The Principle of Duality in Clifford Algebra and Projective Geometry

Conradt, Oliver

doi:10.1007/978-1-4612-1368-0_10

Cited by 9 publications

(15 citation statements)

References 2 publications

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“…V and ∧ n−1 V . Beside the deformation into a Clifford algebra as in (1), one can think about introducing a Clifford product on ∧ n−1 V [2]. Using product co-product duality (see Fig.…”

Section: Bigebra Structurementioning

confidence: 99%

“…Creation is (wedge) concatenation by a grade one element, while annihilation is the (graded) derivation rule given in (1-ii). This is reflected by formulas (2) and (3) where we identified the field operators. But -as we have seen in (2), adding an antisymmetric part does not preserve the grading.…”

Section: Qft Via Clifford Hopf Gebrasmentioning

confidence: 99%

“…This is reflected by formulas (2) and (3) where we identified the field operators. But -as we have seen in (2), adding an antisymmetric part does not preserve the grading. This is the (Hopf algebraic) origin of the need of a Wick normal-ordering which literally does transform to that particular grading which is selected by the propagator (antisymmetric part) [15,19].…”

Section: Qft Via Clifford Hopf Gebrasmentioning

confidence: 99%

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Quantum Clifford Hopf gebra for quantum field theory

Fauser

2003

AACA

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Section: Bigebra Structurementioning

confidence: 99%

Section: Qft Via Clifford Hopf Gebrasmentioning

confidence: 99%

Section: Qft Via Clifford Hopf Gebrasmentioning

confidence: 99%

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Quantum Clifford Hopf gebra for quantum field theory

Fauser

2003

AACA

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“…We emphasized earlier that we did not expect the interpretation and the mathematical aspects of classical Clifford algebras to change in such a transformation. However, see [16] for a far more elaborate application of a similar situation where both gradings are used.…”

Section: Examplementioning

confidence: 99%

On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form

Fauser

Abłamowicz

2000

Clifford Algebras and Their Applications in Mathematical Physics

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Clifford algebras are naturally associated with quadratic forms. These algebras are Z2 -graded by construction. However, only a Zn -gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cℓ(V ) ↔ V and an ordering, guarantees a multivector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Zn -grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a Zn -grading which we now call Clifford algebras of multivectors or quantum Clifford algebras. It turns out, that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonalizability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cℓp,q which can be decomposed in the symmetric case into a tensor product Cℓp−1,q−1 ⊗ Cℓ1,1. The general case used in quantum field theory lacks this feature. Theories with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories. A connection to q -deformed structures through nontrivial vacuum states in quantum theories is outlined.

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“…For a discussion some of the possible issues associated with generalizing Clifford algebras to projective geometry, see Conradt [21,22], as well as Hestenes and Ziegler [23].…”

Section: Bivectors and Bispinorsmentioning

confidence: 99%

Projective geometry and special relativity

Delphenich¹

2006

Ann. Phys.

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Some concepts of real and complex projective geometry are applied to the fundamental physical notions that relate to Minkowski space and the Lorentz group. In particular, it is shown that the transition from an infinite speed of propagation for light waves to a finite one entails the replacement of a hyperplane at infinity with a light cone and the replacement of an affine hyperplane -or rest space -with a proper time hyperboloid. The transition from the metric theory of electromagnetism to the pre-metric theory is discussed in the context of complex projective geometry, and ultimately, it is proposed that the geometrical issues are more general than electromagnetism, namely, they pertain to the transition from point mechanics to wave mechanics.

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The Principle of Duality in Clifford Algebra and Projective Geometry

Cited by 9 publications

References 2 publications

Quantum Clifford Hopf gebra for quantum field theory

Quantum Clifford Hopf gebra for quantum field theory

On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form

Projective geometry and special relativity

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