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arXiv:2412.08154 · Physical review. D/Physical review. D.

Effective description of open quantum dynamics in relativistic scattering

Kaito Kashiwagi, Akira Matsumura

Published Feb 18, 2025• 2 citations
arXivHTMLPDF

Abstract

We investigate the open dynamics of quantum particles in relativistic scattering. Specifically, we consider the scattering process of quantum particles coupled to an environment initially in a vacuum state. Tracing out the environment and using the unitarity of an S operator, we find the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator that describes the evolution of the particles. The GKSL generator is exemplified by focusing on the concrete processes: one is the decay of scalar particle ($\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}$), and the others are the pair annihilation and the $2\ensuremath{\rightarrow}2$ scattering of scalar particles ($\ensuremath{\phi}\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}$ and $\ensuremath{\phi}\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\phi}\ensuremath{\phi}$). The GKSL generator for $\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}$ has a parameter with the coupling between $\ensuremath{\phi}$ and $\ensuremath{\chi}$ and the masses of the fields. The GKSL generator associated with $\ensuremath{\phi}\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}$ is characterized by a Lorentz-invariant function of initial momenta. Notably, in the pair annihilation process, we show that the probability of pair annihilation varies depending on the superposition state of incident scalar $\ensuremath{\phi}$ particles. Furthermore, we observe that the GKSL generators derived in this paper have Poincar\'e symmetry. This means that GKSL generators with Poincar\'e symmetry provide an effective description of the asymptotic behavior of open quantum dynamics in the long-term processes considered here.

DOI
10.1103/physrevd.111.045014

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We investigate the open dynamics of quantum particles in relativistic scattering. Specifically, we consider the scattering process of quantum particles coupled to an environment initially in a vacuum state. Tracing out the environment and using the unitarity of an S operator, we find the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator that describes the evolution of the particles. The GKSL generator is exemplified by focusing on the concrete processes: one is the decay of scalar particle (\ensuremathϕ\ensuremath→\ensuremathχ\ensuremathχ\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}\ensuremathϕ\ensuremath→\ensuremathχ\ensuremathχ), and the others are the pair annihilation and the 2\ensuremath→22\ensuremath{\rightarrow}22\ensuremath→2 scattering of scalar particles (\ensuremathϕ\ensuremathϕ\ensuremath→\ensuremathχ\ensuremathχ\ensuremath{\phi}\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}\ensuremathϕ\ensuremathϕ\ensuremath→\ensuremathχ\ensuremathχ and \ensuremathϕ\ensuremathϕ\ensuremath→\ensuremathϕ\ensuremathϕ\ensuremath{\phi}\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\phi}\ensuremath{\phi}\ensuremathϕ\ensuremathϕ\ensuremath→\ensuremathϕ\ensuremathϕ). The GKSL generator for \ensuremathϕ\ensuremath→\ensuremathχ\ensuremathχ\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}\ensuremathϕ\ensuremath→\ensuremathχ\ensuremathχ has a parameter with the coupling between \ensuremathϕ\ensuremath{\phi}\ensuremathϕ and \ensuremathχ\ensuremath{\chi}\ensuremathχ and the masses of the fields. The GKSL generator associated with \ensuremathϕ\ensuremathϕ\ensuremath→\ensuremathχ\ensuremathχ\ensuremath{\phi}\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}\ensuremathϕ\ensuremathϕ\ensuremath→\ensuremathχ\ensuremathχ is characterized by a Lorentz-invariant function of initial momenta. Notably, in the pair annihilation process, we show that the probability of pair annihilation varies depending on the superposition state of incident scalar \ensuremathϕ\ensuremath{\phi}\ensuremathϕ particles. Furthermore, we observe that the GKSL generators derived in this paper have Poincar'e symmetry. This means that GKSL generators with Poincar'e symmetry provide an effective description of the asymptotic behavior of open quantum dynamics in the long-term processes considered here.
Abstract Avoiding the assumption that relativistic scattering be describable by interacting fields we find in the Schrödinger picture relativistic scattering closely analogue to the non-relativistic case. On the space of scattering states the invariant mass operator M′M'M′ M ′ of the interacting time evolution has to be unitarily equivalent to the invariant mass M=P2M= \sqrt{P^2}M=P2​ M = P 2 where P, acting on many-particle states, is the sum of the one-particle four-momenta. For an observer at rest P0P^0P0 P 0 generates the free time evolution. Poincaré symmetry requires the interacting Hamiltonian H′H'H′ H ′ to Lorentz transform as 0-component of a four-vector and to commute with the four-velocity U=P/MU= P / M U=P/M U = P / M but not with P, else there is no scattering. Even though H′H'H′ H ′ does not commute with P, the scattering matrix does. The four-velocity U generates translations of states as they are seen by shifted observers. Superpositions of nearly mass degenerate particles such as a KlongK_{\text {long}}Klong​ K long are seen by an inversely shifted observer as a shifted KlongK_{\text {long}}Klong​ K long with an unchanged relative phase. In contrast, the four-momentum P generates oscillated superpositions e.g. a shifted KshortK_{\text {short}}Kshort​ K short with a changed relative phase. The probability of scattering of massive particles is shown to be approximately proportional to the spacetime overlap of their position wave functions. This is basic to macroscopic locality and justifies to represent the machinery of actual scattering experiments by the vacuum. In suitable variables the relativistic Hamiltonian of many-particle states is not the sum of a Hamiltonian for the motion of the center and a commuting Hamiltonian for the relative motion but factorizes as their product. They act on different variables of the wave functions.
The year 2026 marks the fiftieth anniversary of the two seminal papers in which Gorini, Kossakowski, and Sudarshan, and independently Lindblad, characterized the generators of completely positive quantum dynamical semigroups. We revisit the conceptual and historical route leading to the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation and examine several of its early developments and consequences. Our aim is to show how several initially separate mathematical and physical ideas converged in the GKLS structure theorem, imme- diately opening the way to manifold consequences for the description of dissipative quantum phenomena and how that theorem, fifty years later, remains both a foundational result and an active framework for open quantum dynamics.
We develop a framework that allows us to describe the dynamics of the total state of an open quantum system and its bosonic environment in the usual Born (weak coupling) and Markov approximation. By shifting the whole time-dependence into an unnormalized s-operator of the open system, the full dynamics is captured by an s-master equation of similar structure than the well-known Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation for the reduced dynamics. By varying the ordering parameter s (0 ≤ s ≤ 1) we obtain the partial Husimi representation (s = 0) and the partial Glauber-Sudarshan representation (s = 1) for the dynamics of the total state. For the reduced density operator the GKSL master equation can be derived easily. The case of s = 1/2, leading to a partial Wigner representation, is helpful to study the overlap of states in the total Hilbert space of system and environment.
An open quantum system can be described by a master equation, of which one of the most popular is the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. We revisit description of driven-dissipative quantum systems focusing on the appropriate choice of the system's basis and the respective transformations. We consider the GKSL equation in different bases and calculate the dynamics for a qubit and for a qudit. An appropriate choice of the basis is a fundamental problem for theoretical consideration of open quantum systems and provides an opportunity to obtain the desired evolution in practice.
A well established approach to describe open quantum system dynamics is the celebrated Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. In this article we aim to shed light on semigroup dynamics that are not of this form. Such dynamics may appear in real physical systems describing asymptotic long-time dynamics. Thus, they correspond to semigroup dynamics applicable to a restricted set of initial states only. As a way of motivating our studies, first we review known results about the single-mode Gaussian Quantum Brownian motion channel and corresponding master equations. Next we discuss a conceptually simpler family of qubit semigroup generators and identify parameter regions of completely positive, merely positive, and non-positive dynamics. We then focus on the latter case to which we will refer to as restricted-positive, as the semigroup generator may well be physically meaningful when applied to a restricted set of admissible initial states.
Open quantum systems are, in general, described by a density matrix that is evolving under transformations belonging to a dynamical semigroup. They can obey the Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space of dimension 2. First, we find final fixed states (called pointers) of an evolution of an open system, and we then obtain a general solution to the FGKLS equation and confirm that it converges to a pointer. After this, we check that the solution has physical meaning, i.e., it is Hermitian, positive and has trace equal to 1, and find a moment of time starting from which the FGKLS equation can be used—the range of applicability of the semigroup symmetry. Next, we study the behavior of a solution for a weak interaction with an environment and make a distinction between interacting and non-interacting cases. Finally, we prove that there cannot exist oscillating solutions to the FGKLS equation, which would resemble the behavior of a closed quantum system.