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  • 1
    Electronic Resource
    Electronic Resource
    Geometric algebra and the causal approach to multiparticle quantum mechanics (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 3327-3340 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is argued that geometric algebra, in the form of the multiparticle spacetime algebra, is well suited to the study of multiparticle quantum theory, with advantages over conventional techniques both in ease of calculation and in providing an intuitive geometric understanding of the results. This is illustrated by comparing the geometric algebra approach for a system of two spin-1/2 particles with the nonrelativistic approach of Holland [Phys. Rep. 169, 294 (1988)]. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532890
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  • 2
    Electronic Resource
    Electronic Resource
    Effects of spin-torsion in gauge theory gravity (1998)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 3303-3321 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The spin-torsion sector of a new gauge-theoretic formulation of gravity is analyzed and the relationship to the Einstein–Cartan–Kibble–Sciama theory of gravity is discussed. The symmetries of the Riemann tensor and the conservation laws of the theory are derived. This formalism is applied to the problem of a Dirac field coupled self-consistently to gravity. The equations derived from a minimally coupled gauge-invariant Lagrangian naturally give the gauge-theoretic analogs of the Einstein–Cartan–Dirac equations. Finally, a semiclassical model for a spinning point-particle moving in a gravitational background with torsion is considered. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532255
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  • 3
    Electronic Resource
    Electronic Resource
    Grassmann calculus, pseudoclassical mechanics, and geometric algebra (1993)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 34 (1993), S. 3683-3712 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A reformulation of Grassmann calculus is presented in terms of geometric algebra—a unified language for physics based on Clifford algebra. In this reformulation, Grassmann generators are replaced by vectors, so that every product of generators has a natural geometric interpretation. The calculus introduced by Berezin [The Method of Second Quantization (Academic, New York, 1966)] is shown to be unnecessary, amounting to no more than an algebraic contraction. This approach is not only conceptually clearer, but it is also computationally more efficient, as demonstrated by treatments of the "Grauss'' integral and the Grassmann Fourier Transform. The reformulation is applied to pseudoclassical mechanics [Ann. Phys. 104, 336 (1977)], where it is shown to lead to a new concept, the multivector Lagrangian. To illustrate this idea, the three-dimensional Fermi oscillator is reformulated and solved, and its symmetry properties discussed. As a result, a new and highly compact formula for generating super-Lie algebras is revealed. The paper ends with a discussion of quantization, outlining a new approach to fermionic path integrals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530053
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  • 4
    Electronic Resource
    Electronic Resource
    An image of the SunyaevZel'dovich effect (1993)
    [s.l.] : Nature Publishing Group
    Nature 365 (1993), S. 320-323 
    Details
    ISSN: 1476-4687
    Source: Nature Archives 1869 - 2009
    Topics: Biology , Chemistry and Pharmacology , Medicine , Natural Sciences in General , Physics
    Notes: [Auszug] In the Rayleigh-Jeans region of the cosmic microwave back-ground radiation (CMBR) spectrum (that is, at frequencies á«200 GHz), the decrease AT in the brightness temperature of the CMBR owing to the Sunyaev-Zel'dovich (S-Z) effect is given by f kBTe Orá«e ---2 J ...
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1038/365320a0
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  • 5
    Electronic Resource
    Electronic Resource
    Massive, Non-ghost Solutions for the Dirac Field Coupled Self-consistently to Gravity (1997)
    Springer
    General relativity and gravitation 29 (1997), S. 1527-1544 
    Details
    ISSN: 1572-9532
    Keywords: COSMOLOGY ; TORSION ; DIRAC FIELDS IN CURVED SPACETIME ; EXACT SOLUTIONS
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We present new, massive, non-ghost solutions for the Dirac field coupled self-consistently to gravity. We employ a gauge-theoretic formulation of gravity which automatically identifies the spin of the Dirac field with the torsion of the gauge fields. Homogeneity of the field observables requires that the spatial sections be flat. Expanding and collapsing singular solutions are given, as well as a solution which expands from a singularity before recollapsing. Torsion effects are only important while the Compton wavelength of the Dirac field is larger than the Hubble radius. We study the motion of spinning point-particles in the background of the expanding solution. The anisotropy due to the torsion is manifest in the particle trajectories.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018830329120
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  • 6
    Electronic Resource
    Electronic Resource
    Imaginary numbers are not real—The geometric algebra of spacetime (1993)
    Springer
    Foundations of physics 23 (1993), S. 1175-1201 
    Details
    ISSN: 1572-9516
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a “geometric product” of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analyzed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics), Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained—results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01883676
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  • 7
    Electronic Resource
    Electronic Resource
    A multivector derivative approach to Lagrangian field theory (1993)
    Springer
    Foundations of physics 23 (1993), S. 1295-1327 
    Details
    ISSN: 1572-9516
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract A new calculus, based upon the multivector derivative, is developed for Lagrangian mechanics and field theory, providing streamlined and rigorous derivations of the Euler-Lagrange equations. A more general form of Noether's theorem is found which is appropriate to both discrete and continuous symmetries. This is used to find the conjugate currents of the Dirac theory, where it improves on techniques previously used for analyses of local observables. General formulas for the canonical stress-energy and angular-momentum tensors are derived, with spinors and vectors treated in a unified way. It is demonstrated that the antisymmetric terms in the stress-energy tensor are crucial to the correct treatment of angular momentum. The multivector derivative is extended to provide a functional calculus for linear functions which is more compact and more powerful than previous formalisms. This is demonstrated in a reformulation of the functional derivative with respect to the metric, which is then used to recover the full canonical stress-energy tensor. Unlike conventional formalisms, which result in a symmetric stress-energy tensor, our reformulation retains the potentially important antisymmetric contribution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01883781
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  • 8
    Electronic Resource
    Electronic Resource
    States and operators in the spacetime algebra (1993)
    Springer
    Foundations of physics 23 (1993), S. 1239-1264 
    Details
    ISSN: 1572-9516
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum σ- and γ-matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar unit imaginary of quantum mechanics is played by a fixed spacetime bivector. The extension to multiparticle systems involves a separate copy of the STA for each particle, and it is shown that the standard unit imaginary induces correlations between these particle spaces. In the STA, spinors and operators can be manipulated without introducing any matrix representation or coordinate system. Furthermore, the formalism provides simple expressions for the spinor bilinear covariants which dispense with the need for the Fierz identities. A reduction to2+1 dimensions is given, and applications beyond the Dirac theory are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01883678
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  • 9
    Electronic Resource
    Electronic Resource
    Electron paths, tunnelling, and diffraction in the spacetime algebra (1993)
    Springer
    Foundations of physics 23 (1993), S. 1329-1356 
    Details
    ISSN: 1572-9516
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract This paper employs the ideas of geometric algebra to investigate the physical content of Dirac's electron theory. The basis is Hestenes' discovery of the geometric significance of the Dirac spinor, which now represents a Lorentz transformation in spacetime. This transformation specifies a definite velocity, which might be interpreted as that of a real electron. Taken literally, this velocity yields predictions of tunnelling times through potential barriers, and defines streamlines in spacetime that would correspond to electron paths. We also present a general, first-order diffraction theory for electromagnetic and Dirac waves. We conclude with a critical appraisal of the Dirac theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01883782
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  • 10
    Electronic Resource
    Electronic Resource
    Quadratic Lagrangians and Topology in Gauge Theory Gravity (2000)
    Springer
    General relativity and gravitation 32 (2000), S. 161-174 
    Details
    ISSN: 1572-9532
    Keywords: Quadratic Lagrangians ; topology ; instantons ; ECKS theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider topological contributions to the action integral in a gauge theory formulation of gravity. Two topological invariants are found and are shown to arise from the scalar and pseudoscalar parts of a single integral. Neither of these action integrals contribute to the classical field equations. An identity is found for the invariants that is valid for non-symmetric Riemann tensors, generalizing the usual GR expression for the topological invariants. The link with Yang–Mills instantons in Euclidean gravity is also explored. Ten independent quadratic terms are constructed from the Riemann tensor, and the topological invariants reduce these to eight possible independent terms for a quadratic Lagrangian. The resulting field equations for the parity non-violating terms are presented. Our derivations of these results are considerably simpler than those found in the literature.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1001856702156
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