Geometric Number Systems and Spinors
Abstract
The real number system is geometrically extended to include three new anticommuting square roots of plus one, each such root representing the direction of a unit vector along the orthonormal coordinate axes of Euclidean 3space. The resulting geometric (Clifford) algebra provides a geometric basis for the famous Pauli matrices which, in turn, proves the consistency of the rules of geometric algebra. The flexibility of the concept of geometric numbers opens the door to new understanding of the nature of spacetime, and of Pauli and Dirac spinors as points on the Riemann sphere, including Lorentz boosts.
AMS Subject Classification: 15A66, 81P16
Keywords: geometric algebra, relative geometric algebra, Riemann sphere, complex Riemann sphere.
1 The geometric algebra of space
The most direct way of obtaining the geometric algebra of space is to extend the real number system to include three new anticommutative square roots of , that represent unit vectors along the respective coordinate axes. Thus, and . The resulting associative geometric algebra , as a real linear space, has the dimensional standard basis
where represent unit bivectors in the three coordinate planes, for , and represents the oriented directed trivector, or pseudoscalar element of space. The geometric numbers of dimensional space are pictured in Figure 1.
To see that the rules of our geometric algebra are consistent, we relate it immediately to the famous Pauli algebra of square matrices over the complex numbers . The most intuitive way of doing this is to introduce the mutually annihilating idempotents , which satisfy the rules
In addition, . All these rules are easily verified and left to the reader. Another important property of the geometric algebra is that the pseudoscalar element is in the center of the algebra, commuting with all elements, and . Thus can take over the roll of the unit imaginary .
By the spectral basis of the geometric algebra , we mean
(1) 
where is taking over the roll of in the usual Pauli algebra. Any geometric number corresponds directly to a Pauli matrix , by the simple rule
For example, the famous Pauli matrices are specified by
(2) 
as can be easily checked. For , we have
This shows that the Pauli algebra and the geometric algebra are fully compatible as algebraically isomorphic algebras over the complex numbers
The geometric product of geometric numbers corresponds to the usual matrix algebra product of the matrices . The great advantage of the geometric algebra over the Pauli algebra , is that the geometric numbers are liberated from their coordinate representations as complex matrices, as well as being endowed with a complete geometric interpretation. On the other hand, the consistency of the rules of the geometric algebra follow from the known consistency of rules of matrix algebra, and matrices offer a computational tool for computing the product of geometric numbers [9].
It is worthwhile to give a summary of the deep relationship between prerelativistic (GibbsHeaviside) vector algebra, and the geometric algebra . The geometric product of two vectors is given by
(3) 
where the inner product , and the outer product has the interpretation (due to Grassmann) of the bivector in the plane of the vectors and . The bivector , where is the vector normal to the plane of , and is its dual.
Another advantage of the geometric product over the inner product and the cross product , is the powerful cancellation rule
provided of course that . It takes knowledge of both and (or ), to uniquely determine the relative directions of the vectors and . Another unique advantage in the the geometric algebra is the Euler formula made possible by (3),
where and similarly for , and is the angle between the vectors and . The unit vector can be defined by .
The Euler formula for the bivector , which has square , is the generator of rotations in the plane of the bivector . Later, when talking about Lorentz boosts, we will also utilize the hyperbolic Euler form
(4) 
where determines the rapidity of the boost in the direction of the unit vector .
A couple more formulas, relating the vector cross and dot products to the geometric product, are
and
The triple vector products and are directly related to the geometric product by the identity
where and . Detailed discussions and proofs of these identities, and their generalizations to higher dimensional geometric algebras, can be found in [8], [10]. Geometric algebra has in recent years become a basic tool for research in quantum mechanics [1], [3], and more generally as a basic language of mathematics and physics [2], [8].
We now return to beautiful results which depend in large part only upon the geometric product. Thus the reader can relax and rely upon the familiar rules of matrix algebra, which are equally valid in the isomorphic geometric algebra.
2 Stereographic projection in
Consider the equation
(5) 
where is a unit vector for the vector . Clearly, this equation is well defined except when . Let us solve this equation for , but first we find that
Returning to equation (5),
(6) 
Equation (5) can be equivalently expressed by
showing that is obtained by a rotation of in the plane of through an angle of where , or equivalently, by a rotation of in the plane of through an angle of .
It is easily shown that the most general idempotent in has the form
(7) 
where
and is the unit pseudoscalar element in . Consider now idempotents of the form , where . Equating , we find that
Changing the parity of this equation, gives
since the parity change (changing the sign of vectors) of is identical to the reverse (reversing the order of products of vectors) of .
We can now solve these last two equations for and in terms of , getting
or
(8) 
where , the plane. From (7), it immediately follows that
(9) 
We also easily find that
A Pauli spinor is a column matrix of two complex components, which we denote by . Each Pauli spinor corresponds to a minimal left ideal , which in turn corresponds to geometric Pauli spinor, or Pauli gspinor in the geometric algebra . We have
(10) 
By factoring out from spinor , we get
where is the idempotent defined above for .
By the norm of the spinor , we mean
(11) 
where means the real number part of the geometric number . More generally we define the sesquilinear inner product between the spinors to be
where means the scalar and pseudoscalar parts of the geometric number . A spinor is said to be normalized if .
From equations (7) and (11), it follows that for ,
where , , and . Equations (5) and (6) have an immediate interpretation on the Riemann sphere centered at the origin. Figure 2 shows a crosssection of the Riemann sphere, taken in the plane of the bivector , through the origin. We see that the stereographic projection from the South pole at the point , to the point on the Riemann sphere, passes through the point of the point onto the plane through the origin with the normal vector . Stereographic projection is just one example of conformal mappings, which have important generalizations to higher dimensions [6].
We can now simply answer a basic question in quantum mechanics. If a spin particle is prepared in a normalized Pauli spin state , what is the probability of finding it in a normalized Pauli gstate immediately thereafter? We calculate
(12) 
This relationship can be more directly expressed in terms of and . Using (6), for and , a short calculation gives the result
(13) 
showing that the probability of finding the particle in that Pauli gstate is directly related to the Euclidean distance between the points and .
3 Dirac spinors
What is missing in the concept of a Pauli spinor is the ability to distinguish between Pauli spinors in different reference frames. Within the geometric algebra , we are able to distinguish different inertial systems. Recall that the rest frame that defined the geometric algebra was , oriented by the property that . A set of three orthonormal geometric numbers , specified by the condition
where each , defines a rest frame, or inertial system of relative vectors moving with a velocity of , with respect to the inertial system defined by . If represents the time and position vector of an event in the inertial system , then the corresponding event in the inertial system is specified by the active Lorentz transformation , [8], [11].
Each inertial system is distinguished by how its observer partitions its geometric numbers into vectors and bivectors, in the same sense that what one observer identifies as a pure electric field, becomes a mixture of an (vector) electric and (bivector) magnetic field in an inertial system not at rest. But all observers have the same pseudoscalar element , representing the unit trivector of space. The relative geometric algebra
consists of the same geometric numbers as in , but with a different observer dependent partition of what elements are identified as (relative) vectors, and what elements are identified as (relative) bivectors. These ideas have been explored by the author more fully in [11], and in the book [8, Chps:2,11].
Consider now Dirac spinors of the form
(15) 
where , and . When , becomes the Pauli spinor given in (8). Indeed, since ,
(16) 
so is just the normalized spinor defined by boosted into the inertial system where .
Starting with (15), we calculate
(17) 
where and . which defines the velocity
in the direction
From (16) and (17), it follows that the spinor boosted in the direction of , with rapidity , is in the same inertial system as the spinor defined by boosted in the direction of with rapidity .
Just as the equations (5) and (6) led to the interpretation of stereographic projection on the Riemann sphere, Figure 2, the generalized equation
where , leads to a stereographic projection of the Riemann sphere, followed by a boost. This is most clearly seen by writing
so the Dirac spinor represented by , maps the North Pole into the point on the Riemann sphere, followed by the boost . See [5] for more details of this construction.
References
 [1] W.E. Baylis, J.D. Keselica, The Complex Algebra of Physical Space: A Framework for Relativity, Advances in Applied Clifford Algebras, Vol. 22, No. 3, pp. 537  561, 2012.
 [2] D. Hestenes, New Foundations for Classical Mechanics, 2nd Ed., Kluwer 1999.

[3]
D. Hestenes, Zitterbewegung in Quantum Mechanics, Found Physics (2010) 40:154.
http://geocalc.clas.asu.edu/pdf/ZBWinQM15**.pdf

[4]
G. Sobczyk, Part I: Vector Analysis of Spinors, (2014)
http://arxiv.org/abs/1507.06608

[5]
G. Sobczyk, Part II: Spacetime Algebra of Dirac Spinors, (2015)
http://arxiv.org/abs/1507.06609
 [6] G. Sobczyk, Conformal Mappings in Geometric Algebra, Notices of the AMS, Volume 59, Number 2, p.264273, 2012.

[7]
G. Sobczyk, Geometry of Spin 1/2 Particles, Revista Mexicana de Física, 61 (2015) 211223.
http://rmf.smf.mx/pdf/rmf/61/3/61_3_211.pdf
 [8] G. Sobczyk, New Foundations in Mathematics: The Geometric Concept of Number, Birkhäuser, New York 2013.
 [9] G. Sobczyk, Geometric Matrix Algebra, Linear Algebra and its Applications, 429 (2008) 11631173.
 [10] G. Sobczyk, Hyperbolic Number Plane, The College Mathematics Journal, Vol. 26, No. 4, pp.268280, September 1995.
 [11] G. Sobczyk, Spacetime Vector Analysis, Physics Letters A, Vol 84A, p.4549, 1981.