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Part I: Vector Analysis of Spinors

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Part I: Vector Analysis of Spinors Part I: Vector Analysis of Spinors Garret Sobczyk Universidad de las Am´ericas-Puebla Departamento de F´ısico-Matem´aticas 72820 Puebla, Pue., M´exico http://www.garretstar.com July 24, 2015 Abstract Part I: The geometric algebra G3 of space is derived by extending the real number system to include three mutually anticommuting square roots of +1. The resulting geometric algebra is isomorphic to the algebra of complex 2 2 matrices, × also known as the Pauli algebra. The so-called spinor algebra of C2, the language of the quantum mechanics, is formulated in terms of the idempotents and nilpotents of the geometric algebra G3, including its beautiful representation on the Riemann sphere, and a new proof of the Heisenberg uncertainty principle. In Part II: Space- time Algebra of Dirac Spinors, the ideas are generalized to apply to 4-component Dirac spinors, and their geometric interpretation in spacetime. Keywords: bra-ket formalism, geometric algebra, Schr¨odinger-Pauli equation, spinor, spacetime algebra, Minkowski spacetime, Riemann sphere. AMS Subject Classication 2010: 15A66, 81P16 1 Introduction Three dimensional Gibbs-Heaviside vector analysis was developed early in the 20th Century, before the development of relativity and quantum mechanics. What is still not widely appreciated is that the Gibbs-Heaviside formalism can be fortified into a far arXiv:1507.06608v1 [math-ph] 21 Jul 2015 more powerful geometric algebra which serves the much more sophisticated needs of relativity theory and quantum mechanics. The associative Geometric algebra is viewed here as the natural completion of the real number system to include the concept of direction. We assume that the real numbers can be always be extended to include new an- ticommuting square roots of +1 and 1. The new square roots of +1 represent or- thogonormal Euclidean vectors along− the xyz-coordinate axes, whereas new square roots of 1 represent orthogonormal pseudo-Euclidean vectors along the coordinate axes of more− general pseudo-Euclidean spacetimes. We shall primarily be interested in 3 the geometric algebra G3 of the ordinary 3-dimensional space R of experience, but the 1 interested reader may pursue how this geometric algebra can be factored into the space- 1,3 time algebra G1,3 of the pseudo-Riemannian space R of 4-dimensional Minkowski spacetime, [1, Ch.11]. One of the most important concepts in quantum mechanics is the concept of spin, and the treatment of spin has led to many important mathematical developments, start- ing with the Pauli and Dirac matrices in the early development of quantum mechanics, to the development of the differential forms, geometric algebras, and other more spe- cialized formalisms, such as the twistor formalism of Roger Penrose [2, Ch.33]. We 3 show here how the geometric algebra G3 of 3-dimensional Euclidean space R has all the algebraic tools necessary to give a clear geometrical picture of the relationship be- tween a classical 2-component spinor in the complex plane, and a pointon the Riemann sphere obtained by stereographic projection from the South Pole. So let’s get started. 2 Geometric algebra of space We extend the real number system R to include three new anticommuting square roots e1,e2, e3 of +1, which we identifyas unit vectorsalong the x- y- and z-axis of Euclidean space R3. Thus, 2 2 2 e = e = e = 1, and e jk := e jek = eke j = ek j 1 2 3 − − for 1 j < k 3. We assume that the associative and distributive laws of multiplica- tion of≤ real numbers≤ remain valid in our geometrically extended number system, and give the new quantities I := e23, J := e13 and K := e12 the geometric interpretation of directed plane segments, or bivectors, parallel to the respective yz-, xz- and xy-planes. Every unit bivector is the generator of rotations in the vector plane of that bivector, and this property generalizes to bivectors of the n-dimensional Euclidean space Rn. We leave it for the reader to check that I,J,K satisfy exactly the same rules as Hamilton’s famous quaternions, but now endowed with the geometric interpretation of oriented bivectors, rather than Hamilton’s original interpretation of these quantities as the unit vectors e1,e2,e3. Whereas the unit bivectors I,J,K satisfy I2 = J2 = K2 = 1, the new quantity − i := e123 := e1e2e3 is a unit trivector, or directed volume element. We easily calculate, with the help of the associative and anticommutitive properties, 2 2 2 2 2 i = (e1e2e3)(e1e2e3)= e e2e3e2e3 = e e e = 1, 1 − 1 2 3 − so the unit trivector i has same square minus one as do the bivectors I,J,K. The important Euler identity θ ei = cosθ + isinθ for θ R, depends only upon the algebraic property that i2 = 1, and so is equally ∈ − valid for the unit bivectors I,J,K. For the unit vectors ek, we have the hyperbolic Euler identities ekφ e = coshφ + ek sinhφ 2 Scalars Vectors a^ b b i a Bivectors c b a a b c Trivectors Geometric numbers of Space Figure 1: Geometric numbers of space for φ R and k = 1,2,3. All of these identities are special cases of the generalalgebraic definition∈ of the exponential function ∞ X n eX ∑ = coshX + sinhX, ≡ n=0 n! [1, Chp. 2] and [3]. 3 The standard basis of the 2 = 8 dimensional geometric algebra G3, with respect 3 to the coordinate frame e1,e2,e3 of the Euclidean space R , is { } 3 G3 := G(R )= span 1,e1,e2,e3,e12,e13,e23,e123 . (1) { } The geometric numbers of 3-dimensional space are pictured in Figure 1. Alternatively, we can obtain the geometric algebra G3 by its representation as a 2 2 matrix algebra MatC(2) over the complex numbers. We define the mutually × 1 anihiliating idempotents u := (1 e3), and note the fundamental relationships ± 2 ± u+u = 0, u+ + u = 1, u+ u = e3, and e1u+ = u e1. (2) − − − − − Since the unit trivector i commutes with all the elements (is in the center) of G3, the spectral basis of G3, over the fomally complex numbers C = spanR 1,i , is specified by { } 1 u+ e1u G3 = span u+ (1 e1 ) = span − . (3) {e1 } {e1u+ u } − The relationship between the standard basis (1) and the spectral basis (3) is directly 3 expressed by 1 10 0 1 u+ e 01 1 0 e u 1 = 1 + . (4) e2 0 i i 0 e1u − − e3 1 0 0 1 u − − For example, using the relationships (2), the spectral basis (3), and the fact that e2 = e123e1e3 = ie1(u+ u ), − − the famous Pauli matrices of the coordinate frame e1,e2,e3 are simply obtained, getting { } 0 1 0 i 1 0 [e1] := , [e2] := − , [e3] := i[e1][e2]= . (5) 1 0 i 0 − 0 1 − Indeed, the matrix representation [g] MatC(2) of any geometric number g G3 is ∈ ∈ 1 g = (1 e1 )u+[g] . (6) e1 For example, the unit vector e2 of the Pauli matrix [e2] in (5), is specified by 0 i 1 e2 = (1 e1 )u+ − = ie1u+ ie1u = ie1(u+ u ) i 0 e1 − − − − in agreement with (4). The proof of the isomorphism of the complex matrix algebra MatC(2) and the geometric algebra G3 is left to the reader. For a further discussion, see [1, p.79]. Any geometric number g G3 can be written in the form ∈ 3 g = ∑ αkek (7) k=0 where e0 := 1, αk = ak + ibk for ak,bk R, i = e123, and where 0 k 3, giving 23 = 8 degrees of freedom. The conjugation,∈ known as the reverse g† of≤ the≤ geometric number g, is defined by reversing the orders of all the products of the vectors that make up g, giving 3 † g := ∑ α kek. (8) k=0 G0 In particular, writing g = s + v + B + T, the sum of a real number s 3, a vector G1 G2 G3 † ∈ v 3, a bivector B 3 and a trivector T 3, g = s + v B T. ∈Two other conjugations∈ are widely used∈ in geometric algebra− .− The grade inversion is obtained by replacing each vector in a product by its negative. It corresponds to an inversion in the origin, otherwise known as a parity inversion. For the geometric number g given in (7), the grade inversion is 3 g− := α0 ∑ αkek. (9) − k=1 4 When g is written as g = s + v + B + T, the grade inversion g = s v + B T. The − − − Clifford conjugation g of the geometric number g G3, defined by ∗ ∈ 3 † g∗ := (g−) = α0 ∑ αkek = s v B + T, (10) − k=1 − − is just the inversion of g followed by the reversion. All other products in the geometric algebra are defined in terms of the geometric product. For example, given vectors a,b G1 R3, ∈ 3 ≡ ab = a b + a b G0+2, (11) · ∧ ∈ 3 1 R 1 where a b := 2 (ab+ba) is the symmetric inner product, and a b := 2 (ab ba) is the antisymmetric· outer∈ product of the vectors a and b, respectively.∧ The outer− prod- uct satisfies a b = i(a b), expressing the duality relationship between the standard Gibbs-Heaviside∧ cross product× a b and the outer product a b G2. A great ad- × ∧ ∈ 3 vantage of the geometric algebra G3 over the Gibbs-Heaviside vector algebra is the cancellation property ab = ac a2b = a2c b = c, ⇐⇒ ⇐⇒ provided a2 = a 2 = 0. The equation ab = ac forces equality of both the scalar and bivector parts of| (11).| 6 The triple products a (b c) and a b c of three vectors are also important. Similar to (11), we write · ∧ ∧ ∧ a(b c)= a (b c)+ a (b c), (12) ∧ · ∧ ∧ ∧ where in this case 1 a (b c) := a(b c) (b c)a = a (b c) G1, · ∧ 2 ∧ − ∧ − × × ∈ 3 and 1 a (b c) := a(b c) + (b c)a = a (b c)i G3.
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