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On an easy transition from operator dynamics to generating functionals by Clifford algebras. (English) Zbl 0935.81039

Summary: Clifford geometric algebras of multivectors are treated in detail. These algebras are built over a graded space and exhibit a grading or multivector structure. The careful study of the endomorphisms of this space makes it clear that opposite Clifford algebras have to be used for obtaining all endomorphisms. Based on these mathematics, the author gives a fully Clifford algebraic account on generating functionals, which is thereby geometric. The field operators are shown to be Clifford and opposite Clifford maps. This picture, relying on geometry, does not need positivity in principle. Furthermore, he proposes a transition from operator dynamics to generating functionals, which is based on the algebraic techniques. As a calculational benefit, this transition is considerably short compared to standard ones. The transition is not injective (unique) and depends additionally on the choice of an ordering. He obtains a direct and constructive connection between orderings and the explicit form of the functional Hamiltonian. These orderings depend on the propagator of the theory and thus on the ground state. This is invisible in path integral formulations.
The method is demonstrated within two examples, a nonlinear spinor field theory and spinor QED. Antisymmetrized and normal-ordered functional equations are derived in both cases.

MSC:

81R25 Spinor and twistor methods applied to problems in quantum theory
15A66 Clifford algebras, spinors
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[1] DOI: 10.1103/PhysRev.85.631 · Zbl 0046.21501 · doi:10.1103/PhysRev.85.631
[2] DOI: 10.1103/PhysRev.85.631 · Zbl 0046.21501 · doi:10.1103/PhysRev.85.631
[3] DOI: 10.1103/PhysRev.85.631 · Zbl 0046.21501 · doi:10.1103/PhysRev.85.631
[4] DOI: 10.1103/PhysRev.94.1362 · doi:10.1103/PhysRev.94.1362
[5] DOI: 10.1103/PhysRev.94.1362 · doi:10.1103/PhysRev.94.1362
[6] DOI: 10.1103/PhysRev.94.1362 · doi:10.1103/PhysRev.94.1362
[7] DOI: 10.1103/PhysRev.94.1362 · doi:10.1103/PhysRev.94.1362
[8] DOI: 10.1103/PhysRev.94.1362 · doi:10.1103/PhysRev.94.1362
[9] DOI: 10.1103/PhysRev.94.1362 · doi:10.1103/PhysRev.94.1362
[10] Freese E., Z. Naturforsch. 8 pp 776– (1951)
[11] DOI: 10.1063/1.530050 · Zbl 0810.15014 · doi:10.1063/1.530050
[12] B. Fauser, ”Clifford geometric parameterization of inequivalent vacua,” preprint (hep-th/9710047). · Zbl 0990.15017
[13] DOI: 10.1063/1.531376 · Zbl 0861.15031 · doi:10.1063/1.531376
[14] Fauser B., Adv. Appl. Clifford Algebr. 6 pp 1– (1996)
[15] DOI: 10.1007/BF02727289 · doi:10.1007/BF02727289
[16] DOI: 10.1007/BF02727289 · doi:10.1007/BF02727289
[17] B. Fauser, ”Hecke algebras as subalgebras of Clifford geometric algebras of multivectors,” preprint (q-alg/9710020).
[18] DOI: 10.1007/BF02731414 · Zbl 0065.21701 · doi:10.1007/BF02731414
[19] Polivanov M. K., Dokl. Akad. Nauk SSSR 100 pp 1061– (1955)
[20] DOI: 10.1103/PhysRev.101.860 · Zbl 0074.22902 · doi:10.1103/PhysRev.101.860
[21] Stumpf H., Z. Naturforsch. 51 pp 1045– (1996)
[22] Stumpf H., Z. Naturforsch. 52 pp 220– (1997)
[23] Stumpf H., Z. Naturforsch. 48 pp 765– (1993)
[24] DOI: 10.1103/PhysRev.128.885 · Zbl 0129.21901 · doi:10.1103/PhysRev.128.885
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