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arXiv:2412.10634 · Advances in Theoretical and Mathematical Physics

Adiabatic definitions of scattering matrix and inclusive scattering matrix

Albert S. Schwarz

Published Jan 1, 2026• 0 citations
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Abstract

The main goal of the present paper is to analyze the adiabatic definition of scattering matrix in the formalism of L-functionals. This definition leads to the notion of inclusive scattering matrix closely related to inclusive cross sections. We discuss this notion and the relation of our techniques to adiabatic quantum computing.
DOI
10.4310/atmp.260521222846

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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.
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The exact expression for the transition rate for quantum-mechanical scattering problems is derived formally without recourse to adiabatically switching on the interaction or averaging the initial state in time. The equivalence is shown of the stationary definition of the SSS matrix and an appropriate definition by the time-dependent method.
The problem of extending quantum-mechanical formal scattering theory to a more general class of models that also includes quantum field theories is discussed, with the aim of clarifying certain aspects of the definition of scattering states. As the strong limit is not suitable for the definition of scattering states in quantum field theory, some other limiting procedure is needed. Two possibilities are considered, the abelian limit and adiabatic switching. Formulas for the scattering states based on both methods are discussed, and it is found that generally there are significant differences between the two approaches. As an illustration of the application and the features of these formulas, S-matrix elements and energy corrections in two quantum field theoretical models are calculated using (generalized) old-fashioned perturbation theory. The two methods are found to give equivalent results.