Quantum field theory in flat spacetime with multiple time directions

TL;DR

The work develops an intrinsic quantum field theory in flat spacetime with multiple time directions $R^{n,d-n}$ by introducing Neumann modes beyond plane waves and defining Neumann and Hankel vacua. It shows that correlators computed within this framework reproduce the standard Minkowski results via analytic continuation, despite the presence of a single conformal boundary that necessitates an $S$-vector instead of an $S$-matrix. The approach relies on a scalar field expansion in hyperspherical harmonics and generalized $S^{n-1}$ Bessel functions, with $q$ acting as the evolution parameter along time. This framework lays groundwork for flat holography and bulk reconstruction while making LSZ and path-integral quantization feasible in spacetimes with multiple time directions.

Abstract

This letter serves as the generalization of the work 2505.16436, where we investigated the quantum field theory in Klein space which has two time directions. We extend studies to the general spacetime $\mathbb{R}^{n,d-n}\,(n,d-n\geq2)$ in a similar manner: the ``length of time'' $q$ is regarded as the evolution direction of the system; additional modes beyond the plane wave modes are introduced in the canonical quantization. We show that this novel formulation is consistent with the analytical continuation of the results in Minkowski spacetime.

Figures (1)

• Figure 1: The Penrose diagram of flat space $\mathbb{R}^{n,d-n}\,(n,d-n\geq2)$. The infinite coordinate ranges $q, r \in [0, \infty)$ are conformally mapped to finite ranges $Q, R \in [0, \pi)$. The conformal boundary emerges at the singularity of $\Omega^{-1}$, with null infinity $\mathscr{I}$ located at $Q + R = \pi$. The endpoints $(Q, R) = (0, \pi)$ and $(Q, R) = (\pi, 0)$ correspond to the spacelike infinity $i_0$ and timelike infinity $i'$, respectively.