Hidden Variables Revisited

Jauch, J. M.; Piron, C.

doi:10.1103/revmodphys.40.228

Cited by 38 publications

(46 citation statements)

References 4 publications

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“…Exponential localization of photons may seem to be in contradiction with the results obtained by Jauch and Piron [19] and Amrein [20]. In particular, it seems to be in disagreement with the statement [20] "The electric field of a localized photon decreases as r 27͞2 at large distances from the volume of localization."…”

Section: Exponential Localization Of Photonsmentioning

confidence: 81%

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Exponential Localization of Photons

1998

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It is shown that photons can be localized in space with an exponential falloff of the energy density and photodetection rates. The limits of localization are determined by the fundamental Paley-Wiener theorem. A direct mathematical connection between the spatial localization of photons and the decay in time of quantum mechanical systems is established. [S0031-9007(98)06370-4] PACS numbers: 03.70. + k, 03.50.De, 03.65.PmThe purpose of this Letter is to prove that, contrary to a widespread belief, photons can be localized in space with an exponential falloff of the photon energy density and of the photodetection rates. In that respect, photons are not much different from massive particles. The only constraint on the photon localizability comes from the fundamental Paley-Wiener theorem [1]. The mathematical mechanism that limits the decrease of the photon energy density is the same as in the case of the decay of quantum systems with time. The exponential falloff of the photon wave function can be of the order exp͓2f͑r͔͒, where f͑r͒ increases with r slower than linearly. I shall show that this exponential localization holds not only for the energy density and for the photodetection rates but also for the modulus of the Landau-Peierls function [2] that is viewed by some as the proper position wave function of the photon. I shall present simple analytic expressions for the photon wave functions that exhibit such an exponentially small tail. I shall also contrast the localization properties of one-photon states with those containing many photons.Adlard, Pike, and Sarkar [3] have recently constructed single photon states that exhibit arbitrarily high powerlaw falloff of the energy density and of the photodetection rates. These states were derived from the solutions of Maxwell equations that have the form of complex rational functions of coordinates. Such solutions were envisaged a long time ago by Gårding [4], Synge [5], and Trautman [6] and were used more recently by Ziolkowski [7] and Hellwarth and Nouchi [8] to describe focused electromagnetic pulses. Incidentally, the arbitrarily high power-law falloff, found in Ref.[3], could have been obtained by simply differentiating the old solutions with respect to space coordinates and by observing that a derivative of a solution of Maxwell equations is also a solution. In the present Letter I go beyond all these algebraic solutions by providing examples of much more tightly localized wave functions and by establishing the absolute limit of the photon localization.The most general one-photon (1ph) state can be described by two complex functions of the wave vectorwhere a y ͑k͒ and b y ͑k͒ are the creation operators of photons with the left-handed and the right-handed polarizations, respectively, normalized to a delta function, ͓a͑k͒, a y ͑k 0 ͔͒ d͑k 2 k 0 ͒ ͓b͑k͒, b y ͑k 0 ͔͒ .In order to allow for the probabilistic interpretation, the two components f 6 ͑k͒ of the photon wave function in momentum representation must be normalized to one,The photon states are conveniently descr...

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Section: Exponential Localization Of Photonsmentioning

confidence: 81%

“…(14) from the Hertz vector (19) describes a single photon state having the form of a spherical shell converging for negative and diverging for positive values of t, attaining the maximal localization at t 0. The function Z and its time derivative at t 0 have the form…”

Section: Exponential Localization Of Photonsmentioning

confidence: 99%

Exponential Localization of Photons

1998

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“…von Neumann's assumption is unjustified for dispersion-free states, so his proof breaks down. The same applies to the proof of Jauch & Piron (1963, 1966, because their apparently reasonable identification of the algebra of the expectations of commuting projectors with the algebra of logical propositions breaks down for dispersion-free states.…”

Section: Hidden Variables and Localitymentioning

confidence: 73%

John Stewart Bell. 28 July 1928 – 1 October 1990

Burke¹,

Percival

1999

Biogr. Mems Fell. R. Soc.

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John Stewart Bell, the second of four children of John and Annie Bell (née Brownlee), was born in Belfast on 28 July 1928. The families of both his parents were Protestant, his father's family coming from Fermanagh and his mother's from Tyrone. Originally, Annie Bell's family had come from Scotland and John Bell's second Christian name was that of her family. At home, John Bell was always known as Stewart, only becoming John when he went to university. John Bell's family were not well off, his father being sent out to work before he was twelve years old to help the family finances. However, John Bell's father and mother always impressed on John, his elder sister Ruby and his two younger brothers David and Robert the importance of education. Later, David, through evening study, became a qualified electrical engineer and a professor at Lambton College in Canada, writing several very successful textbooks; Robert is now a successful Northern Ireland businessman.

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“…Much the same goes for the Jauch-Piron argument [219]. In the shortened form of the argument given by Bell [20] .2) also.…”

Section: -329mentioning

confidence: 80%