Crossing-Symmetry Restrictions on Dispersion Relations, and Sum Rules forππScattering Lengths

1971 ◽  
Vol 4 (4) ◽  
pp. 1046-1052 ◽  
Author(s):  
Samuel Krinsky
Keyword(s):  
Sum Rules ◽  
Dispersion Relations ◽  
Crossing Symmetry

scholarly journals Dispersion relations and exact bounds on CFT correlators

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Miguel F. Paulos
Keyword(s):  
Sum Rules ◽  
Dispersion Relations ◽  
Lower And Upper Bounds ◽  
Equivalent Formulation ◽  
Ising Spin ◽  
Crossing Symmetry ◽  
Free Fields ◽  
Exact Bounds ◽  
Regge Limit ◽  
Spin Correlator

Abstract We derive new crossing-symmetric dispersion formulae for CFT correlators restricted to the line. The formulae are equivalent to the sum rules implied by what we call master functionals, which are analytic extremal functionals which act on the crossing equation. The dispersion relations provide an equivalent formulation of the constraints of the Polyakov bootstrap and hence of crossing symmetry on the line. The built in positivity properties imply simple and exact lower and upper bounds on the values of general CFT correlators on the Euclidean section, which are saturated by generalized free fields. Besides bounds on correlators, we apply this technology to determine new universal constraints on the Regge limit of arbitrary CFTs and obtain very simple and accurate representations of the 3d Ising spin correlator.

scholarly journals Generalized Kramers–Kronig relations and sum rules for moments and powers of degenerate four wave mixing susceptibility

2021 ◽  
Author(s):  
Jarkko J. Saarinen
Keyword(s):  
Sum Rules ◽  
Terahertz Spectroscopy ◽  
Theoretical Models ◽  
Present Theory ◽  
Four Wave Mixing ◽  
Dispersion Relations ◽  
Nonlinear Susceptibility ◽  
Complex Frequency ◽  
Wave Mixing ◽  
Degenerate Four Wave Mixing

AbstractGeneralized Kramers–Kronig (K–K) type dispersion relations and sum rules are derived in the static limit for the moments of the degenerate four wave mixing susceptibility. The degenerate nonlinear susceptibility is different from a typical use of the conventional K–K dispersion relations, which assume absence of complex poles of a function in the upper half of complex frequency plane, whereas degenerate susceptibility has simultaneous poles in both half planes. In the derivation of the generalized K–K relations the poles and their order are taken into account by utilization of the theorem of residues. The conventional K–K relations can be used to estimate the real and imaginary parts of the second and higher powers of the susceptibility as the effect of the poles is reduced due to a faster convergence of the dispersion relations. The present theory is directly applicable to higher order susceptibilities and can be used in testing of theoretical models describing the degenerate four wave mixing susceptibility in nonlinear optical and terahertz spectroscopy.

Dispersion relations and sum rules for the ellipsometric function

1980 ◽  
Vol 96 (1-3) ◽  
pp. A211
Author(s):  
T. Inagaki ◽  
H. Kuwata ◽  
A. Ueda
Keyword(s):  
Sum Rules ◽  
Dispersion Relations
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