Dispersion relations, knot polynomials and the -deformed harmonic oscillator
arXiv:2204.13986 · doi:10.1103/PhysRevD.106.126019
Abstract
We show that the crossing symmetric dispersion relation (CSDR) for 2-2 scattering leads to a fascinating connection with knot polynomials and q-deformed algebras. In particular, the dispersive kernel can be identified naturally in terms of the generating function for the Alexander polynomials corresponding to the torus knot arising in knot theory. Certain linear combinations of the low energy expansion coefficients of the amplitude can be bounded in terms of knot invariants. Pion S-matrix bootstrap data respects the analytic bounds so obtained. We correlate the -deformed harmonic oscillator with the CSDR-knot picture. In particular, the scattering amplitude can be thought of as a -averaged thermal two point function involving the -deformed harmonic oscillator. The low temperature expansion coefficients are precisely the -averaged Alexander knot polynomials.
v4: 10 pages, 4 figures, updated version
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