Schrödinger's Cataplex

We discuss elementary entwiners that cross-weave the variables of certain integrable models: Liouville, sine-Gordon, and sinh-Gordon field theories in two-dimensional spacetime, and their quantum mechanical reductions. First we define a complex time parameter that varies from one energy-shell to another. Then we explain how field propagators can be simply expressed in terms of elementary functions through the combination of an evolution in this complex time and a duality transformation. IT’S COMPLEX TIME One hundred years ago at the close of the 19th century, just before Planck’s discovery of light quanta, H. M. Macdonald [21] considered the mathematical problem of determining zeroes of Bessel functions in the complex plane. He was led to find the lovely integral identity Kν(e )Kν(e ) = ∫ +∞ −∞ dz S (x, y, z) Kν(e ) . The kernel in the integral is a simple, symmetric exponential of exponentials. S (x, y, z) = 1 2 exp (−F (x, y, z)) , F (x, y, z) = 1 2 ( ex+y−z + ex−y+z + e−x+y+z ) . A talk given at Orbis Scientiae, 17 December 1999, Fort Lauderdale, Florida. To appear in the proceedings, “Quantum Gravity, Generalized Theory of Gravitation and Superstring Theory Based Unification”, B. Kursunoglu and A. Perlmutter, Eds., Plenum Press, 2000. [email protected] Curtright quant-ph/0011101 Cataplex 2 The most direct way to prove this result is through the use of the Heine-Schläfli identity that cogently expresses modified Bessel functions as an integral transform (cf. [32], §6.22 and §13.71). Kν(e ) = K−ν(e ) = 1 2 ∫ ∞ −∞ dX exp (−e coshX + νX) . When this is substituted for each of the Bessel functions in the previous bilinear, a simple change of integration variables immediately yields Macdonald’s identity. (For another derivation, which overlaps with many of the standard textbook methods [23, 25] of obtaining similar integral relations for Mathieu functions, see [10].) More recently in the 20th century, the modified Bessel functions in Macdonald’s identity have appeared in a physical context as solutions of the Liouville quantum mechanics. For Liouville quantum mechanics the Hamiltonian is H = p2 + e2x . Coordinate space energy eigenfunctions are then solutions of