Analytic definition of spin structure

Avetisyan, Zhirayr; Fang, Yan-Long; Saveliev, Nikolai; Vassiliev, Dmitri

doi:10.1063/1.4995952

Cited by 6 publications

(11 citation statements)

References 15 publications

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“…11]. See also [38,21,9,32] for a wider picture on the role of gauge transformations in the analysis of systems of PDEs.…”

Section: The Algorithmmentioning

confidence: 99%

Diagonalization of elliptic systems via pseudodifferential projections

Capoferri

2021

Preprint

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Consider an elliptic self-adjoint pseudodifferential operator A acting on m-columns of half-densities on a closed manifold M , whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential projections commuting with A, we construct an almost-unitary pseudodifferential operator that diagonalizes A modulo an infinitely smoothing operator. We provide an invariant algorithm for the computation of its full symbol, as well as an explicit closed formula for its subprincipal symbol. Finally, we give a quantitative description of the relation between the spectrum of A and the spectrum of its approximate diagonalization, and discuss the implications at the level of spectral asymptotics.

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“…11]. See also [38,21,9,32] for a wider picture on the role of gauge transformations in the analysis of systems of PDEs.…”

Section: The Algorithmmentioning

confidence: 99%

Diagonalization of elliptic systems via pseudodifferential projections

Capoferri

2021

Preprint

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“…Indeed, fully relativistic equations of mathematical physics are not always associated with a natural inner product, not even an indefinite non-degenerate one. As a result, in the Lorentzian setting one is often forced to work with equations, as opposed to operators, see [68,69].…”

Section: Construction Of Hadamard Statesmentioning

confidence: 99%

Partial Differential Equations and Quantum States in Curved Spacetimes

Avetisyan

Capoferri

2021

Mathematics

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In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.

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“…This follows by the argument of Section 6.2 and Section 6.3 once the map GL(2, C) → CSO + (3, 1) is replaced by the map U (2) → SO(3). Our analytic definition of spin structure in dimension three, Definition 8.2, is also equivalent to the standard topological one, which follows from [1] with the help of Diagram 6.10.…”

Section: The 3-dimensional Riemannian Casementioning

confidence: 99%

“…for some uniquely defined smooth matrix-function O : M → CSO +(3,1).Suppose now that there exists a matrix-function R : M → GL(2, C) such that S prin = R * S prin R. A straightforward calculation shows that the matrix-function O appearing in (6.2) is expressed via R asO j k = 1 det R| −4/3 tr(s j R * s k R). (6.3)It is convenient to define R := | det R| 2/3 R.(6.4) …”

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confidence: 99%

Classification of first order sesquilinear forms

2020

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A natural way to obtain a system of partial differential equations on a manifold is to vary a suitably defined sesquilinear form. The sesquilinear forms we study are Hermitian forms acting on sections of the trivial C n -bundle over a smooth m-dimensional manifold without boundary. More specifically, we are concerned with first order sesquilinear forms, namely, those generating first order systems. Our goal is to classify such forms up to GL(n, C) gauge equivalence. We achieve this classification in the special case of m = 4 and n = 2 by means of geometric and topological invariants (e.g. Lorentzian metric, spin/spin c structure, electromagnetic covector potential) naturally contained within the sesquilinear form -a purely analytic object. Essential to our approach is the interplay of techniques from analysis, geometry, and topology.

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Analytic definition of spin structure

Cited by 6 publications

References 15 publications

Diagonalization of elliptic systems via pseudodifferential projections

Diagonalization of elliptic systems via pseudodifferential projections

Partial Differential Equations and Quantum States in Curved Spacetimes

Classification of first order sesquilinear forms

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