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arXiv:1904.04050 · Nuclear Physics B

Inclusive scattering matrix and scattering of quasiparticles

Albert S. Schwarz

Published Nov 25, 2019• 7 citations
arXivHTMLPDF

Abstract

The quantum theory can be formulated in the language of positive functionals on Weyl or Clifford algebra (L-functionals). It is shown that this language gives simple understanding of diagrams of Keldysh formalism (that coincide in our case with the diagrams of thermo-field dynamics). The matrix elements of the scattering matrix in the formalism of L-functionals are related to inclusive cross-sections, therefore we suggest the name “inclusive scattering matrix” for this notion. The inclusive scattering matrix can be expressed in terms of on-shell values of generalized Green functions. This notion is necessary if we want to analyze collisions of quasiparticles.

DOI
10.1016/j.nuclphysb.2019.114869

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The quantum theory can be formulated in the language of positive functionals on Weyl or Clifford algebra (L-functionals). It is shown that this language gives simple understanding of diagrams of Keldysh formalism (that coincide in our case with the diagrams of thermo-field dynamics). The matrix elements of the scattering matrix in the formalism of L-functionals are related to inclusive cross-sections, therefore we suggest the name “inclusive scattering matrix” for this notion. The inclusive scattering matrix can be expressed in terms of on-shell values of generalized Green functions. This notion is necessary if we want to analyze collisions of quasiparticles.
We study the scattering of particles and quasiparticles in the framework of algebraic quantum field theory. The main novelty is the construction of inclusive scattering matrix related to inclusive cross-sections. The inclusive scattering matrix can be expressed in terms of generalized Green functions by a formula similar to the LSZ formula for the conventional scattering matrix. The consideration of inclusive scattering matrix is necessary in quantum field theory if a unitary scattering matrix does not exist (if the theory does not have particle interpretation). It is always necessary if we want to consider collisions of quasiparticles.
This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory, including recent results. It is based on the algebraic approach in which the starting point is a star-algebra and on the geometric approach in which the starting point is a convex set of states. Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in the geometric approach. Particles are defined as elementary excitations of the ground state (and quasiparticles as elementary excitations of any translation invariant state). The conventional scattering matrix does not work for quasiparticles (or even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the notion of an inclusive scattering matrix, which is closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) and the inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics. The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of the symbol of an operator; these arguments can be applied in the geometric approach as well. In particular, this result can be used to provide a simple derivation of the diagram technique for generalized Green functions. The notion of an inclusive scattering matrix makes sense in the geometric approach, although it seems that a definition of the conventional scattering matrix cannot be provided in this situation. The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices are able to measure only a part of the observables.
We provide a description of interacting quantum fields in terms of density matrices for any occupation numbers in Fock space in a momentum basis. As a simple example, we focus on a real scalar field interacting with another real scalar field, and present a practicable formalism for directly computing the density matrix elements of the combined scalar–scalar system. For deriving the main formula, we use techniques from non-equilibrium quantum field theory like thermo-field dynamics and the Schwinger–Keldysh formalism. Our results allow for studies of particle creation/annihilation processes at finite times and other non-equilibrium processes, including those found in the theory of open quantum systems.
A tetradic scattering matrix formalism is proposed as a comprehensive framework for describing scattering and relaxation processes of both stationary and non-stationary systems. In the latter case (realized by the scattering of the molecular-maser beam) ordinary dyadic T-matrix formalism involving on-the-energy-shell scattering amplitudes is inadequate, the introduction of off-the-shell elements being required. Expressions are given relating the tetradic T-matrix elements to various scattering and attenuation rates (or cross sections). The relation to the tetradic level-shift operator in relaxation (linear response) theory is discussed. Photon absorption is presented as the successive occurrence of a scattering and a relaxation process, from the point of view of this formalism.
We have analyzed Dyson's UUU-matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the UUU matrix as a dominant term plus an infinite series involving multiple integrals of time. For a certain rather restrictive class of Hamiltonians, our theory is exact for a general time-dependent problem. For other Hamiltonians, we can only obtain approximate expressions for our UUU matrix and hence the wave function. Treating a time-independent problem as a special case of the time-dependent situation with a sudden-switching process, we have shown that our UUU matrix is exact. To demonstrate the working procedures of our theory, we apply it to study the well-known time-independent charged harmonic-oscillator problem and the more general harmonic oscillator with a time-dependent driving force. Compared with other methods, our new theory appears to lead to a result which contains more information than others due to the inclusion of noncommutability properties of operators in the operator Schr\"odinger equation. It has been shown that the classical Feynman path-integral formalism can be deduced from quantum mechanics with the use of the Green's-function operator. It is interesting to note that apart from a step function, the Green's-function operator is the same as that of our U(s){U}^{(s)}U(s) matrix, which is the UUU matrix obtained within the regime of the Schr\"odinger picture for a time-independent Hamiltonian, as a special case of our general time-dependent treatment.