Quantum fractionalism: The Born rule as a consequence of the complex Pythagorean theorem

Everettian Quantum Mechanics, or the Many Worlds Interpretation, lacks an explanation for quantum probabilities. We show that the values given by the Born rule equal projection factors, describing the contraction of Lebesgue measures in orthogonal projections from the complex line of a quantum state to eigenspaces of an observable. Unit total probability corresponds to a complex Pythagorean theorem: the measure of a subset of the complex line is the sum of the measures of its projections on all eigenspaces.Postulating the existence of a continuum infinity of identical quantum universes, all with the same quasi-classical worlds, we show that projection factors give relative amounts of worlds. These appear as relative frequencies of results in quantum experiments, and play the role of probabilities in decisions and inference. This solves the probability problem of Everett's theory, allowing its preferred basis problem to be solved as well, and may help settle questions about the nature of probability.     

玻恩与诺贝尔奖

马克斯·玻恩为20世纪的物理学做出了卓越的贡献,但是他在自己70余岁才获得诺贝尔奖,个中原因多有推测;另外,究竟哪些人是玻恩的诺贝尔奖提名人,也是众说纷纭.根据可靠的文献资料,对以上不可思议的问题做了详细分析,并给出了明确答案.     

On the alleged equivalence of certain field theories

We critically examine some recent claims that certain field theories with and without boson kinetic energy terms are equivalent. We point out that the crucial element in these claims is the finiteness or otherwise of the boson wavefunction renormalisation constant. We show that when this constant is finite, the equivalence proof offered in the literature fails in a direct way. When the constant is divergent, the claimed equivalence is only a consequence of improper use of divergent quantities.     

量子力学80寿诞

因我曾在爱丁堡(1938—1941,1943—1945)玻恩教授处和都柏林(1941—1943,1945—1947)薛定谔教授处做过较长时间的研究工作,兹为文简述量子力学80年前的诞生过程,并说明当时学术思想的演化,同时也借此文表达对薛定谔教授和玻恩教授的怀念.     

精确的量子化条件和不变量

提出并证明了一维量子系统和三维球对称量子系统的一个精确的量子化条件.在此精确量子化条件中, 除了通常的Nπ项外, 还有一积分项, 称为修正项. 发现该修正项正是在超对称量子力学中所谓的有形状不变势的量子系统的一个不变量，它不依赖于波函数的节点数.对这些系统, 可用基态能级和波函数确定此不变量的值, 从而由精确的量子化条件容易算出全部束缚态的能级. 计算得到能级的正确性又反过来验证了在有形状不变势的量子系统中此修正项确实是不变量.计算的有形状不变势的量子系统, 包括一维的有限方势阱、Morse势及其变形、R 关键词： 量子化条件 超对称量子力学 形状不变势 不变量     

On DNA Molecules as Quantum Measurement Devices

In this letter we reconsider the proposal of Ref. [1] about a quantum measurement performed by a DNA molecule in aqueous solution as a tool for illustrating specific difficulties of some approach to quantum measurement problem. Our main result is that, when the interaction of DNA and enzymes with aqueous environment is properly kept into account, no real problem appears for any specific model.     

Consistent histories and the interpretation of quantum mechanics

The usual formula for transition probabilities in nonrelativistic quantum mechanics is generalized to yield conditional probabilities for selected sequences of events at several different times, called consistent histories, through a criterion which ensures that, within limits which are explicitly defined within the formalism, classical rules for probabilities are satisfied. The interpretive scheme which results is applicable to closed (isolated) quantum systems, is explicitly independent of the sense of time (i.e., past and future can be interchanged), has no need for wave function collapse, makes no reference to processes of measurement (though it can be used to analyze such processes), and can be applied to sequences of microscopic or macroscopic events, or both, as long as the mathematical condition of consistency is satisfied. When applied to appropriate macroscopic events it appears to yield the same answers as other interpretative schemes for standard quantum mechanics, though from a different point of view which avoids the conceptual difficulties which are sometimes thought to require reference to conscious observers or classical apparatus.     

Quantum equilibrium and the origin of absolute uncertainty

The quantum formalism is a measurement formalism-a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from Bohmian mechanics, which is what emerges from Schrödinger's equation for a system of particles when we merely insist that particles means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We find that a Bohmian universe, though deterministic, evolves in such a manner that anappearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by = ¦¦ ². A crucial ingredient in our analysis of the origin of this randomness is the notion of the effective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate.This paper is dedicated to the memory of J. S. Bell.     

On the Dirac equation with anomalous magnetic moment term in a beam of electromagnetic wave

The object of the paper is to obtain the solution of the Dirac equation with the Pauli-term in an electromagnetic field depending on the single variable (ct -nr) along the directionn.     

Born’s rule as signature of a superclassical current algebra

We present a new tool for calculating the interference patterns and particle trajectories of a double-, three- and N$N$-slit system on the basis of an emergent sub-quantum theory developed by our group throughout the last years. The quantum itself is considered as an emergent system representing an off-equilibrium steady state oscillation maintained by a constant throughput of energy provided by a classical zero-point energy field. We introduce the concept of a “relational causality” which allows for evaluating structural interdependences of different systems levels, i.e. in our case of the relations between partial and total probability density currents, respectively. Combined with the application of 21st century classical physics like, e.g., modern nonequilibrium thermodynamics, we thus arrive at a “superclassical” theory. Within this framework, the proposed current algebra directly leads to a new formulation of the guiding equation which is equivalent to the original one of the de Broglie–Bohm theory. By proving the absence of third order interferences in three-path systems it is shown that Born’s rule is a natural consequence of our theory. Considering the series of one-, double-, or, generally, of N$N$-slit systems, with the first appearance of an interference term in the double slit case, we can explain the violation of Sorkin’s first order sum rule, just as the validity of all higher order sum rules. Moreover, the Talbot patterns and Talbot distance for an arbitrary N$N$-slit device can be reproduced exactly by our model without any quantum physics tool.     

The Status of the Wave Function in Dynamical Collapse Models

No Heading The idea that in dynamical wave function collapse models the wave function is superfluous is investigated. Evidence is presented for the conjecture that, in a model of a field theory on a 1+1 lightcone lattice, knowing the field configuration on the lattice back to some time in the past, allows the wave function or quantum state at the present moment to be calculated, to arbitrary accuracy so long as enough of the past field configuration is known.     

创立量子力学的睿智才思(续2)——纪念矩阵力学和波动力学诞生80～81周年

1922年前后玻尔-索末菲的旧量子论在原子结构理论上取得辉煌成就的同时，它的致命弱点也开始暴露出来．反常塞曼效应是其一，原子光谱的多重线结构是其二，更不要说氢分子离子H2^＋问题了．一时多少杰出的物理学家为这些问题绞尽脑汁，设计了各种物理模型，提出一些“代用（德文Er-satz）”理论，诸如海森伯和泡利的Zwang之类，到头来还只得愁眉以对，一筹莫展．无怪乎海森伯把反常塞曼效应叫做“光谱项的动物学和塞曼植物学（Term Zoology and Zeeman Botany）”，意即在这个领域里物理学家只能像生物学家那样记录现象和描绘事实，提不出像样的理论．到1925年下半年，人们手里有了矩阵力学，问题是否可以解决了呢？万事俱备，只欠“东风”，那就是“自旋”．     

On the effective non-Hermitian eigenvalue problems for resonant levels

Summary A class of non-Hermitian eigenvalue equations, which are aimed to determine positions and widths of resonant (decaying) levels, is derived quite generally through a projection operator procedure whereby the effects of the continuum states responsible for the decay is represented in an effective manner. The boundary conditions appropriate to these eigenvalue problems are discussed and the wave function renormalization is evaluated. A connection is drawn with the effective eigenvalue problems occurring in the many-body Green's functions theory. Features like the energy dependence of the self-energy for the single-particle Green's function and of the screened Coulomb interaction for the electron-hole Green's function are accordingly interpreted in terms of elimination of continuum channels.

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Riassunto é noto da tempo, soprattutto dai lavori di Feshbach sul metodo delle hamiltoniane efficaci per reazioni nucleari, che tecniche di operatori di proiezione sono particolarmente adatte a costruire hamiltoniane efficaci allo scopo di determinare i parametri rilevanti delle risonanze nella teoria della diffusione. In questo articolo si pone in maggiore risalto l'aspetto quasi-legato degli stati risonanti e si considera un problema modello dove un insieme di stati discreti é accoppiato con un insieme di stati continui, i quali sono poi eliminati dalla trattazione esplicita mediante operatori di proiezione. In questo modo la sottomatrice dell'operatore risolvente nel sottospazio degli stati discreti risulta espressa in termini di un'hamiltoniana efficace che è non hermitiana e dipendente dall'energia. Tale procedimento è poi collegato con la risoluzione della equazione di Schr?dinger soggetta alle condizioni al contorno del tipo Kapur-Peierls che sono appropriate agli stati quasi stazionari. Questo collegamento fornisce, tra l'altro, una relazione tra la rinormalizzazione della funzione d'onda e la dipendenza dall'energia dell'hamiltoniana efficace. I risultati generali così ottenuti sono poi applicati alla teoria delle funzioni di Green a molti corpi. In particolare, si mostra come sia l'equazione di Dyson che determina i livelli a quasi particella sia l'equazione agli autovalori per gli eccitoni con schermaggio dielettrico possano essere interpretate nel senso delle hamiltoniane efficaci.

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Резюме В общем виде выводится класс незрмитовых уравнений для собственных значений, которые преднаэначемы для определения положений и ширин резонансных (распадающих) уровней, используя процедуру операторов проектирования. В таком подходе влияние непрерывных состояний, ответственных за распад, учитывается эффективным образом. Обсуждаются граничные условия, соответствующие этим проблемам собственных значений, и вычисляется перенормировка волновой функции. Отмечается связь с проблемами эффективных собственных значений, которые имеют место в теории многочастичных гриновских функций. Такие особенности, как энергетическая зависимость собственной энергии для одночастичной функции Грина и экранированное кулоновское взаимодействие для электрон-дырочной гриновской функции, интерпретируются соответственно на основе исключения непрерывных каналов.

     

A rigorous criterion to identify the validity of the Born approximation

This paper intends to identify the validity of the orn approximation by a new universal criterion, which is ultimately reduced to the calculation of an operator norm. With the purpose of enabling the criterion to be applicable to general scattering problems, a method is proposed to estimate the norm of the operator concerned. Compared with the conventional criterion, this method excels in its ability to acquire a quantificational upper bound of the relative error by Born approximation as well as to extend its valid frequency to a wider range. Two canonical scattering examples are given as evidence for the validity of the criterion.     

Model of a relativistic oscillator in a generalized Schrödinger picture

In a generalized Schrödinger picture, we consider the motion of a relativistic particle in an external field (like in the case of a harmonic oscillator). In this picture the analogs of the Schrödinger operators of the particle are independent of both the time and the space coordinates. These operators induce operators which are related to Killing vectors of the Anti de Sitter (AdS) space. We also consider the nonrelativistic limit.     

Covariant collapse of the state vector and realism

We point out that the assumption that a quantum-mechanical state vector collapses in a Lorentz-covariant way upon measurement is consistent, but that it implies that the state vector, and any reality which it represents, depends on position. We consider the specific example of collapse along the forward light cone of the measurement.     

玻恩对于中国物理界的贡献

马克斯·玻恩是20世纪最伟大的物理学家之一,他除了自己在多个物理领域做出了卓越的贡献外,还培养了一大批物理人才．这其中有几位著名的中国物理学家,如王福山、彭桓武、程开甲、杨立铭、黄昆等．他的这一贡献值得写入中国物理学史册．回顾他为中国物理界培养杰出人才的贡献,是中国物理界对他诞辰130周年的最好纪念之一．     

New Semiclassical and Numerical Approaches to Locate Zeros of Wave Functions Asiri Nanayakkara

A new semiclassical method is presented for evaluating zeros of wave functions. In this method, locating zeros of the wave functions of Schrodinger equation is converted to finding roots of a polynomial. The coefficients of this polynomial are evaluated using WKB and semi quantum action variable methods. For certain potentials WKB expressions for moments are obtained exactly. Almost explicit formulae for moments are obtained for the potential V (x) = xN. Examples are given to illustrate both methods. Using semi quantum action variable method, complex zeros of the wave functions of the PT symmetric complex system V(x) = x4 iAx are obtained. These zeros exhibit complex version of interlacing.     

半圆和四分之一圆二维弹子球的量子谱分析

采用近来提出的量子谱函数,我们把闭合轨道理论应用到半圆和四分之一圆弹子球系统,这种量子谱函数的傅利叶变换包含了连接任意两点的许多经典轨道的信息.计算表明量子谱的傅立叶变换和经典轨道的长度符合的很好.从这两个体系可以看出半经典理论为经典和量子力学提供了很好的桥梁作用.     

量子力学定态不是驻波

作为一种经典或半经典的观点，可以认为定态是由波的干涉形成的驻波，但在量子力学中，定态本身是基本的，不是驻波。