Inverse Problem for the Yang–Mills Equations

Chen, Xi; Lassas, Matti; Oksanen, Lauri; Paternain, Gabriel P.

doi:10.1007/s00220-021-04006-0

Cited by 15 publications

(34 citation statements)

References 38 publications

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“…The data for the problem arises form observations in a small set and the recovery is achieved on a causal domain where waves can propagate and return. The results here build upon our earlier work in [4,5] where we considered a cubic wave equation for the connection wave operator and the case of pure Yang-Mills theories, but this time we have the distinct objective of retrieving the Higgs field in addition to the connection (Yang-Mills field). The new couplings being introduced make the analysis of the non-linear interaction of waves quite complicated at first glance.…”

Section: Introduction and Overviewmentioning

confidence: 62%

Retrieving Yang--Mills--Higgs fields in Minkowski space from active local measurements

Chen¹,

Lassas²,

Oksanen³

et al. 2022

Preprint

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We show that we can retrieve a Yang-Mills potential and a Higgs field (up to gauge) from source-to-solution type data associated with the classical Yang-Mills-Higgs equations in Minkowski space R 1+3 . We impose natural nondegeneracy conditions on the representation for the Higgs field and on the Lie algebra of the structure group which are satisfied for the case of the Standard Model. Our approach exploits the non-linear interaction of waves generated by sources with values in the centre of the Lie algebra showing that abelian components can be used effectively to recover the Higgs field.

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Section: Introduction and Overviewmentioning

confidence: 62%

Retrieving Yang--Mills--Higgs fields in Minkowski space from active local measurements

Chen¹,

Lassas²,

Oksanen³

et al. 2022

Preprint

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“…We can now define the broken non-abelian X-ray transform. In [CLOP21a] and [CLOP21b], they define it as follows. Consider the set S + ( ) := {(x, y, z) ∈ D 3 : (x, y), (y, z) ∈ L, x < y < z with x, z ∈ , y ∈ }.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Stability estimate for the broken non-abelian X-ray transform in Minkowski space

St-Amant¹

2022

Preprint

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We study the broken non-abelian X-ray transform in Minkowski space. This transform acts on the space of Hermitian connections on a causal diamond and is known to be injective up to an infinite-dimensional gauge. We show a stability estimate that takes into account the gauge, leading to a new proof of the transform's injectivity. Our proof leads us to consider a special type of connections that we call light-sink connections. We then show that we can consistently recover a light-sink connection from noisy measurement of its X-ray transform data through Bayesian inversion.

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“…All these works concern the (1 + 3)-dimensional case. In recent works [16,17,29], the authors have also studied problems of recovering zeroth-and first-order terms for semi-linear wave equations with Minkowski metric. We note that three wave interactions were used in [16,17] to determine the lower order terms in the equations and in modelling nonlinear elastic scattering from discontinuities [20,21].…”

Section: Previous Literaturementioning

confidence: 99%

“…In recent works [16,17,29], the authors have also studied problems of recovering zeroth-and first-order terms for semi-linear wave equations with Minkowski metric. We note that three wave interactions were used in [16,17] to determine the lower order terms in the equations and in modelling nonlinear elastic scattering from discontinuities [20,21]. In [28,69] similar multiple-fold linearisation methods were introduced to study inverse problems for elliptic nonlinear equations; see also [61].…”

Section: Previous Literaturementioning

confidence: 99%

Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers

Feizmohammadi

Lassas

Oksanen

2021

Forum of Mathematics, Pi

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The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension $n+1\geqslant 3$ and with partial data. We consider the case when the set $\Omega _{\mathrm{in}}$ , where the sources are supported, and the set $\Omega _{\mathrm{out}}$ , where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point $p_{in}\in \Omega _{\mathrm{in}}$ and the past of the point $p_{out}\in \Omega _{\mathrm{out}}$ . In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in $\Omega _{\mathrm{in}}$ and observations in $\Omega _{\mathrm{out}}$ , determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.

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Inverse Problem for the Yang–Mills Equations

Cited by 15 publications

References 38 publications

Retrieving Yang--Mills--Higgs fields in Minkowski space from active local measurements

Retrieving Yang--Mills--Higgs fields in Minkowski space from active local measurements

Stability estimate for the broken non-abelian X-ray transform in Minkowski space

Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers

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