Spectral asymptotics for first order systems

We work on a parallelizable time-orientable Lorentzian 4-manifold and prove that in this case the notion of spin structure can be equivalently defined in a purely analytic fashion. Our analytic definition relies on the use of the concept of a non-degenerate two-by-two formally self-adjoint first order linear differential operator and gauge transformations of such operators. We also give an analytic definition of spin structure for the 3-dimensional Riemannian case.

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“…The following result was announced, without proof, in [1,Section 7]. In what follows, we provide a different proof of Theorem 7.2.…”

Section: The 3-dimensional Riemannian Casementioning

confidence: 90%

Analytic definition of spin structure

Avetisyan

Fang

Saveliev

et al. 2017

Journal of Mathematical Physics

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“…Remark 2. The construction presented above can be adapted to cover the case of more general scalar operator, as well as of systems of partial differential equations, under suitable assumptions, see [17][18][19][20][21][22]. We should mention that propagators are an important tool in abstract spectral theory, as they encode asymptotic information on the spectrum of the elliptic operator that generates them, cf.…”

Section: The Wave Propagator On a Riemannian Manifoldmentioning

confidence: 99%

“…with ψ(z) := d dz ln(Γ(z)) being the digamma function. The key idea underpinning the definition of Hadamard states is to prescribe that their 2-point functions possess the same singularity structure as (17). In order to turn this into a mathematically precise statement, we need to introduce further definitions and notation.…”

Section: Definition 4 (Quasifree State) a Statementioning

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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“…1. The difference is that we have now dropped the condition tr L prin (x, p) = 0, replaced the ellipticity condition by the weaker non-degeneracy condition (4) and extended our group of transformations from special unitary to special linear.…”

Section: Spin Structurementioning

confidence: 99%