  The concept of Tomonaga--Luttinger liquids (TLL) on the basis of the
free-boson models is ubiquitous in theoretical descriptions of low-energy
properties in one-dimensional quantum systems. In this work, we develop a
squeezed-field path-integral description for gapless one-dimensional systems
beyond the free-boson picture of the TLL paradigm. In the squeezed-field
description, the parameter of the Bogoliubov transformation for the TL
Hamiltonian becomes a dynamical squeezing field, and its fluctuations give rise
to corrections to the free-boson results. We derive an effective nonlinear
Lagrangian describing the dispersion relation of the squeezing field, and
interactions between the excitations of the TLL and the squeezing modes. Using
the effective Lagrangian, we analyze the imaginary-time correlation function of
a vertex operator in the non-interacting limit. We show that a side-band branch
emerges due to the fluctuation of the squeezing field, in addition to the
standard branch of the free-boson model of the TLL paradigm. Furthermore, we
perturbatively analyze the spectral function of the density fluctuations for an
ultracold Bose gas in one dimension. We evaluate the renormalized values of the
phase velocities and spectral weights of the TLL and side-band branches due to
the interaction between the TLL and the squeezing modes. At zero temperature,
the renormalized dispersion relations are linear in the momentum, but at
nonzero temperatures, these acquire a nonlinear dependence on the momentum due
to the thermal population of the excitation branches.
