Abstract
We develop a field theory with dissipation based on a finite range of wave propagation and associated gapped momentum states in the wave spectrum. We analyze the properties of the Lagrangian and the Hamiltonian with two scalar fields in different representations and show how the new properties of the two-field Lagrangian are related to Keldysh-Schwinger formalism. The proposed theory is non-Hermitian, and we discuss its properties related to $\mathcal{PT}$ symmetry. The calculated correlation functions show a decaying oscillatory behavior related to gapped momentum states. We corroborate this result using path integration. The interaction potential becomes short-ranged due to dissipation. Finally, we observe that the proposed field theory represents a departure from the harmonic paradigm and discuss the implications of our theory for the Lagrangian formulation of hydrodynamics.
Highlights
The basic assumptions and results of statistical physics are related to introducing, and frequently exploiting, the concept of a closed or quasiclosed system or subsystem
We develop a field theory with dissipation based on a finite range of wave propagation and associated gapped momentum states in the wave spectrum
gapped momentum states (GMS) is a well-specified effect and is naturally suited for describing the dissipation in a field theory because the field theory is deeply rooted in the harmonic paradigm involving the propagation of plane waves [30]
Summary
The basic assumptions and results of statistical physics are related to introducing, and frequently exploiting, the concept of a closed or quasiclosed system or subsystem. (We note that calculating these excitations presents an exponentially complex problem because it involves a large number of strongly coupled nonlinear oscillators [21].) In our consideration, a decaying object that loses energy is the harmonic plane wave (phonon) propagating in a disordered system where its not an eigenstate. GMS is a well-specified effect and is naturally suited for describing the dissipation in a field theory because the field theory is deeply rooted in the harmonic paradigm involving the propagation of plane waves [30] Despite its specificity, this effect and the proposed field theory are generally applicable to a wide range of physical phenomena in interacting systems where collective excitations propagate.
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