Andrzej Trautman - Academia.edu (original) (raw)
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Papers by Andrzej Trautman
Complex numbers enter fundamental physics in at least two rather distinct ways. They are needed i... more Complex numbers enter fundamental physics in at least two rather distinct ways. They are needed in quantum theories to make linear differential operators into Hermitian observables. Complex structures appear also, through Hodge duality, in vector and spinor spaces associated with space-time. This paper reviews some of these notions. Charge conjugation in multidimensional geometries and the appearance of Cauchy-Riemann structures on Lorentz manifolds with a congruence of null geodesics without shear are presented in considerable detail.
Journal of Geometry and Physics, 1993
Projective quadrics are known to be conformal compactifications of Euclidean spaces. In particula... more Projective quadrics are known to be conformal compactifications of Euclidean spaces. In particular, the (projective) real quadric QP,Q = (S~x Sq )/~2 is associated, in this manner, with the flat space~P+~endowed with a metric tensor of signature (p, q). For p and q positive, the quadric Qp,q is orientable if p + q is even. The quadric has two natural metrics, invariant with respect to the action of O(p + I) x O(q + 1): a proper Riemannian one and a pseudo-Riemannian metric of signature (p, q). This paper contains an explicit description of spin structures on real, even-dimensional quadrics for both metrics, whenever these structures exist. In particular, it is shown that, for p and q even positive, the proper (pseudo-Riemannian) metric gives rise to two inequivalent spin structures iff p + q 2 (mod 4) (p + q 0 (mod 4)). Ifp and q are odd and > 1, then there is no spin structure for either metric whenever p + q 0 (mod 4); otherwise, there are two spin structures for each of the metrics. There always exist spin structures on real quadrics with a Lorentzian metric, i.e., when p and q are odd and p or q = I.
Journal of Geometry and Physics, 1995
General theorems on pin structures on products of manifolds and on homogeneous (pseudo-)Riemannia... more General theorems on pin structures on products of manifolds and on homogeneous (pseudo-)Riemannian spaces are given and used to find explicitly all such structures on odd-dimensional real projective quadrics, which are known to be non-orientable . It is shown that the product of two manifolds has a pin structure if, and only if, both are pin and at least one of them is orientable. This general result is illustrated by the example of the product of two real projective planes. It is shown how the Dirac operator should be modified to make it equivariant with respect to the twisted adjoint action of the Pin group. A simple formula is derived for the spectrum of the Dirac operator on the product of two pin manifolds, one of which is orientable, in terms of the eigenvalues of the Dirac operators on the factor spaces.
Odd-dimensional Riemannian spaces that are non-orientable, but have a pin structure, require the ... more Odd-dimensional Riemannian spaces that are non-orientable, but have a pin structure, require the consideration of the twisted adjoint representation of the corresponding pin group. It is shown here how the Dirac operator should be modified, also on even-dimensional spaces, to make it equivariant with respect to the action of that group when the twisted adjoint representation is used in the definition of the pin structure. An explicit description of a pin structure on a hypersurface, defined by its immersion in a Euclidean space, is used to derive a "Schroedinger" transform of the Dirac operator in that case. This is then applied to obtain - in a simple manner - the spectrum and eigenfunctions of the Dirac operator on spheres and real projective spaces.
International Journal of Theoretical Physics, 1977
It is shown that the magnetic pole of lowest strength and the pseudoparticle solution of the Yang... more It is shown that the magnetic pole of lowest strength and the pseudoparticle solution of the Yang-Mills equations correspond to natural connections defined on the principal bundlesU(2)/U(1)=S 3 →S 2 andSp(2)/Sp(1)=S 7 →S 4, respectively. This observation leads to a general methods of constructing new, topologically nontrivial solutions of the Maxwell and Yang-Mills equations. Among them is an “electromagnetic instanton” defined over the two-dimensional complex projective space endowed with the Fubini-Study metric.
Complex numbers enter fundamental physics in at least two rather distinct ways. They are needed i... more Complex numbers enter fundamental physics in at least two rather distinct ways. They are needed in quantum theories to make linear differential operators into Hermitian observables. Complex structures appear also, through Hodge duality, in vector and spinor spaces associated with space-time. This paper reviews some of these notions. Charge conjugation in multidimensional geometries and the appearance of Cauchy-Riemann structures on Lorentz manifolds with a congruence of null geodesics without shear are presented in considerable detail.
Journal of Geometry and Physics, 1993
Projective quadrics are known to be conformal compactifications of Euclidean spaces. In particula... more Projective quadrics are known to be conformal compactifications of Euclidean spaces. In particular, the (projective) real quadric QP,Q = (S~x Sq )/~2 is associated, in this manner, with the flat space~P+~endowed with a metric tensor of signature (p, q). For p and q positive, the quadric Qp,q is orientable if p + q is even. The quadric has two natural metrics, invariant with respect to the action of O(p + I) x O(q + 1): a proper Riemannian one and a pseudo-Riemannian metric of signature (p, q). This paper contains an explicit description of spin structures on real, even-dimensional quadrics for both metrics, whenever these structures exist. In particular, it is shown that, for p and q even positive, the proper (pseudo-Riemannian) metric gives rise to two inequivalent spin structures iff p + q 2 (mod 4) (p + q 0 (mod 4)). Ifp and q are odd and > 1, then there is no spin structure for either metric whenever p + q 0 (mod 4); otherwise, there are two spin structures for each of the metrics. There always exist spin structures on real quadrics with a Lorentzian metric, i.e., when p and q are odd and p or q = I.
Journal of Geometry and Physics, 1995
General theorems on pin structures on products of manifolds and on homogeneous (pseudo-)Riemannia... more General theorems on pin structures on products of manifolds and on homogeneous (pseudo-)Riemannian spaces are given and used to find explicitly all such structures on odd-dimensional real projective quadrics, which are known to be non-orientable . It is shown that the product of two manifolds has a pin structure if, and only if, both are pin and at least one of them is orientable. This general result is illustrated by the example of the product of two real projective planes. It is shown how the Dirac operator should be modified to make it equivariant with respect to the twisted adjoint action of the Pin group. A simple formula is derived for the spectrum of the Dirac operator on the product of two pin manifolds, one of which is orientable, in terms of the eigenvalues of the Dirac operators on the factor spaces.
Odd-dimensional Riemannian spaces that are non-orientable, but have a pin structure, require the ... more Odd-dimensional Riemannian spaces that are non-orientable, but have a pin structure, require the consideration of the twisted adjoint representation of the corresponding pin group. It is shown here how the Dirac operator should be modified, also on even-dimensional spaces, to make it equivariant with respect to the action of that group when the twisted adjoint representation is used in the definition of the pin structure. An explicit description of a pin structure on a hypersurface, defined by its immersion in a Euclidean space, is used to derive a "Schroedinger" transform of the Dirac operator in that case. This is then applied to obtain - in a simple manner - the spectrum and eigenfunctions of the Dirac operator on spheres and real projective spaces.
International Journal of Theoretical Physics, 1977
It is shown that the magnetic pole of lowest strength and the pseudoparticle solution of the Yang... more It is shown that the magnetic pole of lowest strength and the pseudoparticle solution of the Yang-Mills equations correspond to natural connections defined on the principal bundlesU(2)/U(1)=S 3 →S 2 andSp(2)/Sp(1)=S 7 →S 4, respectively. This observation leads to a general methods of constructing new, topologically nontrivial solutions of the Maxwell and Yang-Mills equations. Among them is an “electromagnetic instanton” defined over the two-dimensional complex projective space endowed with the Fubini-Study metric.