# Convergence and well-definedness of Lorentzian path integrals

Wick rotation of quantum field theories to Euclidean path integrals with a nonnegative measure everywhere is a wonderful tool. Not so with Lorentzian path integrals. Events far separated in coordinates can have zero or arbitrarily tiny interval separation in relativity. Ultraviolet divergences crop up for infrared separations at arbitrarily high boosts. The integrand becomes highly oscillatory phasewise around null separations. Absolute convergence is nonexistent, only mere convergence which is so sensitive to integral reorderings it raises warning flags. Changing the regulator or the order of limits can change the answer drastically. Are such path integrals even well-defined?

-