diff --git "a/3dAzT4oBgHgl3EQffPzi/content/tmp_files/load_file.txt" "b/3dAzT4oBgHgl3EQffPzi/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/3dAzT4oBgHgl3EQffPzi/content/tmp_files/load_file.txt" @@ -0,0 +1,700 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf,len=699 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='01451v1 [quant-ph] 4 Jan 2023 Reduced dynamics with Poincar´e symmetry in open quantum system Akira Matsumura∗ Department of Physics, Kyushu University, Fukuoka, 819-0395, Japan Abstract We consider how the reduced dynamics of an open quantum system coupled to an environment admits the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The reduced dynamics is described by a dynamical map, which is given by tracing out the environment from the total dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Introducing the notion of covariant map, we investigate the dynamical map which is symmetric under the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Based on the representation theory of the Poincar´e group, we develop a systematic way to give the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using this way, we derive the dynamical map for a massive particle with a finite spin and a massless particle with a finite spin and a nonzero momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We show that the derived map gives the unitary evolution of a particle when its energy is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We also find that the dynamical map for a particle does not have the Poincar´e symmetry when the superposition state of the particle decoheres into a mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ∗Electronic address: matsumura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='akira@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='kyushu-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='jp 1 Contents I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Introduction 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Quantum dynamical map and its symmetry 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Dynamical map with Poincar´e symmetry 5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A model of the dynamical map for a single particle 10 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Conclusion 13 Acknowledgments 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (54),(55),(56),(57),(58) and (59) 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Analysis of a massive particle 15 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Analysis on a massless particle 21 References 27 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' INTRODUCTION It is difficult to isolate a quantum system perfectly, which is affected by the inevitable influence of a surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Such a quantum system is called an open quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since we encounter open quantum systems in a wide range of fields such as quantum information science [1, 2], condensed matter physics [3, 4] and high energy physics [5], it is important to understand their dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In general, the dynamics of an open quantum system, the so-called reduced dynamics, is very complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This is because the environment may have infinitely many degrees of freedom and they are uncontrollable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' One needs the effective theory with relevant degrees of freedom to describe the reduced dynamics of an open quantum system [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' As is well-known, symmetry gives a powerful tool for capturing relevant degrees of freedom in the dynamics of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For example, let us focus on the symmetry in the Minkowski spacetime, which is called the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Imposing the Poincar´e symmetry on a quantum theory, one finds that quantum dynamics in the theory is described by the fundamental degrees of freedom such as a massive particle and a massless particle [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The approach based on symmetries provides 2 a way to get the effective theory of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this paper, we discuss the consequences of the Poincar´e symmetry on the reduced dynamics of an open quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This may give the understanding of the relativistic theories of open quantum systems (for example, [7–14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The present paper is also motivated by the theory of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the unification of quantum mechanics and gravity has not been completed yet, we do not exactly know how gravity is incorporated in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This situation has prompted to propose many models on the gravity of quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the previous work [15], the model with a classical gravitational interaction between quantum systems was proposed, which is called the Kafri-Taylor-Milburn model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In addition, the Diosi-Penrose model [16–18] and the Tilloy-Diosi model [19] were advocated, for which the gravity of a quantum system intrinsically induces decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' They are formulated by the theory of open quantum systems in a non- relativistic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' One may concern how the models are consistent with a relativistic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Our analysis on reduced dynamics with the Poincar´e symmetry would help to obtain a relativistic extension of the above proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For our analysis, we assume that the reduced dynamics of an open quantum system is described by a dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The dynamical map is obtained by tracing out the environment from the total unitary evolution with an initial product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' It is known that the dynamical map is represented by using the Kraus operators [2, 20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The notion of covariant map is adopted for incorporating a symmetry into a dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We derive the condition of a dynamical map with the Poincar´e symmetry in terms of the Kraus operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' With the help of the representation theory of the Poincar´e group, we obtain a systematic way to deduce those Kraus operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Applying the way, we exemplify the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To get the concrete Kraus operators, we focus on the dynamics of a single particle, which is possible to decay into the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Assuming that the particle is a massive particle with a finite spin or a massless particle with a finite spin and a nonzero momentum, we get a model of the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the model, we find the following consequences: (i) if the particle is stable or the energy of particle is conserved, the obtained map turns out to be the unitary map given by the Hamiltonian of particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (ii) If the superposition state of a particle decoheres into a mixed state, the dynamical map for the particle does not have the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' These consequences imply that the Poincar´e symmetry can strongly constraint the reduced dynamics of an open quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='II, we discuss a dynamical map describing the reduced dynamics of an open quantum system and consider how symmetries are introduced in the 3 dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='III, we derive the condition that the dynamical map is symmetric under the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='IV, focusing on the dynamics of a single particle, we present a model of the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We then investigate the properties of the dynamical map in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='V is devoted as the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We use the unit ℏ = c = 1 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' QUANTUM DYNAMICAL MAP AND ITS SYMMETRY In this section, we consider the reduced dynamics of an open quantum system and discuss the symmetry of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The reduced dynamics is given as the time evolution of the density operator of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The time evolution from a time τ = s to τ = t is assumed to be given by ρ(t) = Φt,s[ρ(s)] = TrE[ ˆU(t, s)ρ(s) ⊗ ρE ˆU †(t, s)], (1) where ρ(τ) is the system density operator, ρE is the density operator of an environment and ˆU(t, s) is the unitary evolution operator of the total system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this paper, the map Φt,s is called a dynamical map, which has the property called completely positive and trace-preserving (CPTP) [2, 20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The dynamical map Φt,s is rewritten in the operator-sum representation, Φt,s[ρ(s)] = � λ ˆF t,s λ ρ(s) ˆF t,s † λ , (2) where ˆF t,s λ called the Kraus operators satisfy the completeness condition, � λ ˆF t,s † λ ˆF t,s λ = ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (3) In this notation, λ takes discrete values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' When the label λ is continuous, we should replace the summation � λ with the integration � dµ(λ) with an appropriate measure µ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' It is known that two dynamical maps Φ and Φ′ with Φ[ρ] = � λ ˆFλ ρ ˆF † λ, Φ′[ρ] = � λ ˆF ′ λ ρ ˆF ′† λ , (4) are equivalent to each other (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Φ[ρ] = Φ′[ρ] for any density operator ρ) if and only if there is a unitary matrix Uλλ′ satisfying � λ Uλ1λU∗ λ2λ = δλ1λ2 = � λ Uλλ1U∗ λλ2 and ˆF ′ λ = � λ′ Uλλ′ ˆFλ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (5) This is the uniqueness of a dynamical map [2, 20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We introduce the notion of covariant map [22–24] to impose symmetry on dynamical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A dynamical map Φt,s is covariant under a group G if Φt,s[ ˆUs(g) ρ(s) ˆU † s (g)] = ˆUt(g)Φt,s[ρ(s)] ˆU † t (g), (6) 4 where ˆUs(g) and ˆUt(g) with g ∈ G are the unitary representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this paper, the dynamical map Φt,s satisfying (6) is called symmetric under the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the next section, we will discuss the dynamical map which is symmetric under the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' DYNAMICAL MAP WITH POINCAR´E SYMMETRY In this section, we consider a quantum theory with the Poincar´e symmetry and discuss the general conditions on a dynamical map with the Poincare symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The generators of the unitary representation of the Poincar´e group in the Schr¨odinger picture [6] are given by ˆPµ = � d3x ˆT 0 µ, ˆJµν = � d3x ˆ Mµν0, (7) where ˆTµν is the energy-momentum tensor satisfying ∂µ ˆT µ ν = 0, ˆTµν = ˆTνµ (8) and ˆ Mµνρ with ˆ Mµνρ = xµ ˆT ρ ν − xν ˆT ρ µ (9) is the Noether current associated with the Lorentz transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (8), we can show that ∂ρ ˆ Mµνρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Focusing on each component of the generators, we have ˆH = ˆP 0 = � d3x ˆT 00, ˆP i = � d3x ˆT 0i, (10) ˆJk = 1 2ǫijk ˆJij = � d3x ǫijkxi ˆT 0 j , ˆKk = � d3(xk ˆT 00 − t ˆT 0k), (11) where note that the boost generator ˆKk explicitly depends on a time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' These operators satisfy the commutation relations, [ ˆPi, ˆPj] = 0, (12) [ ˆPi, ˆH] = 0, (13) [ ˆJi, ˆH] = 0, (14) [ ˆJi, ˆJj] = iǫijk ˆJk, (15) [ ˆJi, ˆPj] = iǫijk ˆP k, (16) [ ˆJi, ˆKj] = iǫijk ˆKk, (17) [ ˆKi, ˆPj] = iδij ˆH, (18) [ ˆKi, ˆH] = i ˆPi, (19) [ ˆKi, ˆKj] = −iǫijk ˆJk, (20) 5 which correspond to the Poincar´e algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We consider a dynamical map Φt,s from ρ(s) to ρ(t) = Φt,s[ρ(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The Poincar´e symmetry of the dynamical map requires that ˆUt(Λ, a)Φt,s[ρ(s)] ˆU † t (Λ, a) = Φt,s[ ˆUs(Λ, a)ρ(s) ˆU † s (Λ, a)], (21) where the unitary operator ˆUt(Λ, a) depends on the proper (detΛ = 1) orthochronous (Λ00 ≥ 1) Lorentz transformation matrix Λµν and the real parameters aµ for the spacetime translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The unitary operator ˆUt(Λ, a) generated by ˆH, ˆPi, ˆJi and ˆKi has the group multiplication rule ˆUt(Λ′, a′) ˆUt(Λ, a) = ˆUt(Λ′Λ, a′ + Λ′a), (22) where we used the fact that we can always adopt the non-projective unitary representation of the Poincar´e group [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The explicit time dependence of ˆUt comes from the boost generator ˆKi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using the operator-sum representation, we have ˆUt(Λ, a) � λ ˆF t,s λ ρ(s) ˆF t,s† λ ˆU † t (Λ, a) = � λ ˆF t,s λ ˆUs(Λ, a)ρ(s) ˆU † s (Λ, a) ˆF t,s† λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From the uniqueness of the Kraus operators ˆF t,s λ (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (5)), we obtain ˆU † t (Λ, a) ˆF t,s λ ˆUs(Λ, a) = � λ′ Uλλ′(Λ, a) ˆF t,s λ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (23) We can always choose ˆF t,s λ so that { ˆF t,s λ }λ is the set of linearly independent operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This linear independence and the group multiplication rule of ˆUt(Λ, a) given in (22) lead to the fact that the unitary matrix Uλλ′(Λ, a) satisfies the group multiplication rule � λ′ Uλλ′(Λ′, a′)Uλ′λ′′(Λ, a) = Uλλ′′(Λ′Λ, Λa + a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (24) Hence, the unitary matrix Uλλ′(Λ, a) is a representation of the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Before discussing the condition of symmetry, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (23), we present the useful relation ˆKi = e−i ˆ Ht ˆKi 0 ei ˆ Ht, (25) where ˆKi 0 = � d3x xi ˆT 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (26) According to the Poincar´e algebra, we have ˆUt(Λ, a) = e−i ˆ Ht ˆU0(Λ, a)ei ˆ Ht, (27) 6 where ˆU0(Λ, a) is the unitary representation of the Poincar´e group with the genrators ˆH, ˆP i, ˆKi 0 and ˆJi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the scattering theory, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (27) is consistent with the Poincar´e invariance of the S-operator ˆS(∞, −∞), where ˆS(tf, ti) = ei ˆ H0tfe−i ˆ H(tf−ti)e−i ˆ H0ti and ˆH = ˆH0 + ˆV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This is because ˆU I† tf (Λ, a) ˆS(tf, ti) ˆU I ti(Λ, a) = ei ˆ H0tf ˆU † tf(Λ, a)e−i ˆ H(tf−ti) ˆUti(Λ, a)e−i ˆ H0ti = ei ˆ H0tfe−i ˆ Htf ˆU † 0(Λ, a) ˆU0(Λ, a)ei ˆ Htie−i ˆ H0ti = ˆS(tf, ti), (28) where ˆU I t(Λ, a) = ei ˆ H0t ˆUt(Λ, a)e−i ˆ H0t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (27) also implies that the unitary evolution generated by ˆH is symmetric under the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Indeed, we can show that the unitary map Ut,s[ρ(s)] = e−i ˆ H(t−s) ρ(s) ei ˆ H(t−s) (29) satisfies the condition of symmetry (21) as Ut,s[ ˆUs(Λ, a)ρ(s) ˆU † s (Λ, a)] = e−i ˆ H(t−s) ˆUs(Λ, a) ρ(s) ˆU † s (Λ, a)ei ˆ H(t−s) = e−i ˆ H(t−s) ˆUs(Λ, a)ei ˆ H(t−s) Ut,s[ρ(s)] e−i ˆ H(t−s) ˆU † s(Λ, a)ei ˆ H(t−s) = ˆUt(Λ, a) Ut,s[ρ(s)] ˆU † t (Λ, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (27) helps us to simplify the condition of symmetry, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (23), on the Kraus operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Defining the Kraus operators ˆEt,s λ as ˆEt,s λ = ei ˆ Ht ˆF t,s λ e−i ˆ Hs (30) which have the completeness condition, � λ ˆEt,s† λ ˆEt,s λ = ˆI, (31) we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (23) as ˆU † 0(Λ, a) ˆE ˆU0(Λ, a) = U(Λ, a) ˆE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) Here, we introduced the vector ˆE with the λ component given by ˆEt,s λ and the matrix U(Λ, a) with the (λ, λ′) component given by Uλλ′(Λ, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We define the dynamical map Et,s as Et,s[ρ] = � λ ˆEt,s λ ρ ˆEt,s† λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (33) The condition (32) gives the fact that the map Et,s is symmetric under the Poincar`e group in the sense that ˆU0(Λ, a)Et,s[ρ] ˆU † 0(Λ, a) = Et,s[ ˆU0(Λ, a) ρ ˆU † 0(Λ, a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (34) 7 The dynamical map Φt,s is written by the unitary map Ut,s and the dynamical map Et,s as Φt,s[ρ] = � λ ˆF t,s λ ρ ˆF t,s† λ = e−i ˆ Ht � λ ˆEt,s λ ei ˆ Hsρe−i ˆ Hs ˆEt,s† λ ei ˆ Ht = e−i ˆ HtEt,s[ei ˆ Hsρe−i ˆ Hs]ei ˆ Ht = e−i ˆ H(t−s)Et,s[ρ]ei ˆ H(t−s) = Ut,s ◦ Et,s[ρ], (35) where in the fourth equality we used the symmetric condition (34) noticing that ei ˆ Hs is the unitary transformation of the time translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Our task is to determine ˆE satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) (or Et,s satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='(34)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) is decomposed into equations for each irreducible representation subspace, the irreducible unitary representations of the Poincar´e group is useful for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Let us present how to classify the unitary representations of the Poincar´e group [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We consider the standard momentum ℓµ and the Lorentz transformation matrix (Sq)µν with qµ = (Sq)µνℓν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (36) The unitary matrix U(Λ, a) is written as U(Λ, a) = U(I, a)U(Λ, 0) = T (a)V(Λ), (37) where I is the identity matrix, U(I, a) = T (a) = e−iPµaµ and U(Λ, 0) = V(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We define the vector vq,ξ as vq,ξ = NqV(Sq)vℓ,ξ, (38) where Pµvℓ,ξ = ℓµvℓ,ξ, Nq is the normalization and the label ξ describes the degrees of freedom other than them determined by ℓµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We obtain the following transformation rules for the vector vq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ: T (a)vq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = Nqe−iP µaµV(Sq)vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(Sq)e−i(Sq)µνP νaµvℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(Sq)e−i(Sq)µνℓνaµvℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(Sq)e−iqµaµvℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = e−iqµaµvq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ (39) 8 and V(Λ)vq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(Λ)V(Sq)vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(ΛSq)vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(SΛq)V(S−1 Λq ΛSq)vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(SΛq) � ξ′ Dξ′ξ(Q(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' q))vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='s′ = Nq NΛq � ξ′ Dξ′ξ(Q(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' q))vΛq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (40) where Q(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' q) = S−1 Λq ΛSq is an element of the little group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' which satisfies Qµνℓν = ℓµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' and Dξ′ξ(Q) is the unitary representation of the little group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The irreducible unitary representations of the Poincar´e group are classified by the standard momentum ℓµ and the irreducible unitary represen- tations of the little group which does not change ℓµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' standard momentum ℓµ little group composed of Qµν with Qµνℓν = ℓµ (a) ℓµ = [M, 0, 0, 0], M > 0 SO(3) (b) ℓµ = [−M, 0, 0, 0], M > 0 SO(3) (c) ℓµ = [κ, 0, 0, κ], κ > 0 ISO(2) (d) ℓµ = [−κ, 0, 0, κ], κ > 0 ISO(2) (e) ℓµ = [0, 0, 0, N], N 2 > 0 SO(2,1) (f) ℓµ = [0, 0, 0, 0] SO(3,1) TABLE I: Classification of the standard momentum ℓµ and the little group associated with ℓµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For simplicity, ξ is regarded as the label of basis vectors of the irreducible representation sub- spaces of the little group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Other degeneracies not represented by q and ξ will be reintroduced in the form of the dynamical map Φt,s, which we will see in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We investigate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) restricted on each irreducible representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For convenience, we separately focus on the Lorentz transformation and the spacetime translation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='(32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The unitary operator ˆU0(Λ, a) is written as ˆU0(Λ, a) = ˆU0(I, a) ˆU0(Λ, 0) = ˆT(a) ˆV (Λ), (41) where ˆU0(I, a) = ˆT(a) = e−i ˆPµaµ with ˆP µ = [ ˆH, ˆP 1, ˆP 2, ˆP 3] and ˆU0(Λ, 0) = ˆV (Λ) with the genera- tors ˆJi and ˆKi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) for Λ = I, we have ˆT †(a) ˆE ˆT(a) = T (a) ˆE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (42) 9 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) for aµ = 0 gives ˆV †(Λ) ˆE ˆV (Λ) = V(Λ) ˆE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (43) Introducing ˆEq,ξ = v† q,ξ ˆE, we obtain the following equations from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (42) and (43): ˆT †(a) ˆEq,ξ ˆT(a) = e−iqµaµ ˆEq,ξ (44) and ˆV †(Λ) ˆEq,ξ ˆV (Λ) = N ∗ q N ∗ Λ−1q � ξ′ D∗ ξ′ξ(Q(Λ−1, q)) ˆEΛ−1q,ξ′, (45) where we used Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (39) and (40), and Q(Λ, q) = S−1 Λq ΛSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The label ξ can take discrete or contin- uous values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the continous case, the summation � ξ is replaced with the integration � dµ(ξ) with a measure µ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Focusing on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (45) for Λ = Sq, we get ˆV †(Sq) ˆEq,ξ ˆV (Sq) = N ∗ q ˆEℓ,ξ, (46) where note that Nℓ = 1 and Q(S−1 q , q) = S−1 S−1 q qS−1 q Sq = S−1 ℓ = I hold by the definition of vq,ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (46) tells us that the Kraus operators ˆEq,ξ is determined from the Kraus operators ˆEℓ,ξ with the standard momentum ℓµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' All we have to do is to give the form of the Kraus operators ˆEℓ,ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To this end, we present the following equations given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (44) for qµ = ℓµ and by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (45) for qµ = ℓµ and Λ = W with W µνℓν = ℓµ, respectively: ˆT †(a) ˆEℓ,ξ ˆT(a) = e−iℓµaµ ˆEℓ,ξ, (47) ˆV †(W) ˆEℓ,ξ ˆV (W) = � ξ′ D∗ ξ′ξ(W −1) ˆEℓ,ξ′, (48) where Q(Λ−1, q) = Q(W −1, ℓ) = S−1 W −1ℓW −1Sℓ = W −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the next section, we construct a model of the dynamical map with the Poincar´e symmetry to describe the reduced dynamics of a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A MODEL OF THE DYNAMICAL MAP FOR A SINGLE PARTICLE In this section, based on Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (47) and (48), we give a model of the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To simplify the analysis, we consider the Hilbert space H0 ⊗ H1, where H0 is the one-dimensional Hilbert space with a vacuum state |0⟩ and H1 is the irreducible subspace with one-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Any state vector |Ψ⟩ in H1 ( |Ψ⟩ ∈ H1 ) is given by |Ψ⟩ = � d3q � σ Ψ(p, σ) ˆa†(p, σ)|0⟩, (49) 10 where |0⟩ is the vacuum state satisfying ˆa(p, σ)|0⟩ = 0, Ψ(p, σ) with the momentum p and the spin σ is the wave function, ˆa(p, σ) and ˆa†(p, σ) are the annihilation and creation operators satisfying [ˆa(p, σ), ˆa(p′, σ′)]± = 0 = [ˆa†(p, σ), ˆa†(p′, σ′)]±, [ˆa(p, σ), ˆa†(p′, σ′)]± = δ3(p − p′)δσσ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (50) In the above notation, [ ˆA, ˆB]± is defined as [ ˆA, ˆB]± = ˆA ˆB ± ˆB ˆA, in which the signs − and + apply bosons and fermions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [6, 26], the transformation rules of ˆa†(p, σ) are given by ˆT(a)ˆa†(p, σ) ˆT †(a) = e−ipµaµˆa†(p, σ), (51) ˆV (Λ)ˆa†(p, σ) ˆV †(Λ) = � EpΛ Ep � σ′ Dσ′σ(Q(Λ, p))ˆa†(pΛ, σ′), (52) where Ep = p0, EpΛ = (Λp)0 and pΛ is the vector with the component (pΛ)i = (Λp)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The matrix Q(Λ, p) = S−1 Λp ΛSp is the element of the little group which satisfies Q(Λ, p)µνkν = kµ, where kµ is the standard momentum for a massive particle (kµ = [m, 0, 0, 0], m > 0) or a massless particle (kµ = [k, 0, 0, k], k > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The momentum pµ and the standard momentum kµ are connected with (Sp)µνkν = pµ, and Dσ′σ(Q(Λ, p)) is the irreducible unitary representation of the little group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We consider the Kraus operators ˆEℓ,ξ acting on the Hilbert space H0 � H1, that is, ˆEℓ,ξ : H0 � H1 → H0 � H1, which have the following form ˆEℓ,ξ = Aℓ,ξˆI + � d3p � σ Bℓ,ξ(p, σ)ˆa(p, σ) + � d3p′d3p � σ′,σ Cℓ,ξ(p′, σ′, p, σ)ˆa†(p′, σ′)ˆa(p, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (53) The dynamical map given by these operators describes the reduced dynamics of a single particle, which can possibly decay into the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Substituting the above operators into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (47) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (48), we obtain the equations Aℓ,ξ = e−iℓµaµAℓ,ξ, (54) Bℓ,ξ(p, σ)e−ipµaµ = Bℓ,ξ(p, σ)e−iℓµaµ, (55) Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cℓ,ξ(p′, σ′, p, σ)e−iℓµaµ, (56) and Aℓ,ξ = � ξ′ D∗ ξ′ξ(W −1)Aℓ,ξ′, (57) � EpW Ep � σ Bℓ,ξ(pW , σ)D∗ σ′σ(Q) = � ξ′ D∗ ξ′ξ(W −1)Bℓ,ξ′(p, σ′), (58) � Ep′ W EpW Ep′Ep � σ′,σ Cℓ,ξ(p′ W, σ′, pW , σ)D¯σ′σ′(Q′)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(W −1)Cℓ,ξ′(p′, ¯σ′, p, ¯σ), (59) 11 where Q = Q(W −1, Wp) and Q′ = Q(W −1, Wp′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The derivation of these equations is devoted in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We can analyze the explicit form of Aℓ,ξ, Bℓ,ξ(p, σ) and Cℓ,ξ(p′, σ′, p, σ) for a massive particle and a massless particle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the analysis, we assume that the massive particle has a finite spin and the massless particle has a finite spin and a nonzero momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Through the long computations presented in Appendices B and C, we get the following dynamical map with the Poincar´e symmetry, Φt,s[ρ(s)] = Ut,s◦Et,s[ρ(s)], Et,s[ρ] = � j � β(j) t,s � d3p � σ ˆa(p, σ)ρˆa†(p, σ)+α(j) t,s �ˆI+γ(j) t,s ˆN � ρ �ˆI+γ(j)∗ t,s ˆN �� , (60) where α(j) t,s , β(j) t,s and γ(j) t,s are the parameters depending on time, ˆN is the number operator defined as ˆN = � d3p � σ ˆa†(p, σ)ˆa(p, σ), (61) and Ut,s is the unitary map given in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the form of the dynamical map Φt,s, we recovered the labels j which represent the degeneracies other than the labels q and ξ appearing in the Kraus operators ˆEq,ξ defined around (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The parameters α(j) t,s, β(j) t,s and γ(j) t,s in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (60) satisfy 0 ≤ α(j) t,s ≤ 1, � j α(j) t,s = 1, 0 ≤ β(j) t,s , 0 ≤ � j β(j) t,s ≤ 1 � j � β(j) t,s + α(j) t,s � γ(j) t,s + γ(j)∗ t,s + |γ(j) t,s |2�� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (62) These conditions come from the completeness condition of the Kraus operators (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the com- putation of the completeness condition, note that the number operator ˆN satisfies ˆN 2 = ˆN on the Hilbert space H0 ⊗ H1, since we assume that the dynamical map describes the reduced dynamics of a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From the transformation rules of the creation and the annihilation operators, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (51) and (52), it is easy to check that the map Et,s satisfies the condition of symmetry given in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the unitary map Ut,s is symmetric under the Poincar´e group, which is checked around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (29), we can confirm that Φt,s is also symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Let us consider the case where there is no decay under the dynamical map Φt,s and focus on the dynamics of one-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this case, the parameter � j β(j) t,s vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the density operator ρ given by one-particle states satisfies ˆNρ = ρ = ρ ˆN, we have Φt,s[ρ(s)] = Ut,s ◦ Et,s[ρ(s)] = � j α(j) t,s |1 + γ(j) t,s |2Ut,s[ρ(s)] = Ut,s[ρ(s)], (63) 12 where we used the condition (62) with � j β(j) t,s = 0 in the third equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This means that the dynamical map with the Poincar´e symmetry for a non-decaying particle is the unitary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The result corresponds to a non-perturbative extension of the analysis in [25], which gives an implication on the particle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For example, if the superposition state of a particle decoheres under a non-unitary evolution, the Poincar´e symmetry breaks in the particle dynamics described by a dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We discuss the energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The expectation value of ˆHn at a time t, where n is a natural number, is computed as Tr[ ˆHnρ(t)] = � j � β(j) t,s Tr[ ˆHn � d3p � σ ˆa(p, σ)ρ(s)ˆa†(p, σ)] + α(j) t,s Tr[ ˆHn(ˆI + γ(j) t,s ˆN)ρ(s)(ˆI + γ(j)∗ t,s ˆN)] � = (1 − � j β(j) t,s )Tr[ ˆHnρ(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (64) In the reduced dynamics by the dynamical map Φt,s, the energy of a single particle is not conserved unless � j β(j) t,s is a constant, even when the map is symmetric under the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Such a deviation between symmetry and conservation law was discussed in, for example, Refs [23] and [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' If the parameter � j β(j) t,s is a constant, then � j β(j) t,s = � j β(j) s,s = 0 and the dynamical map Φt,s is reduced to the unitary map Ut,s as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' CONCLUSION We discussed what a dynamical map describing the reduced dynamics of an open quantum system is realized under the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The unitary representation theory of the Poincar´e group refines the condition for the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For a massive particle and a massless particle, we derived a concrete model of the dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the model, the particle can decay into the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' If there is no decay process, the dynamical map describes the unitary evolution generated by the Hamiltonian of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This means that the non-decaying single particle does not decohere as long as the dynamical map for the particle has the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this way, it was exemplified that the Poincar´e symmetry strongly constrains the possible dynamics of an open quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this paper, we assumed an open system with a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Our analysis is possible to be extended to the case with many particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Considering interactions among many particles, we can understand more general effective theories in terms of the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the particles interacting via gravity, the models which induce intrinsic gravitational decoherence have 13 been proposed [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' These models are written in the theory of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the weak field regime of gravity, the Poincar´e symmetry may provide a guidance for establishing the theory of an open quantum system with gravitating particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This paper has a potential to develop the theory of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To describe the reduced dynamics of an open quantum system, a quantum master equation is often adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' It has been discussed how the quantum Markov dynamics given by the equation is consistent with a relativistic theory [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Applying the present approach, it will be possible to discuss the quantum Markov dynamics with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' It is hoped that this paper paves the way to understand a relativistic theory of open quantum systems and to study the interplay between quantum theory and gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Acknowledgments We thank Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Kuramochi for useful discussions and comments related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' was supported by 2022 Research Start Program 202203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Appendix A: Derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (54),(55),(56),(57),(58) and (59) We present the transformation rules of Aℓ,ξ, Bℓ,ξ and Cℓ,ξ given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (54),(55),(56),(57),(58) and (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using the assumed form of the Kraus operators ˆEℓ,ξ defined by (53), we can compute the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (47) as ˆT †(a) ˆEℓ,ξ ˆT(a) = Aℓ,ξˆI + � d3p � σ Bℓ,ξ(p, σ)e−ipµaµˆa(p, σ) + � d3p′d3p � σ′,σ Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµˆa†(p′, σ′)ˆa(p, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (47), we have Aℓ,ξ = e−iℓµaµAℓ,ξ, (A1) Bℓ,ξ(p, σ)e−ipµaµ = Bℓ,ξ(p, σ)e−iℓµaµ (A2) Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cℓ,ξ(p′, σ′, p, σ)e−iℓµaµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) 14 The right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (48) is evaluated as ˆV †(W) ˆEℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆV (W) = Aℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξˆI + � d3p � σ Bℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) � EpW −1 Ep � σ′ D∗ σ′σ(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p))ˆa(pW −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′) + � d3p′d3p � σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='σ Cℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) × � Ep′ W −1 Ep′ � EpW −1 Ep � ¯σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ′ D¯σ′σ′(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p′))D∗ ¯σσ(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p))ˆa†(p′ W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ′)ˆa(pW −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ) = Aℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξˆI + � d3p � σ Bℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(pW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) � EpW Ep � σ′ D∗ σ′σ(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Wp))ˆa(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′) + � d3p′d3p � σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='σ Cℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p′ W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' pW ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) × � Ep′ W Ep′ � EpW Ep � ¯σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ′ D¯σ′σ′(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Wp′))D∗ ¯σσ(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Wp))ˆa†(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ′)ˆa(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' where note that the Lorentz invariant measure is d3p/Ep and hence f(p)d3p = Epf(p)d3p/Ep = EpΛf(pΛ)d3p/Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (48), we have Aℓ,ξ = � ξ′ D∗ ξ′ξ(W −1)Aℓ,ξ′, (A4) � EpW Ep � σ Bℓ,ξ(pW, σ)D∗ σ′σ(Q) = � ξ′ D∗ ξ′ξ(W −1)Bℓ,ξ′(p, σ′) (A5) � Ep′ W EpW Ep′Ep � σ′,σ Cℓ,ξ(p′ W , σ′, pW, σ)D¯σ′σ′(Q′)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(W −1)Cℓ,ξ′(p′, ¯σ′, p, ¯σ), (A6) where Q = Q(W −1, Wp) and Q′ = Q(W −1, Wp′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Appendix B: Analysis of a massive particle We assume that the spectrum of ˆP µ on any state |Ψ⟩ in the Hilbert space of one-particle states, H1, satisfies ˆP µ ˆPµ|Ψ⟩ = −m2|Ψ⟩, ⟨Ψ| ˆP 0|Ψ⟩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B1) The above equations are equivalent to the fact that the Hamiltonian ˆH = ˆP 0 has the form ˆH = � ˆPk ˆP k + m2, which implies that |Ψ⟩ is the state of a massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this appendix, we derive the form of the dynamical map Et,s for a massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 15 Case (a) ℓµ = [M, 0, 0, 0], M > 0 or (b) ℓµ = [−M, 0, 0, 0], M > 0 : We focus on the spectrum ℓµ = [±M, 0, 0, 0], M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [a, 0, 0, 0], we have Aℓ,ξ = e±iMaAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a] leads to Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ) ∴ Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0], we get Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iMa, and combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we obtain Bℓ,ξ(σ)eima = Bℓ,ξ(σ)e±iMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the mass m is positive, to get a nontrivial result, we should choose +M with M = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A5) for Q = R ∈ SO(3) and adopting the result Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we find � σ Bℓ,ξ(σ)D∗ σ′σ(R−1) = � ξ′ D∗ ξ′ξ(R−1)Bℓ,ξ′(σ′), where note that Q = Q(W −1, Wp) = Q(R−1, Rℓ) = S−1 ℓ R−1SRℓ = R−1 for ℓµ = [m, 0, 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the representations Dσ′σ and Dξ′ξ are irreducible and unitary, by Schur’s lemma we have Bℓ,ξ(σ) = Bℓ uξσ, where uξσ is a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a], we deduce Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ) ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0] leads to Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iMa, and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p) into the above equation, we have Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(p, σ′, σ)e±iMa ∴ Cℓ,ξ(p, σ′, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The above results imply that the Kraus operator ˆEℓ,ξ with ℓµ = [m, 0, 0, 0] has the following form, ˆEℓ,ξ = Bℓ � σ uξσˆa(0, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 16 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (46) tells us that ˆEq,ξ = N ∗ q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = N ∗ q Bℓ � Eq m � σ uξσˆa(q, σ), where Eq = (Sq ℓ)0 and qi = (Sq ℓ)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To determine the normalization Nq, the inner product v† q,ξvq,ξ is assumed to be v† q′,s′vq,ξ = δ3(q′ − q)δξ′ξ, which leads to Nq = � m/Eq up to a phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For this normalization, the following complete- ness condition is given as � d3q � s vq,ξv† q,ξ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Under the completeness condition, we derive a part of the dynamical map Et,s as Et,s[ρ(s)] ⊃ |Bℓ|2 � d3q � σ ˆa(q, σ)ρ(s)ˆa†(q, σ), (B2) where we used the fact that uξσ is the unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Case (c) ℓµ = [κ, 0, 0, κ], κ > 0 or (d) ℓµ = [−κ, 0, 0, κ], κ > 0 : We consider the spectrum ℓµ = [±κ, 0, 0, κ], κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [a, 0, 0, 0], we have Aℓ,ξ = e±iκaAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a], we get Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a ∴ Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), where ℓ = [0, 0, κ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0] leads to Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iκa, and from the equation Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we obtain Bℓ,ξ(σ)ei √ κ2+m2a = Bℓ,ξ(σ)e±iκa ∴ Bℓ,ξ(σ) = 0, where Eℓ = √ ℓ2 + m2 = √ κ2 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a] gives Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0], we get Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iκa, 17 and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ)e±iκa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Noticing the fact that Ep−ℓ − Ep ± κ ̸= 0, we get the result Cℓ,ξ(p, σ′, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Combined with the above analysis, the Kraus operator ˆEℓ,ξ vanishes: ˆEℓ,ξ = 0 ∴ ˆEq,ξ = Nq ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B3) Case (e) ℓµ = [0, 0, 0, N], N 2 > 0 : We focus on the spectrum ℓµ = [0, 0, 0, N], N 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [0, a], we have Aℓ,ξ = e−iℓ·aAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0] leads to Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ) ∴ Bℓ,ξ(p, σ) = 0, where note that Eq = � q2 + m2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a], we deduce Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ), where ℓ = [0, 0, N]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0], we get Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ), and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ) ∴ Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(σ′, σ)δ3(p − ℓ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Combined with the above analysis, the function Cℓ,ξ(p′, σ′, p, σ) is Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(σ′, σ)δ3(p′ + ℓ/2)δ3(p − ℓ/2), and the Kraus operator ˆEℓ,ξ is written as ˆEℓ,ξ = � σ′,σ Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' By the completeness condition of the Kraus operators, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (31), the above Kraus operator ˆEℓ,ξ should satisfy ˆE† ℓ,ξ ˆEℓ,ξ ≤ ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Concretely, ˆE† ℓ,ξ ˆEℓ,ξ is evaluated as ˆE† ℓ,ξ ˆEℓ,ξ = � ¯σ′,¯σ C∗ ℓ,ξ(¯σ′, ¯σ)ˆa†(ℓ/2, ¯σ)ˆa(−ℓ/2, ¯σ′) � σ′,σ Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ) = δ3(0) � σ′ � � ¯σ Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) �† � σ Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ), 18 where the term given by the linear combination of ˆa†ˆa†ˆaˆa vanishes on H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To satisfy ˆE† ℓ,ξ ˆEℓ,ξ ≤ ˆI, we find that � σ′ � � ¯σ Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) �† � σ Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ) = 0 ∴ Cℓ,ξ(σ′, σ) = 0 The consequence of Cℓ,ξ(σ′, σ) = 0 is that the Kraus operator ˆEℓ,ξ vanishes as ⟨Φ| ˆEℓ,ξ|Ψ⟩ = 0 for all |Ψ⟩, |Φ⟩ ∈ H0 � H1, and hence ˆEq,ξ = N ∗ q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0, (B4) on the Hilbert space H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Case (f) ℓµ = [0, 0, 0, 0] : We consider the case where ℓµ = [0, 0, 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the following, we drop the label ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) is identical for all aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the little group associated with ℓµ is SO(3, 1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A4) for W = Λ ∈ SO(3, 1) is given as Aξ = � ξ′ D∗ ξ′ξ(Λ−1)Aξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B5) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0] gives the condition Bξ(p, σ)eiEpa = Bξ(p, σ) ∴ Bξ(p, σ) = 0, where note that Eq = � q2 + m2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ, we obtain Cξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cξ(p′, σ′, p, σ) ∴ Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A6) for W = Λ ∈ SO(3, 1) is written as � Ep′ ΛEpΛ Ep′Ep � σ′,σ Cξ(p′ Λ, σ′, pΛ, σ)D¯σ′σ′(Q′)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(Λ−1)Cξ′(p′, ¯σ′, p, ¯σ), where Q = Q(Λ−1, Λp) and Q′ = Q(Λ−1, Λp′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From the equation Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p) and noticing the fact that the invariant delta function is Epδ3(p − p′), we get the condition � σ′,σ Cξ(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(Λ−1)Cξ′(p, ¯σ′, ¯σ), (B6) where Q′ = Q(Λ−1, Λp′) turns out to be Q = Q(Λ−1, Λp) by the presence of the delta function δ3(p − p′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' It is known that the dimension of irreducible unitary representations Dξ′ξ of SO(3,1) 19 is one or infinite [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the one-dimensional representation, dropping the label ξ, we find that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B5) trivially holds and that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B6) is reduced to � σ′,σ C(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ ¯σσ(Q) = C(p, ¯σ′, ¯σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For p = 0 and Λ = R ∈ SO(3), we get � σ′,σ C(0, σ′, σ)D¯σ′σ′(R−1)D∗ ¯σσ(R−1) = C(0, ¯σ′, ¯σ) ∴ C(0, σ′, σ) = Cδσ′σ, where this holds by the Schur’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Choosing p = 0 and Λ = Sp with (Sp)µνkν = pµ for kµ = [m, 0, 0, 0], we have C(p, σ′, σ) = C(0, σ′, σ), where we used Q = Q(S−1 p , p) = S−1 k S−1 p Sp = I and Dσ′σ(I) = δσ′σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Hence C(p, σ′, σ) = Cδσ′σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the infinite dimensional representation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B5) leads to Aℓ,ξ = 0, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B6) for p = 0 and Λ = R ∈ SO(3) gives � σ′,σ Cξ(0, σ′, σ)D¯σ′σ′(R−1)D∗ ¯σσ(R−1) = � ξ′ D∗ ξ′ξ(R−1)Cξ′(0, ¯σ′, ¯σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Assuming that the massive particle has a finite spin and using the Schur’s lemma, we get Cξ′(0, ¯σ′, ¯σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B6) for p = 0 and Λ = Sp with (Sp)µνkν = pµ for kµ = [m, 0, 0, 0] pro- vides Cξ(p, σ′, σ) = � ξ′ D∗ ξ′ξ(S−1 p )Cξ′(0, σ′, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The above analysis on ℓµ = [0, 0, 0, 0] tells us the following Kraus operator ˆE = AˆI + C ˆN, where ˆN is the number operator defined in (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A part of the dynamical map Et,s is given as Et,s[ρ(s)] ⊃ � AˆI + C ˆN � ρ(s) � AˆI + C ˆN �† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B7) The above results given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B2), (B3), (B4) and (B7) provide the following form of Et,s: Et,s[ρ(s)] = |Bℓ|2 � d3q � σ ˆa(q, σ)ρ(s)ˆa†(q, σ) + � AˆI + C ˆN � ρ(s) � AˆI + C ˆN �† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B8) Recovering other degeneracies labeled by j differently from q and ξ, introducing � j and redefining the parameters as |A|2 = α(j) t,s , C/A = γ(j) t,s and |Bℓ|2 = β(j) t,s , we get the form of the dynamical map Et,s given in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 20 Appendix C: Analysis on a massless particle We assume that the spectrum of ˆP µ on any state |Ψ⟩ in the Hilbert space of one-particle states, H1, satisfies ˆP µ ˆPµ|Ψ⟩ = 0, ⟨Ψ| ˆP 0|Ψ⟩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C1) The above equations leads to the fact that the Hamiltonian ˆH = ˆP 0 has the form ˆH = � ˆPk ˆP k, which means that |Ψ⟩ is the state of a massless particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this appendix, we derive the form of the dynamical map Et,s for a massless particle with nonzero momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Case (a) ℓµ = [M, 0, 0, 0], M > 0 or (b) ℓµ = [−M, 0, 0, 0], M > 0 : We focus on the spectrum ℓµ = [±M, 0, 0, 0], M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [a, 0, 0, 0] gives Aℓ,ξ = e±iMaAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a] leads to Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ) ∴ Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0], we get Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iMa, and combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we obtain Bℓ,ξ(σ) = Bℓ,ξ(σ)e±iMa ∴ Bℓ,ξ(σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a], we deduce Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ) ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0] leads to Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iMa, and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p) into the above equation, we have Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(p, σ′, σ)e±iMa ∴ Cℓ,ξ(p, σ′, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The above results imply that the Kraus operator ˆEℓ,ξ vanishes and has the following form, ˆEq,ξ = N ∗ q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C2) 21 Case (c) ℓµ = [κ, 0, 0, κ], κ > 0 or (d) ℓµ = [−κ, 0, 0, κ], κ > 0 : We consider the spectrum ℓµ = [±κ, 0, 0, κ], κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [a, 0, 0, 0], we have Aℓ,ξ = e±iκaAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a] gives Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a ∴ Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), where ℓ = [0, 0, κ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0] leads to Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iκa, and from the equation Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we have Bℓ,ξ(σ)eiκa = Bℓ,ξ(σ)e±iκa, where Eℓ = √ ℓ2 = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To get a nontrivial result, we should choose +κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A5) for Q = L ∈ ISO(2) and adopting the result Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we find � σ Bℓ,ξ(σ)D∗ σ′σ(L−1) = � ξ′ D∗ ξ′ξ(L−1)Bℓ,ξ′(σ′), where note that Q = Q(W −1, Wp) = Q(L−1, Lℓ) = S−1 ℓ L−1SLℓ = L−1 for ℓµ = [κ, 0, 0, κ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the representations Dσ′σ and Dξ′ξ are irreducible and unitary, by Schur’s lemma we get Bℓ,ξ(σ) = Bℓ uξσ, where uξσ is a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a], we deduce Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0] leads to Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iκa, and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ)e±iκa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The condition of Ep−ℓ − Ep + κ = 0 is written by p⊥ = [p1, p2] = 0 and p3 ≥ κ, and the condition of Ep−ℓ − Ep − κ = 0 is given by p⊥ = 0 and p3 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Hence, the form of Cℓ,ξ(p, σ′, σ) is Cℓ,ξ(p, σ′, σ) = � C+ ℓ,ξ(p3, σ′, σ)θ(p3 − κ) + C− ℓ,ξ(p3, σ′, σ)θ(−p3) � δ2(p⊥) 22 Combined with the above analysis, the Kraus operator ˆE+ ℓ,ξ for +κ is ˆE+ ℓ,ξ = Bℓ � σ uξσˆa(ℓ, σ) + � dp3 � σ,σ′ C+ ℓ,ξ(p3, σ′, σ)θ(p3 − κ)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ), and the Kraus operator ˆE− ℓ,ξ for −κ is ˆE− ℓ,ξ = � dp3 � σ,σ′ C− ℓ,ξ(p3, σ′, σ)θ(−p3)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' By the completeness condition of the Kraus operators, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (31), the above Kraus operator ˆE± ℓ,ξ should satisfy ˆE±† ℓ,ξ ˆE± ℓ,ξ ≤ ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Concretely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ˆE+† ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆE+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ is evaluated as ˆE+† ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆE+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = � Bℓ � σ uξσˆa(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) + � dp3 � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='σ′ C+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)θ(p3 − κ)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3 − κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) �† × � Bℓ � σ uξσˆa(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) + � dp3 � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='σ′ C+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)θ(p3 − κ)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3 − κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) � = |Bℓ|2 � σ u∗ sσˆa†(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) � σ′ usσ′ˆa(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′) + δ2(0) � dp3 � σ′ � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ C+∗ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)C+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ)θ(p3 − κ)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' where the term given by the linear combination of ˆa†ˆaˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ˆa†ˆa†ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' and ˆa†ˆa†ˆaˆa vanishes on H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To satisfy ˆE+† ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆE+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ≤ ˆI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' we find � dp3 � σ′ � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ C+∗ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)C+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ)θ(p3−κ)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ) = 0 ∴ C+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ) = 0 In the same manner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' we have ˆE−† ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆE− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = δ2(0) � dp3 � σ′ � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ C−∗ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)C− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ)θ(−p3)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' and to satisfy ˆE−† ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆE− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ≤ ˆI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' we obtain � dp3 � σ′ � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ C−∗ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)C− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ)θ(−p3)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ) = 0 ∴ C− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' These analyses give the following form of the Kraus operators, ˆE+ ℓ,ξ = Bℓ � σ uξσˆa(ℓ, σ), ˆE− ℓ,ξ = 0, on the Hilbert space H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (46), we get ˆE+ q,ξ = N ∗ q ˆV (Sq) ˆE+ ℓ,ξ ˆV †(Sq) = N ∗ q Bℓ � Eq κ � σ uξσˆa(q, σ), ˆE− q,ξ = N ∗ q ˆV (Sq) ˆE− ℓ,ξ ˆV †(Sq) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 23 Setting that the inner product v† q,ξvq,ξ is v† q′,s′vq,ξ = δ3(q′ − q)δξ′ξ, the normalization Nq is given as Nq = � κ/Eq up to a phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For this normalization, we get the following completeness condition as � d3q � s vq,ξv† q,ξ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Taking account for the completeness, we can derive a part of the dynamical map Et,sas Et,s[ρ(s)] ⊃ |Bℓ|2 � d3q � σ ˆa(q, σ)ρ(s)ˆa†(q, σ), (C3) where we used the fact that uξσ is the unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Case (e) ℓµ = [0, 0, 0, N], N 2 > 0 : We focus on the spectrum ℓµ = [0, 0, 0, N], N 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [0, a], we have Aℓ,ξ = e−iℓ·aAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a] leads to Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a ∴ Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (55) for all aµ = [a, 0, 0, 0] gives Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ), and then combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we get Bℓ,ξ(σ)eiκa = Bℓ,ξ(σ) ∴ Bℓ,ξ(σ) = 0 where we used Eℓ = √ ℓ2 = κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Adopting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a], we deduce Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ), where ℓ = [0, 0, N]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0], we get Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ), and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ) ∴ Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(σ′, σ)δ3(p − ℓ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 24 Combined with the above analysis, the function Cℓ,ξ(p′, σ′, p, σ) is Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(σ′, σ)δ3(p′ + ℓ/2)δ3(p − ℓ/2), and the Kraus operator ˆEℓ,ξ is written as ˆEℓ,ξ = � σ′,σ Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' By the completeness condition of the Kraus operators, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (31), the above Kraus operator ˆEℓ,ξ should satisfy ˆE† ℓ,ξ ˆEℓ,ξ ≤ ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Explicitly, ˆE† ℓ,ξ ˆEℓ,ξ is evaluated as ˆE† ℓ,ξ ˆEℓ,ξ = � ¯σ′,¯σ C∗ ℓ,ξ(¯σ′, ¯σ)ˆa†(ℓ/2, ¯σ)ˆa(−ℓ/2, ¯σ′) � σ′,σ Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ) = δ3(0) � σ′ � � ¯σ Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) �† � σ Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ), where the term associated with the linear combination of ˆa†ˆa†ˆaˆa vanishes on H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To satisfy ˆE† ℓ,ξ ˆEℓ,ξ ≤ ˆI, we find that � σ′ � � ¯σ Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) �† � σ Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ) = 0 ∴ Cℓ,ξ(σ′, σ) = 0 Hence, the Kraus operator ˆEℓ,ξ vanishes, and we have that ˆEq,ξ = N ∗ q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0, (C4) on the Hilbert space H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Case (f) ℓµ = [0, 0, 0, 0] : We focus on the spectrum ℓµ = [0, 0, 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the following, we do not write the label ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) is identical for all aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the little group associated with ℓµ is SO(3, 1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A4) for W = Λ ∈ SO(3, 1) is given as Aξ = � ξ′ D∗ ξ′ξ(Λ−1)Aξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C5) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a] gives the condition Bξ(p, σ)e−ip·a = Bξ(p, σ) ∴ Bξ(p, σ) = Bξ(σ)δ3(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This equation makes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0] and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A5) for W = Λ ∈ SO(3, 1) trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This form Bξ(p, σ) = Bξ(σ)δ3(p) leads to ˆEℓ,ξ ⊃ � σ Bξ(σ)ˆa(0, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' However, this operator vanishes on the Hilbert space of massless particles since we assumed that there are no states with zero momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ gives us the condition Cξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cξ(p′, σ′, p, σ) ∴ Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 25 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A6) for W = Λ ∈ SO(3, 1) is written as � Ep′ ΛEpΛ Ep′Ep � σ′,σ Cξ(p′ Λ, σ′, pΛ, σ)D¯σ′σ′(Q′)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(Λ−1)Cξ′(p′, ¯σ′, p, ¯σ), where Q = Q(Λ−1, Λp) and Q′ = Q(Λ−1, Λp′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From the equation Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p) and noticing the fact that the invariant delta function is Epδ3(p − p′), we get the condition � σ′,σ Cξ(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(Λ−1)Cξ′(p, ¯σ′, ¯σ), (C6) where note that the delta function δ3(p − p′) leads to Q′ = Q(Λ−1, Λp′) = Q(Λ−1, Λp) = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The (proper orthochronous) Lorentz group SO(3, 1) has one and infinite dimensional unitary irreducible representations [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Choosing the one-dimensional representation of Dξ′,ξ and dropping the label ξ, we find that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C5) trivially holds and that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C6) is reduced to � σ′,σ C(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ ¯σσ(Q) = C(p, ¯σ′, ¯σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For p = ℓ = [0, 0, κ] and Λ = L ∈ ISO(2), we get � σ′,σ C(ℓ, σ′, σ)D¯σ′σ′(L−1)D∗ ¯σσ(L−1) = C(ℓ, ¯σ′, ¯σ) ∴ C(ℓ, σ′, σ) = Cδσ′σ, where we used the Schur’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Choosing p = ℓ and Λ = Sp with (Sp)µ��ℓν = pµ for ℓµ = [κ, 0, 0, κ], we have C(p, σ′, σ) = C(ℓ, σ′, σ), where we used Q = Q(S−1 p , p) = S−1 k S−1 p Sp = I and Dσ′σ(I) = δσ′σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Hence C(p, σ′, σ) = Cδσ′σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' If we adopt the infinite dimensional representation of Dξ′ξ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C5) leads to Aℓ,ξ = 0 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C6) for p = ℓ and Λ = L ∈ ISO(2) gives � σ′,σ Cξ(ℓ, σ′, σ)D¯σ′σ′(L−1)D∗ ¯σσ(L−1) = � ξ′ D∗ ξ′ξ(L−1)Cξ′(ℓ, ¯σ′, ¯σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Assuming that the massless particle has a finite spin and using the Schur’s lemma, we get Cξ′(ℓ, ¯σ′, ¯σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C6) for p = ℓ and Λ = Sp with (Sp)µνℓν = pµ for ℓµ = [κ, 0, 0, κ] pro- vides Cξ(p, σ′, σ) = � ξ′ D∗ ξ′ξ(S−1 p )Cξ′(ℓ, σ′, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 26 The above analysis tells us that the Kraus operator has the following form ˆE = AˆI + C ˆN, where ˆN is the number operator defined in (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A part of the dynamical map Et,s with the Poincar´e symmetry is given as Et,s[ρ(s)] ⊃ � AˆI + C ˆN � ρ(s) � AˆI + C ˆN �† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C7) Gathering the above results (C2), (C3), (C4) and (C7), we have the following form of Et,s: Et,s[ρ(s)] = |Bℓ|2 � d3q � σ ˆa(q, σ)ρ(s)ˆa†(q, σ) + � AˆI + C ˆN � ρ(s) � AˆI + C ˆN �† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C8) In the same manner performed around (B8), we obtain the form of the dynamical map Et,s given in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Breuer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Laine, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Piilo, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Vacchini, “Colloquium: Non-Markovian Dynamics in Open Quantum Systems”, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 88, 021002 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Breuer and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Petruccione, “The Theory of Open Quantum Systems” (Oxford University Press, New York, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Feynman and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Vernon, “The theory of a general quantum system interacting with a linear dissipative system”, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 24, 118 (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Caldeira and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Leggett, “Path integral approach to quantum Brownian motion”, Physica A 121, 587 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Calzetta and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Hu, “Nonequilibrium Quantum Field Theory” (Cambridge University Press, Cambridge, England, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Weinberg, “The Quantum Theory of Fields, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' I” (Cambridge University Press, Cambridge, Eng- land, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Jones, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Guaita, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Bassi, “Impossibility of extending the Ghirardi-Rimini-Weber model to relativistic particles”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A 103, 042216 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Jones, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Gasbarri, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Bassi, “Mass-coupled relativistic spontaneous collapse models”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A 54, 295306 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Bedingham, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' D¨urr, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Ghirardi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Goldstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Tumulka, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Zangh`ı, “Matter Density and Relativistic Models of Wave Function Collapse”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 154, 623 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Bedingham and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Pearle “Continuous-spontaneous-localization scalar-field relativistic collapse model”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Research 1, 033040 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Pearle, “Relativistic dynamical collapse model”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' D 91, 105012 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 27 [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Kurkov and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Franke, “Local Fields Without Restrictions on the Spectrum of 4-Momentum Operator and Relativistic Lindblad Equation”, Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 41, 820 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Wang, “Relativistic quantum field theory of stochastic dynamics in the Hilbert space”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' D 105, 115037 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Ahn, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Lee, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Hwang, “Lorentz-covariant reduced-density-operator theory for relativistic-quantum-information processing”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A 67, 032309 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Kafri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Taylor, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Milburn, “A classical channel model for gravitational decoherence”, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 16, 065020 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Di´osi, “A universal master equation for the gravitational violation of quantum mechanics”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 120 377–381 (1987) [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Di´osi, “Models for universal reduction of macroscopic quantum fluctuations”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 40, 1165 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Penrose, “On Gravity’s role in Quantum State Reduction”, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' and Gravt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 28, 581–600 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Tilloy and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Di´osi, “Sourcing semiclassical gravity from spontaneously localized quantum matter”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' D 93, 024026 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [20] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Davies, “Quantum Theory Of Open Systems” (Academic, New York, 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Holevo, “Statistical Structure of Quantum Theory”, Lecture Notes in Physics Monographs (Springer-Verlag, Berlin, Heidelberg, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Keyl, “Fundamentals of quantum information theory”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 369, 431 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Cˆırstoiu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Korzekwa, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Jennings, “Robustness of Noether’s Principle: Maximal Disconnects between Conservation Laws and Symmetries in Quantum Theory”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' X 10, 041035 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [24] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Marvian and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Spekkens, “Extending Noether’s theorem by quantifying the asymmetry of quantum states”, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 5, 3821 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Toroˇs, “Constraints on the spontaneous collapse mechanism:theory and experiments”, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='D thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Peres and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Terno, “Quantum information and relativity theory”, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 76, 93 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Alicki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Fannes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Verbeure, “Unstable particles and the Poincare semigroup in quantum field theory”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A 19, 919 (1986) [28] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Di´osi, “Is there a relativistic Gorini-Kossakowski-Lindblad-Sudarshan master equation?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' D 106, L051901 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [29] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Bargmann, “Irreducible Unitary Representations of the Lorentz Group”, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 48, 568 (1947).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'}