Clifford Algebras and Dirac Operators in Harmonic Analysis

Gilbert, John E.; Murray, Margaret

doi:10.1017/cbo9780511611582

Cited by 556 publications

(498 citation statements)

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“…Bilyalov's generalized eigenvector formalism works more generally [Bilyalov, 2002], but still requires that 'time' be listed first among the coordinates; otherwise negative eigenvalues can appear, yielding a complex square root of the metric. Note that such coordinate restrictions do not arise in the positive definite case considered by Gilbert and Murray [Gilbert and Murray, 1991] and by Bourguignon and Gauduchon [Bourguignon and Gauduchon, 1992], which seem to be rather rare examples of the consideration of the symmetric square root of the metric by mathematicians.…”

Section: Nonlinear Geometric Objectsmentioning

confidence: 99%

The nontriviality of trivial general covariance: How electrons restrict ‘time’ coordinates, spinors (almost) fit into tensor calculus, and of a tetrad is surplus structure

Pitts

2012

Studies in History and Philosophy of Science Part B: Studies in

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It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus.On the other hand, the physics literature often claims that spinors as such cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is (nearly) Ockhamized, with most of the fluff removed. As developed * Forthcoming in Studies in History and Philosophy of Modern Physics 1 nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix diag(−1, 1, 1, 1), the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and values of the fields involved.Apart from coordinate order and the usual spinorial two-valuedness, (densitized) Ogievetsky-Polubarinov spinors form, with the (conformal part of the) met- The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited.

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Section: Nonlinear Geometric Objectsmentioning

confidence: 99%

The nontriviality of trivial general covariance: How electrons restrict ‘time’ coordinates, spinors (almost) fit into tensor calculus, and of a tetrad is surplus structure

Pitts

2012

Studies in History and Philosophy of Science Part B: Studies in

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“…A few prerequisites from Clifford analysis and spin geometry are briefly summarized in the first two subsections. As excellent introductions to these specific research fields concerned with the study of Dirac, Cauchy-Riemann, and Laplace operators in either a Euclidean, Hermitian, or a real or complex Dirac vector bundle setting, we refer to the monographs by Berline, Getzler, and Vergne [2], Brackx, Delanghe, and Sommen [5], Gilbert and Murray [15], and Lawson and Michelsohn [18].…”

Section: Spin Operator Theorymentioning

confidence: 99%

Seminormal systems of operators in Clifford environments

Martin

2017

OpMath

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Abstract. The primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex C * -algebras that enable one to set up counterparts of the geometric Bochner-Weitzenböck and Bochner-Kodaira-Nakano curvature identities for systems of elements of a C * -algebra. The so derived self-commutator identities in conjunction with Bochner's method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities.

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“…The Dirac operator reduces to ∂ x = ∂ x 1 g (11) e 1 = ∂ x 1 e 1 . The matrix associated to the system, which is elliptic, can be formally obtained from (12) by setting y 1 = 0. Consider the Clifford algebra R 2,2 generated by e 1 , e 2 , ε 1 , ε 2 and the dual basis e 1 , e 2 , ε 1 , ε 2 satisfying the relations (see Proposition 4.5)…”

Section: The General Casementioning

confidence: 99%

“…It is worth noticing that the idea to consider a local metric dependence has already been exploited for Clifford analysis on manifolds and goes back to [12]. However, the novelty of [5] and of this paper is that we consider a global metric tensor whereby the components are interpreted as variables.…”

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confidence: 99%