Extrapolation dictionary in flat holography

Determine the extrapolation dictionary that maps operators in the boundary field theory to bulk fields in flat holography for flat spacetimes R^{n,d−n} with n, d−n ≥ 2, thereby specifying how boundary data on the single null conformal boundary correspond to bulk quantum fields in this multiple-time-direction framework.

Background

The paper develops a canonical quantization of scalar quantum field theory in flat spacetimes with multiple time directions R^{n,d−n} (n, d−n ≥ 2), using the “length of time” coordinate q as the evolution parameter and introducing Neumann-function modes in addition to plane waves to achieve a consistent quantization. Two distinct vacua are defined: a Neumann vacuum at the origin and a Hankel vacuum at asymptotic infinity.

Because these spacetimes possess only a single null boundary, they are proposed as natural arenas for flat holography where an S-vector replaces the conventional S-matrix. Within this context, the authors highlight a key unresolved issue: an extrapolation dictionary between boundary operators and bulk fields analogous to constructions familiar from AdS/CFT has not yet been established in flat holography.

The authors suggest that the novel modes associated with Neumann functions may be important for bulk reconstruction, indicating a potential pathway to address the dictionary, but the explicit mapping remains to be determined.