Gauss’s Law, the Manifestations of Gauge Fields, and Their Impact on Local Observables
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Cited by 4 publications
References 16 publications
The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity
Given a globally hyperbolic spacetime $$M={\mathbb {R}}\times \Sigma $$ M = R × Σ of dimension four and regularity $$C^\tau $$ C τ , we estimate the Sobolev wavefront set of the causal propagator $$K_G$$ K G of the Klein–Gordon operator. In the smooth case, the propagator satisfies $$WF'(K_G)=C$$ W F ′ ( K G ) = C , where $$C\subset T^*(M\times M)$$ C ⊂ T ∗ ( M × M ) consists of those points $$(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })$$ ( x ~ , ξ ~ , y ~ , η ~ ) such that $$\tilde{\xi },\tilde{\eta }$$ ξ ~ , η ~ are cotangent to a null geodesic $$\gamma $$ γ at $$\tilde{x}$$ x ~ resp. $$\tilde{y}$$ y ~ and parallel transports of each other along $$\gamma $$ γ . We show that for $$\tau >2$$ τ > 2 , $$\begin{aligned} WF'^{-2+\tau -{\epsilon }}(K_G)\subset C \end{aligned}$$ W F ′ - 2 + τ - ϵ ( K G ) ⊂ C for every $${\epsilon }>0$$ ϵ > 0 . Furthermore, in regularity $$C^{\tau +2}$$ C τ + 2 with $$\tau >2$$ τ > 2 , $$\begin{aligned} C\subset WF'^{-\frac{1}{2}}(K_G)\subset WF'^{\tau -\epsilon }(K_G)\subset C \end{aligned}$$ C ⊂ W F ′ - 1 2 ( K G ) ⊂ W F ′ τ - ϵ ( K G ) ⊂ C holds for $$0<\epsilon <\tau +\frac{1}{2}$$ 0 < ϵ < τ + 1 2 . In the ultrastatic case with $$\Sigma $$ Σ compact, we show $$WF'^{-\frac{3}{2}+\tau -\epsilon }(K_G)\subset C$$ W F ′ - 3 2 + τ - ϵ ( K G ) ⊂ C for $$\epsilon >0$$ ϵ > 0 and $$\tau >2$$ τ > 2 and $$WF'^{-\frac{3}{2}+\tau -\epsilon }(K_G)= C$$ W F ′ - 3 2 + τ - ϵ ( K G ) = C for $$\tau >3$$ τ > 3 and $$\epsilon <\tau -3$$ ϵ < τ - 3 . Moreover, we show that the global regularity of the propagator $$K_G$$ K G is $$H^{-\frac{1}{2}-\epsilon }_{loc}(M\times M)$$ H loc - 1 2 - ϵ ( M × M ) as in the smooth case.
The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity
Given a globally hyperbolic spacetime $$M={\mathbb {R}}\times \Sigma $$ M = R × Σ of dimension four and regularity $$C^\tau $$ C τ , we estimate the Sobolev wavefront set of the causal propagator $$K_G$$ K G of the Klein–Gordon operator. In the smooth case, the propagator satisfies $$WF'(K_G)=C$$ W F ′ ( K G ) = C , where $$C\subset T^*(M\times M)$$ C ⊂ T ∗ ( M × M ) consists of those points $$(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })$$ ( x ~ , ξ ~ , y ~ , η ~ ) such that $$\tilde{\xi },\tilde{\eta }$$ ξ ~ , η ~ are cotangent to a null geodesic $$\gamma $$ γ at $$\tilde{x}$$ x ~ resp. $$\tilde{y}$$ y ~ and parallel transports of each other along $$\gamma $$ γ . We show that for $$\tau >2$$ τ > 2 , $$\begin{aligned} WF'^{-2+\tau -{\epsilon }}(K_G)\subset C \end{aligned}$$ W F ′ - 2 + τ - ϵ ( K G ) ⊂ C for every $${\epsilon }>0$$ ϵ > 0 . Furthermore, in regularity $$C^{\tau +2}$$ C τ + 2 with $$\tau >2$$ τ > 2 , $$\begin{aligned} C\subset WF'^{-\frac{1}{2}}(K_G)\subset WF'^{\tau -\epsilon }(K_G)\subset C \end{aligned}$$ C ⊂ W F ′ - 1 2 ( K G ) ⊂ W F ′ τ - ϵ ( K G ) ⊂ C holds for $$0<\epsilon <\tau +\frac{1}{2}$$ 0 < ϵ < τ + 1 2 . In the ultrastatic case with $$\Sigma $$ Σ compact, we show $$WF'^{-\frac{3}{2}+\tau -\epsilon }(K_G)\subset C$$ W F ′ - 3 2 + τ - ϵ ( K G ) ⊂ C for $$\epsilon >0$$ ϵ > 0 and $$\tau >2$$ τ > 2 and $$WF'^{-\frac{3}{2}+\tau -\epsilon }(K_G)= C$$ W F ′ - 3 2 + τ - ϵ ( K G ) = C for $$\tau >3$$ τ > 3 and $$\epsilon <\tau -3$$ ϵ < τ - 3 . Moreover, we show that the global regularity of the propagator $$K_G$$ K G is $$H^{-\frac{1}{2}-\epsilon }_{loc}(M\times M)$$ H loc - 1 2 - ϵ ( M × M ) as in the smooth case.
Where Charged Sectors are Localizable: A Viewpoint from Covariant Cohomology
Given a Haag–Kastler net on a globally hyperbolic spacetime, one can consider a family of regions where quantum charges are supposed to be localized. Assuming that the net fulfils certain minimal properties (factoriality of the global observable algebra and relative Haag duality), we give a geometric criterion that the given family must fulfil to have a superselection structure with charges localized on its regions. Our criterion is fulfilled by all the families used in the theory of sectors (double cones, spacelike cones, diamonds, hypercones). In order to take account of eventual spacetime symmetries, our superselection structures are constructed in terms of covariant charge transporters, a novel cohomological approach generalizing that introduced by J. E. Roberts. In the case of hypercones, with the forward light cone as an ambient spacetime, we obtain a superselection structure with Bose–Fermi parastatistics and particle-antiparticle conjugation. It could constitute a candidate for a different description of the charged sectors introduced by Buchholz and Roberts for theories including massless particles.
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