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arXiv:2312.04069 · Physical Review A

Markovian quantum master equation with Poincaré symmetry

Kaito Kashiwagi, Akira Matsumura

Published May 13, 2024• 2 citations
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Abstract

We investigate what kind of Markovian quantum master equation (QME) in the Gorini-Kossakowski-Sudarshan-Lindblad form is realized under Poincar\'e symmetry. The solution of the Markovian QME is given by a quantum dynamical semigroup, for which we introduce invariance under Poincar\'e transformations. Using the invariance of the dynamical semigroup and applying the unitary representation of the Poincar\'e group, we derive the Markovian QME for a relativistic massive spin-0 particle. Introducing the field operator of the massive particle and examining its evolution, we find that the field follows a dissipative Klein-Gordon equation. In addition, we show that any two local operators for spacelike separated regions commute with each other. This means that the microcausality condition is satisfied for the dissipative model of the massive particle.

DOI
10.1103/physreva.109.052214

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We investigate what kind of Markovian quantum master equation (QME) in the Gorini-Kossakowski-Sudarshan-Lindblad form is realized under Poincar'e symmetry. The solution of the Markovian QME is given by a quantum dynamical semigroup, for which we introduce invariance under Poincar'e transformations. Using the invariance of the dynamical semigroup and applying the unitary representation of the Poincar'e group, we derive the Markovian QME for a relativistic massive spin-0 particle. Introducing the field operator of the massive particle and examining its evolution, we find that the field follows a dissipative Klein-Gordon equation. In addition, we show that any two local operators for spacelike separated regions commute with each other. This means that the microcausality condition is satisfied for the dissipative model of the massive particle.
The celebrated GKLS master equation, widely called just the Lindblad equation, is the universal dynamical equation of nonrelativistic open quantum systems in their Markovian approximation. It is not necessary and perhaps impossible that GKLS equations possess sensible relativistic forms. In 2017, in a lucid talk on black hole information loss paradox, David Poulin argued for a Lorentz invariant GKLS master equation proposed by Alicki, Fannes, and Verbeure in 1986. The equation is really puzzling. A closer look uncovers a smartly hidden defect that leaves us without Lorentz invariant Markovian master equations. They, in view of the present author, should not exist.
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.
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 present a derivation of the Markovian master equation by the renormalization group method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then by applying the renormalization group method, we show that the dependence of the specific initial time on the perturbative solution can be removed and the Markovian semigroup master equation in the Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) form is obtained in the weak coupling limit.
We give a new nonstandard proof of the well-known theorem that the generator LLL of a quantum dynamical semigroup exp⁡(tL)\exp(tL)exp(tL) on a finite-dimensional quantum system has a specific form called a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator (also known as a Lindbladian) and vice versa. The proof starts from the Kraus representation of the quantum channel exp⁡(δtL)\exp (δt L)exp(δtL) for an infinitesimal hyperreal number δt>0δt>0δt>0 and then estimates the orders of the traceless components of the Kraus operators. The jump operators naturally arise as the standard parts of the traceless components of the Kraus operators divided by δt\sqrt{δt}δt​. We also give a nonstandard proof of a related fact that close completely positive maps have close Kraus operators.
The theory of open quantum systems is one of the most essential tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) master equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and straightforward as possible to be useful to readers with no previous knowledge of this field.