Quantum Mechanics of the electron particle-clock David Hestenes Department of Physics, Arizona State University, Tempe, Arizona 85287-1504∗ Understanding the electron clock and the role of complex numbers in quantum mechanics is grounded in the geometry of spacetime, and best expressed with Spacetime Algebra (STA). The efficiency of STA is demonstrated with coordinate-free applications to both relativistic and non- relativistic QM. Insight into the structure of Dirac theory is provided by a new comprehensive analysis of local observables and conservation laws. This provides the foundation for a comparative analysis of wave and particle models for the hydrogen atom, the workshop where Quantum Mechanics was designed, built and tested. PACS numbers: 10,03.65.-w Keywords: pilot waves, zitterbewegung, spacetime algebra I. INTRODUCTION The final Section argues for a realist interpretation of the Dirac wave function describing possible particle De Broglie always insisted that Quantum Mechanics is paths. a relativistic theory. Indeed, it began with de Broglie`s relativistic model for an electron clock. Ironically, when Schr¨odingerintroduced de Broglie`s idea into his wave II. SPACETIME ALGEBRA equation, he dispensed with both the clock and the rela- tivity! Spacetime Algebra (STA) plays an essential role in the This paper aims to revitalize de Broglie's idea of an formulation and analysis of electron theory in this paper. electron clock by giving it a central role in physical inter- Since thorough expositions of STA are available in many pretation of the Dirac wave function. In particular, we places [2{4], a brief description will suffice here, mainly aim for insight into structure of the wave function and to establish notations and define terms. fibrations of particle paths it determines. This opens up STA is an associative algebra generated by spacetime new questions about physical interpretation. vectors with the property that the square of any vector We begin with a synopsis of Spacetime Algebra (STA), is a (real) scalar. Thus for any vector a we can write which is an essential tool in all that follows. 2 2 Section III applies STA in a review of Real Dirac theory a = aa = "jaj ; (1) and provides a rigorous new analysis of its physical inter- where " is the signature of a and jaj is a positive scalar. pretation in terms of local observables. That facilitates As usual, we say that a is timelike, lightlike or spacelike if the study of electron paths in subsequent sections. its signature is positive (" = 1), null (" = 0), or negative Section IV discusses a new approach to Born's statis- (" = −1). tical interpretation of the Dirac wave function dubbed From the geometric product ab of two vectors it is con- Born-Dirac theory. It includes a relativistic extension of venient to define two other products. The inner product de Broglie-Bohm Pilot Wave theory to interpret the Dirac a · b is defined by wave function as describing a fibration (or ensemble) of possible particle paths. Spin dependence of the so-called a · b = 1 (ab + ba) = b · a ; (2) Quantum potential is made explicit and generalized. 2 Section V presents a new relativistic particle model while the outer product a ^ b is defined by for electron states in the hydrogen atom. This corrects errors and deficiencies of Old Quantum Theory to make it 1 a ^ b = 2 (ab − ba) = −b ^ a : (3) consistent with the Dirac equation. It invites comparison with the standard Darwin model for hydrogen discussed The three products are therefore related by in Section IVD. Thereby it opens up new possibilities for experimental test and theoretical analysis. ab = a · b + a ^ b : (4) Section VI discusses modifications of Dirac theory to fully incorporate electron zitter. This serves as an intro- This can be regarded as a decomposition of the product duction to a more definitive model of the electron clock ab into symmetric and skewsymmetric parts, or alterna- developed in a subsequent paper [1]. tively, into scalar and bivector parts. For physicists unfamiliar with STA, it will be helpful to note its isomorphism to Dirac algebra over the reals. To that end, let fγµ; 0; 1; 2; 3g be a right-handed orthonor- ∗Electronic address: [email protected]; URL: http://geocalc. mal frame of vectors with γ0 in the forward light cone. clas.asu.edu/ The symbols γµ have been selected to emphasize direct PREPRINT July 2018 2 1 correspondence with Dirac's γ-matrices. In accordance F ^ a = 2 (F a + aF ); (12) with (2), the components gµν of the metric tensor are given by so that 1 gµν = γµ · γν = 2 (γµγν + γν γµ) : (5) F a = F · a + F ^ a (13) This will be recognized as isomorphic to a famous formula expresses a decomposition of F a into vector and pseu- of Dirac's. Of course, the difference here is that the γµ dovector parts. For F = b ^ c it follows that are vectors rather than matrices. The unit pseudoscalar i for spacetime is related to the (b ^ c) · a = b(c · a) − c(b · a) : (14) frame fγν g by the equation Many other useful identities can be derived to facilitate i = γ0γ1γ2γ3 = γ0 ^ γ1 ^ γ2 ^ γ3 : (6) coordinate-free computations. They will be introduced as needed throughout the paper. It is readily verified from (6) that i2 = −1, and the geo- Any fixed timelike vector such as fγ0g defines an iner- metric product of i with any vector is anticommutative. tial frame that determines a unique separation between By multiplication the γµ generate a complete basis of 4 space and time directions. Algebraically, this can be ex- k-vectors for STA, consisting of the 2 = 16 linearly in- pressed as the \spacetime split" of each vector x designat- dependent elements ing a spacetime point (or event) into a time component 1; γ ; γ ^ γ ; γ i; i : (7) x · γ0 = ct and a spatial position vector x ≡ x ^ γ0 as µ µ ν µ specified by the geometric product Obviously, this set corresponds to 16 base matrices for the Dirac algebra, with the pseudoscalar i corresponding xγ0 = ct + x : (15) to the Dirac matrix γ5. The entire spacetime algebra is obtained from linear We call this a γ0-split when it is important to specify the combinations of basis k-vectors in (7). A generic element generating vector. The resulting quantity ct + x is called M of the STA, called a multivector, can thus be written a paravector in the expanded form This \split" maps a spacetime vector into the STA subalgebra of even multivectors where, by \regrading," 4 the bivector part can be identified as a spatial vector. X M = α + a + F + bi + βi = hMik ; (8) Accordingly, the even subalgebra is generated by a frame k=0 of \spatial vectors" fσk ≡ γkγ0; k = 1; 2; 3g, so that where α and β are scalars, a and b are vectors, and F σ σ σ = γ γ γ γ = i: (16) is a bivector. This is a decomposition of M into its k- 1 2 3 0 1 2 3 vector parts, with k = 0; 1; 2; 3; 4, where h:::ik means \k- Obviously, this rendition of the STA even subalgebra is vector part." Of course, hMi0 = α, hMi1 = a, hMi2 = F , isomorphic to the Pauli algebra, though the Pauli alge- hMi3 = bi, hMi4 = βi. It is often convenient to drop the bra is not a subalgebra of the Dirac algebra because the subscript on the scalar part, writing hMi = hMi0: matrix dimensions are different. We say that a k-vector is even (odd) if the integer k We use boldface letters exclusively to denote spatial is even (odd). Accordingly, any multivector can be ex- vectors determined by a spacetime split. Spatial vectors pressed as the sum of even and odd parts. A multivector generate a coordinate-free spatial geometric algebra with is said to be \even" if its parts are even k-vectors. The the geometric product even multivectors compose a subalgebra of the STA. We will be using the fact that spinors can be represented as ab = a · b + a ^ b = a · b + ia × b; (17) even multivectors. Computations are facilitated by the operation of re- where a×b = −i(a^b) is the usual vector cross product. version. For M in the expanded form (8) the reverse Mf For the even part hMi+ = Q of the multivector M, a can be defined by spacetime split gives us Mf = α + a − F − bi + βi : (9) Q = z + F; (18) For arbitrary multivectors M and N where scalar and pseudoscalar parts combine in the form of a complex number (^MN) = NeM:f (10) z = α + iβ; (19) It is useful to extend the definitions of inner and outer products to multivectors of higher grade. Thus, for bivec- and the bivector part splits into the form of a \complex tor F and vector a we can define inner and outer products vector" 1 ~ F · a = 2 (F a − aF ); (11) F = E + iB = −F: (20) PREPRINT July 2018 3 Thus, the even subalgebra in STA has the formal struc- Note that the cross product on the right is distinguished ture of complex quaternions. However, the geometric from the commutator product on the left of this equation interpretation of the elements is decidedly different from by a boldface of the product symbol. Also, it should the usual one assigned to quaternions. Specifically, the be understood that the equivalence of commutator and bivector iB corresponds to a \real vector" in the quater- outer products in this equation does not generally obtain nion literature. This difference stems from a failure to for arbitrary multivectors.