By Amalia Torre

Ray, wave and quantum innovations are critical to various and possible incompatible types of sunshine. each one version particularizes a particular ''manifestation'' of sunshine, after which corresponds to sufficient actual assumptions and formal approximations, whose domain names of applicability are well-established. hence every one version includes its personal set of geometric and dynamic postulates with the pertinent mathematical means.At a easy point, the publication is a whole creation to the Wigner optics, which bridges among ray and wave optics, supplying the optical section house because the atmosphere and the Wigner functionality dependent strategy because the mathematical equipment to house among the 2 contrary extremes of sunshine illustration: the localized ray of geometrical optics and the unlocalized wave functionality of wave optics.At a parallel point, the analogies with different branches of either classical and quantum physics, like classical and quantum mechanics, quantum optics, sign concept in addition to magnetic optics, are evidenced via pertinent reviews and/or rigorous arithmetic. So, the Lie algebra and staff tools are brought and defined during the simple optical platforms inside either the ray and wave optics contexts, the previous being on the topic of the symplectic workforce and the latter to the metaplectic team. In a like demeanour, the Wigner functionality is brought through following the unique factor to individualize a part house illustration of quantum mechanics, that is reflected through the problem to individualize a neighborhood frequency spectrum in the sign idea context. the fundamental analogy with the optics of charged debris inherently underlying the ray-optics photo in part area can also be evidenced in the wave-optics photo within the Wigner section area. ?· amalgamation of loads of contributions having witnessed the section area photo of optics over the last 30 years?· introduces summary strategies via concrete systems?· hosts of figures and logical diagrams to favour instinct and to introduce mathematics?· emphasis at the interrelations with quantum optics, sign conception and magnetic optics ?· feeds a sense for real concerns in greater arithmetic and theoretical physics

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**Extra info for Linear Ray and Wave Optics in Phase Space: Bridging Ray and Wave Optics via the Wigner Phase-Space Picture**

**Example text**

36) we can individualize a correspondence of {f(q, p)} to the set of the associated Lie operators {ki}" A A {f(q,p)} ~ {kf}. 41) A A A to obtain yet a Lie operator. Notably, the operators {kl} span a Lic algebra with respect to the commutation between operators. We recall that the commutator of two operators F and G is defined by the antisyinmetric product A A A A A A IF, G] - A A FG - A GF. , {f,g}} - A - k{:,,} - k{g,f}. 43) The cominutator of two Lie operators is therefore a Lie operator as well.

Il~g t,l~e ray l)atl~ i~ a, l)laa~(~ tl~r()~gl~ the ()t)ti(:al axis. 17) in t11(; llm,ri(li()lla,1 1)lml( ~, ill(livi(llm,liz(~,(l t)y t,ll( ~, illl)llt ray. F. lysis ()f a r(~,fi'a,(:l,illg sl)lm, re 1)y lisillg a, t)()wer s(wies exl)a,llSi()]l ill l,lle ()if-axis (lista,n(:es an(t a,ngh~,s. xia,l ()I)ti(:s ()r first-()r(ter ()t)ti(:s. tion of para,xia,1 (Ilmri(ti()lm,1) ra,ys t h r o u g h simt)le astignmti(: and stigma,ti(: optical devi(:es can t)e trea,te(t as a, 1D d y n a m i c a l prot)lem, with the "plane of the motion" t)eing respe(:tively any one of the two pla,nes of s y m m e t r y of the s y s t e m ()r any ()ne of the infinite meridional pla, nes.

We consider now the set of dynamical variables {f(q, p)}, which, as previously remarked, configures as a Lie algebra under the Poisson bracket operation. 36) we can individualize a correspondence of {f(q, p)} to the set of the associated Lie operators {ki}" A A {f(q,p)} ~ {kf}. 41) A A A to obtain yet a Lie operator. Notably, the operators {kl} span a Lic algebra with respect to the commutation between operators. We recall that the commutator of two operators F and G is defined by the antisyinmetric product A A A A A A IF, G] - A A FG - A GF.