Champs spinoriels et propagateurs en relativité générale

Lichnerowicz, André

doi:10.24033/bsmf.1604

“…We show that our elegant theory agrees with the standard one developed for the so-called covariant Dirac spinor fields as developed, e.g., in (Lichnerowicz, 1964(Lichnerowicz, , 1984Choquet-Bruhat et al, 1982).…”

Section: Introductionsupporting

confidence: 65%

Dirac-Hestenes spinor fields on Riemann-Cartan manifolds

Rodrigues

¹

,

Souza

²

,

Vaz

³

et al. 1996

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Abstract. In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named Spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to the famous Geroch's theorem concerning to the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and compare these with DHSF, showing that all the three kinds of spinor fields contain the same mathematical and physical information. We clarify also the notion of (Crumeyrolle's) amorphous spinors (Dirac-Kähler spinor fields are of this type), showing that they cannot be used to describe fermionic fields. We develop a rigorous theory for the covariant derivatives of Clifford fields (sections of the Clifford bundle (CB)) and of Dirac-Hestenes spinor fields. We show how to generalize the original Dirac-Hestenes equation in Minkowski spacetime for the case of a RCST. Our results are obtained from a variational principle formulated through the multiform derivative approach to Lagrangian field theory in the Clifford bundle.

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“…A spinor u is a solution if and only if u+ is a solution. We immediately note the following important identity [16] (1.5) (~iV +m)(iW +m)u = (□ -\R + m2)u where □ is the spinor wave operator given by (1.6) (□«)" = ^VaVbuA.…”

Section: The Dirac Equationmentioning

confidence: 99%

“…We develop the mathematical concepts of spin structures, spin connections, etc., which are needed to formulate the classical Dirac equation. Our treatment roughly follows the original work of Lichnerowicz [16].…”

mentioning

confidence: 99%

Dirac quantum fields on a manifold

Dimock¹

1982

Trans. Amer. Math. Soc.

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Abstract. On globally hyperbolic Lorentzian manifolds we construct field operators which satisfy the Dirac equation and have a causal anticommutator. Ambiguities in the construction are removed by formulating the theory in terms of C* algebras of local observables. A generalized form of the Haag-Kastler axioms is verified.Introduction. A Lorentzian manifold is a four dimensional manifold TV/ together with a pseudo-Riemannian metric g of signature ( + , -, -, -). Such manifolds are widely used as models for space-time. In particular in the absence of boundaries and gravitational fields one uses M = R4 and g = tj = Minkowski metric which has components {r)ab} = diag(l, -1,-1,-1).We are interested in formulating quantum field theories on general Lorentzian manifolds. This means giving up many ideas familiar from the usual Minkowski space treatments. Thus there are no Poincaré transformations on M, no vacuum, no particle states, and so forth. All one is left with are the field equations. These are to be solved taking as data a representation of the CCR or CAR over a space-like hypersurface (CCR = canonical commutation relations, CAR = canonical anticommutation relations). For linear equations this quantum problem can be reduced to a corresponding classical problem for which solutions can be constructed in favorable cases. In the nonlinear case very singular formal solutions can be exhibited which one might hope to make rigorous.The procedure sketched above is fraught with ambiguities. What hypersurface should one take? what representation of the CCR/CAR? and so forth. In general there is no natural choice and one should show that all choices lead to the same theory. To do this in a fundamental way it seems necessary to abandon the field

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“…Spinors. We review the geometric setting for the Dirac equation (see [4], [16]). Our starting point is a representation of the Dirac-Clifford algebra as given by 4x4 matrices y0, ■ ■ ■ , y3 satisfying i1-1) Jayb + YfrYfl = Mob1-Given any two such representations ya, y'a there is a nonsingular matrix M (unique up to a multiple of the identity) such that y'a = MyaM~x [19].…”

Section: The Dirac Equationmentioning

confidence: 99%

Dirac Quantum Fields on a Manifold

Dimock¹

1982

Transactions of the American Mathematical Society

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Abstract. On globally hyperbolic Lorentzian manifolds we construct field operators which satisfy the Dirac equation and have a causal anticommutator. Ambiguities in the construction are removed by formulating the theory in terms of C* algebras of local observables. A generalized form of the Haag-Kastler axioms is verified.Introduction. A Lorentzian manifold is a four dimensional manifold TV/ together with a pseudo-Riemannian metric g of signature ( + , -, -, -). Such manifolds are widely used as models for space-time. In particular in the absence of boundaries and gravitational fields one uses M = R4 and g = tj = Minkowski metric which has components {r)ab} = diag(l, -1,-1,-1).We are interested in formulating quantum field theories on general Lorentzian manifolds. This means giving up many ideas familiar from the usual Minkowski space treatments. Thus there are no Poincaré transformations on M, no vacuum, no particle states, and so forth. All one is left with are the field equations. These are to be solved taking as data a representation of the CCR or CAR over a space-like hypersurface (CCR = canonical commutation relations, CAR = canonical anticommutation relations). For linear equations this quantum problem can be reduced to a corresponding classical problem for which solutions can be constructed in favorable cases. In the nonlinear case very singular formal solutions can be exhibited which one might hope to make rigorous.The procedure sketched above is fraught with ambiguities. What hypersurface should one take? what representation of the CCR/CAR? and so forth. In general there is no natural choice and one should show that all choices lead to the same theory. To do this in a fundamental way it seems necessary to abandon the field

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Champs spinoriels et propagateurs en relativité générale

Cited by 109 publications

References 3 publications

Dirac-Hestenes spinor fields on Riemann-Cartan manifolds

Dirac-Hestenes spinor fields on Riemann-Cartan manifolds

Dirac quantum fields on a manifold

Dirac Quantum Fields on a Manifold

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