Quantum field theory with scale-dependent weighted measures: nonlinear momentum representations, vacuum fluctuations, and causal structure

Abstract

We reconsider the construction of free quantum fields through momentum-space weighted measures, introducing an independent energy scale that restores the conventional theory at low momenta. The resulting family is characterized by a dimensionless spectral function and includes, as a distinguished case, the spatial weight associated with the Euclidean Snyder algebra. We discuss its relation to modified dispersion relations and nonlinear momentum coordinates, and introduce an observer-covariant spectral weight defined using a timelike four-velocity. For the complex Klein-Gordon field we obtain the equal-time algebra, conserved weighted observables, vacuum fluctuations, and two-point functions in arbitrary spatial dimension. The ultraviolet conditions for their finiteness are established, and the nonlocal causal structure is analyzed in terms of a spatial convolution of the ordinary Pauli-Jordan distribution. An illustrative interacting extension and numerical profiles of the kernels and amplitudes are also presented.