Discrete model of spacetime in terms of inverse spectra of the $T_0$ Alexandroff topologica

———————————————收稿日期：2021-05-06基金项目：国家重点研发计划（2020YFB2007600）模块化桁架机构动力源的设计和优化方法李志勇1，林秋红2，盆洪民*,1，曹子振1（1.天津航天机电设备研究所，天津 300458；2.北京空间飞行器总体设计部，北京 100094） 摘要：模块化桁架机构其动力源系统由于需兼顾多自由度、最简原则、可扩展性和空间环境，需要在应用中进行优化设计。

本文针对一种模块化的多自由度空间可展开桁架机构，分析了模块化机构特性，包括单模块机构自由度和多模块机构组合特性，具体介绍了动力源的设计过程和优化方法，基于动力学虚拟样机技术进行了有源动力源与无源动力源的参数匹配性优化设计，提高了模块化机构的展开动力学特性和稳定性，降低了对主动动力源的功率需求。

关键词：模块化；展开机构；动力学仿真；动力源 中图分类号：V211.3 文献标志码：Adoi ：10.3969/j.issn.1006-0316.2022.02.008文章编号：1006-0316 (2022) 02-0054-07Optimization Design of the Power Source for Modular Truss MechanismLI Zhiyong 1，LIN Qiuhong 2，PEN Hongmin 1，CAO Zizhen 1( 1.Tianjin Institute of Aerospace Mechanical and Electrical Equipment, Tianjin 300458, China;2.Beijing Institute of Spacecraft System Engineering, Beijing 100094, China )Abstract ：Power source system of modular truss mechanism needs to be optimized in application for catering its multi-degree of freedom, minimalist principle, scalability, and space environment requirements. In this paper, the design process and optimization method of power source for a modular deployable truss mechanism with multi-degree of freedom is introduced. Based on dynamic virtual prototype technique, the optimization design of active power source and passive power source for acquiring matched parameters is performed. The results show that proposed method can improve the deployable dynamic characteristics and stability of modular mechanism and reduce the power demand of active power source.Key words ：modular ；deployable mechanism ；dynamics simulation ；power source随着卫星SAR （Synthetic Aperture Radar ，合成孔径雷达）天线口径的日益增大，平面天线展开机构的规模和复杂程度也迅速提高，模块化桁架机构以其单一构型、大收纳比、良好可扩展性和高集成效率成为大型空间展开机构的热点[1-3]，而模块化机构展开动力源的设计由于存在多自由度、系统柔性大引起模块间动力传递延迟等问题成为研究难点之一[4]。

discrete ordinates(do)模型公式英文版Discrete Ordinates (DO) Model FormulaThe Discrete Ordinates (DO) model is a numerical method used in radiation transport calculations, particularly in the field of computational fluid dynamics. It solves the radiative transfer equation, which governs the propagation of radiation energy through a medium. The DO model discretizes the angular domain, allowing for the computation of radiation intensity at various directions.The basic DO model formula can be expressed as:I(s,Ω)= I0(s,Ω)e-∫s0κ(s')ds' + ∫s0κ(s')∫Ω'4πp(Ω'→Ω)I(s',Ω')e-∫ss'κ(s'')ds''ds'where:I(s,Ω) is the radiation intensity at position s and direction Ω.I0(s,Ω) is the incident radiation intensity at position s and direction Ω.κ(s) is the absorption coefficient at position s.p(Ω'→Ω) is the probability of a photon being scattered from direction Ω' to direction Ω.The integrals represent the accumulation of radiation intensity along the path from the source to the point of interest.The DO model uses a finite number of discrete ordinates (or directions) to approximate the angular dependence of the radiation intensity. This approximation allows for efficient numerical solutions, especially in complex geometries where analytical solutions are not feasible.The DO model is widely used in various applications such as combustion modeling, solar radiation analysis, and radiation heat transfer in participating media. It provides a computationally efficient means to model radiation transport in complex systems.中文版离散坐标（DO）模型公式离散坐标（DO）模型是一种用于辐射传输计算的数值方法，特别是在计算流体动力学领域。

When writing an essay in English about My Model,its important to consider the context in which the term model is being used.Here are a few different approaches you might take,depending on the specific meaning of model in your essay:1.A Role Model:Begin by introducing who your role model is and why they are important to you. Discuss the qualities and achievements of your role model that you admire. Explain how their actions or life story has influenced your own life or goals.Example Paragraph:My role model is Malala Yousafzai,a Pakistani activist for female education and the youngest Nobel Prize laureate.Her courage and determination to fight for girls education rights in the face of adversity have deeply inspired me.Malalas story has taught me the importance of standing up for what I believe in,even when it is difficult.2.A Fashion Model:Describe the physical attributes and style of the model.Discuss the impact they have had on the fashion industry or their unique contributions to it.Explain why you find their work or presence in the industry notable.Example Paragraph:Kendall Jenner is a fashion model who has made a significant impact on the industry with her unique style and presence.Her tall and slender physique,combined with her ability to carry off diverse looks,has made her a favorite among designers and fashion enthusiasts alike.I admire her for her versatility and the way she uses her platform to promote body positivity.3.A Model in Science or Technology:Introduce the model as a theoretical framework or a practical tool used in a specific field.Explain the principles behind the model and how it is applied.Discuss the benefits or limitations of the model and its implications in the real world.Example Paragraph:The Standard Model in physics is a theoretical framework that describes three of the four known fundamental forces excluding gravity and classifies all known elementary particles.It has been instrumental in understanding the behavior of subatomic particles and predicting the existence of new particles,such as the Higgs boson.However,the models inability to incorporate gravity or dark matter has led to ongoing research for amore comprehensive theory.4.A Model in Business or Economics:Introduce the business or economic model and its purpose.Explain how the model works and the strategies it employs.Discuss the success or challenges associated with the model and its potential for future growth.Example Paragraph:The subscriptionbased business model has become increasingly popular in recent years, particularly in the software panies like Adobe have transitioned from selling packaged software to offering services on a subscription basis,allowing for continuous revenue streams and a more predictable income.This model has been successful in fostering customer loyalty and providing a steady income,although it requires ongoing innovation to maintain customer interest.5.A Model in Art or Design:Describe the aesthetic or functional qualities of the model.Discuss the creative process or design principles that inform the model.Explain the cultural or historical significance of the model and its influence on contemporary art or design.Example Paragraph:The Eames Lounge Chair,designed by Charles and Ray Eames,is a model of modern furniture that has become an icon of midcentury design.Its elegant form,made from molded plywood and leather,exemplifies the designers commitment to blending comfort with aesthetics.The chairs timeless appeal has made it a staple in both residential and commercial settings,influencing countless furniture designs that followed. Remember to structure your essay with a clear introduction,body paragraphs that develop your points,and a conclusion that summarizes your main e specific examples and evidence to support your claims,and ensure your writing is clear,concise, and engaging.。

Vol. 41, No. 2航 天 器 环 境 工 程第 41 卷第 2 期244SPACECRAFT ENVIRONMENT ENGINEERING2024 年 4 月https:// E-mail: ***************Tel: (010)68116407, 68116408, 68116544航天器典型产品性能试验数据标准化管理体系刘佳琳，唐小军*，穆 城，严振刚，田 欣，回天力（北京卫星制造厂有限公司，北京 100086）摘要：文章基于航天器产品高可靠性要求和小子样的特点，依照产品顶层研制测试需求为主题模板，提出了一套全新的试验数据标准化管理模型，具体可细分为数据置信度管理模型、小子样产品数据筛选模型和性能数据管理模型；并以航天器供配电二次电源产品为应用对象，实现了二次电源产品的性能试验数据标准模型的建立，将现有非结构化测试数据进行了汇总，统一了量纲，可为产品的批产化横向对比和代系发展纵向对比，以及数据的后续利用提供高效可靠的试验数据信息。

关键词：产品性能试验；标准化数据存储；数据管理模型；数据筛选；数据置信度评估中图分类号：V416; TP274文献标志码：A文章编号：1673-1379(2024)02-0244-07 DOI: 10.12126/see.2023065Standardized management system for performance test data oftypical spacecraft productsLIU Jialin, TANG Xiaojun*, MU Cheng, YAN Zhengang, TIAN Xin, HUI Tianli(Beijing Spacecraft Manufactory Co., Ltd., Beijing 100086, China)Abstract: Based on the high reliability requirements of spacecraft products and characteristics of small samples, and according to the top-level development and testing requirements of products as the theme template, a set of novel standardized models for test data management were proposed. They include data confidence management model, small sample product data screening model, as well as performance data management model. The secondary power supply product of spacecraft was taken as the application object to establish the standardized models for performance test data. The previous fragmented test data were summarized and the dimensions were unified. The proposed study can be used for horizontal comparison of batch production and vertical comparison of generation development for spacecraft, so as to offer reliable information for the subsequent use of test data.Keywords: product performance test; standardized data storage; data management model; data screening; data confidence evaluation收稿日期：2023-05-09；修回日期：2024-04-01基金项目：北京市科技新星计划项目（编号：2022095）引用格式：刘佳琳, 唐小军, 穆城, 等. 航天器典型产品性能试验数据标准化管理体系[J]. 航天器环境工程, 2024, 41(2): 244-250LIU J L, TANG X J, MU C, et al. Standardized management system for performance test data of typical spacecraft products[J]. Spacecraft Environment Engineering, 2024, 41(2): 244-2500 引言为满足我国载人航天和商业航天等新任务的需求，以及日益增多的航天产品生产计划，加快航天产品生产线的数字化转型成为必然。

离散选择模型的基本原理及其发展演进评介聂冲贾生华(浙江大学管理学院)=摘要>离散选择模型的研究真正兴起于19世纪50年代末,属于微观计量经济学的范畴。

该模型能够对个体和家庭行为进行经验性的统计分析,因而在经济学和其他社会科学中得到广泛的应用。

本文从离散选择模型的基本性质及效用最大化的理论背景出发,指出logit模型虽然使用的是最早并且最为广泛的离散选择模型,但是其存在着三大局限性:不能表示随机口味的变化、暗含成比例的替代形式和不能处理不可观测因素在不同期间相关的情形。

GEV(含嵌套logit)、pr obit和混合logit模型等其他的离散选择模型,很大程度上都是为了避免这些限制而产生并发展起来的。

关键词离散选择logit局限性中图分类号F06114文献标识码AResearch on The Theoretical Basis and Evolutionof Discrete Choice ModelsAbstract:The boom of discrete choice models,which belong to the area of microeconometrics,were really beginning from the end of1950s1T he models canbe used to analyse the behaviors of private and family experientially,and they are widely applied to economics and other social sciences1T his paper begins with the theoretical basis and utility maximization,then demonstrates that there are thr ee limitations of logit:it can not represent r andom taste variation;it implies pr opor2 tional substitution acr oss alter natives and cannot handle situations wher e unob2ser ved factors are corr elated over time,though logit is by far the easiest and most widely used discr ete choice model1Other models,such as GEV(including nested logit),probit and mixed logit,have arisen largely to avoid these limitations1 Key words:Discrete Choice Model;Logit;Limitation导言2000年10月11日瑞典皇家科学院宣布,2000年度诺贝尔经济学奖将授予美国芝加哥大学的詹姆斯#赫克曼(James H eckman)教授和美国加州大学伯克利分校的丹尼尔#麦克法登(Daniel McFadden)教授,以表彰他们在微观经济计量学领域所做出的贡献。

a r X i v :g r -q c /0312125v 1 31 D e c 2003MODERNAS TEORIAS SOBRE ELTIEMPO DISCRETOMiguel Lorente Universidad de Oviedo 1Introducci´o n En las explicaciones f´ısicas del Universo el tiempo aparece como una magni-tud fundamental que es imprescindible para describir los procesos naturales.Cl´a sicamente el tiempo era una entidad absoluta,independiente de las cosas y que de alguna manera acompa˜n aba su teor´ıa de la relatividad introdujo un car´a cter relativo en el tiempo depen-diente del observador,aunque esto no supon´ıa un rechazo de la existencia de un tiempo independiente de las cosas.Modernamente se han propuesto teor´ıas al-ternativas del tiempo como un concepto derivado de las relaciones entre las cosas de modo que niega toda entidad al tiempo que no sea la misma realidad de las cosas.Aunque estas teor´ıas puedan parecer nuevas,hay que remontarse hasta Leib-niz (y a´u n m´a s lejos)ya que ´e ste propuso y defendi´o ac´e rrimamente contra los disc´ıpulos de Newton una teor´ıa relacional del tiempo [1].Obviamente estas teor´ıas del tiempo est´a n unidas a las del espacio por lo que es pr´a cticamente imposible hablar de unas sin mencionar a las otras [2].Siguiendo un orden epistemol´o gico nos parece m´a s conveniente empezar por las teor´ıas f´ısicas que describen el tiempo como una magnitud discreta,cuya finalidad es puramente pragm´a tica:resolver un conjunto de problemas f´ısico-matem´a ticos sin preocuparse de su interpretaci´o n filos´o fica.Otro bloque de teor´ıas intentan dar un contenido f´ısico a los modelos discretos,buscando una estructura relacional de las cosas como substrato material a las propiedades espacio-temporales.Por ´u ltimo se encuentran las posturas filos´o ficas que tratan de dar una fun-damentaci´o n ontol´o gica a las teor´ıas relacionales del espacio-tiempo.2Modelos f ´ısicos con tiempo discretoEl uso de ret´ıculos espaciales ha sido muy utilizado en F´ısica para poder contar las part´ıculas en cada celdilla y describir as´ılas propiedades estad´ısticas de un sistema con un n´u mero muy elevado de elementos.Hoy d´ıa se est´a poniendo de moda el introducir valores discretos de las coordenadas espacio-temporales enlas ecuaciones que rigen la evoluci´o n de las funciones de onda que representan part´ıculas elementales en estado libre o en interacci´o n.Estos modelos tienen un inter´e s matem´a tico:resolver por m´e todos num´e ricos ecuaciones diferenciales que no tienen una soluci´o n anal´ıtica exacta y por otra parte evitar los valores infinitos que aparecen en los desarrollos perturbativos de dichas literatura reciente en esta direcci´o n es muy amplia.Para fen´o menos locales donde se usa la relatividad especial,encontramos las teor´ıas gauge en ret´ıculos rectangulares espacio-temporales que han sido muy ´u tiles para calcular factores de forma y masas de part´ıculas elementales.El trabajo pionero en esta l´ınea fue escrito por K.Wilson en1974que in-tentaba explicar el confinamiento de los quarks en los hadrones y fue seguido por otros f´ısicos en la descripci´o n de las interacciones fuertes y electrod´e biles.Al extender estos modelos a las interacciones gravitacionales,el ret´ıculo c´u bico resulta insuficiente ya que es necesario adaptar al modelo las propiedades geom´e-tricas de los espacios curvos riemannianos usados en la relatividad general.Uno de los primeros en proponer una teor´ıa de espacio-tiempo discreto fue Wheeler [3]que introdujo la discretizaci´o n del espacio-tiempo para abordar el problema de la gravitaci´o n c´u antica.Entre sus m´a s inmediatos seguidores se encuentran Ponzano y Regge[4]que construyen un modelo de espacio-tiempo discreto,donde una red de tri´a ngulos adyacentes da lugar a superficies de curvatura arbitraria. Recientemente han proliferado los autores que han seguido el c´a lculo de Regge [5].La t´e cnica general de estos modelos consiste en aproximar una superficie Rie-manniana por triangulaciones hechas defiguras simpliciales de lados iguales(un simplicial es un conjunto de puntos donde cada uno de ellos est´a relacionado por aristas con todos los dem´a s).Cada capa formada de simpliciales est´a enlazada a otra capa pr´o xima de la misma estructura de modo que hay una conexi´o n uno a uno entre los puntos correspondientes de cada aplicaci´o n sucesiva de las diferentes capas define una l´ınea del universo(en la terminolog´ıa relativista) cuyo par´a metro para enumerar las diferentes capas toma valores discretos y se puede identificar con el tiempo[6].3Teor´ıas relacionales del espacio-tiempoEstas teor´ıas dan un paso m´a s en la explicaci´o n del Universo.Intentan encontrar un modelo donde el espacio-tiempo es un concepto derivado de las propiedades de las cosas y de las relaciones entre ellas.Tambi´e n aqu´ıWheeler[7]fue precursor con su Pregeometr´ıa basada en un conjunto de constantes fundamentales con las que se describen las interacciones entre los“ladrillos”o entidades b´a sicas del Universo.En la misma direcci´o n Marlow[8]ha desarrollado una teor´ıa axiom´a tica de la relatividad general cu´a ntica basada en el c´a lculo de proposiciones donde el tiempo toma valores discretos.Un autor que recogi´o las sugerencias de Wheeler fue Penrose[9]que utiliz´o las entidades b´a sicas como unidades provistas de un valor de spin,que interaccionan entre s´ısiguiendo la ley de suma de dos momentos angulares.El resultado es unared que no requiere un substrato espacial,porque ella es el mismo substrato.El tiempo es un par´a metro que enumera las diferentes interacciones que se producen sucesivamente.El programa de Penrose se prolonga en redes muy complejas que dan lugar a objetos matem´a ticos quebautiz´o con el nombre de“twistors”y que´e l mismo utiliza en el formalismo de la teor´ıa de la gravitaci´o n.Un enfoque paralelo se puede encontrar en las ideas de D.Finkelstein[10].Recientemente Penrose ha manifestado su intenci´o n de no renunciar a su modelo de espacio-tiempo basado en una red de spines:“Tengo todav´ıa aspiraciones en mis ideas que he desarrollado hace varios a˜n os con la teor´ıa de la red de spines(1971,Quantum theory and Beyond;1972,Magic without magic).Los experimentos ideales del tipo de Bohm, Einstein,Podolsky y Rosen han jugado un papel muy importante en esa teor´ıa, y la idea fue construir los conceptos de espacio y tiempo como una estructura l´ımite impl´ıcita cuando el n´u mero de part´ıculas se hace muy grande.Sin embargo, ni la teor´ıa de los‘twistors’ni la teor´ıa de las‘redes de spin’tienen entre sus ingredientes una asimetr´ıa temporal.Por eso me resulta evidente que es necesario una idea esencialmente nueva”[11].Garc´ıa Sucre y Bunge tambi´e n han introducido una teor´ıa relacional del es-pacio tiempo[12].Las entidades fundamentales son todas las cosas del Universo cuyas relaciones son descritas con un formalismo basado en la teor´ıa de s unidades fundamentales o prepart´ıculas se describen por elementos de un conjuntofinito.El papel crucial que juega el tiempo se representa por la sucesi´o n de subconjuntos enlazados por la relaci´o n l´o gica de la inclusi´o n que implica un orden entre los mismos.El espacio no es m´a s que la suma de todas las cadenas de subconjuntos ordenados por la inclusi´o n y que pueden tomar todas las configu-raciones posibles.Si escogemos unas determinadas l´ıneas entre todas las cadenas posibles habremos definido un sistema de referencia determinado.El concepto de tiempo y de espacio emana de una manera natural de un determinado sistema referencial.Klapunosky y Weinstein[13]han propuesto recientemente una teor´ıa de cam-pos cuantificados en los que los valores de las coordenadas espacio-temporales son n´u meros enteros y no representan ninguna referencia al espacio tiempo,sino unos par´a metros para distinguir los valores del s interacciones entre los campos est´a n producidas por acoplamientos entre los campos fundamentales, de manera que´e stos son la´u nica realidad subyacente y las relaciones producidas por las conexiones entre ellos da lugar a un ret´ıculo de estructura simplicial que se nos presenta a los sentidos como una“ilusi´o n”que llamamos espacio-tiempo.El autor de este trabajo tambi´e n ha propuesto una teor´ıa relacional del espacio-tiempo[14]con elfin de justificar de una manera axiom´a tica los fundamentos de la geometr´ıa,m´a s all´a todav´ıa de los postulados formulados por Hilbert.Siguien-do el esp´ıritu de este matem´a tico,seg´u n el cual,se deben considerar los puntos, l´ıneas y superficies como sillas,mesas y jarros de cerveza,(es decir,sin referencia a una intuici´o n espacial)se postula un ret´ıculo n-dimensional c´u bico donde cada punto est´a relacionado con2n-puntos diferentes y solamente con´e stos,de cuyo ´u nico postulado se deducen l´o gicamente todos los axiomas de Hilbert en su libro Fundamentos de la Geometr´ıa.4Concepciones ontol´o gicas subyacentes a las teor´ıas relacionales del espacio-tiempoLas teor´ıas relacionales mencionadas anteriormente se pueden analizar a un nivel puramente l´o gico(el tiempo que percibimos,se puede interpretar como la im-presi´o n sensible que nos produce la sucesi´o n temporal de relaciones entre los objetos f´ısicos).Pero tambi´e n se puede preguntar sobre el substrato ontol´o gico de estas relaciones,que no suponga ninguna entidad fuera de las cosas mismas.Se puede citar a Leibniz entre los que han propuesto una explicaci´o nfilos´ofica de la teor´ıa relacional del espacio-tiempo.En su obra Initia rerum mathemati-carum metaphysica,defiende que el tiempo es el orden de las cosas existentes que no son simult´a neas,mientras que el espacio es el orden de las cosas que coexisten o el orden de las cosas existentes que son simult´a neas.El fundamento del orden temporal es la conexi´o n causal.Cuando una cosa es el principio de otra,aquella se dice anterior y´e sta posterior.Esta idea la vuelve a repetir en su Monadolog´ıa y en numerosas cartas.Max Jammer[15]indica que Leibniz se inspir´o para su Monadolog´ıa en la Gu´ıa de perplejos de Maim´o nides y recientemente Pannenberg[16]recuerda la influencia de losfil´o sofos´a rabes en la teor´ıa atomista del tiempo de Leibniz. Seg´u n estosfil´o sofos la creencia en la creaci´o n implicaba que no exist´ıa nada antes de la creaci´o n y que los primeros seres existentes ser´ıan unos´a tomos o part´ıculas indivisibles.El tiempo comenz´o tambi´e n en ese instante pero no como distinto de la materia sino concomitante con ella.“El tiempo,dice Maim´o nides[17],consta de instantes,a saber,que hay mucha unidad de temporaneidad,los cuales por su ef´ımera duraci´o n,excluyen la divisi´o n”.Hablar de´a tomos de materia es hablar de´a tomos de tiempo y por consiguiente,a˜n ade Maim´o nides“el tiempo se inserta en‘instantes’que no admiten divisi´o n”.Dos autores contempor´a neos que se pueden adscribir a una concepci´o n atom-ista del tiempo son Whitehead y Weizs¨a ecker.Whitehead[18],el colaborador m´a s estrecho de Russel,desarrolla en su edad madura unafilosof´ıa del Cosmos, donde la´u ltima verdad es el atomismo metaf´ıs entidades actuales—denominadas tambi´e n ocasiones actuales—son las´u ltimas cosas de que est´a compuesto el s entidades actuales se relacionan entre s´ıpor nexos extr´ınsecos e intr´ınsecos para formar estructuras m´a s complejas a trav´e s de las prehensiones o sentires f´ıs entidades actuales producen su tiempo y su entidad actual es regi´o n que ocupa la entidad actual es divisible s´o lo mentalmente.El tiempo y el espacio son una abstracci´o n a partir de las actualidades y las relaciones entre ellos.Para Weizs¨a ecker[19]los conceptos de espacio y tiempo son una consecuencia de las relaciones entre las entidades m´a s fundamentales del Universo:los procesos que el observador percibe como una simple alternativa(experimento si-no).Los procesos constituyen un entramado de simples alternativas(“urs”)y el tiempo es un par´a metro que diversifica la realidad presente y futura de estos procesos.Para Weizs¨a ecker la estructura actual del Universo est´a compuesta de un n´u merofinito de procesos elementales pero el n´u mero de posibilidades de inter-acciones entre estos entes elementales es infinita,de donde se sigue el car´a cter discreto para la descripci´o n de los entes actuales y continuo para las leyes de evoluci´o n de estos procesos.Durante varios a˜n os Weizs¨a ecker ha organizado unos Encuentros para trabajar en la unificaci´o n de la Mec´a nica C´u antica y la Relativi-dad de modo que los postulados de la´u ltima se derivan de los postulados de la primera.Como resumen de esta exposici´o n podemos decir que la hip´o tesis de un tiempo discreto,como consecuencia de un concepto relacional del espacio-tiempo,se ha desarrollado recientemente por numerosos autores en sus aspectos epistemol´o gicos y ontol´o gicos,como alternativa a la concepci´o n absolutista,y que se corrobora con la extensa bibliograf´ıa.Las consecuencias f´ısicas de esta hip´o tesis est´a n todav´ıa muy lejos de ser com-probadas experimentalmente aunque han progresado los modelos matem´a ticos que permitir´ıan hacer una predicci´o n detectable,por lo menos indirectamente, de la hip´o tesis,y ciertamente mucho m´a s cercana a los hechos que las primitivas especulaciones defendidas por Leibniz.Referencias1J.Earman,World enough and Space-time:Absolute versus Relational Theories of Space and Time,MIT Press,Cambridge1989.2M.Lorente,“Modernas teor´ıas sobre la estructura del espacio-tiempo”en Actas de la Reuni´o n Matem´a tica en honor de A.Dou,Ed.Universidad Complutense,Madrid1989,pp.353–363.3A.Wheeler,Geometrodynamics,Academic Press,N.Y.1962.4G.Ponzano,R.Regge,en Spectroscopy and Group Theoretical Methods in Physics(ed.F.Bloch),North Holland,Amsterdam1968.fave,A Step Toward Pregeometry I:Ponzano-Regge Spin Networks and the Origin of Space-time Structure in Four Dimensions(preprint), Houston,Texas1993.6Y.Shamir,Dynamical-Space Regular-Time Lattice and Induced Gravity, (preprint)Weizmann Institute of Science,Israel1994.7A.Wheeler,Quantum Theory and Gravitation(ed.A.R.Marlow)Aca-demic Press,N.Y.1980.8A.R.Marlow,“An axiomatic general relativity quantum theory”Cfr.[7] p.35.9R.Penrose,“Angular Momentum:an Approach to Combinatorial Space-Time”en Quantum Theory and Beyond(T.Bastin,ed.)Cambridge U.Press,1971.R.Penrose,“On the nature of quantum geometry”en Magic without magic(J.R.Klauder ed.)Freeman1972.R.Penrose,“On the origin of twistor theory”en Gravitation and Geom-etry(ed.W.Rindler and A.Trautman)Bibliopolis,Naples1986.10D.Finkelstein,E.Rodr´ıguez,“Quantum Time-Space and Gravity”en Quantum Concepts in Space and Time(ed.R.Penrose,C.J.Isham) Clarendon Press,Oxford1986.11R.Penrose,“Newton,quantum theory and reality”en Three hundred years of gravitation(S.W.Hawking and W.Israel,ed.)Cambridge U.Press 1987.12G.Sucre,“Quantum Statistics in a Simple Model of Space-Time”,Int.J.of Theor.Phys.24,441–445(1985).M.Bunge,“Una teor´ıa relacional del espacio f´ısico,en Controversias en F´ısica,Tecnos Madrid1983.13V.Kaplunosky,M.Weinstein,“Space-time,Arena or Illusion”,Phys.Rev.D31,1879–1898(1985).14M.Lorente,“Quantum Processes and the Foundation of Relational The-ories of Space and Time”.Encuentros Relativistas Espa˜n oles1993(ser´a publicado en Ed.Lumi´e re,Par´ıs1994).15M.Jammer,Concepts of Space,The History of theories of Space in Physics, Harvard U.Press,1969,p.64.Cambridge1969.16Pannenberg,Systematische Theologie,G¨o ttingen1991.17M.Maimonides,Gu´ıa de Perplejos(edici´o n preparada por D.Gonz´a lez Maeso),Ed.Nacional Madrid1983,p.213.18A.Whitehead,The Concept of Nature,Cambridge U.Press,1920.Sci-ence and the Modern World,McMillan1925.Process and Reality,McMillan 1929.19K.F.Weizs¨a ecker,Die Einheit der Natur(Hauser1971)Quantum The-ory and the Structure of Space and Time6vol.(Hauser1986)Aufbau der Physik,Hauser1985.Coloquio a la comunicaci´o n de M.LorenteEn el coloquio subsiguiente,Alberto Dou manifest´o su simpat´ıa por estas teor´ıas modernas del tiempo discreto,porque eran coherentes con la progresiva cuantificaci´o n que la ciencia ha ido ampliando en su descripci´o n de la natu-raleza.Primero el atomismo de la materia fue introduciendo unidades naturales en la composici´o n de los cuerpos,que se ha ido completando con las teor´ıas de part´ıculas elementales,´u ltimos elementos invisibles de la materia.Por otro lado, la mec´a nica cu´a ntica ha introducido valores discretos naturales en ciertas magni-tudes f´ısicas,como la carga,el momento angular,la acci´o n.A.Dou describi´o a continuaci´o n un modelo de estructura de la materia,de acuerdo con la hip´o tesis de un espacio-tiempo discreto.Un conjunto de l´a mparas luminosas est´a n conectadas entre s´ıformando una estructura c´u bica.Utilizando procedimientos digitales de encendido y apagado de las l´a mparas se puede obtener se˜n ales luminosas que se propagan por el ret´ıculo y que pod´ıa interpretarse como la funci´o n de onda que utiliza la mec´a nica cu´a ntica para la descripci´o n de un sistema elemental.Tambi´e n Alberto Galindo se interes´o por la comunicaci´o n haciendo dos pre-guntas:1)¿Se han de dar en el ret´ıculo,antes de tomar el l´ımite continuo,las propiedades de simetr´ıa y leyes de conservaci´o n que se demuestran en el modelo continuo de las leyes f´ısicas?El autor respondi´o que parece razonable que tambi´e n en el modelo discreto se den unas simetr´ıas an´a logas,y se refiri´o a los trabajos de investigaci´o n que est´a realizando sobre subgrupos discretos de los grupos de Lie.2)¿Qu´e papel pueden jugar en estos espacios discretos las teor´ıas recientes de geometr´ıa no conmutativa?El autor respondi´o que se est´a n publicando recien-temente hip´o tesis f´ısicas donde las coordenadas espacio-temporales no conmu-tan entre s´ı(cfr.A.Connes,“Geometrie non-commutative”,Intereditions,Paris 1990).En esta obra Connes propone la idea de un espacio no conmutativo con el objeto de suprimir las divergencias ultravioletas introduciendo un corte(cut-off) natural.Pero esta hip´o tesis lleva a la deformaci´o n de los grupos de simetr´ıa por los grupos cu´a nticos.Precisamente las realizaciones de los grupos cu´a nticos para construir las ecuaciones de onda llevan a introducir de una manera natural los operadores diferenciasfinitas equivalentes a las que el autor ha empleado en sus modelos discretos.。

2-dimensional space3D mapabstractaccess dataAccessibilityaccuracyacquisitionad-hocadjacencyadventaerial photographsAge of dataagglomerationaggregateairborneAlbers Equal-Area Conic projection (ALBER alignalphabeticalphanumericalphanumericalalternativealternativealtitudeameliorateanalogue mapsancillaryANDannotationanomalousapexapproachappropriatearcarc snap tolerancearealAreal coverageARPA abbr.Advanced Research Projects Agen arrangementarrayartificial intelligenceArtificial Neural Networks (ANN) aspatialaspectassembleassociated attributeattributeattribute dataautocorrelationautomated scanningazimuthazimuthalbar chartbiasbinary encodingblock codingBoolean algebrabottombottom leftboundbreak linebufferbuilt-incamouflagecardinalcartesian coordinate system cartographycatchmentcellcensuscentroidcentroid-to-centroidCGI (Common Gateway Interface) chain codingchainscharged couple devices (ccd) children (node)choropleth mapclass librariesclassesclustercodecohesivelycoilcollinearcolumncompactcompasscompass bearingcomplete spatial randomness (CSR) componentcompositecomposite keysconcavityconcentricconceptual modelconceptuallyconduitConformalconformal projectionconic projectionconnectivityconservativeconsortiumcontainmentcontiguitycontinuouscontourcontour layercontrol pointsconventionconvertcorecorrelogramcorrespondencecorridorCostcost density fieldcost-benefit analysis (CBA)cost-effectivecouplingcovariancecoveragecoveragecriteriacriteriacriterioncross-hairscrosshatchcross-sectioncumbersomecustomizationcutcylindrical projectiondangledangle lengthdangling nodedash lineDATdata base management systems (DBMS) data combinationdata conversiondata definition language (DDL)data dictionarydata independencedata integritydata itemdata maintenancedata manipulationData manipulation and query language data miningdata modeldata representationdata tabledata typedatabasedateDBAdebris flowdebugdecadedecibeldecision analysisdecision makingdecomposededicateddeductiveDelaunay criterionDelaunay triangulationdelete(erase)delineatedemarcationdemographicdemonstratedenominatorDensity of observationderivativedetectabledevisediagonaldictatedigital elevation model (DEM)digital terrain model (DTM) digitizedigitizedigitizerdigitizing errorsdigitizing tablediscrepancydiscretediscretedisparitydispersiondisruptiondissecteddisseminatedissolvedistance decay functionDistributed Computingdividedomaindot chartdraftdragdrum scannersdummy nodedynamic modelingeasy-to-useecologyelicitingeliminateellipsoidellipticityelongationencapsulationencloseencodeentity relationship modelingentity tableentryenvisageepsilonequal area projectionequidistant projectionerraticerror detection & correctionError Maperror varianceessenceet al.EuclideanEuclidean 2-spaceexpected frequencies of occurrences explicitexponentialextendexternal and internal boundaries external tablefacetfacilityfacility managementfashionFAT (file allocation table)faultyfeaturefeaturefeedbackfidelityfieldfield investigationfield sports enthusiastfields modelfigurefile structurefillingfinenessfixed zoom infixed zoom outflat-bed scannerflexibilityforefrontframe-by framefreefrom nodefrom scratchfulfillfunction callsfuzzyFuzzy set theorygantrygenericgeocodinggeocomputationgeodesygeographic entitygeographic processgeographic referencegeographic spacegeographic/spatial information geographical featuresgeometricgeometric primitive geoprocessinggeoreferencegeo-relational geosciences geospatialgeo-spatial analysis geo-statisticalGiven that GNOMONIC projection grain tolerance graticulegrey scalegridhand-drawnhand-heldhandicaphandlehand-written header recordheftyheterogeneity heterogeneous heuristichierarchical hierarchicalhill shading homogeneoushosthouseholdshuehumichurdlehydrographyhyper-linkedi.e.Ideal Point Method identicalidentifiable identification identifyilluminateimageimpedanceimpedanceimplementimplementimplicationimplicitin excess of…in respect ofin terms ofin-betweeninbuiltinconsistencyincorporationindigenousinformation integration infrastructureinherentinheritanceinlandinstanceinstantiationintegerintegrateinteractioninteractiveinteractiveinternet protocol suite Internet interoperabilityinterpolateinterpolationinterrogateintersectintersectionIntersectionInterval Estimation Method intuitiveintuitiveinvariantinventoryinvertedirreconcilableirreversibleis adjacent tois completely withinis contained iniso-iso-linesisopleth mapiterativejunctionkeyframekrigingKriginglaglanduse categorylatitudelatitude coordinatelavalayerlayersleaseleast-cost path analysisleftlegendlegendlegendlength-metriclie inlightweightlikewiselimitationLine modelline segmentsLineage (=history)lineamentlinearline-followinglitho-unitlocal and wide area network logarithmiclogicallogicallongitudelongitude coordinatemacro languagemacro-like languagemacrosmainstreammanagerialmanual digitizingmany-to-one relationMap scalemarshalmaskmatricesmatrixmeasured frequencies of occurrences measurementmedialMercatorMercator projectionmergemergemeridiansmetadatameta-datametadatamethodologymetric spaceminimum cost pathmirrormis-representmixed pixelmodelingmodularmonochromaticmonolithicmonopolymorphologicalmosaicmovemoving averagemuiticriteria decision making (MCDM) multispectralmutually exclusivemyopicnadirnatureneatlynecessitatenestednetworknetwork analysisnetwork database structurenetwork modelnodenodenode snap tolerancenon-numerical (character)non-spatialnon-spatial dataNormal formsnorth arrowNOTnovicenumber of significant digit numeric charactersnumericalnumericalobject-based modelobjectiveobject-orientedobject-oriented databaseobstacleomni- a.on the basis ofOnline Analytical Processing (OLAP) on-screen digitizingoperandoperatoroptimization algorithmORorderorganizational schemeoriginorthogonalORTHOGRAPHIC projectionortho-imageout ofoutcomeoutgrowthoutsetovaloverdueoverheadoverlapoverlayoverlay operationovershootovershootspackagepairwisepanpanelparadigmparent (node)patchpath findingpatternpatternpattern recognitionperceptionperspectivepertain phenomenological photogrammetric photogrammetryphysical relationships pie chartpilotpitpixelplanarplanar Euclidean space planar projection platformplotterplotterplottingplug-inpocketpoint entitiespointerpoint-modepointspolar coordinates polishingpolygonpolylinepolymorphism precautionsprecisionpre-designed predeterminepreferences pregeographic space Primary and Foreign keys primary keyprocess-orientedprofileprogramming tools projectionprojectionproprietaryprototypeproximalProximitypseudo nodepseudo-bufferpuckpuckpuckPythagorasquadquadrantquadtreequadtree tessellationqualifyqualitativequantitativequantitativequantizequasi-metricradar imageradii bufferrangelandrank order aggregation method ranking methodrasterRaster data modelraster scannerRaster Spatial Data Modelrating methodrational database structureready-madeready-to-runreal-timerecordrecreationrectangular coordinates rectificationredundantreference gridreflexivereflexive nearest neighbors (RNN) regimeregisterregular patternrelationrelationalrelational algebra operators relational databaseRelational joinsrelational model relevancereliefreliefremarkremote sensingremote sensingremote sensingremotely-sensed repositoryreproducible resemblanceresembleresemplingreshaperesideresizeresolutionresolutionrespondentretrievalretrievalretrievalretrieveridgerightrobustrootRoot Mean Square (RMS) rotateroundaboutroundingrowrow and column number run-length codingrun-length encoded saddle pointsalientsamplesanitarysatellite imagesscalablescalescanscannerscannerscannerscarcescarcityscenarioschemascriptscrubsecurityselectselectionself-descriptiveself-documentedsemanticsemanticsemi-automatedsemi-major axessemi-metricsemi-minor axessemivariancesemi-variogram modelsemi-varogramsensorsequencesetshiftsillsimultaneous equations simultaneouslysinusoidalskeletonslide-show-stylesliverslope angleslope aspectslope convexitysnapsnapsocio-demographic socioeconomicspagettiSpatial Autocorrelation Function spatial correlationspatial dataspatial data model for GIS spatial databaseSpatial Decision Support Systems spatial dependencespatial entityspatial modelspatial relationshipspatial relationshipsspatial statisticsspatial-temporalspecificspectralspherical spacespheroidsplined textsplitstakeholdersstand alonestandard errorstandard operationsstate-of-the-artstaticSTEREOGRAPHIC projection STEREOGRAPHIC projection stereoplotterstorage spacestovepipestratifiedstream-modestrideStructured Query Language(SQL) strung outsubdivisionsubroutinesubtractionsuitesupercedesuperimposesurrogatesurveysurveysurveying field data susceptiblesymbolsymbolsymmetrytaggingtailoredtake into account of … tangencytapetastefullyTelnettentativeterminologyterraceterritorytessellatedtextureThe Equidistant Conic projection (EQUIDIS The Lambert Conic 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（源自）航空的,空中的艾伯特等面积圆锥投影匹配，调准，校直字母的字母数字的字母数字混合编制的替换方案替代的海拔，高度改善,改良,改进模拟地图，这里指纸质地图辅助的和注解不规则的，异常的顶点方法适合于…弧段弧捕捉容限来自一个地区的、 面状的面状覆盖范围(美国国防部)高级研究计划署排列，布置数组，阵列人工智能人工神经网络非空间的方面， 方向， 方位， 相位，面貌采集，获取关联属性属性属性数据自动扫描方位角，方位，地平经度方位角的条状图偏差二进制编码分块编码布尔代数下左下角给…划界断裂线缓冲区分析内置的伪装主要的，重要的，基本的笛卡儿坐标系制图、制图学流域，集水区像元，单元人口普查质心质心到质心的公共网关接口链式编码链电荷耦合器件子节点地区分布图类库类群编码内聚地线圈在同一直线上的列压缩、压紧罗盘， 圆规， 范围 v.包围方位角完全空间随机性组成部分复合的、混合的复合码凹度，凹陷同心的概念模型概念上地管道,导管,沟渠,泉水,喷泉保形（保角）的等角投影圆锥投影连通性保守的，守旧的社团,协会,联盟包含关系相邻性连续的轮廓,等高线,等值线等高线层控制点习俗，惯例，公约，协定转换核心相关图符合，对应走廊, 通路费用花费密度域，路径权值成本效益分析有成本效益的，划算的结合协方差面层，图层覆盖，覆盖范围标准，要求标准，判据，条件标准，判据，条件十字丝以交叉线作出阴影截面麻烦的用户定制剪切圆柱投影悬挂悬挂长度悬挂的节点点划线数据文件的扩展名数据库管理系统数据合并数据变换数据定义语言数据字典与数据的无关数据的完整性数据项数据维护数据操作数据操作和查询语言数据挖掘数据模型数据表示法数据表数据类型数据库日期数据库管理员泥石流调试十年，十，十年期分贝决策分析决策，判定分解专用的推论的，演绎的狄拉尼准则狄拉尼三角形删除描绘划分人口统计学的说明分母，命名者观测密度引出的，派生的可察觉的发明，想出对角线的，斜的要求数字高程模型数字地形模型数字化数字化数字化仪数字化误差数字化板，数字化桌差异，矛盾不连续的，离散的不连续的，离散的不一致性分散，离差中断，分裂，瓦解，破坏切开的，分割的发散，发布分解距离衰减函数分布式计算分割域点状图草稿，起草拖拽滚筒式扫描仪伪节点动态建模容易使用的生态学导出消除椭球椭圆率伸长包装，封装围绕编码实体关系建模实体表进入,登记想像,设想,正视,面对希腊文的第五个字母ε等积投影等距投影不稳定的误差检查和修正误差图误差离散，误差方差本质，本体，精华以及其他人，等人欧几里得的，欧几里得几何学的欧几里得二维空间期望发生频率明显的指数的延伸内外边界外部表格(多面体的)面工具设备管理样子，方式文件分配表有过失的，不完善的（地理）要素，特征要素反馈诚实，逼真度，重现精度字段现场调查户外运动发烧友场模型外形， 数字，文件结构填充精细度以固定比例放大以固定比例缩小平板式扫描仪弹性，适应性，机动性，挠性最前沿逐帧无…的起始节点从底层完成，实现函数调用模糊的模糊集合论构台,桶架, 跨轨信号架通用的地理编码地理计算大地测量地理实体地理（数据处理）过程地理参考地理空间地理信息，空间信息地理要素几何的，几何学的几何图元地理（数据）处理过程地理坐标参考地理关系的地球科学地理空间的地学空间分析地质统计学的假设心射切面投影颗粒容差地图网格灰度栅格,格网手绘的手持的障碍，难点处置、处理手写的头记录重的,强健的异质性异构的启发式的层次层次的山坡（体）阴影图均匀的、均质的主机家庭色调腐植的困难，阻碍水文地理学超链接的即，换言之，也就是理想点法相同的可识别的、标识识别阐明图像，影像全电阻，阻抗阻抗实现，履行履行,实现牵连,暗示隐含的超过…关于根据…在中间的嵌入的，内藏的不一致性，矛盾性结合，组成公司(或社团)内在的,本土的信息集成基础设施固有的继承，遗传， 遗产内陆的实例，例子实例，个例化整数综合，结合相互作用交互式的交互式的协议组互操作性内插插值询问相交交集、逻辑的乘交区间估值法直觉的直觉的不变量存储，存量反向的，倒转的，倒置的互相对立的不能撤回的，不能取消的相邻完全包含于包含于相等的，相同的线族等值线图迭代的接合,汇接点主帧克里金内插法克里金法标签，标记间隙，迟滞量土地利用类别纬度 （B）纬度坐标熔岩，火山岩图层图层出租，租用最佳路径分析左图例图例图例长度量测在于小型的同样地限制，限度，局限线模型线段谱系，来源容貌，线性构造线性的，长度的，直线的线跟踪的岩性单元局域和广域网对数的逻辑的逻辑的经度 （L）经度坐标宏语言类宏语言宏主流管理人的， 管理的手工数字化多对一的关系地图比例尺排列,集合掩膜matrix 的复数矩阵实测发生频率量测中间的合并墨卡托墨卡托投影法合并合并，融合子午线元数据元数据，也可写为 metadata元数据方法学，方法论度量空间最佳路径镜像错误表示混合像素建模模块化的单色的,单频整体的垄断， 专利权， 专卖形态学镶嵌, 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a rX iv:physics /061234v2[physics.ge n-ph]26Nov26Abstract Heisenberg’s uncertainty relation means that one observer cannot know an exact position and velocity for another (finite mass)observer.By con-trast,the Poincare transformation of classical special relativity assumes that one observer knows the other’s position and velocity exactly.The present paper describes a simple-minded way to consider the issue using a semiclassical discussion of spacetime diagrams,and draws out some possi-ble implications.Uncertainties arise in transformations.A consideration is raised regarding the use of light-cone coordinates.1Introduction In an early paper on quantum mechanics [1],Heisenberg noted that his uncer-tainty principle applies to observers as well as to observed systems.Heisenberg noted that the quantum uncertainties in an observer O1’s position and mo-mentum,in another observer O2’s coordinates,are subject to his uncertainty principle:∆x i O 1∆p i O 1≥¯h2Preliminary discussionThis paper uses the basic Heisenberg uncertainty relation.The relation for position and momentum is a fuzzy relation:∆x i O1∆p i O1≥¯h.(3)2We now drop the label i,and consider a spacetime with1+1dimensions. Using the relativistic relation v=p/2(p2+m2O1)3(5)2m O1Hypothetically,if O2could measure O1’s velocity with zero uncertainty,∆x O1would be infinite.One might think that this could correspond to some extent with the classical relativistic picture of an infinitely extended observer with clocks and rods extending throughout spacetime,if spacetime has infinite extent.But by the positions of O1and O2,we really mean the positions of the origins of those reference frames;O2has infinite uncertainty about where O1’s origin is.Alternatively,if one can consider a Feynman path integral approach[2] to the motion of O1,even if two path endpoints are preciselyfixed,there are many possible paths consistent with those endpoints.A classical limit would pick out the path that extremises the action,but in quantum mechanics one has a superposition of all possible paths,each of which contributes to an interference pattern.3Classical spacetime diagramLet O1and O2be the origins of classical inertial observers,in a spacetime with one spatial dimension and one time dimension.We will call O2’s coordinates x2and t2.Suppose that the origin of O1passes through x2=b when t2=a, and that O1’s velocity relative to O2is v in the positive x2direction.Relativity texts show O1’s and O2’s t and x axes in a simple spacetime diagram.For technical reasons graphs are not included in this paper,but the description of the spacetime diagram is simple.O2’s axes t2and x2are drawn2perpendicular to each other.O1’s axes t1and x1are lines drawn in thefirst quadrant of the diagram.The classical Poincare transformation relating the co-ordinates of O2to those of O1is x2=γ(x1+vt1)+b,t2=γ(t1+vx1)+a,√whereγ=1/The two most interesting of these intersection points are at(−∆x2,+∆t2) and(+∆x2,−∆t2).Let us call these points P and Q.At each point P and Q,a pair of axes for O1could be drawn for the velocity v+∆v,and a pair of axes can be drawn for v−∆v.But we will focus on the axes for velocity v.It will be assumed that the speed of light is constant and equal for O2and O1.The time axis for O1that goes through point Q intersects the spatial axis for O1that goes through point P.These two lines roughly define a fuzzy region near O2’s(0,0)point,a spacetime region that could be considered as a kind of no man’s land,where O2cannot say much if anything about what O1perceives as time and what O1perceives as space.The time axis for O1that goes through point Q-let us call this line L1-is given by the equationx2=v(t2+∆t2)+∆x2(8) The spatial axis for O1that goes through point P-let us call this line L2-is given by the equationc2x2=1−v2+∆t2(1+v2c21−v2length of the order of the Planck length,which has arisen from a generalised uncertainty principle.Also,as v→c the”average”spacetime axes for O1, going through O2’s origin of coordinates,grow closer to the null line x2=ct2. The extent to which these”average axes”fall within the region defined by P,Q and(t c,x c)increases as v increases.It seems that in a graded way,O1’s concept of time and space becomes to some extent less accessible to O2,the faster O1is moving relative to O2.By contrast,classical relativity makes a sharp distinction between inertial observers with v<c and entities moving at the speed of light.Classically,any inertial observer with v<c can compute another such observers coordinates exactly, but this cannot be done for entities moving at the speed of light.What might an actual value for x c be,for observers with high relative speed? If∆t2is zero,x c≈∆x2/(1−v),which could generate large x c values.One might think that this would generate a conflict with experiments;[8]has noted that these show Lorentz invariance down to10−18m.But there need be no conflict.Nothing in the present paper necessarily influences Lorentz or Poincare invariance;the issue here is rather how to describe transformations between reference systems to begin with.However,it might be interesting to consider whether quantum uncertainties involved in observers’reference systems might mask Lorentz invariance violations arising from other theories.It might also be interesting to consider the implications(if any)of these results for the use of light cone coordinates in physics.If the use of each such coordinate is tantamount to the formal limit of a Poincare transformation multi-plied by an overall scale factor,it may be necessary to consider quantum uncer-tainties in the coordinate transformation;non-commutative geometry probably does this automatically.This comment is subject to the proviso that the above analysis is a simple semi-classical one,and may not capture all of the relevant effects.It might however be suggestive.6The zero v limitIn the nonrelativistic v→0limit,we can estimate x c and t c by substituting (v+∆v)for v in the equations above.Wefind:2∆x2∆vt c(v→0)≈∆t2+(12)m O1c2So t c is nonzero as v→0,even if∆t2and∆x2are zero;note that they will be at least the relevant Planck scales.For x c we have:¯h∆vx c(v→0)≈∆x2+2∆t2∆v+Using the Heisenberg uncertainty principle,we have:x c(v→0)≥∆x2+¯hm O1c2(14)We get a strange term with v in the denominator.I propose to replace v in the denominator by∆v.This can be considered to reflect thatfluctuations of order ∆v will dominate over v=0.Then we have:x c(v→0)≥∆x2+¯hm O1c2(15)orx c(v→0)≥3∆x2+1(m O1c)2(16)The above relations,if correct,imply a nonzero minimum value:x c(v→0)≥√m O1c(17)7Speculative discussionThis paper does not attempt to construct a consistent theory of quantum me-chanics and special relativity.Perhaps an observer could be described by a quantum mechanical superposition of classical inertial reference frames;each frame and amplitude in a superposition might be labelled by a pair(a,p),in 1+1dimensions.The Poincare transformation might be adaptable accordingly.A superposition of transformations has been discussed in[4].The relevant con-cepts may depend on how observers are defined.The sections above considered a spacetime with one spatial dimension.It may be interesting to speculate about behaviour in two or more spatial dimen-sions.If the simple working above is correct,O2cannot completely distinguish O1’s t and x coordinates.In2spatial dimensions,it seems likely that O2can-not completely distinguish O1’s t and y coordinates.Perhaps O2cannot then completely distinguish O1’s x and y coordinates,and perhaps a species of non-commutative geometry can reflect this.It is also to be noted that rotations are part of the Poincare group.Although non-commutative geometry is generally considered as applicable at high energies,at least one author has previously noted that it may be useful in lower-energy contexts.The sections above assumed that the observer O2could be taken to have a classical inertial reference frame.However,O2will in practice be affected by quantumfluctuations.This would most likely smear out the overall picture even more.O2will also be subject to a backreaction from its measurement of O1. These factors have not been considered in the present paper.The results in this paper may indicate a need to take care when making coordinate transformations.Rather than simply taking the classical form,some coordinate transformations may inherently add quantum mechanical uncertain-ties.In any particular case,the physical content of the transformation may6be the determining factor,e.g.whether one is trying to describe what another observer perceives,or not.On the other hand,it may be that for any given observer,the calculations required of that observer to transform its own coordi-nates(within its own reference frame/system)might be considered to produce entropy,which in some cases may correspond to the uncertainties discussed above in this paper.These comments should probably be taken to be limited by what is presently known about the possible interpretations of quantum me-chanics.Relativity could perhaps be modified by considering more realistic observers and configurations of measuring devices than that assumed for classical inertial reference frames.Such observers might have limited access to information;they might not have passive instantaneous access to all information about events at a distance and their coordinates,as assumed by the classical picture of an observer with access to readings of an infinite number of clocks and rods.Perhaps a Lagrangian(density)could combine contributions from the observed system,first-tier observing devices,and second-tier observers such as aspects of human observation,in some parametrisation.A second-tier observer might extrapolate or derive coordinates for spacetime as a whole using information fromfirst-tier devices in a way that uses an explicit model for the second-tier observer’s notion of spacetime.Regarding Lagrangians,it is noteworthy that work is ongoing in relativistic two-body and many-body classical mechanics,e.g.[7]. References[1]W.Heisenberg,Zeitschrift fur Physik,43,172-98(1927).English translationreproduced in Quantum Theory and Measurement,ed.J.Wheeler and W.H.Zurek,Princeton University Press,1983.[2]Quantum Mechanics and Path Integrals,R.Feynman and A.R.Hibbs,Mc-Graw-Hill,1965.[3]Y.S.Kim and M.E.Noz,”Can you do quantum mechanics without Einstein?”,quant-ph/0609127.[4]Wen-ge Wang,”Entanglement and Disentanglement,Probabilistic Interpre-tation of Statevectors,and Transformation between Intrinsic Frames of Ref-erence”,quant-ph/0609093.[5]S.Bartlett,T.Rudolph and R.Spekkens,”Reference frames,superselectionrules,and quantum information”,quant-ph/0610030.[6]E.Rosinger,”Covariance and Frames of Reference”,quant-ph/0511112.[7]D.Alba,H.Crater and L.Lusanna,”Hamiltonian Relativistic Two-BodyProblem:Center of Mass and Orbit Reconstruction”,hep-th/0610200. [8]M.Maziashvili,”Quantumfluctuations of space-time”,hep-ph/0605146.7。