Section III. Implications for Quantum Mechanics
Section III explores implications of GA for the interpretation of
quantum mechanics. The basic fact is that GA reveals a hidden geometric
structure in the Dirac electron theory, including a geometric intepretation
of the unit imaginary that links it to spin. A "zitterbewegung interpretation"
of the Dirac theory is proposed to explain the geometric structure as
an expression of particle kinematics. It is argued that a point particle
model of the electron is completely consistent with all features
of the Dirac theory. Analogous arguments by Bohm and others, that a
point particle interpretation is completely consistent with Schroedinger's
theory, have gained wider currency among physicists recently. Of course,
the Dirac theory tells us a lot more about the electron than
Schroedinger's theory. Several papers speculate on the possibility
of a deeper particle theory of the electron to which the Dirac theory gives clues.
- D. Hestenes, Physics and Probability: Essays in Honor of Edwin T. Jaynes,1993, 153-160, W.T. Grandy & P.W. Miloni, Cambridge U. Press, Cambridge.
: The Dirac theory has a hidden geometric structure. This talk traces the conceptual steps taken to uncover that structure and points out significant implications for the interpretation of quantum mechanics. The unit imaginary in the Dirac equation is shown to represent the generator of rotations in a spacelike plane related to the spin. This implies a geometric interpretation for the generator of electromagnetic gauge transformations as well as for the entire electroweak gauge group of the Weinberg-Salam model. The geometric structure also helps to reveal closer connections to classical theory than hitherto suspected, including exact classical solutions of the Dirac equation.
D. Hestenes, in: Clifford Algebras and their Applications in Mathematical Physics, 321-346, 1986 (J.S.R. Chisholm and A.K. Common, Eds.), Kluwer Academic Publishers, Dordrecht.
- Abstract: The zitterbewegung is a local circulatory motion of the electron presumed to be the basis of the electron spin and magnetic moment. A reformulation of the Dirac theory shows that the zitterbewegung need not be attributed to interference between positive and negative energy states as originally proposed by Schroedinger. Rather, it provides a physical interpretation for the complex phase factor in the Dirac wave function generally. Moreover, it extends to a coherent physical interpretation of the entire Dirac theory, and it implies a zitterbewegung interpretation for the Schroedinger theory as well.
- D. Hestenes, Foundations of Physics, 20, No. 10 (Oct. 1990).
- Abstract: We explore the possibility that zitterbewegung is the key to a complete understanding of the Dirac theory of electrons. We note that a literal interpretation of the zitterbewegung implies that the electron is the seat of an oscillating bound electromagnetic field similar to de Broglie's pilot wave. This opens up new possibilities for explaining two major features of quantum mechanics as consequences of an underlying physical mechanism. On this basis, qualitative explanations are given for electron diffraction, the existence of quantized radiationless states, the Pauli principle, and other features of quantum mechanics.
- D. Hestenes, Foundations of Physics, 15, No. 1 (Jan. 1985).
- Abstract: The zitterbewegung is a local circulatory motion of the electron presumed to be the basis of the electron spin and magnetic moment. A reformulation of the Dirac theory shows that this interpretation can be sustained rigorously, with the complex phase factor in the wave function describing the local frequency and phase of the circulatory motion directly. This reveals the zitterbewegung as a mechanism for storing energy in a single electron, with many implications for radiative processes.
- D. Hestenes, in: The Electron, 21-36, 1991 (D. Hestenes and A. Weingartshofer, Eds.), Kluwer Academic Publishers, Dordrecht.
- Abstract: A means for separating subjective and objective aspects of the electron wave function is suggested, based on a reformulation of the Dirac Theory in terms of Spacetime Algebra. The reformulation admits a separation of the Dirac wave function into a two parameter probability factor and a six parameter kinematical factor. The complex valuedness of the wave function as well as its bilinearity in observables have perfect kinematical interpretations independent of any probabilistic considerations. Indeed, the explicit unit imaginary in the Dirac equation is automatically identified with the electron spin in the reformulation. Moreover, the canonical momentum is seen to be derived entirely from the rotational velocity of the kinematical factor, and this provides a geometrical interpretation of energy quantization. Exact solutions of the Dirac equation exhibit circular zitterbewegung in exact agreement with the classical Wessenhoff model of a particle with spin. Thus, the most peculiar features of quantum mechanical wave functions have kinematical explanations, so the use of probability theory in quantum mechanics should not differ in any essential way from its use in classical mechanics.
- D. Hestenes, in: Maximum Entropy and Bayseian Methods, 161-183, 1990 (P.F. Fougere, Ed.), Kluwer Academic Publishers, Dordrecht.
- Abstract: Guidelines for constructing point particle models of the electron with zitterbewegung and other features of the Dirac theory are discussed. Such models may at least be useful approximations to the Dirac theory, but the more exciting possibility is that this approach may lead to a more fundamental reality.
- D. Hestenes, Foundations of Physics, 23, No. 3 (Mar. 1993).
- Abstract: The generator of electromagnetic gauge transformations in the Dirac equation has a unique geometric interpretation and a unique extension to the generators of the gauge group SU(2) X U(1) for the Weinberg-Salam theory of weak and electromagnetic interactions. It follows that internal symmetries of the weak interactions can be interpreted as space-time symmetries of spinor fields in the Dirac algebra. The possibilities for interpreting strong interaction symmetries in a similar way are highly restricted.
- D. Hestenes, Foundations of Physics, 12, No. 153-168 (1982).