R. de la Madrid
This paper is a contribution to the problem of particle localization in non-relativistic Quantum Mechanics. Our main results will be (1) to formulate the problem of localization in terms of invariant subspaces of the Hilbert space, and (2) to show that the rigged Hilbert space incorporates particle localization in a natural manner.
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The combination of quantum theory and special relativity leads to structures that differ in several respects from non-relativistic quantum mechanics of particles. These differences are quite familiar to practitioners of Algebraic Quantum Field Theory but less well known outside this community. The paper is intended as a concise survey of some selected aspects of relativistic quantum physics, in particular regarding localization and entanglement.
The conflict between relativistic causality and localizability is analyzed in the light of the existence of unsharp localization observables. A theorem due to S. Schlieder is generalized, showing that the assumption of local commutativity implies the localization observable in question to be unsharp in a strong sense. Furthermore, a recent generalization of a theorem of Lüders is applied to demonstrate that local commutativity is a necessary consequence of Einstein causality even in the case of unsharp observables if they admit local measurements. These findings raise the question whether loca…
Spatial Localization of Relativistic Quantum Systems: The Commutativity Requirement and the Locality Principle. Part I: A General Analysis
本文探讨了在闵氏时空中的相对论局部性是否需要交换性来表示局部化观测量。通过Busch的操作分析，指出粒子系统中交换性不必然由基本局部性原理推导出，且局部可探测性原则下局部化观测量不局限于任意小时空区域。
In this paper, I provide a formal set of assumptions and give a natural criterion for a quantum field theory to admit particles. I construct a naïve approach to localization for a free bosonic quantum field theory and show how this localization scheme, as a consequence of the Reeh-Schlieder theorem, fails to satisfy this criterion. I then examine the Newton-Wigner concept of localization and show that it fails to obey strong microcausality and thus is subject to a more general version of the Reeh-Schlieder theorem. I review approaches to quantum field theoretic explanations of particle detecti…
Many of the "counterintuitive" features of relativistic quantum field theory have their formal root in the Reeh-Schlieder theorem, which in particular entails that local operations applied to the vacuum state can produce any state of the entire field. It is of great interest, then, that I.E. Segal and, more recently, G. Fleming (in a paper entitled "Reeh-Schlieder Meets Newton-Wigner") have proposed an alternative "Newton-Wigner" localization scheme that avoids the Reeh-Schlieder theorem. In this paper, I reconstruct the Newton-Wigner localization scheme and clarify the limited extent to which…
Constructing observables that describe the localization of relativistic particles is an important foundational problem in relativistic quantum field theory (QFT). The description of localization in terms of single-time observables leads to conflict with the requirement of causality. In this paper, we describe particle localization in terms of time-of-arrival observables, defined in terms of the interaction between a quantum field and a measuring apparatus. The resulting probabilities are linear functionals of QFT correlation functions. Specializing to the case of a scalar field, we identify se…
Localization of relativistic particles and their position-momentum uncertainty relations are not yet fully understood. We discuss two schemes of photon localization that are based on the energy density. One scheme produces a positive operator-valued measure for localization. It coincides with the number density operator and reproduces the effective 3x3 polarization density matrix. Another scheme results in a probability distribution that is conditioned on the detection. In both schemes the uncertainty relations for transversal position and momentum approach the Heisenberg bound $Δp Δz= \hbar/2…
We calculate correlation functions for a relativistic Einstein-Podolsky-Rosen-type experiment with massive Dirac particles. We take the influence of the Newton-Wigner localization into account and perform the calculations for a couple of physically interesting states.
Spatial Localization of Relativistic Quantum Systems: The Commutativity Requirement and the Locality Principle. Part II: A Model from Local QFT
在标准量子场论中，利用应力-能量-动量张量与测试函数的涂抹，构造了闵可夫斯基时空中的正能相对论空间局域化可观测量，给出了条件局域化可观测量的交换性恢复。
A new proposal for a Lorentz-invariant spontaneous localization theory is presented. It is based on the choice of a suitable set of macroscopic quantities to be stochastically induced to have definite values. Such macroscopic quantities have the meaning of a time-integrated amount of a microscopically defined quantity called stuff related to the presence of massive particles.
The interaction of a moving charged particle with its coherent electromagnetic field is analysed in the framework of non-relativistic quantum mechanics. It is shown that, when this interaction is taken into account, a spatially localized state may have a mean energy lower than the one corresponding to a delocalized state.
On the Relativistic Spatial Localization for massive real scalar Klein-Gordon quantum particles
本文严格分析了Terno提出的Klein-Gordon粒子空间局域化POVM，证明其为Newton-Wigner PVM的运动学变形，保留其优良性质并避免Hegerfeldt定理的非物理特征，同时验证了其满足Castrigiano因果性要求。
We make a critical comparison of relativistic and non-relativistic classical and quantum mechanics of particles in inertial frames and of the open problems in particle localization at the two levels. The solution of the problems of the relativistic center of mass, of the clock synchronization convention needed to define relativistic 3-spaces and of the elimination of the relative times in the relativistic bound states leads to a description with a decoupled non-local (non-measurable) relativistic center of mass and with only relative variables for the particles (single particle subsystems do n…
Energy diffusion due to spontaneous localization (SL) for a relativistically-fast moving particle is examined. SL is an alternative to standard quantum theory in which quantum state reduction is treated as a random physical process which is incorporated into the Schrödinger equation in an observer-independent way. These models make predictions in conflict with standard quantum theory one of which is non conservation of energy. On the basis of proposed relativistic extensions of SL it is argued that for a single localized particle, non-relativistic SL should remain valid in the rest frame of th…
Space-Time-Matter: Some Notes on the Localization Problem in Relativistic Quantum Theory
本文探讨相对论量子理论中正能量假设的脆弱性及其与局域化问题的关联，通过分析负能量波函数与粒子对产生的关系，并从玻姆量子场论视角反思，深化了对局域化问题的理解。
The purity of a reduced state for spins that is pure in the rest frame will most likely appear to degrade because spin and momentum become mixed when viewed by a moving observer. We show that such a boost-induced decrease in spin purity observed in a moving reference frame is intrinsically related to the spatial localization properties of the wave package observed in the rest frame. Furthermore, we prove that, for any localized pure state with separable spin and momentum in the rest frame, its reduced density matrix for spins inevitably appears to be mixed whenever viewed from a moving referen…
We discuss relations between several relativistic spin observables and derive a Lorentz-invariant characteristic of a reduced spin density matrix.A relativistic position operator that satisfies all the properties of its nonrelativistic analog does not exist. Instead we propose two causality-preserving positive operator-valued measures (POVMs) that are based on projections onto one-particle and antiparticle spaces, and on the normalized energy density. They predict identical expectation values for position. The variances differ by less than a quarter of the squared de Broglie wavelength and coi…
Relativistic model of spontaneous wave-function localization induced by nonHermitian colored noise
本文基于随机非厄米作用量发展相对论量子场论，通过带纯虚数项的狄拉克作用量耦合费米子密度算符与普适有色噪声，推导波包定域化长度并发现其随可观测宇宙增大而减小，为相对论自发波函数坍缩提供潜在框架。
The thesis is devoted to the phase space representation of relativistic quantum mechanics. For a class of observables with matrix-valued Weyl symbols proportional to the identity matrix, the Weyl-Wigner-Moyal formalism is proposed. The evolution equations are found to coincide with their counterparts in relativistic quantum mechanics with non-local Hamiltonian. The difference between the theories is connected with peculiarities of the constraints on the initial conditions. Effective increase in coherence between eigenstates of the Hamiltonian is found. Relativistic coherent states that take in…
The localization problem: an antinomy between measurability and causal dynamics
针对相对论量子理论中存在七十余年、易被忽视且困扰学界的局域化问题，本文通过历史回顾厘清文献误解，揭示可局域性与相对论因果性的根本冲突，凸显因果动力学与局域可观测量的二律背反。
We consider the use of the energy density for describing a localization of relativistic particles. This method is consistent with the causality requirements. The related positive operator valued measure is presented. The probability distributions for one particle states are given explicitly.
We study the motion of M particles performing a quantum walk on the line. Under various conditions on the initial coin states for quantum walkers controlled by the Hadamard operator, we give theoretical criterion to observe the quantum walkers at an initial location with high probability.
Exact solutions of two-particle relativistic equations of quantum field theory describing the scattering $s$-states and the bound $s$-states are found in the cases of delta-shell potential and superposition of delta-shell potentials. Some properties of obtained relativistic wave functions, scattering amplitudes and quantization conditions are investigated. The resonance character of scattering processes is demonstrated by the behavior of amplitudes. It is shown that the non-relativistic limits of these relativistic values coincide with respective non-relativistic ones obtained from the Schrödi…
The significance of the de Broglie-Bohm hidden-particle position in the relativistic regime is addressed, seeking connection to the (orthodox) single-particle Newton-Wigner position. The effect of non-positive excursions of the ensemble density for extreme cases of positive-energy waves is easily computed using an integral of the equations of motion developed here for free spin-0 particles in 1+1 dimensions and is interpreted in terms of virtual-like pair creation and annihilation beneath the Compton wavelength. A Bohmtheoretic description of the acausal explosion of a specific Newton-Wigner-l…
We put forward an interpretation of scalar quantum field theory as relativistic quantum mechanics by curing well known problems related to locality. A probabilistic interpretation of quantum field theory similar to quantum mechanics is difficult if particle localization is defined using the Newton-Wigner position operator as it is non-local and non-covariant. An alternative bilinear covariant position operator is discussed which incorporates a time operator that can be exponentiated to a unitary operator. Moreover, it satisfies an algebra that unifies special relativity and quantum mechanics a…
We predict the quantum correlations between non-interacting particles evolving simultaneously in a disordered medium. While the particle density follows the single-particle dynamics and exhibits Anderson localization, the two-particle correlation develops unique features that depend on the quantum statistics of the particles and their initial separation. On short time scales, the localization of one particle becomes dependent on whether the other particle is localized or not. On long time scales, the localized particles show oscillatory correlations within the localization length. These effect…
We show that there are significant conceptual differences between QM and QFT which make it difficult to view the latter as just a relativistic extension of the principles of QM. At the root of this is a fundamental distinction between Born-localization in QM (which in the relativistic context changes its name to Newton-Wigner localization) and modular localization which is the localization underlying QFT, after one liberates it from its standard presentation in terms of field coordinates. The first comes with a probability notion and projection operators, whereas the latter describes causal pr…
Using relativistic tensor-bispinorial equations proposed in hep-th/0412213 we solve the Kepler problem for a charged particle with arbitrary half-integer spin interacting with the Coulomb potential.
The general and explicit relation between the phase time and the dwell time for quantum tunneling of a relativistically propagating particle is investigated and quantified. In analogy with previously obtained non-relativistic results, it is shown that the group delay can be described in terms of the dwell time and a self-interference delay. Lessons concerning the phenomenology of the relativistic tunneling are drawn.
Free evolution for quantum particle in general ultrametric space is considered. We find that if mean zero wave packet is localized in some ball in the ultrametric space then its evolution remains localized in the same ball.
Correlation of interacting particles is studied in their dynamics and localization in ideal and disordered lattice systems with the help of numerical tools. Both 1D and 2D systems are considered. In 1D lattices with long-range hopping, differences between dynamics of attractively and repulsively interacting particles are noted. For calculations of 2D systems, a recursion based numerical algorithm for two-particle Greens functions is implemented which provides accurate results. Using this algorithm, spectral properties of 2D Hubbard and Hofstadter models is computed, along with localization par…
Positivity of the Hamiltonian alone is used to show that particles, if initially localized in a finite region, immediately develop infinite tails.
We describe a previously unexplored effect of the continuous spontaneous localization model whereby a correlation develops in the distributions of two nearby non-interacting particles following a period of diffusion. We propose the use of this effect as an experimental test differentiating between the continuous spontaneous localization model and standard quantum theory. The test involves building a joint probability distribution for the locations of the two particles by repeatedly releasing them from two nearby traps and subsequently measuring their positions after a brief period of time. We …
We propose a feedback scheme to control the vibrational motion of a single trapped particle based on indirect measurements of its position. It results the possibility of a motional phase space uncertainty contraction, correponding to cool the particle close to the motional ground state.
We study bipartite entanglement in a general one-particle state, and find that the linear entropy, quantifying the bipartite entanglement, is directly connected to the paricitpation ratio, charaterizing the state localization. The more extended the state is, the more entangled the state. We apply the general formalism to investigate ground-state and dynamical properties of entanglement in the one-dimensional Harper model.
Theorems (most notably by Hegerfeldt) prove that an initially localized particle whose time evolution is determined by a positive Hamiltonian will violate causality. We argue that this apparent paradox is resolved for a free particle described by either the Dirac equation or the Klein-Gordon equation because such a particle cannot be localized in the sense required by the theorems.
We investigate the effect of coupling between translational and internal degrees of freedom of composite quantum particles on their localization in a random potential. We show that entanglement between the two degrees of freedom weakens localization due to the upper bound imposed on the inverse participation ratio by purity of a quantum state. We perform numerical calculations for a two-particle system bound by a harmonic force in a 1D disordered lattice and a rigid rotor in a 2D disordered lattice. We illustrate that the coupling has a dramatic effect on localization properties, even with a s…
The well known relation of Einstein relativistic energy for a free particle is extended to cover the total relativistic energy of a bound particle by calculating the relativistic potential energy. A non dissipative harmonic oscillator (NDHO) is a fundamental bound system. Therefore, the potential energy of an NDHO is analytically extended from the non relativistic to the relativistic regime for the first time. This study is essentially concerned with the relativistic mass, where the Lorentz factor transforms the state of the second-order differential equation of an NDHO from linear into nonlin…
We describe how quasiclassical relative positions of particles emerge in an initially delocalized quantum system as scattering of a probe beam is observed. We show that in the multiparticle case this localization in position space occurs via intermediate states that are quantum superpositions of spatial configurations. These superpositions are robust to consecutive scattering events, and scattering alone does not lead to their decoherence. Instead the free evolution of the system combines with scattering to destroy these superpositions, leading the particles to adopt classical-like positions r…
An analytical expression for the relativistic corrections to the energy spectra of particles completely confined in an one-dimensional limited length in real space is given, based upon the wave property of particles, the relativistic energy-momentum relation and two mathematical equations.
We demonstrate that, in a many-particle system, particles can be strongly confined to their sites. The localization is obtained by constructing a sequence of on-site energies that efficiently suppresses resonant hopping. The time during which a many-particle state remains strongly localized in an infinite chain can exceed the reciprocal hopping frequency by $\agt 10^6$ already for a narrow energy bandwidth. The results show viability of quantum computers with time-independent qubit coupling.
The properties of the s-wave for a quasi-free particle with position-dependent mass(PDM) have been discussed in details. Differed from the system with constant mass in which the localization of the s-wave for the free quantum particle around the origin only occurs in two dimensions, the quasi-free particle with PDM can experience attractive forces in $D$ dimensions except D=1 when its mass function satisfies some conditions. The effective mass of a particle varying with its position can induce effective interaction which may be attractive in some cases. The analytical expressions of the eigenf…
We demonstrate the onset of strong on-site localization in a one-dimensional many-particle system. The localization is obtained by constructing, in an explicit form, a bounded sequence of on-site energies that eliminates resonant hopping between both nearest and remote sites. This sequence leads to quasi-exponential decay of the single-particle transition amplitude. It also leads to strong localization of stationary many-particle states in a finite-length chain. For an {\it infinite} chain, we instead study the time during which {\it all} many-particle states remain strongly localized. We show…
With the help of von Neumann entropy, we study numerically the localization properties of two interacting particles (TIP) with on-site interactions in one-dimensional disordered, quasiperiodic, and slowly varying potential systems, respectively. We find that for TIP in disordered and slowly varying potential systems, the spectrum-averaged von Neumann entropy <E_v> first increases with interaction U until its peak, then decreases as U gets larger. For TIP in the Harper model[S. N. Evangelou and D. E. Katsanos, Phys. Rev. B 56, 12797(1997)], the functions of <E_v> versus U are different for part…
We present a scheme to describe the dynamics of accelerating discrete-time quantum walk for one- and two-particle in position space. We show the effect of acceleration in enhancing the entanglement between the particle and position space in one-particle quantum walk and in generation of entanglement between the two unentangled particle in two-particle quantum walk. By introducing the disorder in the form of phase operator we study the transition from localization to delocalization as a function of acceleration. These inter-winding connection between acceleration, entanglement generation and lo…
In several recent papers on entanglement in relativistic quantum systems and relativistic Bell's inequalities, relativistic Bell-type two-particle states have been constructed in analogy to non-relativistic states. These constructions do not have the form suggested by relativistic invariance of the dynamics. Two relativistic formulations of Bell-type states are shown for massive particles, one using the standard Wigner spin basis and one using the helicity basis. The construction hinges on the use of Clebsch-Gordan coefficients of the Poincaré group to reduce the direct product of two unitary …
Stable interaction-induced Anderson-like localization embedded in standing waves
本研究发现无无序超晶格中玻色子可通过相互作用产生稳定类安德森自局域化，推导了有效哈密顿量并验证其能级间距服从泊松统计，提出动力学检测方案，为平移不变无无序系统实现类安德森局域化提供实验可行路径。
I discuss topics in Particle Physics applying the novel ontological formulation of Relativistic Quantum Field Theory due to David Bohm. I argument that particle physicists might too benefit from this truly novel way of thinking Physics.
The one-dimensional Klein-Gordon equation is investigated with the most general Lorentz structure for the external potentials. The analysis of the scattering of particles in a step potential with an arbitrary mixing of vector and scalar couplings reveals that the scalar coupling contributes for increasing the threshold energy for the particle-antiparticle pair production. Furthermore, it is shown that the pair production is only feasible whether the vector coupling exceeds the scalar one. An apparent paradox concerning the localization of a particle in an arbitrarily small region of space, due…
We study the quantum correlation dynamics of two relativistic particles which is transmitted through one of the Pauli channels $ σ_{x}, σ_{y},$ and $σ_{z}$. We compare sudden death and robustness of entanglement and geometric discord and quantum discord of two relativistic particles under noisy Pauli channels. we find out geometric discord and quantum discord may be more robust than entanglement against decoherence.
The nonrelativistic singlet state average $\langle ψ|{\vec a}\cdot\vec σ\otimes {\vec b}\cdot\vec σ|ψ\rangle =-\vec a\cdot\vec b $ can be relativistically generalized if one defines spin {\it via\/} the relativistic center-of-mass operator. The relativistic correction is quadratic in $v/c$ and can be measured in Einstein-Podolsky-Rosen-Bohm-type experiments with massive spin-1/2 particles. A deviation from the nonrelativistic formula would indicate that for relativistic nonzero-spin particles centers of mass and charge do not coincide.
We show that, in contrast to immediate intuition, Anderson localization of noninteracting particles induced by a disordered potential in free space can increase (i.e., the localization length can decrease) when the particle energy increases, for appropriately tailored disorder correlations. We predict the effect in one, two, and three dimensions, and propose a simple method to observe it using ultracold atoms placed in optical disorder. The increase of localization with the particle energy can serve to discriminate quantum versus classical localization.
The notion of Einstein causality, i.e. the limiting role of the velocity of light in the transmission of signals, is discussed. It is pointed out that Nimtz and coworkers use the notion of signal velocity in a different sense from Einstein and that their experimental results are in full agreement with Einstein causality in its ordinary sense. We also show that under quite general assumptions instantaneous spreading of particle localization occurs in quantum theory, relativistic or not, with fields or without. We discuss if this affects Einstein causality.
As a serious attempt for constructing a new foundation for describing micro-entities from a causal standpoint, it was explained before in [1, 2, 3] that by unifying the concepts of information, matter and energy, each micro-entity is assumed to be composed of a probability field joined to a particle called a particle-field or PF system. The relativistic generalization of this theory and its invariance under Lorentz transformation has been proved. In this essay, based on the relativistic generalization of Schrodinger equation derived in [4], we solve the relativistic Schrodinger equation for re…
We study the classical dynamics of non-relativistic particles endowed with spin. Non-vanishing Zitterbewegung terms appear in the equation of motion also in the small momentum limit. We derive a generalized work-energy theorem which suggests classical interpretations for tunnel effect and quantum potential.
Proximity-measurement induced random localization in quantum fluids
本研究探究随机邻近测量对多可分辨粒子量子流体的影响，发现测量会诱导部分粒子随机空间局域化且宏观保持均匀，最终全粒子局域化，局域长度分布由测量速率控制并遵循常见标度形式。
We study the scaling of the localization length of two interacting particles in a one-dimensional random lattice with the single particle localization length. We obtain several regimes, among them one interesting weak Fock space disorder regime. In this regime we derive a weak logarithmic scaling law. Numerical data support the absence of any strong enhancement of the two particle localization length.
We show, using quasi-exact numerical simulations, that Anderson localization of one-dimensional particles in a disordered potential survives in the presence of attractive interaction between particles. The localization length of the composite particle can be computed analytically for weak disorder and is in good agreement with the quasi-exact numerical observations using Time Evolving Block Decimation. Our approach allows for simulation of the entire experiment including the final measurement of all atom positions.
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