On the orthogonality of the MacDonald's functions

Passian, Ali; Simpson, Henry C.; Kouchekian, Sherwin; Yakubovich, Semyon

doi:10.1016/j.jmaa.2009.06.067

Cited by 19 publications

(11 citation statements)

References 12 publications

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“…We find that our inner product is given by Our remaining task reduces to solving for solutions ( ) to our extension of the Klein-Gordon equation (60) . From Appendix F in Appendix S1 which utilizes [45] , [46] , [47] , [48] , and [49] , we find where and . is the spherical harmonic of degree and order .…”

Section: Discussionmentioning

confidence: 99%

Re-Examination of Globally Flat Space-Time

Feldman

2013

PLoS ONE

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In the following, we offer a novel approach to modeling the observed effects currently attributed to the theoretical concepts of “dark energy,” “dark matter,” and “dark flow.” Instead of assuming the existence of these theoretical concepts, we take an alternative route and choose to redefine what we consider to be inertial motion as well as what constitutes an inertial frame of reference in flat space-time. We adopt none of the features of our current cosmological models except for the requirement that special and general relativity be local approximations within our revised definition of inertial systems. Implicit in our ideas is the assumption that at “large enough” scales one can treat objects within these inertial systems as point-particles having an insignificant effect on the curvature of space-time. We then proceed under the assumption that time and space are fundamentally intertwined such that time- and spatial-translational invariance are not inherent symmetries of flat space-time (i.e., observable clock rates depend upon both relative velocity and spatial position within these inertial systems) and take the geodesics of this theory in the radial Rindler chart as the proper characterization of inertial motion. With this commitment, we are able to model solely with inertial motion the observed effects expected to be the result of “dark energy,” “dark matter,” and “dark flow.” In addition, we examine the potential observable implications of our theory in a gravitational system located within a confined region of an inertial reference frame, subsequently interpreting the Pioneer anomaly as support for our redefinition of inertial motion. As well, we extend our analysis into quantum mechanics by quantizing for a real scalar field and find a possible explanation for the asymmetry between matter and antimatter within the framework of these redefined inertial systems.

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Section: Discussionmentioning

confidence: 99%

Re-Examination of Globally Flat Space-Time

Feldman

2013

PLoS ONE

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“…(3.14) and (3.18). Somewhat earlier, Yakubovich [7] and Passian et al [8] proved the validity of the relation (3.22) exploiting more involved mathematical techniques.…”

Section: Summary Of Relevant Properties Of the Whittaker Functions Ofmentioning

confidence: 93%

An orthogonality relation for the Whittaker functions of the second kind of imaginary order

Szmytkowski

Bielski

2010

Integral Transforms and Special Functions

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“…x, x ⊥ ), (4.16) 19 Eqs. (4.13a) and (4.13b) follow from the following identities for the modified Bessel function of the second kind (also known as the Macdonald function) with imaginary index (see, e.g., [39][40][41]):…”

Section: Representation Of the Lie Algebra So(2 D) In The So(1 1) Bmentioning

confidence: 99%

Intertwining operator in thermal CFTd

Ohya

2017

Int. J. Mod. Phys. A

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It has long been known that two-point functions of conformal field theory (CFT) are nothing but the integral kernels of intertwining operators for two equivalent representations of conformal algebra. Such intertwining operators are known to fulfill some operator identities -the intertwining relations -in the representation space of conformal algebra. Meanwhile, it has been known that the S-matrix operator in scattering theory is nothing but the intertwining operator between the Hilbert spaces of in-and out-particles. Inspired by this algebraic resemblance, in this paper, we develop a simple Lie-algebraic approach to momentum-space two-point functions of thermal CFT living on the hyperbolic space-time H 1 × H d−1 by exploiting the idea of Kerimov's intertwining operator approach to exact S-matrix. We show that in thermal CFT on H 1 × H d−1 , the intertwining relations reduce to certain linear recurrence relations for two-point functions in the complex momentum space. By solving these recurrence relations, we obtain the momentum-space representations of advanced and retarded two-point functions as well as positive-and negative-frequency two-point Wightman functions for a scalar primary operator in arbitrary space-time dimension d ≥ 3.

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On the orthogonality of the MacDonald's functions

Cited by 19 publications

References 12 publications

Re-Examination of Globally Flat Space-Time

Re-Examination of Globally Flat Space-Time

An orthogonality relation for the Whittaker functions of the second kind of imaginary order

Intertwining operator in thermal CFTd

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