Quantum squeezing sidesteps the limits on mechanical transducers
Article excerpt
From detecting the ripples of colliding black holes to imaging individual chemical bonds, mechanical transducers have repeatedly transformed our understanding of the universe. So far, however, the sensitivity of these devices has been intrinsically limited by the laws of quantum mechanics itself.
"Colored" knot polynomials satisfy difference equation w.r.t. the highest weights of the underlying representation, which in the case of symmetrically colored Jones are named "quantum $A$-polynomials". In the double scaling quasiclassical (Kashaev) limit, when representation size $r\sim \hbar^{-1}$, there are different phases, in one of them the classical action vanishes and in another one it is a deformation of hyperbolic volume (of a knot complement in $S^3$). This corresponds to a splitting of the non-homogeneous version of the quantum $A$-polynomial into two pieces, which we illustrate by more examples than just a figure-eight knot $4_1$ in the original paper. From the point of view of quasiclassics, hyperbolic volume is just an integration constant, which is not fully determined by the $A$-polynomial equation, and actually remains ambiguous in this formalism. As a byproduct, we expect that classical $A$-polynomial at $L=1$ becomes proportional to Alexander: $A^{\cal K}(1,M)\sim Δ^{\cal K}(M)$, this seems true, but $A$ should be consistent with the polynomiality of {\it non-homogeneous quantum} ${\cal A}$-polynomial, what sometime implies that it is not minimal.