in the next section, this is not quite By a geometric spinor |αi with respect to correct. u+ = (1/2)(1 + e3), we mean ˆ Given a second idempotent b+, we calculate √ |αi := 2(α0 + α1e1)u+ ⇐⇒ hα| 1 √ ˆ ˆ † 2haˆ+b+i0 = h(1 + aˆ)(1 + b)i0 := |αi = 2u (α + α e ) (26) 2 + 0 1 1 1 where α , α ∈ G0+3. The spinor |αi is said to be non- = (1 + aˆ · bˆ), (21) 0 1 3 2 degenerate if in addition α0α0 + α1α1 6= 0. Spinors have

Rev. Mex. Fis. 61 (2015) 211–223 1 GEOMETRY OF SPIN 2 PARTICLES 215 the important superposition property that if |αi and |βi are and the unit vector aˆ = meˆ 3mˆ = a1e1 + a2e2 + a3e3 is spinors with respect to u+ then specified by 0+3 |ωi := λ1|αi + λ2|βi for λ1, λ2 ∈ G3 (27) α0α1 + α0α1 i(α0α1 − α1α0) a1 = , a2 = , is also a spinor with respect to u+. This is precisely the prop- α0α0 + α1α1 α0α0 + α1α1 erty that naked idempotents m+, n+ do not have. α0α0 − α1α1 a3 = . (38) Given two spinors |αi and |βi, the geometric product α0α0 + α1α1 hα||βi = 2u+(α0 + α1e1)(β0 + β1e1)u+ For a normalized spinor (31), = 2(α0β0 + α1β1)u+ (28) 2 α α + α α 1 m2 = = 0 0 1 1 = . (39) is also a spinor. The inner product hα|βi is defined to be the 1 + aˆ · e3 α0α0 α0α0 scalar and pseudoscalar parts of the geometric product of the Using the above information (34) and (35) for the idem- spinors, D E potent A, the spinor |αi takes the canonical form hα|βi := hα||βi = α0β0 + α1β1. (29) √ √ √ 0+3 2 |αi = 2α0A = 2α0mu+ = 2α0m aˆ+u+. (40) An important observation which will be used later is that D E ¡ ¢†¡ ¢ The matrix [aˆ+] of aˆ+ is the density matrix of the pure state hα||βi hα||βi = 2hα|βihα|βi, (30) 0+3 |αi. By taking the reverse of the spinor equation (40), we get as can be easily verified. The spinor |αi is said to be a nor- √ √ hα| = 2α u m = 2α m2u aˆ , (41) malized spinor if 0 + 0 + +

hα|αi = α0α0 + α1α1 = 1. (31) and find that Traditionally, a 2-component column spinor is 1 |αihα| = α α m2mˆ u mˆ = aˆ . (42) µ ¶ 2 0 0 + + α0 [α]2 := where α0, α1 ∈ C. (32) α1 It is interesting to note that a Cartan spinor is specifed by √ In general, for a spinor |αi := 2(α0 + α1e1)u+ with ∗ − |αie1|αi = |αie1hα| α0 6= 0, we can write √ ³ ´ √ 2 2 2 2 α1 = (α0 − α1)e1 + i(α0 + α1)e2 − 2α0α1e3 |αi = 2α0 1 + e1 u+ = 2α0A (33) α0 ³ ´ α0 2 = − mˆ (e1 + ie2)mˆ = − 2 aˆ+meˆ 1mˆ for A = 1 + (α1/α0)e1 u+. It is easily checked that A is α0 α0 an idempotent, so that A can be expressed in the cannonical and represents a null complex vector, [8,p.41]. form (20), ³ ´ There are several other operations on spinors which are 1 α1 α1 important. By multiplying the spinor equation (40) on the A = 1 + e1 − i e2 + e3 2 α0 α0 left and right by e3, we find that 1 1 √ = (1 + m + i n) = m (1 + aˆ) = mu , (34) e3 2 2 + |αi : = e3|αie3 = 2α0e3me3u+ √ 2 2 since aˆ = (m + i mn/m ) = e3. Another closely related = 2α0m e3aˆ+e3u+. (43) canonical form for the idempotent A, with the help of (23), is 2 Multiplying the spinor equation (40) on the left and right A = mu+ = mmˆ mˆ u+mu+ = m aˆ+u+, (35) by e1, we get where we have redefined the unit vector aˆ := meˆ 3mˆ . √ e1 All of the above quantities can be expressed in terms of |αi : = e1|αie1 = 2α0e1me1u− α0 and α1. We find that √ = 2α m2e aˆ e u . (44) 1 ³ 0 1 + 1 − m = (α1α0 + α0α1)e1 2α α e3 0 0 The spinor |αi is called the e3-conjugate of |αi, and ´ e1 the spinor |αi is called the e1-conjugate of |αi. Using (11) + i(α0α1 − α1α0)e2 + e3, (36) − and (40) the grade inversion |αi of the spinor |αi ∈ G3 is 1 ³ √ ³ ´− √ n = i(α0α1 − α1α0)e1 − 2α0α0 |αi : = 2 (α0 + α1e1)u+ = 2(α0 − α1e1)u− ´ √ √ − (α α + α α )e , (37) 2 0 1 1 0 2 = − 2α0mu− = 2α0m aˆ−u−. (45)

Rev. Mex. Fis. 61 (2015) 211–223 216 G. SOBCZYK

Given normalized spinors |αi, |βi, we calculate with the Using (48) and (49), we also calculate help of (40) and (23), ˆ ˆ h0|b|0i = b3, and h1|b|1i = −b3, (50) hα|βi = 2αβ0u+mnu+ ³ ´ and = 2α0β0 m · n + i(m × n) · e3 u+ = 0 ˆ h1|b|0i = b+ = b1 + ib2, and iff m·n = 0 and (m×n)·e3 = 0. Expressed in terms of the ˆ ˆ ˆ associated spinor directions m, n, the inner product hα|βi is h0|b|1i = b− = b1 − ib2. (51) quite different than the component expression (29). A great √ simplification occurs when we calculate hα|βihα|βi, finding Often an electron in the spinor state |0i = 2√u+ is said to be with the help of (30) and (39), up, and an electron in the spinor state |1i = 2e1u+ is said D E 1 ¡ ¢†¡ ¢ to be down. Using (50), the average value of the observable hα|βihα|βi = hα||βi hα||βi ˆ ˆ 2 0+3 b in the state |0i is b3, and the average value of b in the state |1i is −b . 1D ¡ ¢ E 3 = hβ| |αihα| |βi We can also carry out the above calculation using spinor 2 0+3 operators. Following [10,p.63], given the spinor |αi, the D E 1 ˆ spinor operator ψ of |αi is defined by = hβ|aˆ+|βi = (1 + aˆ · b). 0+3 2 1 ³ ´ Given an observable aˆ, suppose that an electron has been ψ : = √ |αi + |αi− prepared in the pure spin state |αi. Then the probability of 2 finding the electron in the pure spin state |βi is = m(α0u+ − α0u−) ∈ SU(2), (52)

P rob(|αi, |βi) : = hα|βihα|βi where the inversion |αi− has been defined in (45). The spinor + 1 operator is an even multivector in G3 and hence invarient un- = (1 + aˆ · bˆ) = 2haˆ bˆ i , (46) 2 + + 0 der a parity swap. It retains the normalization property that ψψ† = det[ψ] = 1, and allows us to calculate the associated just as we have already seen in (21) and (22). Immediately direction aˆ of |αi as a rotation of the unit vector e . With the after the measurment is carried out, the electron will be in the 3 help of (22), (38) and (52), we find that pure state |βi. Identity (46) is independent of the basis spinor u |αi + that was used in the definition (26) of the spinors and ψe ψ† = m(α u − α u )e (α u − α u )m |βi. Clearly, the spinor states |αi and |βi are orthogonal iff 3 0 + 0 − 3 0 + 0 − ˆ aˆ = −b. = m(α0u+ + α0u−)(α0u+ − α0u−)m Given a spinor state |αi, the average value of the observ- = α α m2mˆ (u − u )mˆ = aˆ, able bˆ in that state is defined by hα|bˆ|αi. Using (22), (23), 0 0 + − (53) (39), (40), we calculate √ √ and also ˆ ˆ ˆ ˆ 1 b|αi = 2α0bmu+ = 2α0(b · m + b ∧ m)u+, (47) ψu ψ† = ψ(1 + e )ψ† = aˆ . (54) + 2 3 + and The relationship (54) should be compared with (42). ˆ ˆ ˆ hα|b|αi : = 2hu+(mˆ bmˆ )u+i0+3 = (mˆ bmˆ ) · e3 ˆ ˆ = b · (meˆ 3mˆ ) = b · aˆ. (48) 3. Geometric algebra of spacetime

It follows that the average value lies in between the values The geometric algebra G3 of the coordinate frame 3 ±1. It is important to note that the final values in both (46) {e1, e2, e3} in R can be extended to the geometric alge- 1,3 and (48) in no way depend upon the idempotent u+, so u+ bra G1,3 of the pseudoeuclidean space R of Minkowski could be replaced by any other idempotent cˆ+ defined by any spacetime, by extending G3 by an additional square root γ0 unit vector cˆ, but we would then also have to redefine |αi. of +1 which anticommutes with each of the space vectors 2 In the special cases where e1, e2, e3. So γ0 satisfies γ0 = 1 and γ0ek = −ekγ0 for √ √ k = 1, 2, 3. Equivalently, the spacelike vectors |αi = |0i := 2u+ or |αi = |1i := 2e1u+, γ := e γ ⇐⇒ e = γ γ = γ , we have k k 0 k k 0 k0 √ ˆ together with the timelike vector γ , define the spacetime al- b|0i = 2(b3 + b+e1)u+ and 0 √ gebra ˆ b|1i = 2(b− − b3e1)u+. (49) G1,3 = gen{γ0, γ1, γ2, γ3}

Rev. Mex. Fis. 61 (2015) 211–223 1 GEOMETRY OF SPIN 2 PARTICLES 217 of the orthonormal spacetime frame {γµ| 0 ≤ µ ≤ 3} in and similarly γ31 = e13, it follows that R1,3. In summary, the spacetime vectors obey the rules e13u++ = u+−e13, e3u++ 2 2 γ = 1, γ = −1, γµγν = −γν γµ 0 k = u−+e3, e1u++ = u−−e1, (56) for µ 6= ν, µ, ν = 0, 1, 2, 3, and k = 1, 2, 3. Note also that where pseudoscalar 1 u : = (1 + γ )(1 − iγ ), +− 4 0 12 γ = γ γ γ = e 0123 10 20 30 123 1 u−+ : = (1 − γ0)(1 + iγ12), of G1,3 is the same as the pseudoscalar of the rest frame 4 {e1, e2, e3} of G3, and it anticommutes with each of the 1 u−− : = (1 − γ0)(1 − iγ12). spacetime vectors γµ for µ = 0, 1, 2, 3. 4

The forgoing shows that the geometric algebra G3 is nat- The idempotents u++, u+−, u−+, u−− are mutually anni- + urally isomorphic to the even subalgebra G1,3 of the space- hiliating in the sense that the product of any two of them is time algebra G1,3 generated by the bivectors γµγν of G1,3, zero, and partition unity [9]. Any element g ∈ G1,3 can be expressed in the form u++ + u+− + u−+ + u−− = 1. (57) g = G1 + G2γ0 where G1,G2 ∈ G3. In particular, a unit vector mˆ = m e + m e + m e in the geometric algebra 1 1 2 2 3 3 By the spectral basis of the Dirac algebra G1,3 we mean G3 of the rest frame {e1, e2, e3} becomes the unit spacetime the elements of the matrix bivector   1 2 e13 mˆ = (m1γ1 + m2γ2 + m3γ3)γ0 ∈ G1,3,   u++(1 − e13 e3 e1)  e3  1 e with the spacelike vector m1γ1 + m2γ2 + m3γ3 ∈ G1,3 as a 1 factor.   u++ −e13u+− e3u−+ e1u−− In the above, we have carefully distinguished the rest   e13u++ u+− e1u−+ −e3u−−  frame {e1, e2, e3} in the spacetime algebra G1,3. Any =   . (58) 0 0 0 e3u++ e1u+− u−+ −e13u−− other rest frame {e1, e2, e3} can be obtained by an ordi- e1u++ −e3u+− e13u−+ u−− nary space rotation of the rest frame {e1, e2, e3} followed by a Lorentz boost. In the spacetime algebra G1,3, this Any geometric number g ∈ G1,3 can be written in the is equivalent to defining a new frame of spacetime vectors form   0 1 {γµ| 0 ≤ µ ≤ 3} ⊂ G1,3, and the corresponding rest frame 0 −e  {e0 = γ 0γ 0| k = 1, 2, 3} R3 g = (1 e e e )u [g]  13 (59) k k 0 of a Euclidean space moving 13 3 1 ++  e  with respect to the Euclidean space R3 defined by the rest 3 e1 frame {e1, e2, e3}. The way we introduced the geometric where [g] is the Dirac matrix corresponding to the geometric algebras G3 and G1,3 may appear novel, but they perfectly number g. In particular, reflect all the common relativistic concepts [5,Chp.11].   The well-known Dirac matrices can be obtained as a sub- 1 0 0 0 algebra of the 4×4 matrix algebra Mat (4) over the complex  0 1 0 0  C [γ ] =   , numbers. We first define the idempotent 0  0 0 −1 0  0 0 0 −1 1 u : = (1 + γ )(1 + iγ )   ++ 4 0 12 0 0 0 −1  0 0 −1 0  1 [γ ] =   , = (1 + iγ12)(1 + γ0), (55) 1   (60) 4 0 1 0 0 √ 1 0 0 0 where the unit imaginary i = −1 is assumed to com- and mute with all elements of G1,3. Whereas it would be nice   to identify this unit imaginary i with the pseudoscalar ele- 0 0 0 i   ment γ0123 = e123 as we did in G3, this is no longer possible  0 0 −i 0  [γ2] =   , since γ0123 anticommutes with the spacetime vectors γµ as 0 −i 0 0 previously mentioned. i 0 0 0   Noting that 0 0 −1 0  0 0 0 1  γ12 = γ1γ0γ0γ2 [γ ] =   . 3  1 0 0 0  = e2e1 = e21 0 −1 0 0

Rev. Mex. Fis. 61 (2015) 211–223 218 G. SOBCZYK   It is interesting to see what the representation is of the 1   basis vectors of G3. We find that for k = 1, 2, 3, −e13 ψ = (1 e13 e3 e1)u++[ψ]α   µ ¶  e3  [0]2 [ek]2 e [ek]4 = [γk][γ0] = and 1 [ek] [0]2 2 + µ ¶ = α1 + α2e13 + α3e3 + α4e1 ∈ G1,3. (64) [0]2 [1]2 [e123]4 = i , To establish the last equality above, with the help of (57), [1] [0]2 2 note that where the outer subscripts denote the order of the matrices ψ = ψu++ + ψu+− + ψu−+ + ψu−−, and, in particular, [0]2, [1]2 are the 2 × 2 zero and unit matrices, respectively. The last relationship shows that the i occur- and then succesively show that ³ ´ ing in the Pauli matrix representation (15), which represents√ the oriented unit of volume, is different than the i = −1 ψu±± = α1 + α2e13 + α3e3 + α4e1 u±±, which occurs in the complex matrix representation (60) of for each of different sign combinations ++, +−, −+, −−. the of Dirac algebra. + As an even geometric number in G1,3, ψ generates A Dirac spinor is a 4-component column matrix [ϕ]4, Lorentz boosts in addition to ordinary rotations in the   Minkowski space R1,3. The amazing property of (64) is ϕ 1 that where as we started by formally introducing the com-  ϕ  √ [ϕ] :=  2  for ϕ ∈ C. (61) plex number i = −1, in order to represent the Dirac 4  ϕ  k 3 gamma matrices, we have ended up with a spinor operator ϕ4 + √ ψ ∈ G1,3 in which the role of i = −1 is taken over by Following [10,p.143], we use the Dirac spinor to construct the γ21 = e12 ∈ G3. This is the key idea in the Hestenes the (matrix) spinor operator representation of the Dirac equation [18].   Although we√ have formally eliminated the need for the ϕ1 −ϕ2 ϕ3 ϕ4 artificial i = −1 in the Dirac spinor by replacing it with  ϕ ϕ ϕ −ϕ  [ψ] =  2 1 4 3  . (62) the bivector γ21 = e12 in the spinor operator, such a quan-  ϕ ϕ ϕ −ϕ  3 4 1 2 tity occuring in a real√ geometric algebra is unacceptable. ϕ4 −ϕ3 ϕ2 ϕ1 The imaginary i = −1 has the effect of complexifying the geometric algebra G to the complex geometric alge- Unlike the Dirac spinor [ϕ] , the spinor operator [ψ] is invert- 1,3 4 bra G (C)=eMat (C), [10,p.217]. We might, instead, search ible iff det[ψ] 6= 0. We find that 4 4 for a larger real geometric algebra containing the geometric det[ψ] = r2 + 4a2 ≥ 0, (63) algebra G1,3, just as we enlarged the geometric algebra G3 of space to the geometric algebra G1,3 of spacetime. The where real geometric algebra G2,3, obtained from the geometric algebra G1,3 by assuming the existence of a new timelike r = |ϕ |2 + |ϕ |2 − |ϕ |2 − |ϕ |2 and 0 1 2 3 4 vector γ0 which anticommutes with all the spacetime vec- ¡ ¢ tors γµ ∈ G1,3, has the required properties. The unit pseu- a = = ϕ1ϕ3 + ϕ2ϕ4 . 0 0 doscalar i := γ0γ0γ1γ2γ3 ∈ G2,3 has√ all the desired alge- Where-as the spinor operator [ψ] obviously contains the braic properties of the imaginary i = −1, commuting with same information as the Dirac spinor [ϕ]4, we will shortly all the elements of G1,3 and having square −1. All of the pre- see that it acquires the geometric interpretation of an even vious arguments in the complex algebra G4(C) remain valid 0 multivector in G+ . in G2,3, but now i is the oriented element of volume of the 1,3 2,3 Noting that 5-dimentional pseudoeclidean space R . However, the real geometric algebras of the pseudoeu- 1 clidean spaces R4,1 and R0,5 also fit the bill, as is seen in u++γ21 = (1 + γ0)(γ21 + iγ12γ21) 4 Table I in [10, p.217]. In [11, p.326], Hestenes argues that 4,1 1 the geometric algebra G4,1 of R might be an even bet- = (1 + γ0)(i − γ12) = iu++ = γ21u++, + 4 ter choice because the even subalgebra G4,1 is isomorphic to G in exactly the same way that G+ is isomorphic to G . it follows that u ϕ = u <(ϕ ) + u iγ =(ϕ )i = 1,3 1,3 3 ¡ ++ k ¢ ++ k ++ 12 k In the paper, “Vector Analysis of Spinors” [12], the author u <(ϕ ) + γ =(ϕ ) and hence ++ k 21 k explores the interpretation of a geometric Pauli spinor on the

u++[ψ] = u++[ψ]α, Riemann sphere, ideas which he then generalizes in another paper to geometric Dirac spinors in the complex geometric where each of the elements αk in [ψ]α is deﬁned by αk = algebra G3(C) [13]. These real and complex geometric alge- + ϕk|i→γ21 . Using (59), the geometric number ψ ∈ G1,3 cor- bras open up many new possibilities for the interpretation of responding to the matrix spinor operator [ψ] is relativistic quantum mechanics.

Rev. Mex. Fis. 61 (2015) 211–223 1 GEOMETRY OF SPIN 2 PARTICLES 219 4. Tensor products of geometric numbers where we are using the qubit notation

Tensor products of geometric numbers, and their corresponding geometric algebras, are necessary for describing multi- |00i := 2u+ ⊗ u+, |01i := 2u+ ⊗ e1 u+ particle systems used, for example, in the construction of a quantum computer, or even in the much simpler “double slit” and, similarly, for the deﬁnitions of |10i and |11i. Taking the experiments. The resulting quantum entanglement, which reverse of the product state |αi ⊗ |βi gives implies the “spooky action-at-a distance” that even Einstein couldn’t stomach, is discussed in Sec. 5. ³ ´† By the tensor product G3 ⊗ G3 of the geometric alge- |αi ⊗ |βi = hα| ⊗ hβ| bra G3 with itself, we essentially mean two distinguishable copies of the same geometric algebra G3. We can express and this as ³ ´³ ´ G3 ⊗ G3 = {g1 ⊗ g2| g1, g2 ∈ G3}. (65) hα| ⊗ hβ| |αi ⊗ |βi := hα|αihβ|βi. In addition, the subspace consisting of everything which commutes with the entire algebra, called the center of G , 3 The product state |αi ⊗ |βi represents the state of two may be identiﬁed with the complex numbers unentangled electrons, where each electron is prepared in-

C ≡ span{1, i = e123}. dependently, and can be measured independently. A direct consequence of (48) is that for any values s, t ∈ [−1, 1] cho- The tensor products of g1, g2, g3, g4 ∈ G3 and α, β ∈ C sat- sen, directions mˆ and nˆ can always be found resulting in the isﬁes the rules average values

α(g1 ⊗ g2) = (αg1) ⊗ g2 = g1 ⊗ (αg2), (66) hα|mˆ |αi = s, and hβ|nˆ|βi = t. (αg1 + βg2) ⊗ g3 = α(g1 ⊗ g3) + β(g2 ⊗ g3), (67) and The states |S±i = |01i ± |10i are entangled states. In an entangled state if one of the electrons is measured to be up g ⊗ (αg + βg ) = α(g ⊗ g ) + β(g ⊗ g ). (68) 1 2 3 1 2 1 3 along a given direction, then the other will deﬁnitely be down Either the standard basis (8) or the spectral basis (13) can without the need for measurement. For any direction mˆ , the be tensored with itself to get the respective standard basis or averages of the states |S±i measured in the directions mˆ ⊗ 1 spectral basis of the tensor algebra G3 ⊗ G3 over C. By the and 1 ⊗ mˆ is 0, meaning that measurements of the ﬁrst and product of tensors g1 ⊗ g2 and g3 ⊗ g4, we mean second electrons along this direction are equally likely to be ³ ´³ ´ ±1. With the help of (49), we easily calculate g1 ⊗ g2 g3 ⊗ g4 = (g1g3) ⊗ (g2g4). (69)

The tensor product (65) satisﬁes the same algebraic rules (mˆ ⊗ 1)|S±i = mˆ |ui ⊗ |1i ± mˆ |1i ⊗ |0i (67) and (68) as the geometric product itself. This suggests ³ ´ = m |0i + m |1i ⊗ |1i extending the tensor product to commuting copies of the geo- 3 + ³ ´ metric algebra G1,3 of spacetime. In [14,15] the authors iden- ± − m3|1i + m−|0i ⊗ |0i tify the tensor product of N copies of G3 with commuting + ³ ´ copies of G1,3 in the geometric algebra GN,3N to represent the spinor operators of an N-qubit quantum computer. As ex- = m3|01i + m+|11i plained in Ref. 14, in order to get the usual tensor products of ³ ´ complex matrices that is used for multi-distinguishable spin ± − m3|10i + m−|00i , (1/2) particles, one must introduce a commuting idempotent to project down to only a single set of complex numbers. from which we ﬁnd that ³ ´ 5. Non-entangled and entangled states hS±|mˆ ⊗ 1|S±i = h01| ± h10| √ √ ³ ´ If |αi = 2(α0 +α1)u+ and |βi = 2(β0 +β1)u+ are normalized states, then the product state is given by the tensor m3|01i + m+|11i ∓ m3|10i ± m−|00i product = m − m = 0, ³√ ´ ³√ ´ 3 3 |αi ⊗ |βi = 2(α0 + α1e1)u+ ⊗ 2(β0 + β1e1)u+ and a similar calculation shows that hS±|1 ⊗ mˆ |S±i = 0. = α0β0|00i + α0β1|01i + α1β0|10i + α1β1|11i, (70) For a further insightful discussion of these results, see [1].

Rev. Mex. Fis. 61 (2015) 211–223 220 G. SOBCZYK

6. The Schrodinger,¨ Pauli, and Dirac Equa- for the electric field potential V and the magnetic field B = tions −i(∇ ∧ A) = ∇ × A of a particle of mass m with electric charge e, and where i = e123 is the unit pseudoscalar of G3. Spin and quantum entanglement have been studied by many Expanding Eq. (76), we find that authors. The Susskind Lectures [1] give a fascinating ac- 1 ³ e2 count, presenting clearly the issues that separate quantum i~∂ |αi = − ~2∇2 + A2 t 2mc c2 and classical mechanics. Another elementary presentation is e¡ ¢´ given in [16]. Of course, no account of quantum mechan- + i~ ∇A + A∇ |αi + eV |αi ics can be complete without at least a cursory presentation of c ³ the fundamental Schrodinger¨ and Dirac equations, the foun- 1 e2 e¡ ¢ = − ~2∇2 + A2 + i~ ∇ · A + 2A · ∇ dation upon which all of quantum mechanics is built. 2mc c2 c The Schrodinger¨ equation is obtained from the classical e ´ − ~ B |αi + eV |αi expression for the total energy H of a particle at the position c x = xe1 + ye2 + ze3, This is equivalent to the classical 2 × 2 matrix form of this equation, 1 2 1 2 H = mv + V = p + V, (71) ∂[α] 1 ³ e ´2 2 2m i~ 2 = i~[∇] + [A] [α] + eV [α] , (77) ∂t 2mc c 2 2 where p = pxe1 + pye2 + pze3 is the momentum vector of the particle and V is the potential, by way of the substitution obtained by taking the matrix of both sides of (76) and extracting the first column, where [ek] are the Pauli matrices p → −i~∇, (72) given in (14) and [ϕ]2 is the 2-component spinor (32). The matrix Eq. (77) can be expanded in the same way that we where ∇ := ∇x is the standard gradient with respect to the expanded Eq. (76). See reference [17,eqn(A.4)]. The reader 3 position x ∈ R , ~ = h/2π and h is Planck’s constant. The is referred to the Appendix for more technical details.√ ¡ Schrodinger¨ equation thus becomes Recall¢ that the spinor operator (52) is ψ = (1/ 2) |αi+ − ³ ´ |αi . If |αi satisfies the Schrodinger-Pauli¨ equation (76), ∂ϕ ~ − i~ = Hϕˆ = − ∇2 + V ϕ, (73) then |αi will satisfy the parity inversion of this equation, ∂t 2m ³ ´2 − 1 e − − −i~∂t|αi = i~∇ − A |αi + eV |αi . (78) for the Hamiltonian Hˆ = −(~/2m)∇2 + V , where 2mc c ϕ := ϕ(x, t) ∈ C. It then follows that the operator√ spinor ψ will satisfy the equa- When the potential V is independent of time, there will tion obtained by taking (1/ 2) times the sum of equations be a complete set of stationary states (76) and (78), giving the parity invarient Schrodinger-Pauli-¨ Hestenes equation −iEn(x)t/~ ϕn(x, t) = e 1 ³ e ´2 i~∂tψe3 = i~∇ + A ψu+ where the spatial wave function ϕ (x) satisfies the time in- 2mc c n ³ ´ dependent Schrodinger¨ equation 1 e 2 + i~∇ − A ψu− + eV ψ ³ ´ 2mc c ~ 2 ³ 2 ´ − ∇ + V ϕn(x) = Enϕn(x), (74) 1 e 2m = − ~2∇2 + A2 ψ 2mc c2 in terms of which the time dependent Schrodinger¨ equation i~e ¡ ¢ has the general solution + ∇ · A + 2A · ∇ ψe 2mc2 3 X −iEnt/~ ~e ϕ(x, t) = cnϕn(x)e , (75) − Bψ + eV ψ, n 2mc2 [17, eqn.(1.8)]. [16,p.122]. According to the Born interpretation of the wave The Schrodinger¨ equation (73) applies to all quantum function, the probability of finding the particle in the in- phenomena except magnetism and relativity. The magnetism finitesimal volume and spin of an electron are taken care of in the Schrodinger-¨ |d3x| = dxdydz is |ϕ(x, t)|2|d3x|. Pauli equation (76), or equivalently (77). In order to extend quantum mechanics to special relativity, we start with the ex- The Schrodinger¨ equation (73) is generalized to include pression for relativistic energy (E2/c2) − p2 = m2c2. Fol- spin in the Schrodinger-Pauli¨ equation [10,p.51]. Expressed lowing [19, (5.5)] and [10, p.136], we express the Dirac equa- in terms of our spinor (26) with respect to u+, tion in terms of the Dirac spinor operator (64), getting what is known as the Dirac-Hestenes equation 1 ³ e ´2 i~∂t|αi = i~∇ + A |αi + eV |αi (76) ~∂ψγ − eAψ = mψγ , 2mc c 21 0 (79)

Rev. Mex. Fis. 61 (2015) 211–223 1 GEOMETRY OF SPIN 2 PARTICLES 221 P 3 µ µ 1 where ∂ = γ (∂/∂x ), and an interaction with a elec- e1 µ=0 √ [|βi ] = [(β0 + β1e1)u−] tromagnetic field is included by way of the spacetime poten- 2 P3 µ µ ¶ µ ¶ µ ¶ tial A = γµA . µ=0 0 β1 0 0 0 β1 Just like for the Shrodinger-Pauli¨ equation (76), the clas- = = 0 β0 0 1 0 β0 sical Dirac equation can be retrieved from its spinor oper- µ ¶ β ator form (79). First, multiply both sides of the Eq. (76) so the right column spinor 1 is represented by the matrix on the right by the primitive idempotent u++, using that µ ¶ β0 0 β u++γ0 = u++ and u++γ21 = iu++, and using the spec- 1 . Unlike with ordinary 2-component spinors (32), tral basis relations (58), (59), and (60). Finally, extracting the 0 β0 first column on each side of the resulting matrix equation, we the full matrix [g] of a geometric number g ∈ G3 can be obtain recovered from the corresponding matrices of left and right column spinors, i~[∂][ϕ]4 − e[A][ϕ]4 = m[ϕ]4 (80) √ 1 e1 where [ϕ] is the Dirac spinor (61) and i = −1. To show g = gu+ + gu− = √ (|αi + |βi ) 4 2 the full equivalence of (80) and (79), one must also show µ ¶ µ ¶ that (80) implies (79), which is taken up in the Appendix. α0 β1 1 = (1 e1)u+ . In [17-19], and in many other papers, Hestenes explores new α1 β0 e1 geometric features of the Dirac theory of the electron made Taking the matrix representation of (28), we find that possible by the spinor operator form (79) of the Dirac equation. [hα||βi] = 2[u+][(α0 + α1e1)][(β0 + β1e1)][u+]

= 2(α0β0 + α1β1)[u+] Appendix or µ ¶ µ ¶ In this Appendix, we explore in greater detail how our pre- 1 0 α0 α1 sentation of quantum mechanics is related to the more usual 0 0 α1 α0 µ ¶ µ ¶ µ ¶ approaches. β β 1 0 α β + α β 0 Multiplying the spectral basis (13) on the right by 0 1 = 0 0 1 1 , β1 β0 0 0 0 0 u+ = (1/2)(1 + e3) gives the left spinor basis from which it follows that µ ¶ µ ¶ D E 1 u+ e1u− u+(1 e1)u+ = u+ hα|βi = [hα||βi] e1 e1u+ u− 0+3 µ ¶ µ ¶ u+ 0 α0β0 + α1β1 0 = (81) = T r = α0β0 + α1β1. e1u+ 0 0 0 The matrix of the spinor operator (52) is given by Similarly, multiplying the spectral basis (13) on the right 1 ¡ ¢ by u− = (1/2)(1 − e3) gives the right spinor basis [ψ] = √ [|αi] + [|αi−] µ ¶ µ ¶ 2 1 u+ e1u− µ ¶ µ ¶ u+(1 e1)u− = u− α0 0 0 −α1 e1 e1u+ u− = + µ ¶ α1 0 0 α0 0 e1u− µ ¶ = . (82) α0 −α1 0 u− = α1 α0 Taking the matrix of each side of Eq. (26), we get √ Recalling (40), |αi = 2α0mu+, the calculation (42), is 1 equivalent to the matrix calculation √ [|αi] = [(α + α e )u ] 0 1 1 + 1 2 [|αihα|] = α α [m][u ][m] = [aˆ ] µ ¶ µ ¶ µ ¶ 2 0 0 + + α α 1 0 α 0 = 0 1 = 0 α α 0 0 α 0 where the matrices [m], [u+] and [aˆ+] are speciﬁed by 1 0 1 µ ¶ µ ¶ 1 α1 [m] = α0 , α0 α1 −1 so the left column spinor is represented by the matrix α0 µ ¶ α1 µ ¶ α0 0 √ 1 0 . The factor of (1/ 2) is inserted as a matter of [u+] = , α1 0 0 0 convenience in the way that we deﬁned the Hermitian inner µ ¶ 1 1 + a3 a− product (29). Similarly, taking the matrix of each side of the [aˆ+] = . 2 a+ 1 − a3 e1-conjugate (44), we get

Rev. Mex. Fis. 61 (2015) 211–223 222 G. SOBCZYK

Analogously to (82) for the spectral basis (13) of the Pauli corresponds to the Dirac spinor (61). algebra, we can extract any column of the spectral basis (58) The classical Dirac equation (80) is equivalent to the of the Dirac algebra. For example, to extract the third column Dirac-Hestenes equation (79) multiplied on the right by u++, we multiply (58) on the right by u−+ to get ³ ´   i~∂ψ − eAψ − mψ u++ = 0. (83) u++ −e13u+− e3u−+ e1u−−   e13u++ u+− e1u−+ −e3u−− Taking the spacetime inversion of this equation, by replac-   u−+  e3u++ e1u+− u−+ −e13u−− ing all spacetime vectors by their negatives, gives the parity e1u++ −e3u+− e13u−+ u−− related equation   0 0 e u 0 ³ ´ 3 −+ i~∂ψ − eAψ + mψ u = 0.  0 0 e u 0  −+ =  1 −+  .  0 0 u 0  −+ Two more equations are obtained, which are parity related to 0 0 e u 0 13 −+ (83), by taking the complex conjugate of both of these last two equations, giving Multiplying the spinor operator (64) on the right by u++ we obtain ³ ´ i~∂ψ + eAψ + mψ u+− = 0,

ψu++ = ϕ1u++ + ϕ2e13u++ + ϕ3e3u++ + ϕ4e1u++, and ³ ´ whose spinor matrix i~∂ψ + eAψ − mψ u−− = 0.   But the sum of these four parity related equations is exactly ϕ1 0 0 0   the parity invarient Dirac-Hestenes equation (79).  ϕ2 0 0 0    [ψu++] =  ϕ3 0 0 0    ϕ4 0 0 0 Acknowledgement

  I thank Professor Melina Gomez Bock of the Universidad de α1 0 0 0 Las Americas-Puebla for many fruitful hours discussing the    α2 0 0 0  intricacies of quantum mechanics. I am also grateful to Dr. =   Mod(u++) α3 0 0 0 Timothy Havel for invaluable remarks leading to extensive α4 0 0 0 revisions of an earlier version of this work.

∗. http://www.garretstar.com 10. P. Lounesto, Clifford Algebras and Spinors, 2nd Edition. Cam- 1. L. Susskind, 9 YouTube Lectures: Quantum Entanglements, bridge University Press, Cambridge, 2001. Part 1, (Stanford University 2008). 11. D. Hestenes, Clifford Algebra and the Interpretation of Quan- 2. T. Dantzig, NUMBER: The Language of Science, Fourth Edi- tum Mechanics, in Clifford Algebras and Their Applications tion, (Free Press, 1967). in Mathematical Physics, edited by J.S.R. Chisholm and A.K. Common, NATO ASI Series C: Mathematical and Physical Sci- 3. W.K. Clifford, On the Classiﬁcation of Geometric Algebras, in ences Vol. 183, D. Reidel (Publishing Company 1985). Mathematical Papers by William Kingdon Clifford, edited by Robert Tucker, London, Macmillan and Co., 1882. 12. G. Sobczyk, Vector Analysis of Spinors (to appear), http:// www. garretstar.com/nyuvas3-10-15.pdf 4. D. Hestenes, New Foundations for Classical Mechanics, 2nd Ed. (Kluwer 1999). 13. G. Sobczyk, Spacetime Algebra of Dirac Spinors, (2015) (to 5. G. Sobczyk, New Foundations in Mathematics: The Geometric appear), http://www.garretstar.com/diracspin03-24-15.pdf Concept of Number, Birkhauser,¨ (New York 2013). 14. T.F. Havel, J.L. Doran, Geometric Algebra in Quantum Infor- http://www.garretstar.com/ mation Processing, Contemporary Mathematics, ISBN-10: 0- 6. G. Sobczyk, Hyperbolic Number Plane, The College Mathe- 8218-2140-7, Vol. 305 (2002). matics Journal 26, No. 4 (1995) 268-280. 15. C. Doran, A. Lasenby, Geometric Algebra for Physicists, (Cam- 7. G. Sobczyk, Geometric Matrix Algebra, Linear Algebra and its bridge 2007). Applications, 429 (2008) 1163-1173. 16. D.J. Grifﬁths, Introduction to Quantum Mechanics, (Prentice 8. E. Cartan, The Theory of Spinors, Dover Publications, (New Hall, Inc. 1995). York 1981). 17. D. Hestenes and R. Gurtler, Local Observables in Quantum 9. D. Hestenes, Spacetime Algebra, (Gordon and Breach 1966). Theory, Am. J. Phys. 39 (1971) 1028-1038.

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18. D. Hestenes, Zitterbewegung in Quantum Mechanics, Found ORY”, in: A Symposium on the Mathematics of Phys- Physics 40 (2010) 1-54. http://geocalc.clas.asu.edu/pdf/ZBWin ical Space-Time, Facultad de Quimica, Universidad Na- QM15**.pdf http://geocalc.clas.asu.edu/pdf-preAdobe8/Loc cional Autonoma de Mexico, Mexico City, 67-96, (1981). ObsinQT.pdf http://geocalc.clas.asu.edu/pdf/Geom Dirac.pdf 19. D. Hestenes, “GEOMETRY OF THE DIRAC THE-

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