Quantum Field Theory I

论文《电磁场的动力学理论》[编辑]

主条目：电磁场的动力学理论

于1864年，麦克斯韦发表了论文《电磁场的动力学理论》^[11]。这篇论文的第三节的标题为电磁场一般方程，在这节里，麦克斯韦写出了二十个未知量的二十个方程；其中，有十八个方程可以用六个矢量方程集中表示（对应于每一个直角坐标轴，有一个方程），另外两个是标量方程。所以，以现代矢量标记，麦克斯韦方程组可以表示为八个方程，分别为
（A）总电流定律、（B）磁场方程、（C）安培环流定律、（D）洛伦兹力方程、（E）电弹性方程、（F）欧姆定律、（G）高斯定律、（H）连续方程。
在这篇论文里，麦克斯韦推导出光波是一种电磁现象。在他的导引里，他并没有用法拉第感应定律，而是用方程（D）来解释电磁感应作用。现代教科书大多是用法拉第感应定律来解释电磁感应作用。事实上，他的八个方程里，并没有包括法拉第感应方程在内。

教科书《电磁通论》[编辑]

英语维基文库中与本条目相关的原始文献：电磁通论

发行于1873年，麦克斯韦亲自著作的《电磁通论》是一本电磁学教科书。在这本书内，方程被收集成两组。第一组是
、；
其中，是电势，是磁矢势。

第二组是
、。
从第一组的两个方程，分别取旋度和散度，则可得到法拉第感应定律和高斯磁定律的方程：
、。
宏观麦克斯韦方程组[编辑]

束缚电荷和束缚电流[编辑]

主条目：束缚电荷和束缚电流



假设，施加外电场于介电质，响应这动作，介电质的分子会形成一个微观的电偶极子，显示出伴随的电偶极矩。分子的原子核会朝着电场的方向稍微迁移位置，而电子则会朝着相反方向稍微迁移位置。这形成了介电质的电极化。如右图的理想状况所示，虽然，所有涉及的电荷都仍旧束缚于其原本的分子，由于这些微小迁移所造成的电荷分布，变得好像是在介电质的一边形成了一薄层正表面电荷，在另一边又形成了一薄层负表面电荷。电极化强度定义为介电质内部的的电偶极矩密度，也就是单位体积的电偶极矩。在介电质内部，假设电极化强度是均匀的，则宏观的面束缚电荷只会出现于介电质表面，进入或离开介电质之处；否则，假设是不均匀的，则介电质内部也会出现束缚电荷^[18]。

与静电学有些类似，在静磁学里，假设施加外磁场于物质，响应这动作，物质会被磁化，组成的原子会显示出磁矩。在本质上，这磁矩与原子的各个亚原子粒子的角动量有关，其中，响应最显著的是电子。这角动量的连结，不禁令人联想到一副图画，在图画中，磁化物质变成了一群微观的束缚电流回路。虽然每一个电荷只是移动于其原子的微观回路，一群微观的束缚电流回路聚集在一起会形成宏观的面束缚电流循环流动于物质的表面。这些束缚电流可以用磁化强度来描述。磁化强度定义为磁偶极矩在一个磁化物质内的密度，也就是单位体积的磁偶极矩^[19]。

对于许多案例，原子行为和电子行为的微观细节，可以使用较简易的方法来处理。这样，很多精密尺度的细节，对于研究物质的宏观行为并不重要，因此可以被忽略。这解释了为什么要区分出束缚与自由的物理行为。

这些非常复杂与粗糙的束缚电荷与束缚电流的物理行为，在宏观尺度，可以分别以电极化强度与磁化强度来表达。电极化强度与磁化强度分别将这些束缚电荷与束缚电流以恰当的尺度做空间平均，这样，可以除去单独整体原子形成的凹凸粗糙结构，但又能够显示出强度随着位置而变化的物理性质。由于所有涉及的矢量场都已做过恰当体积的空间平均，宏观麦克斯韦方程组忽略了微观尺度的许多细节，对于了解物质的宏观尺度性质，这些细节可能不具什么重要性。

本构关系[编辑]

为了要应用宏观麦克斯韦方程组，必须分别找到场与场之间，和场与场之间的关系。这些称为本构关系（constitutive relations）的物理性质，设定了束缚电荷和束缚电流对于外场的响应。它们实际地对应于，一个物质响应外场作用而产生的电极化或磁化。

本构关系式的基础建立于场与场的定义式：
、；
其中，是电极化强度，是磁化强度。

在解释怎样计算电极化强度与磁化强度之前，最好先检视一些特别案例。

自由空间案例[编辑]

假设，在自由空间（即理想真空）里，就不用考虑介电质和磁化物质，本构关系式变得很简单：
、。
将这些本构关系式代入宏观麦克斯韦方程组，则得到的方程组很像微观麦克斯韦方程组，当然，在得到的高斯定律方程和麦克斯韦-安培方程内，总电荷密度和总电流密度分别被自由电荷密度和自由电流密度替代。这符合期待的结果，因为，在自由空间里，没有束缚电荷、束缚电流和电极化电流。

线性物质案例[编辑]

对于线性、各向同性物质，本构关系式也很直接：
、；
其中，是物质的电容率，是物质的磁导率。

将这些本构关系式代入宏观麦克斯韦方程组，可以得到方程组

对于线性、各向同性物质的表述[th]名称[/th][th]微分形式[/th][th]积分形式[/th]

高斯定律
高斯磁定律
法拉第感应定律
麦克斯韦-安培定律

除非这物质是均匀物质，不能从微分式或积分式内提出电容率和磁导率。通量的方程为
。
这方程组很像微观麦克斯韦方程组，当然，在得到的高斯定律方程和麦克斯韦-安培方程内，自由空间的电容率和磁导率分别被物质的电容率和磁导率替代；还有，总电荷密度和总电流密度分别被自由电荷密度和自由电流密度替代。这符合期待的结果，因为，在均匀物质内部，没有束缚电荷、束缚电流和电极化电流，虽然由于不连续性，可能在表面会有面束缚电荷、面束缚电流或面电极化电流。

一般案例[编辑]

对于实际物质，本构关系并不是简单的线性关系，而是只能近似为简单的线性关系。从场与场的定义式开始，要找到本构关系式，必需先知道电极化强度和磁化强度是怎样从电场和磁场产生的。这可能是由实验得到（建立于直接测量），或由推论得到（建立于统计力学、传输力学（transport phenomena）或其它凝聚态物理学的理论）。所涉及的细节可能是宏观或微观的。这都要视问题的层级而定。

虽然如此，本构关系式通常仍旧可以写为
、。
不同的是，和不再是简单常数，而是函数。例如，

• 色散或吸收：和是频率的函数。因果论不允许物质具有非色散性，例如，克拉莫-克若尼关系式。场与场之间的相位可能不同相，这导致和为复值，也导致电磁波被物质吸收^[20]。
  
  
• 非线性：和都是电场与磁场的函数。例如，克尔效应（Kerr effect）^[21]和波克斯效应（Pockels effect）。
  
  
• 各向异性：例如，双折射或二向色性（dichroism）。和都是二阶张量^[22]：
  
  

、。

• 双耦合各向同性（Bi-isotropy）或双耦合各向异性（Bi-anisotropy）：在双耦合各向同性物质里，场与场分别各向同性地耦合于场与场^[22]：
  
  

、；其中，与是耦合常数，每一种介质的内禀常数。在双耦合各向异性物质里，场与场分别各向异性地耦合于场与场，系数、、、都是张量。

• 在不同位置和时间，场与场分别跟场、场有关：这可能是因为“空间不匀性”。例如，一个磁铁的域结构、异质结构或液晶，或最常出现的状况是多种材料占有不同空间区域。这也可能是因为随时间而改变的物质或磁滞现象。对于这种状况，场与场计算为^[23][24]
  
  

、；其中，是电极化率，是磁化率。
实际而言，在某些特别状况，一些物质性质给出的影响微乎其微，这允许物理学者的忽略。例如，在低场强度状况，光学非线性性质可以被忽略；当频率局限于狭窄带宽内时，色散不重要；对于能够穿透物质的波长，物质吸收可已被忽略；对于微波或更长波长的电磁波，有限电导率的金属时常近似为具有无穷大电导率的完美金属（perfect metal），形成电磁场穿透的趋肤深度为零的硬障碍。

随着材料科学的进步，材料专家可以设计出具有特定的电容率或磁导率的新材料，像光子晶体。

本构关系的演算[编辑]

通常而言，感受到局域场施加的洛伦兹力，介质的分子会有所响应，从相关的理论计算，可以得到这介质的本构关系式。除了洛伦兹力以外，可能还需要给出其它作用力的理论模型，像涉及晶体内部晶格振动的键作用力，将这些作用力纳入考量，一并计算。

在介质内部任意分子的位置，其邻近分子会被电极化和磁化，从而造成其局域场会与外场或宏观场不同。更详尽细节，请参阅克劳修斯-莫索提方程。真实介质不是连续性物质，其局域场在原子尺度的变化相当剧烈，必需经过空间平均，才能形成连续近似。

这连续近似问题时常需要某种量子力学分析，像应用于凝聚态物理学的量子场论。请参阅密度泛函理论和格林-库波关系式（Green–Kubo relations）等等案例。物理学者研究出许多近似传输方程，例如，玻尔兹曼传输方程（Boltzmann transport equation）、佛克耳-普朗克方程（Fokker–Planck equation）和纳维-斯托克斯方程。这些方程已经广泛地应用于流体动力学、磁流体力学、超导现象、等离子模型（plasma modeling）等等学术领域。一整套处理这些艰难问题的物理工具已被成功地发展出来。另外，从处理像砾岩（conglomerate）或叠层材料（laminate）一类物质的传统方法演变出来的“均质化方法”，是建立于以“均质有效介质”来近似“非均质介质”的方法^[25]。当激发波长超大于非均质性的尺度时，这方法正确无误^[26][27][28]。

理论得到的答案必须符合实验测量的数据。许多真实物质的连续近似性质，是靠着实验测量而得到的^[29]。例如，应用椭圆偏振技术得到的薄膜的介电性质。

自由空间[编辑]

主条目：电磁波方程

在自由空间里，不需要考虑介电质或磁化物质的问题。假设源电流和源电荷为零，则麦克斯韦方程组变为^{[注 1]}
、、、。
对于这方程组，平面行进正弦波是一组解。这解答波的电场和磁场相互垂直，并且分别垂直于平面波行进的方向。电场与磁场同相位地以光速传播^{[注 2]}：
。
仔细地观察麦克斯韦方程组，就可以发现这方程组很明确地解释了电磁波怎样传播于空间。根据法拉第感应定律，时变磁场会生成电场；根据麦克斯韦-安培定律，时变电场又生成了磁场。这不停的循环使得电磁波能够以光速传播于空间。

1856年，威廉·韦伯和鲁道夫·科尔劳施（Rudolf Kohlrausch），从他们的莱顿瓶实验，计算出的数值，发觉这数值非常接近于，先前从天文学得到的，光波传播于行星际空间的速度^[30]。从这实验结果，麦克斯韦正确地断定光波就是一种电磁辐射。

磁单极子[编辑]

主条目：磁单极子

麦克斯韦方程组将电场、磁场与电荷的运动相连结。在方程组中，他有给电荷安排位置，但并没有给磁荷（磁单极子）安排位置。在粒子物理学里，并没有类比于电子的磁粒子。虽然如此，包括磁荷与磁流在内的麦克斯韦方程组是一门很热门的理论研究题目^[31]。根据最新实验结果，科学家发现，有一种称为自旋冰（spin ice）的晶态物质，其宏观物理行为很像磁单极子的物理行为^[32]。请注意，这发现并没有违背磁荷从未被观察到和可能不存在的事实。除了磁荷这例外，麦克斯韦方程组拥有对称的形式。实际而言，当所有电荷等于零时，可以写出对称的方程组。请参阅前面的自由空间段落。

假设允许磁荷存在的可能，则也可以写出完全对称的方程组。麦克斯韦方程组内会增添两个新的变量，磁荷和磁流。采用厘米-克-秒制，延伸的麦克斯韦方程组表示为

麦克斯韦方程组（厘米-克-秒制）[th]名称[/th][th]磁单极子不存在[/th][th]磁单极子存在[/th]

高斯定律
高斯磁定律
法拉第感应定律
麦克斯韦-安培定律
请注意，删除因子，即可得到无单位的形式。

假若，磁荷不存在，或者假若它们不存在于某一个区域，则新增添的两个变量和都等于零，对称的方程组约化为一般形式的麦克斯韦方程组。

边界条件[编辑]

主条目：边值问题和初值问题

就像其它微分方程组，假若没有合适的边界条件^[33]与初始条件^[34]，则无法给出麦克斯韦方程组的唯一解答。

特别而言，在一个不含有任何自由电荷和自由电流的区域内的电磁场，必定是来自于其它区域。当解析这状况时，通过适当的边界条件或初始条件，可以将电磁场引进这区域。举一个电磁波散射的例子，一个来自于散射区域之外的电磁波，遭遇到散射区域内的一个靶子，被这靶子散射出去。在这散射过程里，由于电磁波与靶子之间相互作用，散射的电磁波含有很多与这靶子性质相关的资料。经过仔细地分析，将这些资料萃取出来，就可以更详细地了解这靶子的性质^[35]。

对于某些案例，譬如波导或空腔共振器（resonator），因为像金属墙壁一类的隔离设施，解答区域大部份孤立于外部世界。在金属墙壁位置的边界条件决定了解答区域的电磁场。在解答区域以外的外部世界，只能靠着边界条件来影响内部的状况^[36]。对于另外一些案例，像光导纤维或薄膜，解答区域时常会被分割为几个亚区域，每个亚区域都有其简单独自的性质。通过亚区域与亚区域之间界面的边界条件，可以将每一个亚区域的解答连结起来^[37]。

应用边界条件，有时也可以简化问题，使得问题更容易被了解。例如，均匀物体的电极化可以被更换为在这物体外表的一层面电荷分布^[18]，或者，均匀物体的磁化被更换为在这物体外表的一层面电流分布^[38]。详尽细节，请参阅束缚电荷和束缚电流段落。

以下列出一些重要的边界条件：斯徒姆-刘维边界条件（Sturm-Liouville boundary condition）、狄利克雷边界条件、诺伊曼边界条件、混合边界条件（mixed boundary condition）、柯西边界条件（Cauchy boundary condition）、索末菲辐射条件（Sommerfeld radiation condition）。在解析问题时，必须选择适当的边界条件，才可得到正确的答案^[39]。

高斯单位制[编辑]

厘米-克-秒单位制的三个基本单位是长度单位厘米、质量单位克、时间单位秒。在经典力学里，厘米-克-秒单位制的单位是一致的；但在电磁学里，则出现了几种变型。高斯单位制是其中一种变形。在高斯单位制里，麦克斯韦方程组的形式为^[3]
、、、。
在自由空间里，假设不存在任何电荷和电流，则方程组简化为
、、、。
采用这单位制，电位移、电场和电极化强度的关系式为
，
B场、H场和磁化强度的关系式为
。
对于线性物质，电极化率 和磁化率分别定义为
、。
电容率和磁导率分别为
、。
所以，电位移和B场分别为
、。
在自由空间里，方程组变得相当简单：
、、。
根据洛伦兹力定律，一个以速度移动于电场和磁场的带电粒子，所感受到的洛伦兹力为
。
这形式与先前国际单位制的形式稍微有点不同。特别注意，电位移、电场和电极化强度、B场、H场和磁化强度的单位相同。

关于怎样正确地从一个单位制变换到另外一个单位制，请参阅高斯单位制。

进阶表述[编辑]

主条目：电磁场的数学表述

麦克斯韦方程组的协变形式[编辑]

主条目：经典电磁学与狭义相对论和经典电磁理论的协变形式

麦克斯韦方程组与狭义相对论之间的关系密切。不只是因为麦克斯韦方程组对于狭义相对论的初始发展，做了相当大的贡献，也因为狭义相对论激荡出一种更简洁的表述，能以协变张量来表达麦克斯韦方程组。

自由空间的麦克斯韦方程组的形式，对于任意惯性坐标系，都是一样的。在狭义相对论里，为了要更明确地表达出这论点，必须以四维矢量和张量写出协变形式的麦克斯韦方程组。这表述的一个构成要素为电磁张量。这张量是一个结合了电场和磁场在一起的二阶反对称协变张量^[40]：
 。
使用闵可夫斯基度规，
 ，
将下标拉高为上标，可以得到反变张量：
 。
给予一个阶反对称协变张量，则其阶对偶张量（dual tensor）是一个反对称反变张量：
；
其中，是维列维-奇维塔符号。

根据这定义，的二阶对偶张量是 ^[41]
 。
换一种方法，将的项目做以下替换：、，也可以得到二阶对偶张量。

另外一个要素是四维电流密度：
；
其中，是电荷密度，是电流密度。

借着这些要素，采用爱因斯坦求和约定，麦克斯韦方程组可以写为^[41]
、 ;
其中，是四维梯度（Four-gradient）。

这两个张量方程等价于麦克斯韦方程组。第一个张量方程表达两个非齐次麦克斯韦方程，高斯定律和麦克斯韦-安培定律。第二个张量方程表达两个齐次麦克斯韦方程，高斯磁定律和法拉第感应定律。

势场表述[编辑]

在高等经典力学里，采用势场表述，以电势与磁矢势来表达麦克斯韦方程组，有时候可能对解析问题很有助益。在量子力学里，这是必需手段。电势与磁矢势分别如此定义：
、。
从这两个定义式，两个齐次麦克斯韦方程自动成立，另外两个非齐次方程变为
、。
这两个势场方程组合起来，具有与原本麦克斯韦方程组同样的功能和完备性。由于电场和磁场各有三个分量，原本的麦克斯韦方程组需要解析六个分量。势场表述只需要解析四个分量，因为电势只有一个分量，磁矢势有三个分量。可是，势场表述涉及了二次微分，方程也比较冗长。

许多不同的与数值组可以得到同样的电场与磁场。因此，这些数值组相互物理等价，可以自由选择。这性质称为规范自由。恰当的选择可以简化方程的形式，或者，可以专门适用于某特别状况。

协变形式[编辑]

主条目：四维势

采用洛伦茨规范，势场的两个矢量方程可以约化为单独一个具有洛伦兹不变性的四维矢量方程。四维电流密度乃是由电流密度和电荷密度共同组成，以方程定义为
。
四维势乃是由磁矢势和电势共同组成，以方程定义为
。
十九世纪初，阿诺·索末菲提出了四维矢量方程，这是波恩哈德·黎曼先前想出的一个方程的推广，因此，知名为“黎曼-索莫菲方程”^[42]，或麦克斯韦方程的势场表述的协变形式^[43]：
；
其中，是达朗白算符，又称为“四维拉普拉斯算符”。

弯曲时空[编辑]

主条目：弯曲时空中的麦克斯韦方程组

物质和能量会造成时空弯曲。这是广义相对论的主题。时空弯曲会影响电动力学的物理。一个电磁场所拥有的能量和动量也会造成时空弯曲。将平直时空的方程组中的偏导数，替换为协变导数，就可以得到弯曲时空中的麦克斯韦方程组。采用高斯单位制，麦克斯韦方程组表达为
、；
其中，是表征时空弯曲的克里斯托费尔符号。

参阅[编辑]

阿布拉罕-洛伦兹力 制动辐射 时域有限差分法 菲涅耳方程 移动中的磁铁与导体问题

杰斐缅柯方程 电磁场的数学表述 非齐次的电磁波方程 双缝实验中光子的动力学 光子偏振（Photon polarization）

注释[编辑]

1. ^ 术语“真空”时常用于这案例。但是，请注意，在这里，自由空间指的是一种理想的，实际不可能体现的参考状态，迥然不同于任何可以实际体现的真空，像实验室内制造的超高真空（Ultra high vacuum）或外太空，或任何从理论方面体现的真空，像量子真空（quantum vacuum）或量子色动力真空（QCD vacuum）。
   
   
2. ^ 国际标准化组织建议使用为自由空间光速的国际标准标记（ISO 31-5）。参阅美国国家标准与科技学院（NIST）的特刊国际单位制
   
   

The Method of Stationary Phase
The method of stationary phase has its roots in wave optics. Nowadays
it is a magic tool in the hands of Edward Witten for studying quantumfield
models in terms of the Feynman functional integral and for producing
deep topological invariants. Here, the Feynman functional integral plays
the role of a generating functional.
Folklore


Stationary phase approximation

From Wikipedia, the free encyclopedia


[ltr]In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to oscillatory integrals

taken over n-dimensional space ℝⁿ where i is the imaginary unit. Here fand g are real-valued smooth functions. The role of g is to ensure convergence; that is, g is a test function. The large real parameter k is considered in the limit as k → ∞.
This method originates from the 19th century, and is due to George Gabriel Stokes and Lord Kelvin.^[1]

[/ltr]

1 Basics

2 An example

3 Reduction steps

4 One-dimensional case

5 See also

6 References

7 Notes

8 External links

[size][ltr]

Basics[edit]
The main idea of stationary phase methods relies on the cancellation of sinusoids with rapidly varying phase. If many sinusoids have the same phase and they are added together, they will add constructively. If, however, these same sinusoids have phases which change rapidly as the frequency changes, they will add incoherently, varying between constructive and destructive addition at different times.
An example[edit]
Consider a function
.
The phase term in this function, ϕ = k(ω) x − ω t, is stationary when

or equivalently,
.
Solutions to this equation yield dominant frequencies ω₀ for some x and t. If we expand ϕ as a Taylor series about ω₀ and neglect terms of order higher than (ω − ω₀)²,

where k″ denotes the second derivative of k. When x is relatively large, even a small difference (ω − ω₀) will generate rapid oscillations within the integral, leading to cancellation. Therefore we can extend the limits of integration beyond the limit for a Taylor expansion. If we double the real contribution from the positive frequencies of the transform to account for the negative frequencies,
.
This integrates to
.
Reduction steps[edit]
The first major general statement of the principle involved is that the asymptotic behaviour of I(k) depends only on the critical points of f. If by choice of g the integral is localised to a region of space where f has no critical point, the resulting integral tends to 0 as the frequency of oscillations is taken to infinity. See for example Riemann-Lebesgue lemma.
The second statement is that when f is a Morse function, so that the singular points of f are non-degenerate and isolated, then the question can be reduced to the case n = 1. In fact, then, a choice of g can be made to split the integral into cases with just one critical point P in each. At that point, because the Hessian determinant at P is by assumption not 0, theMorse lemma applies. By a change of co-ordinates f may be replaced by
.
The value of j is given by the signature of the Hessian matrix of f at P. As for g, the essential case is that g is a product of bump functions of x_i. Assuming now without loss of generality that P is the origin, take a smooth bump function h with value 1 on the interval [−1, 1] and quickly tending to 0 outside it. Take
,
then Fubini's theorem reduces I(k) to a product of integrals over the real line like

with f(x) = ±x². The case with the minus sign is the complex conjugate of the case with the plus sign, so there is essentially one required asymptotic estimate.
In this way asymptotics can be found for oscillatory integrals for Morse functions. The degenerate case requires further techniques. See for example Airy function.
One-dimensional case[edit]
The essential statement is this one:
.
In fact by contour integration it can be shown that the main term on the right hand side of the equation is the value of the integral on the left hand side, extended over the range [−∞, ∞]. Therefore it is the question of estimating away the integral over, say, [1, ∞].^[2]
This is the model for all one-dimensional integrals I(k) with f having a single non-degenerate critical point at which f has second derivative > 0. In fact the model case has second derivative 2 at 0. In order to scale using k, observe that replacing k by c k where c is constant is the same as scalingx by √c. It follows that for general values of f″(0) > 0, the factor √(π / k)becomes
.
For f″(0) < 0 one uses the complex conjugate formula, as mentioned before.
See also[edit]
[/ltr][/size]

• Method of steepest descent
  
• Common integrals in quantum field theory
  

[size][ltr]
References[edit]
[/ltr][/size]

• Bleistein, N. and Handelsman, R. (1975), Asymptotic Expansions of Integrals, Dover, New York.
  
• Victor Guillemin and Shlomo Sternberg (1990), Geometric Asymptotics, (see Chapter 1).
  
• Hörmander, L. (1976), Linear Partial Differential Operators, Volume 1, Springer-Verlag, ISBN 978-3-540-00662-6.
  
• Aki, Keiiti; & Richards, Paul G. (2002). "Quantitative Seismology" (2nd ed.), pp 255–256. University Science Books, ISBN 0-935702-96-2
  
• Wong, R. (2001), Asymptotic Approximations of Integrals, Classics in Applied Mathematics, Vol. 34. Corrected reprint of the 1989 original. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. xviii+543 pages, ISBN 0-89871-497-4.
  

[size][ltr]
Notes[edit]
[/ltr][/size]

1. Jump up^ Courant, Richard; Hilbert, David (1953), Methods of mathematical physics 1 (2nd revised ed.), New York: Interscience Publishers, p. 474,OCLC 505700
   
2. Jump up^ See for example Jean Dieudonné, Infinitesimal Calculus, p. 119.
   

[size][ltr]
External links[edit]
[/ltr][/size]

• Hazewinkel, Michiel, ed. (2001), "Stationary phase, method of the",Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
  

拉普拉斯方法[编辑]

本条目翻译品质不佳。  翻译者可能不熟悉中文或原文语言，也可能使用了机器翻译，请协助翻译本条目或重新编写。明显的机器翻译请改挂 代码: {{Delete|G13}} 提交删除。

[ltr]
在数学上，以皮埃尔-西蒙·拉普拉斯命名的拉普拉斯方法是用于得出下列积分形式的近似解的方法：

其中的 ƒ(x) 是一个二次可微函数， M 是一个很大的数，而积分边界点 a 与 b 则允许为无限大。此外，函数 ƒ(x) 在此积分范围内的 全域极大值 所在处必须是唯一的并且不在边界点上。则它的近似解可以写为

其中的 x₀ 为极大值所在处。这方法最早是拉普拉斯在 (1774, pp. 366–367) 所发表。(待考查)[/ltr]

• 1 拉普拉斯方法的想法概论
  
• 2 其他形式
  
• 3 拉普拉斯方法的推广：最速下降法
  
• 4 更进一步一般化
  
• 5 复数积分
  
• 6 例子1：斯特灵公式
  
• 7 例子2：贝氏网络 ，参数估计与概率推理
  
• 8 相关维基百科文章
  
• 9 参考文献
  

[ltr]

拉普拉斯方法的想法概论[编辑][/ltr]

[th] [展开] 1. 相对误差[/th][th] [展开] 2. 函数  在极大值附近的行为，当  很大时会趋近于 [/th][th] [展开] 3. 当  增加，则相对应的  范围变小[/th][th] [展开] 4.若拉普拉斯方法所用的积分会收敛，它的相对误差相对应的非极大值附近的其余部分的积分就会随  增大而趋近于0[/th][ltr]
基于上述四点，就有办法证明拉普拉斯方法的可靠性。而 Fog(2008) 又将此方法推广到任意精确。 ***待考查***
此方法的正式表述与证明：
假设  是一个在  这点满足 (1)  ，(2)唯一全域最大，(3)  附近为二阶可微且  (4) 当拉普拉斯方法的积分范围为无限大时，此积分会收敛，
则，
[/ltr]

[th] [展开] 证明：[/th][ltr]
其他形式[编辑]
有时拉普拉斯方法也会被写成其他形式，如：

其中 为正 (好像不必要)。
重要的是，这方法精确度与函数  和  有关。 ^[1] ***待考查***[/ltr]

[th] [展开] 相对误差的推导[/th][ltr]
至于多维的情形，让我们令  是一个  维向量，而  则是一个标量函数，则此拉普拉斯方法可以写成

其中的  是一个在 取值的 海森矩阵，而  则是指矩阵的 行列式 ；此外，与单变量的拉普拉斯方法类似，这里的 海森矩阵 必须为 负定矩阵，即该矩阵的所有本征值皆小于0，这样才会是极大值所在。.^[2]
拉普拉斯方法的推广：最速下降法[编辑]
此拉普拉斯方法可以被推广到 复分析 里头使用，搭配 留数定理 ，以找出一个过最速下降点的 contour (翻路径的话，会与path integral 相冲，所以，这里还是以英文原字称呼) 的 曲线积分 ，用来取代原有的复数空间的 contour积分。因为有时  的一阶微分为0的点未必就在实数轴上，而就算真在实数轴上，也未必二阶微分在  方向上为小于0 的实数；此种情况下，就得使用最速下降法了。由于最速下降法中，已经利用另一条通过最速下降的鞍点来取代原有的 contour 积分，经过变量变换后就会变得有如拉普拉斯方法，因此，我们可以透过这新的 contour ，找到原本的积分的渐进近似解，而这将大大的简化整个计算。就好像原本的路径像是在蜿蜒的山路开车，而新的路径就像干脆绕过这座山开，反正目的只是到达目的地而已，留数定理已经帮我们把中间的差都算好了。请读 Erdelyi (2012)与 Arfken & Weber (2005) 的书里有关 steepest descents 的章节。
以下就是该方法在z 平面下的形式：

其中 z₀ 就是新的路径通过的鞍点。 注意，开根号里的负号是用来指定最速下降的方向，千万别认为取  的绝对值来取代这个负号，若然，那就大错特错了。 另外要注意的是，如果该被积函数是 在手动语言转换规则中检测到错误 ，就有必要将被新旧 contour 包到的极点所贡献的留数给加入，范例请参考 Okounkov 的文章 arXiv:math/0309074 的第三章。
更进一步一般化[编辑]
最速下降法还可以更进一步的推广到所谓的 非线性稳定相位/最速下降法 (nonlinear stationary phase/steepest descent method)。 这方法主要用在解非线性偏微分方程，透过将非线性偏微分方程转换为求解柯西变换(Cauchy transform)的积分形式，就可以借由最速下降法的想法来得到非线性解的渐进解。
以 艾里方程(线性)  为例，它可以写成积分形式如下：

由这条积分式，我们就可以借由最速下降法(若  指的是负实数轴，那么就回到此拉普拉斯方法了)来得到它的渐进解了。
然而，若方程式如 KdV方程  是个非线性偏微分方程，想要找到它的解相对应的一个复数 contour 积分的话，就没那么简单，在非线性稳定相位/最速下降法里所用到的概念主要是基于散射逆转换 (inverse scattering transform) 的处理方式，即先将原本的非线性偏微分方程变成 Lax 对 ，其中一个像是线性的 薛定谔方程式 ，其位能障为我们要找的  ，本征值为  ，波函数为  (不过，它并非我们所要的  )；因此可以解它的散射矩阵，若利用解析延拓将原本的波函数由实数  延拓到复数空间时，就可以得到黎曼希尔伯特问题(RHP)的形式。利用这个黎曼希尔伯特问题(RHP) ，我们可以解得  的柯西变换的积分形式，再利用此线性薛定谔方程的特性，就可以反推出  的复数 contour 积分 形式了。
而 Lax 对 的另一个偏微分方程则是决定每个  随时间变化的行为，由于  在  时被要求为0 ，会发现整个偏微分方程会变得十分简单，并且只决定  里的  的值，不过，条件是  必须是指由正负无限远入射或反射波的解。这样，我们所得到的这个只与时间与本征值有关的系数  就可以直接被应用在上述的黎曼希尔伯特问题(RHP)里的跃变矩阵里了。
接着就是非线性稳定相位/最速下降法所要做的工作，即找出 鞍点 来，在该点附近基于最速下降法的精神做近似。不过，这近似因着考虑到收敛性，需要将原本的 contour 变形，与将原本的黎曼希尔伯特问题(RHP)作转换，所以有再比原本的最速下降法多出一些步骤来。
这整个方法最早由 Deift 与 Zhou 在 1993 基于 Its 之前的工作所提出的，后续又有许多人加以改进，主要的应用则有 孤波 理论，可积模型等。
复数积分[编辑]
以下的积分常用在 拉普拉斯变换#拉普拉斯逆变换 里，

假定我们想要得到该积分在  时的结果(若  为时间，通常就是在找经过够久时间后达稳定的结果)，我们可以透过 解析延拓 的概念，先将这时间换成虚数，如 t = iu 并且一并做  的变换，则我们可以将上式转换为如下的 拉普拉斯变换#双边拉普拉斯变换

这里就可以套用此拉普拉斯方法求渐进解，最后，再利用 u = t / i 把 t 换回来，就可以得到该拉普拉斯逆变换的渐进解了。
例子1：斯特灵公式[编辑]
拉普拉斯方法可以用在推导 斯特灵公式 上；当 N 很大时，

证明：
由 Γ函数 的积分定义，我们可以得到

接着让我们做变量变换，

因此

将这些代回 Γ函数 的积分定义里，我们可以得到

经由此变量变换后，我们有了拉普拉斯方法所需要的 为

而它乃为二次可微函数，且

因此， ƒ(z) 的极大值出现在 z₀ = 1 而且在该点的二次微分为  。因此，我们得到

例子2：贝氏网络 ，参数估计与概率推理[编辑]
关于概率推理的简介，请参考 http://doc.baidu.com/view/e9c1c086b9d528ea81c77989.html 。 而在 Azevedo-Filho & Shachter 1994 的文章里，则回顾了如何应用此拉普拉斯方法 (无论是单变量或者多变量) 如何应用在 概率推理 上，以加速得到系统的 后验矩 (数学) (posterior moment) ， 贝氏参数 等，并举医学诊断上的应用为例。
相关维基百科文章[编辑][/ltr]

• https://en.wikipedia.org/wiki/Stationary_phase_approximation
  

支撑集[编辑]



[ltr]在数学中，一个定义在集合上的实值函数的支撑集，或简称支集，是指的一个子集，满足恰好在这个子集上非。最常见的情形是，是一个拓扑空间，比如实数轴等等，而函数在此拓扑下连续。此时，的支撑集被定义为这样一个闭集：在中为，且不存在的真闭子集也满足这个条件，即，是所有这样的子集中最小的一个。拓扑意义上的支撑集是点集意义下支撑集的闭包。
特别地，在概率论中，一个概率分布是随机变量的所有可能值组成的集合的闭包。
紧支撑[size=13][编辑]
一个函数被称为是紧支撑于空间的，如果这个函数的支撑集是中的一个紧集。例如，若是实数轴，那么所有在无穷远处消失的函数都是紧支撑的。事实上，这是函数必须在有界集外为的一个特例。在好的情形下，紧支撑的函数所构成的集合，在所有在无穷远处消失的函数构成的集合中，是稠密集的，当然在给定的具体问题中，这一点可能需要相当的工作才能验证。例如对于任何给定的，一个定义在实数轴上的函数在无穷远处消失，可以粗略通过通过选取一个紧子集来描述：

其中表示的指示函数。
注意，任何定义在紧空间上的函数都是紧支撑的。
当然也可以更一般地，将支撑集的概念推广到分布，比如狄拉克函数：定义在直线上的。此时，我们考虑一个测试函数，并且是光滑的，其支撑集不包括。由于（即作用于）为，所以我们说的支撑集为。注意实数轴上的测度（包括概率测度）都是分布的特殊情况，所以我们也可以定义一个测度支撑集。
奇支集[编辑]
在傅立叶分析的研究中，一个分布的奇支集或奇异支集有非常重要的意义。 直观地说，这个集合的元素都是所谓的奇异点，即使得这个分布不能局部地看作一个函数的点。
例如，单位阶跃函数的傅立叶变换，在忽略常数因子的情况下，可以被认为是，但这在时是不成立的。所以很明显地，是一个特殊的点，更准确地说，这个分布的傅立叶变换的奇支集是，即对于一个支撑集包括的测试函数而言，这个分布的作用效果不能表示为某个函数的作用。当然这个分布可以表示为一个柯西主值意义下的瑕积分。
对于多变量的分布，奇支集也可以更精确地被描述为波前集，从而可以利用数学分析来理解惠更斯原理。奇支集也可以用来研究分布理论中的特殊现象，如在试图将分布'相乘'时候导致的问题（狄拉克函数的平方是不存在的，因为两个相乘的分布的奇支集必须不相交）。
支撑族[编辑]
支撑族是一个抽象的拓扑概念，昂利·嘉当在一个层中定义了这个概念。在将庞加莱对偶性推广到非紧的流形上的时候，在对偶的一个方面上引入紧支撑的概念是自然的。
Bredon的书《Sheaf Theory》(第二版 1997)中给出了这些定义。的一组闭子集是一个支撑族，如果它是下闭的并且它的有限并也是闭的。它的扩张是的并。一个仿紧化(paracompactifying)的支撑族对于任何，在子空间拓扑意义下是一个仿紧空间，并且存在一些是一个邻域。如果是一个局部紧空间，并且是豪斯多夫空间，所有的紧子集组成的族满足上的条件，那么就是仿紧化的。[/ltr]
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动量映射[编辑]

当前条目的内容正在依照[:en:Moment map]的内容进行翻译。（2014年6月27日） 如果您熟知条目内容并擅长翻译，欢迎协助改善或校对这篇条目，长期闲置的非中文内容可能会被移除。目前的翻译进度为： 7%

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在数学，尤其在辛几何中，动量映射是一个与辛流形上的李群的哈密顿作用有关的工具，可用于构造作用的守恒量。动量映射推广了经典的 动量和角动量。它在各种辛流形的建立中是一个重要的部分，包括将会在后面讨论的symplectic (Marsden–Weinstein) quotients，以及symplectic cuts和sums。

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1 正式定义

2 哈密顿群作用

3 例子

4 Symplectic quotients

5 See also

6 Notes

7 References

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正式定义[编辑]
令 M 是一个配有辛形式 ω 的流形。假定一个李群 G 通过辛同胚作用在 M 上（也就是每个 G 中的 g 保持 ω ）。令  是 G 上的李代数， 是它的对偶，且令

是两者间的pairing。任一中的ξ诱导了 M 上的一个向量场 ρ(ξ) 以描述ξ的无限小作用。更精确地说，向量场  在M上一点x是

其中  是指数映射并且  表示 M 上的 G-作用。^[1]令  表示 向量场与 ω 的缩并。由于 G 通过辛同胚作用，它意味着对于  中所有的ξ， 是闭形式。
一个在(M，ω)上的 G-作用的动量映射是一个映射 ，对于  中所有的ξ满足

。这里  是通过  定义的从 M 到 R 的函数。动量映射在差一个积分的常数的程度上是唯一定义的。
一个动量映射经常也要求是 G-等价的，这里 G 通过余伴随作用作用在  上。如果群是紧的或半单的，那么总是选择积分常数使动量映射是余伴随等价的; 但是通常余伴随作用必须被修正以使映射等价(this is the case for example for the Euclidean group). The modification is by a 1-cocycle on the group with values in ，as first described by Souriau (1970).
哈密顿群作用[编辑]
动量映射的定义要求 是闭形式。在实际中一个更强的假定是有用的。G-作用被称作是哈密顿的当且仅当当以下的条件满足。首先，对于  中的每一个ξ，1-形式  是恰当的，这意味着它对于一些光滑函数

等于  。 如果这成立，那么我们可以选择  使映射  为线性。第二个使G-作用是哈密顿的的要求是映射  是一个从  到 M 在泊松括号下的光滑函数的代数的李代数同态。
如果 G 在(M，ω)上的作用在这个意义上是哈密顿的，那么一个动量映射是映射  ，这样  定义了一个李代数同态  满足 . 这里  是一个由哈密顿函数  通过

定义的向量场。
例子[编辑]
In the case of a Hamiltonian action of the circle G = U(1)，the Lie algebra dual  is naturally identified with R，and the 动量映射 is simply the Hamiltonian function that generates the circle action.
Another classical case occurs when M is the cotangent bundle of R³ and G is the Euclidean group generated by rotations and translations. That is，G is a six-dimensional group，the semidirect product of SO(3) and R³. The six components of the 动量映射 are then the three angular momenta and the three linear momenta.
Symplectic quotients[编辑]
Suppose that the action of a compact Lie group G on the symplectic manifold (M，ω) is Hamiltonian，as defined above，with 动量映射 . From the Hamiltonian condition it follows that  is invariant under G.
Assume now that 0 is a regular value of μ and that G acts freely and properly on . Thus  and its quotient  are both manifolds. The quotient inherits a symplectic form from M; that is，there is a unique symplectic form on the quotient whose pullback to  equals the restriction of ω to . Thus the quotient is a symplectic manifold，called the Marsden–Weinstein quotient，symplectic quotient or symplectic reduction of M by G and is denoted . Its dimension equals the dimension of M minus twice the dimension of G.
See also[编辑]
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• Poisson-Lie group
  
• Toric manifold
  
• Geometric Mechanics
  
• Kirwan map
  

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Notes[编辑]
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1. ^ The vector field ρ(ξ) is called sometimes the Killing vector field relative to the action of the one-parameter subgroup generated by ξ. See，for instance，（Choquet-Bruhat & DeWitt-Morette 1977）
   

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References[编辑]
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• J.-M. Souriau, Structure des systèmes dynamiques, Ma?trises de mathématiques, Dunod, Paris, 1970. ISSN 0750-2435.
  
• S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford Science Publications, 1990. ISBN 0-19-850269-9.
  
• Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, Oxford Science Publications, 1998. ISBN 0-19-850451-9.
  
• Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile, Analysis, Manifolds and Physics, Amsterdam: Elsevier. 1977, ISBN 978-0-7204-0494-4
  
• Ortega, Juan-Pablo; Ratiu, Tudor S. Momentum maps and Hamiltonian reduction. Progress in Mathematics 222. Birkhauser Boston. 2004. ISBN 0-8176-4307-9.
  

泊松流形[编辑]



[ltr]在数学中，泊松流形（Poisson manifold）是一个微分流形 M 使得 M 上光滑函数代数 C^∞(M) 上装备有一个双线性映射称为泊松括号，将其变成泊松代数。
每个辛流形是泊松流形，反之则不然。

[/ltr]

1 定义

2 泊松双向量

3 泊松映射

4 乘积流形

5 辛叶子

6 例子

7 复结构

8 另见

9 参考文献

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定义[编辑]
M 上一个泊松结构（Poisson structure）是一个双线性映射

使得这个括号反对称：

服从雅可比恒等式：

是 C^∞(M) 关于第一个变量的导子：
 对所有 
上一个性质有多种等价的表述。取定一个光滑函数 g ∈ C^∞(M)，我们有映射  是 C^∞(M) 上一个导子。这意味着存在 M 上哈密顿向量场 X_g 使得

对所有 f ∈ C^∞(M)。这说明这个括号只取决于 f 的微分。从而，任何泊松结构有一个相伴的从 M 的余切丛 T^∗M 到切丛 TM 的映射

将 df 映为 X_f。
泊松双向量[编辑]
余切丛与切丛之间的映射意味着 M 上存在一个双向量场 η，泊松双向量（Poisson bivector），一个反对称 2 张量 ，使得

这里  是切丛与其对偶之间的配对。反之，给定 M 上一个双向量场 η，这个公式可用来定义一个关于第一个变量为导子的反对称括号。这个括号服从雅可比恒等式，从而定义了一个泊松结构当且仅当斯豪滕–尼延黑斯括号 [η,η] 等于 0。
在局部坐标中，双向量在一点 x = (x₁, ..., x_m) 有表达式

从而

对一个辛流形，η 不过是由辛形式 ω 诱导的余切丛与切丛之间的配对，存在性是其非退化保证。辛流形与泊松流形的差别在于辛形式必须无处奇异，而泊松双向量不必处处都满秩。当泊松双向量处处为零时，称流形有平凡泊松结构。
泊松映射[编辑]
泊松映射（Poisson map）定义为光滑映射 ，从一个泊松流形 M 映到泊松流形 N，保持括号积：

这里 { , }_M 与 { , }_N 分别是 M 与 N 上的泊松括号。
乘积流形[编辑]
给定两个泊松流形 M 与 N，可以在乘积流形上定义一个泊松括号。设 f₁ 与 f₂ 是定义在乘积流形 M × N 上两个光滑函数，利用在因子流形上的括号 { , }_M 与 { , }_N 定义乘积流形上的括号{ , }_M×N：

这里 x ∈ M 与 y ∈ N 都是常数；这就有，当

则蕴含着

与

辛叶子[编辑]
一个泊松流形可以分成一族辛叶子（symplectic leaves）。每一片叶子是泊松流形的一个子流形，每片叶子自身是一个辛流形。两个点在同一片叶子上如果他们由一个哈密顿向量场的积分曲线连接。即，哈密顿向量场的积分曲线在这个流形上定义了一个等价关系。这个等价关系的等价类就是辛叶子。
例子[编辑]
如果  是一个有限维李代数， 是其对偶空间，则李括号在  上诱导了一个泊松结构。令 f₁ 与 f₂ 是  上两个函数， 是一点，可定义

这里 ，而 [ , ] 是李括号。如果 e_k 是李代数  上的局部坐标，则泊松双向量由

给出，这里  是李代数的结构常数（structure constant）。
复结构[编辑]
一个复泊松流形（complex Poisson manifold）是一个具有复结构或殆复结构 J 的泊松流形使得复结构保持双向量：

复泊松流形的辛叶子是伪凯勒流形（pseudo-Kähler manifold）。
另见[编辑]
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• 泊松-李群（Poisson-Lie group）
  
• 泊松超流形（Poisson supermanifold）
  
• 南部-泊松流形（Nambu-Poisson manifold）
  

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参考文献[编辑]
[/ltr][/size]

• A. Lichnerowicz, "Les variétès de Poisson et leurs algèbres de Lie associées", J. Diff. Geom. 12 (1977), 253-300.
  
• A. A. Kirillov, "Local Lie algebras", Russ. Math. Surv. 31 (1976), 55-75.
  
• V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, Cambridge Univ. Press 1984.
  
• P. Liberman, C.-M. Marle, Symplectic geometry and analytical mechanics, Reidel 1987.
  
• K. H. Bhaskara, K. Viswanath, Poisson algebras and Poisson manifolds, Longman 1988, ISBN 0-582-01989-3.
  
• I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, 1994. See also the review by Ping Xu in the Bulletin of the AMS.
  

巴塔林-维尔可维斯基代数[编辑]



[ltr]Batalin-Vilkovisky代数（Batalin-Vilkovisky algebra，简称BV代数）是Batalin和Vilkovisky在研究规范场的量子化过程中发现的一种代数结构^[1][2]。他们所提出的量子化方法（称为BV formailism或者BV quantization），是一种十分普遍而且有效的量子化方法，正受到越来越多的量子场论学家和弦理论家的重视和应用，而BV代数也越来越受到数学家们的重视。

[/ltr]

1 定义

2 例子

3 背景

4 参考文献

[size][ltr]

定义[编辑]
设是数域上的一个分次(graded)线性空间。上的一个BV代数结构是三元组，满足以下两个关系：
[/ltr][/size]

1. 是上的分次、交换、结合的代数(algebra)；
   
2. 是关于的二阶微分算子，即的度数为1，，并且对任给的,
   

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在上面的定义中，如果令
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则可以验证，形成一个Gerstenhaber代数。因此可以说，BV代数是一类特殊的Gerstenhaber代数。不仅如此，还是关于的导子(derivation)，即
[/ltr][/size][size][ltr]
使得形成一个微分分次李代数(differential graded Lie algebra, DGLA)。
例子[编辑]
迄今为止所发现的BV代数的例子几乎都与数学物理有关。
[/ltr][/size]

1. 设是一个奇的辛流形(odd symplectic manifold)，记为上光滑函数组成的集合。我们有形成一个分次交换结合的代数，记其乘法为。设为上的一组Darboux坐标，令
   
   
   则可以验证，形成一个BV代数，参见^[3][4]；
   
2. 田刚(G. Tian)在关于卡拉比-丘流形(Calabi-Yau manifold)的复结构的形变空间是光滑的证明中，实际上证明了控制复结构形变的微分分次李代数是一个BV代数^[5]；
   
3. B. Lian和G. Zuckerman证明了量子场论的数学背景(background，指从量子场论中抽象出来的代数结构)有一个BV代数结构^[6]；
   
4. E. Getzler用不同于Lian和Zuckerman的方法证明，一个二维拓扑共形场论(TCFT，此处采用Segal的定义)的同调群有一个自然的BV代数结构^[7]；
   
5. M. Chas和D. Sullivan证明，一个流形的自由环路空间(free loop space)的同调群上有一个BV代数结构^[8]。
   

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背景[编辑]
正如上面所述，BV代数跟量子场论有密切的联系。事实上，对一些数学物理学家来说，一个量子场论就指一个BV代数以及其中一个元素，该元素满足以下方程：
[/ltr][/size][size][ltr]
称为Master方程，有时候必须满足所谓的量子Master方程，即
[/ltr][/size][size][ltr]
另外，BV代数跟弦理论里面的镜像对称(Mirror Symmetry)也有密切的关系。事实上，镜像对称的A模型和B模型都有一个BV代数，而它们相应的Master方程的解空间上都有一个所谓弗罗贝尼乌斯流形的结构。镜像对称的一种表述就是，这两个Frobenius流形是同构的。
BV代数的研究是目前数学特别是数学物理中一个比较活跃的领域，关于它的研究仍在进行之中。
参考文献[编辑]
[/ltr][/size]

1. ^ I.A. Batalin and G.A. Vilkovisky, Gauge algebra and quantization. Phys. Lett. B 102 (1981), no. 1, 27-31.
   
   
2. ^ I.A. Batalin and G.A. Vilkovisky, Quantization of gauge theories with linearly dependent generators. Phys. Rev. D (3) 28 (1983), no. 10, 2567-2582.
   
   
3. ^ A. Schwarz, Geometry of Batalin-Vilkovisky quantization, arxiv: hep-th/9205088
   
   
4. ^ D. Fiorenza, An introduction to the Batalin-Vilkovisky formalism, arxiv: math.QA/0402057
   
   
5. ^ G. Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory (San Diego, Calif., 1986), 629-646, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987.
   
   
6. ^ B. Lian and G. Zuckerman, New perspectives on the BRST-algebraic structure of string theory. Comm. Math. Phys. 154 (1993), no. 3, 613-646.
   
   
7. ^ E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories. Comm. Math. Phys. 159 (1994), no. 2, 265-285.
   
   
8. ^ M. Chas and D. Sullivan, String topology, arxiv: math-GT/9911159.
   
   

We now want to discuss that
There exists a duality between wave fronts and light rays which can be
traced back to Huygens (1625–1695).
This duality was fully developed by the following authors:
• Cauchy (1789–1857) (solution theory for first-order partial differential equations
via characteristic curves),
• Hamilton (1805–1865) (canonical equations),
• Jacobi (1804–1851) (Hamilton–Jacobi partial differential equation),
• Lie (1842–1899) (contact transformations),
• Hilbert (1862–1943) (invariant integral),
• Poincar´e (1854–1912) and ´Elie Cartan (1859–1951) (integral invariants),
• Carath´eodory (1873–1950) (field theory and the royal road to the calculus of
variations), and
• H¨older (1901–1990) (H¨older’s contact transformation and the Huygens principle).
We will study this in Volume II. It turns out that:
• Wave fronts are related to the Hamilton–Jacobi partial differential equation
(eikonal equation), whereas
• light rays are described by the Euler–Lagrange ordinary differential equations to
the Fermat variational principle (and, alternatively, by the Hamilton canonical
equations.) The eikonal is the minimal time that is need by light in order to pass
from a fixed point (e.g., the origin) to all the other points.
• In the 1950s, this duality played a fundamental role in the foundations of optimal
control for dynamical systems. Bellman (1920–1984) based his dynamic
programming on the notion of wave fronts (the Hamilton-Jacobi–Bellman equation),
whereas Pontryagin (1908–1988) invented his maximum principle which is
related to light rays (the canonical Hamilton–Pontryagin equations).20
In terms of mathematics,
• the wave fronts of the Maxwell equations correspond to solutions of the characteristic
equation of the Maxwell system, and
• the light rays correspond to the solutions of the bicharacteristic system, which
is Cauchy’s characteristic system to the characteristic equation of the Maxwell
system.

Diffraction of Light
Diffraction problems for light were studied by Fraunhofer (1787–1826),
Fresnel (1788–1827), Helmholtz (1821–1894), Kelvin (1824–1907), Kirchhoff
(1824–1887), Rayleigh (1842–1919), Poincar´e (1854–1912), and Sommerfeld
(1868–1951). In his famous lectures on light, Poincar´e usedKelvin’s
method of stationary phase. In the 20th century, the rigorous mathematical
treatment of diffraction problems was a challenge for the theory of
integral equations. The Kirchhoff–Green representation formula is closely
related to the Born approximation and the Lippmann–Schwinger integral
equation for scattering processes in quantum physics. The Feynman path
integral from the 1940s generalizes wave optics.
Folklore

Fourier integral operators play a fundamental role in quantum field theory
for describing the propagation of physical effects.

12.6 Multiplication of Distributions
In a na ive setting of quantum field theory, one encounters the square δ(x)δ(x) of
Dirac’s delta function when computing cross sections of scattering processes. In
1954 Laurent Schwartz showed that it is not possible to construct a perfect theory
of products FG for distributions F,G ∈ D‘(R) without leaving the space D’(R) of
distributions.

13. Basic Strategies in Quantum Field Theory
If one does not sometimes think the illogical, one will never discover new
ideas in science.
Max Planck (1858–1947)
Mathematics is not a deductive science – that’s a clich´e. When you try
to prove a theorem, you don’t just list the hypotheses, and then start to
reason. What you do is trial-and-error, experimentation, and guess work.
Paul Halmos (born 1916)
“I think, this is so”, says Cicha, “in the fight for new insights, the breaking
brigades are marching in the front row. The vanguard that does not look
to left nor to right, but simply forges ahead – those are the physicists.
And behind them there are following the various canteen men, all kinds
of stretcher bearers, who clear the dead bodies away, or simply put, get
things in order. Well, those are the mathematicians.”
From the criminal novel Death Loves Poetry of the
Czech physicist Jan Klima (born in 1938)1
Whatever the future may bring, it is safe to assert that the theoretical advances
made in the unravelling of the constitution of matter since World
War II (1939–1945) comprise one of the greatest intellectual achievements
of mankind. They were based on the ground secured by Tomonaga, Bethe,
Schwinger, Feynman, and Dyson to quantum field theory and renormalization
theory in the period from 1946 to 1951.
Silvan Schweber, 1994
QED and the Men Who Made it
The mathematical language of physicists is formal, like a short-hand writing,
but this is extremely useful for getting very quickly the desired results
that are related to the outcome of physical experiments. It is then the hard
task of mathematicians to give rigorous proofs for the heuristic arguments of
physicists. The flow of ideas from physics to mathematics is an indispensable
source of inspiration for mathematicians.
Warning to the reader. In the following chapters, we will summarize
the most important heuristic formulas used in quantum field theory.
These heuristic formulas are not to be understood in the sense of
rigorous mathematics.
Nevertheless, this approach should help the reader to understand the language
of physicists and to find his/her own way in the jungle of literature,
which is full of inconsistencies and pseudo-proofs.
In Chaps. 14 and 15, the heuristic formulas are motivated by applying
the formal continuum limit to the rigorous finite-dimensional
approach from Chap. 7.
The elegance of the mnemonic language of physicists. Mathematicians
should note that the language of physicists is optimal from the
mnemonic point of view. Therefore, physicists are not willing to give up their
language despite the mathematical shortcomings.
In each order of perturbation theory, the heuristic formulas used by
physicists can be given a rigorous mathematical meaning.
This will be investigated in the following volumes of this monograph. Unfortunately,
as a rule, the rigorous mathematical approach is equipped with
technical details which obscure the basic ideas, in contrast to the language
of physicists. Therefore, it is important for mathematicians to know the language
of physicists as a guide to the rigorous approach. We will use the following
two approaches in order to pass from heuristic quantum field theory
to rigorous quantum field theory in each order of perturbation theory.
(D) Discretization: The idea is to pass from the heuristic continuum formulas
to rigorous formulas by discretization of space, time, and momentum.
We then supplement the finite formulas by additional terms called counterterms
in order to guarantee the existence of the continuum limit. This
second step is called renormalization.
(G) Generalized functions: We avoid discretization by using tempered distributions
from the very beginning. This is the so-called Epstein–Glaser
approach.
Survey on different approaches. Roughly speaking, there exist the
following approaches to quantum field theory:
(i) the response approach based on the generating functional for correlation
functions, the magic quantum action reduction formula, and the magic
LSZ reduction formula for the S-matrix (Sects. 14.2.4 and 14.2.5);
(ii) the response approach based on the global quantum action principle via
functional integrals, which implies (i) (Sect. 14.2.7);
(iii) the operator approach based on Dyson’s S-matrix along with creation
and annihilation operators for particles (Chap. 15);
(iv) gauge fields, functional integrals, Faddeev–Popov ghosts, and BRST
symmetry (Chap. 16);
(v) functional integrals and quantum field theory at finite temperature (Sect.
13.;
(vi) the rigorous Epstein–Glaser approach via the S-matrix as a tempered
operator-valued distribution (Sect. 15.4.4);
(vii) the rigorous global approach of axiomatic and algebraic quantum field
theory (Sect. 15.6);
(viii) the Ashtekar program based on the transport of quantum information
along loops and holonomy (loop quantum gravity).3
In (i) through (vi), we get formal power series expansions which can be regarded
as asymptotic series, by the classical Ritt theorem (Sect. 15.5.2). In
higher-order perturbation theory, the expressions are rather involved from the
analytical point of view. These expressions can be represented graphically by
Feynman diagrams. This helps very much to get insight into the structure of
perturbation theory and its renormalization. In this context,
• Zimmermann’s forest formula and
• Kreimer’s Hopf algebra play a crucial role (see Sect. 15.4.6).
Surprisingly enough, as a rule of thumb, the experience of physicists shows
that the apparently different approaches (i) through (vi) above yield the same
numerical results when applied to the measurements of concrete physical
effects. It is the task for the future to understand this equivalence in the
framework of a general mathematical theory. Hints for further reading on
quantum field theory can be found on page 907ff.
Historical remarks. Originally, Heisenberg and Pauli started quantum
field theory in 1929 by representing quantum fields as operator-valued functions
ϕ = ϕ(x, t)
depending on the position vector x and time t. Here, the value ϕ(x, t) lies in
a Hilbert space. However, it turned out that this approach is full of contradictions
caused, for example, by not knowing the right commutation relations
for interacting quantum fields from the very beginning.
Therefore, Feynman and Schwinger moved to a pragmatic point of
view and tried to completely avoid the notion of operators in Hilbert
space.
The approaches (i) and (ii) above will be formulated independently of operator
theory.
The idea is to relate the classical principle of critical action to correlation
functions which describe the correlations of the quantum field
at different space points and time points.
In Wightman’s axiomatic approach to quantum field theory from 1956, quantum
fields are tempered distributions with values in a Hilbert space. In algebraic
quantum field theory dating back to Segal in 1947, the fundamental
notions are
• observables (elements of an operator algebra), and
• states (positive functionals on the operator algebra).
This notion was generalized by Haag and Kastler in 1967, by passing to socalled
nets of local operator algebras. As the standard textbook, we recommend
R. Haag, Local Quantum Physics: Fields, Particles, Algebras, Springer,
New York, 1996. Roughly speaking, a local operator net assigns an operator
algebra to each open subset of the 4-dimensional space-time manifold. This
operator algebra represents physical observables. It is important that this
universal approach can be extended to many-particle systems in statistical
physics and to curved space-time manifolds in order to include quantum gravitation.
In this setting, it is possible to mathematically introduce two types
of distinguished physical states, namely,
• Kubo–Martin–Schwinger (KMS) states which describe thermodynamic
equilibrium states of many-particle systems, and
• Hadamard states which play a fundamental role in curved space–time manifolds.
For example, the theory of KMS states is equivalent to the Tomita–Takesaki
theory for von Neumann algebras. It is also possible to formulate Einstein’s
principle of general relativity (or the covariance principle) in terms of both
• algebraic quantum field theory and
• the Ashtekar program.
In the framework of string theory, quantum field theory is a low-energy approximation
of
• vibrating strings and
• higher-dimensional vibrating membranes called D-branes.
This is closely related to the theory of minimal sur faces and Riemannian
geometry in mathematics. We will study this in Volume VI.

The Method of Moments and Correlation
Functions
The family of moments knows all about a given random phenomenon in
nature.
Folklore
We expect that a quantum field possesses a random structure depending on
space and time with an infinite number of degrees of freedom. The idea of
physicists is to describe the quantum field by its local moments which depend
on position and time. For the local moments, physicists use the following
notions synonymously:
• n-point correlation function,
• n-point Green’s function, and
• n-point Feynman propagator.
To be honest, the formal definitions of these notions differ in the literature.
But, roughly speaking, the physics behind these notions is the same in each
order of perturbation theory. This follows from the nontrivial fact that the
computations based on Feynman diagrams and the corresponding Feynman
rules are the same. Observe the following. We have to carefully distinguish
between
• free quantum fields, and
• full quantum fields.
Here, free quantum fields are free of interactions, whereas full quantum fields
describe interactions. Naturally enough, the mathematics of free quantum
fields is much simpler than the mathematics of full quantum fields. From the
physical point of view, our main interest is to understand full quantum fields.
As with quantum fields, we distinguish between
• free correlation functions, and
• full correlation functions.

Intuitively, the n-point Green’s functions (and hence the correlation functions)
describe vacuum fluctuations of the quantum field.
The philosophy is that vacuum fluctuations know all about the quantum
field.
Advantages and disadvantages of different approaches. The advantage
of the correlation functions C1, C2, . . . is that they only depend on the
classical action S, but not on operators in Hilbert space. The disadvantage is
that functional integrals are beautiful mnemonic tools, but not well-defined
mathematical objects, as a rule.
The advantage of the Green’s functions G1,G2, . . . is that no ill-defined
functional integrals appear, but operators in Hilbert space. The disadvantage
is that the operators ϕ(x) are highly singular objects. Furthermore, the
commutation (resp. anticommutation) relations are not known a priori for
interacting quantum fields, but they have to be determined. In particular,
the na ive assumption that the commutation (resp. anticommutation) relations
for interacting quantum fields are the same as for free quantum fields
does not hold.
The idea of local averaging over space and time. It turns out that,
as a rule, the correlation functions Cn are not well-defined as classical local
functions of the space-time variables.More precisely, the
n-point correlation functions are distributions. Note that distributions are
linear mathematical objects and the multiplicative structure of distributions
is subtle, according to Laurent Schwartz. Therefore, the case of nonlinear interacting
quantum fields has to be handled very carefully. In this connection,
the Epstein–Glaser approach works successfully.
The philosophy behind the use of averages is that physical experiments
are based on measurement devices which are only able to measure
mean values.

Correlation functions represent a basic tool for studying physical processes
in both elementary particle physics and solid state physics. There exists a
wealth of physical phenomena which have to be explained. Physicists use
both
• the mean field ϕmean and
• the effective quantum action V (ϕmean)
as approximations for describing quantum corrections.

The Beauty of Functional Integrals
The action knows all about a quantum system via functional integrals.
Folklore
In modern elementary particle physics, most physicists prefer the use of functional
integrals because of their mnemonic elegance. Functional integrals allow
us the economical formulation of basic principles in quantum field theory.
S[ϕ] = critical!


Here, the temperature T is equal to h
/k(t−s) where k denotes the Boltzmann
constant. In other words, the passage from the propagator to the partition
function corresponds to the replacement
i(t − s)/h⇒ 1/kT (13.12)
which sends imaginary time to inverse temperature. Planck’s quantum of
action,
, and the Boltzmann constant, k, guarantee that the quantities have
the correct physical dimensions.
The fundamental transformation (13.12) is responsible for the close
connection between quantum field theory and statistical physics.
Summarizing, the three key formulas (13.9), (13.10), and (13.11) above relate
the action S of the classical field ϕ to the Hamiltonian operator H of the
corresponding quantized field. Observe that the key formulas do not give us
the Hamiltonian operator H itself, but only the transition probabilities as
an averaging over classical fields. However, this is very useful for computing
physical effects. A special model will be considered in Sect. 13.8.3.
The reader should observe that in quantum field theory, one has to distinguish
between
(i) processes which are independent of temperature (e.g., scattering processes
in particle accelerators) and
(ii) processes which critically depend on the temperature (e.g., processes in
stars or in the early universe).
In case (ii), we speak of quantum fields at finite temperature.

13.8 Quantum Field Theory at Finite Temperature
In mathematics and physics, many-particle systems are described by generating
functions. Physicists call them partition functions.
Folklore
What happens when ordinary matter is so greatly compressed that the
electrons form a relativistic degenerate gas, as in a white dwarf? What
happens when the matter is compressed even further so that atomic nuclei
overlap to form superdense nuclear matter, as in a neutron star? What
happens when nuclear matter is heated to such great temperatures that
the nucleons and pions melt into quarks and gluons, as in high-energy
nucleus-nucleus collisions? What happened to the spontaneous symmetry
breaking of the unified theory of the weak and electromagnetic interactions
during the big bang? Questions such as these have been fascinated me for
the past ten years. One reason is that a study of such systems involves statistical
physics, elementary particle physics, nuclear physics, astrophysics,
and cosmology, all of which I find interesting.
Joseph Kapusta, 1993
Finite-Temperature Field Theory7
Quantum statistics concerns many-particle systems (e.g., gases, systems of
elementary particles, condensed matter). We want to show that the theory of
such systems is governed by a single function, namely, the partition function
Z. Both
• the generating functional Z(J) (see Sect. 13.7.2) and
• the partition function Z (see Sect. 13.8.3)
can be represented by functional integrals. This underlines the close relation
between quantum field theory and statistical physics. In order to emphasize
this relationship, physicists use the same symbol Z.8 Therefore, generating
functionals Z(J) are also called partition functionals. Interesting physical
applications will be considered in the later volumes.
We have seen in Chap. 6 that the notion of partition function also plays
a fundamental role in mathematics, for example, in number theory. Edward
Witten uses partition functions in an ingenious manner in order to get deep
insight into mathematics by using models motivated by physics.
13.8.1 The Partition Function
The general scheme of quantum statistics. Many-particle quantum systems
are described by two operators in a Hilbert space, namely,
• the energy operator H (also called Hamiltonian), and
• the particle number operator N.
In addition, we use the following three real parameters:
• the absolute temperature T ,
• the chemical potential μ, and
• the size (volume) V of the system.
More generally, we want to describe s species of particles (e.g., electrons,
photons, etc.) Therefore, we use s particle number operators
N1,N2, . . . , Ns
along with the chemical potentials μ1, . . . , μs. Here, Nj refers to the jth
species of particles.

14. The Response Approach
All dynamical information about a quantum system may be extracted by
studying the response of the ground state (vacuum state) of the field to
an arbitrary external source J.
Julian Schwinger, 1970
The two magic formulas in quantum field theory. It is fascinating that
the huge field of quantum field theory can be based on two magic formulas,
namely,
(QA) the quantum action reduction formula for full correlation functions,
(LSZ) and the Lehmann–Szymanzik–Zimmermann reduction formula for the
scattering matrix (also called S-matrix).
Since the 1950s, physicists have discovered different ways of formulating these
two magic formulas.
In this chapter, our basic strategy is to generalize the rigorous finitedimensional
results from Sect. 7.24 to infinite dimensions by carrying
out a formal limit.
In Sect. 7.24 on page 438, we started with discrete functional integrals in
order to derive the two magic formulas (QA) and (LSZ) by means of the
principle of stationary phase.
In this chapter, we will start with the two magic formulas (QA) and
(LSZ) as basic principles.
The situation is similar to Newton’s equation of motion in mechanics and
to Maxwell’s equations in electrodynamics. It is possible to motivate these
equations by using physical and formal mathematical arguments; but one
can also postulate the validity of these equations as the starting point of the
theory. The prototypes of (QA) and (LSZ) can be found in the following two
basic papers:
M. Gell-Mann and F. Low, Bound states in quantum field theory. Phys.
Rev. 84 (1951), 350–354.
H. Lehmann, K. Szymanzik, and W. Zimmermann, The formulation of
quantized field theories, Nuovo Cimento 1 (1955), 205–225 and 6 (1957),
319–333.
In the first paper, Gell-Mann and Low reduced the computation of the full
correlation functions to the free correlation functions. The paper by Lehmann,
Szymanzik, and Zimmermann showed how to reduce Heisenberg’s S-matrix to
the full correlation functions. In fact, this chapter combines several important
contributions made to quantum field theory by Feynman, Schwinger, Dyson,
Gell-Mann and Low. We will proceed as follows.
(i) We start with the classical principle of critical action
S[ϕ] = critical!
Here, the action depends on an additional source term which describes the
influence of an external source on the quantum system. For our approach,
it is crucial that the action of the free field is a quadratic form. This allows
us then to apply the methods of Gaussian integrals.
(ii) From (i) we get the classical field equation (Euler–Lagrange equation)
Dϕ = −κLint(ϕ) − J
by using the classical calculus of variations.
(iii) Switching off the interaction by setting κ = 0, we get the so-called
response equation,
Dϕ = −J, (14.1)
which tells us the response ϕ of the classical field ϕ in the presence of an
arbitrary external source J. We replace this by the regularized equation
(D + iεI)ϕ = −J (14.2)
with the small regularization parameter ε > 0, and we assume that the
inverse operator (D + iεI)−1 exists. Setting Rε := −(D + iεI)−1, the
unique solution of the response equation reads as ϕ = RεJ.

The physical idea behind this approach is the following one. The physics of a
quantum field differs from the physics of the corresponding classical field by
additional quantum fluctuations of the ground state of the quantum field.
Physicists use the following intuitive picture: quantum fluctuations
are caused by virtual particles which jump from the ground state (also
called vacuum state) of the quantum field to the real world and back
to the ground state. These particles are called virtual, since they violate
energy-momentum conservation. In the language of Feynman
diagrams, the virtual particles correspond to internal lines.

克莱因-戈尔登方程[编辑]



[ltr]克莱因-戈尔登方程（Klein-Gordon equation）是相对论量子力学和量子场论中的最基本方程，它是薛定谔方程的相对论形式，用于描述自旋为零的粒子。克莱因-戈尔登方程是由瑞典理论物理学家奥斯卡·克莱因和德国人沃尔特·戈尔登于二十世纪二三十年代分别独立推导得出的。

[/ltr]

1 陈述

2 相对论量子力学下的形式推导

3 量子场论下的形式推导

4 自由粒子解

5 行波解

6 参见

7 参考资料

[size][ltr]

陈述[编辑]
克莱因-戈尔登方程为
。
很多时候会用自然单位（c=ħ=1）写成

由于平面波为此方程已知的一组解，所以方程形式由它决定：

遵从狭义相对论的能量动量关系式

跟薛定谔方式不同，每一个k在此都对应着两个，只有通过把频率的正负部份分开，才能让方程描述到整个相对论形式的波函数。若方程在时间流逝下不变，则其形式为
。
相对论量子力学下的形式推导[编辑]
自由粒子的薛定谔方程是非相对论量子力学的最基本方程：

其中是动量算符。
薛定谔方程并非相对论协变的，意味着它不满足爱因斯坦的狭义相对论。
利用狭义相对论中四维动量的不变性导出的相对论动量能量关系，相对论能量

替换薛定谔方程左边自由粒子的动能，
并最终得到它的协变形式

其中
达朗贝尔算符
从相对论量子力学的观点来看，达朗贝尔算符的出现意味着克莱因-戈尔登方程是一个量子力学的波方程。
量子场论下的形式推导[编辑]
场论中，对于自旋为零的场（标量场），拉格朗日量被写成

这里依照量子场论的习惯选取了自然单位，将光速和普朗克常数都取作1。
代入欧拉-拉格朗日方程可直接得到克莱因-戈尔登方程。
从量子场论的观点来看，以上推导过程都在经典场论的范围之内，因此克莱因-戈尔登方程只是一个经典场的场方程。
自由粒子解[编辑]
相对论量子力学中自由粒子只是一个理想化的概念，但形如克莱因-戈尔登方程这样的波方程仍然具有形式上的波包解：

其中
从克莱因-戈尔登方程得出的能量本征值为

因而克莱因-戈尔登方程的解包含了负能量。同时，由这个解导出相应的概率密度也不能保证是正值。这两个问题使得克莱因-戈尔登方程在很长一段时间里被认为是缺乏物理意义的.英国物理学家保罗·狄拉克为了确保概率密度具有物理意义建立了狄拉克方程，但这个方程仍然没有避免出现负能量。从那时起物理学家们逐渐意识到负能量的出现实际上意味着反粒子的存在。
行波解[编辑]
克莱因-戈尔登方程有行波解^[1]
[/ltr][/size]

Klein Gordon equation traveling wave plot4  Klein Gordon equation traveling wave plot5  Klein Gordon equation traveling wave plot6

[size][ltr]
参见[编辑]
[/ltr][/size]

• 狄拉克方程
  
• 量子场论
  

[size][ltr]
参考资料[编辑]
[/ltr][/size]

1. ^ 83.Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple p64-72 Springer
   
   

• Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley.ISBN 0-201-06710-2.
  
• Greiner, W. (1990). Relativistic Quantum Mechanics. Springer-Verlag.ISBN 3-540-67457-8.
  

[th] [显示] 查  论  编 量子力学[/th]

[th] [显示] 查  论  编 量子场论[/th]

The Response Function and the Feynman Propagator
In physics, use the right approximations and the right limits.
The golden rule

The singularities of the correlation function C2,free are related to causality.

The Extended Quantum Action Functional
The extended quantum action functional knows all about the quantum field.
Folklore

This implies that the magic formula (14.39) is well-defined
as a formal power series expansion with respect to the coupling constant κ.
This means that, in each order of perturbation theory with respect to
the coupling κ, we get well-defined correlation functions.
However, these functions depend on the regularization parameter ε and the
cut-off set Ω.
It remains to study the two limits ε → +0 and Ω → R4.
It turns out that this limit leads to divergent expressions. Therefore, physicists
invented the method of renormalization. The idea is
• to add counterterms to the Lagrangian density κLint and
• to replace the bare mass m0 by an renormalized (or effective) mass mren.
The goal is to get convergent expressions in each order of perturbation theory,
as ε → +0 and Ω → R4. We refer to Sects. 15.3ff on page 847ff.
The experience of physicists shows that one only needs the quantum
action axiom and the LSZ axiom below, combined with the procedure
of renormalization, in order to successfully compute a large number
of physical processes for elementary particles such that the computed
values coincide with the values measured in experiments.
Applications to the computation of concrete physical effects will be considered
in Volume II and the later volumes.

杜哈梅积分[编辑]



[ltr]在振动理论中，杜哈梅积分（Duhamel's integral）是求解线性系统在任意外载激励下响应的一种方法。

[/ltr]

1 概要介绍



• 1.1 问题背景
  
• 1.2 结论导出
  



2 参考文献

[size][ltr]

概要介绍[编辑]
问题背景[编辑]
受随时间变化的外载p(t)和粘性阻尼作用下的线性单自由度（SDF）系统的运动方程是一个二阶常微分方程，可写为

其中m为等效振子的质量，x代表系统振幅，t代表时间，c是粘性阻尼系数，k是系统刚度。
若初始静止于平衡位置的系统在t=0时刻受到一个单位冲击载荷作用，即p(t)是一个狄拉克δ函数δ(t)，，可以解得系统响应（称为单位脉冲响应函数）为

其中称为系统的阻尼比，是系统在无阻尼状态下振动的固有圆频率，是系统在当前存在的阻尼c作用下的实际振动圆频率。推广到任意时刻τ时受到冲击载荷作用的脉冲响应为
，
结论导出[编辑]
将任意载荷p(t)视为一系列脉冲激励的迭加：

那么根据线性性质可知，系统的响应同样可以表示成对这一系列脉冲激励的响应函数的迭加：

在时，连续求和转化为积分，此时上面的等式是严格成立的

将h(t-τ)的表达式代入即得杜哈梅积分的一般形式：

参考文献[编辑]
[/ltr][/size]

• 倪振华 编著，《振动力学》，西安交通大学出版社，西安，1990
  
• R. W. Clough, J. Penzien, Dynamics of Structures, Mc-Graw Hill Inc., New York, 1975.（中文版：R.W.克拉夫，J.彭津 著，王光远等 译，《结构动力学》，科学出版社，北京，1981）
  
• Anil K. Chopra, Dynamics of Structures - Theory and applications to Earthquake Engineering, Pearson Education Asia Limited and Tsinghua University Press, Beijing, 2001
  
• Leonard Meirovitch, Elements of Vibration Analysis, Mc-Graw Hill Inc., Singapore, 1986
  

Propagator

From Wikipedia, the free encyclopedia


[ltr]This article is about Quantum field theory. For plant propagation, see Plant propagation.
[/ltr][th]Quantum field theory[/th]

Feynman diagram

History

Background[show]

Symmetries[show]

Tools[show]

Equations[show]

Standard Model[show]

Incomplete theories[show]

Scientists[show]

v t e

[size][ltr]
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. In Feynman diagrams, which calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the diagram. They also can be viewed as the inverse of the wave operator appropriate to the particle, and are therefore often calledGreen's functions.

[/ltr][/size]

1 Non-relativistic propagators



• 1.1 Basic Examples: Propagator of Free Particle and Harmonic Oscillator
  



2 Relativistic propagators



• 2.1 Scalar propagator
  
• 2.2 Position space
  
  
  
  • 2.2.1 Causal propagator
    
    
    
    • 2.2.1.1 Retarded propagator
      
    • 2.2.1.2 Advanced propagator
      
    
    
    
  • 2.2.2 Feynman propagator
    
  
  
  
• 2.3 Momentum space propagator
  
• 2.4 Faster than light?
  
• 2.5 Propagators in Feynman diagrams
  
• 2.6 Other theories
  



3 Related singular functions



• 3.1 Solutions to the Klein–Gordon equation
  
  
  
  • 3.1.1 Pauli–Jordan function
    
  • 3.1.2 Positive and negative frequency parts (cut propagators)
    
  • 3.1.3 Auxiliary function
    
  
  
  
• 3.2 Green's functions for the Klein-Gordon equation
  



4 References

5 External links

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Non-relativistic propagators[edit]
In non-relativistic quantum mechanics the propagator gives the probability amplitude for a particle to travel from one spatial point at one time to another spatial point at a later time. It is the Green's function (fundamental solution) for the Schrödinger equation. This means that, if a system has Hamiltonian H, then the appropriate propagator is a function

satisfying

where H_x denotes the Hamiltonian written in terms of the x coordinates, δ(x) denotes the Dirac delta-function, Θ(x) is the Heaviside step function and K(x, t ;x′, t′) is the kernelof the differential operator in question, often referred to as the propagator instead of G in this context, and henceforth in this article. This propagator can also be written as

where Û(t, t′) is the unitary time-evolution operator for the system taking states at time t to states at time t′.
The quantum mechanical propagator may also be found by using a path integral,

where the boundary conditions of the path integral include q(t) = x, q(t′)=x′. Here L denotes the Lagrangian of the system. The paths that are summed over move only forwards in time.
In non-relativistic quantum mechanics, the propagator lets you find the state of a system given an initial state and a time interval. The new state is given by the equation

If K(x,t;x',t') only depends on the difference x − x′, this is a convolution of the initial state and the propagator.
Basic Examples: Propagator of Free Particle and Harmonic Oscillator[edit]
For a time-translationally invariant system, the propagator only depends on the time difference t − t′, so it may be rewritten as

The propagator of a one-dimensional free particle, with the far-right expression obtained via saddle-point methods,^[1] is then

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Similarly, the propagator of a one-dimensional quantum harmonic oscillator is the Mehler kernel,

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The latter may be obtained from the previous free particle result upon *** use of van Kortryk's SU(2) Lie-group identity,

valid for operators  and  satisfying the Heisenberg relation .
For the N-dimensional case, the propagator can be simply obtained by the product

Relativistic propagators[edit]
In relativistic quantum mechanics and quantum field theory the propagators are Lorentz invariant. They give the amplitude for a particle to travel between two spacetime points.
Scalar propagator[edit]
In quantum field theory the theory of a free (non-interacting) scalar field is a useful and simple example which serves to illustrate the concepts needed for more complicated theories. It describes spin zero particles. There are a number of possible propagators for free scalar field theory. We now describe the most common ones.
Position space[edit]
The position space propagators are Green's functions for the Klein–Gordon equation. This means they are functions G(x, y) which satisfy

where:
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• x, y are two points in Minkowski spacetime.
  
•  is the d'Alembertian operator acting on the x coordinates.
  
• δ(x − y) is the Dirac delta-function.
  

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(As typical in relativistic quantum field theory calculations, we use units where the speed of light, c, and Planck's reduced constant, ħ, are set to unity.)
We shall restrict attention to 4-dimensional Minkowski spacetime. We can perform a Fourier transform of the equation for the propagator, obtaining

This equation can be inverted in the sense of distributions noting that the equation xf(x)=1 has the solution, (see Sokhotski-Plemelj theorem)

with ε implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements.
The solution is

where

is the 4-vector inner product.
The different choices for how to deform the integration contour in the above expression lead to different forms for the propagator. The choice of contour is usually phrased in terms of the  integral.
The integrand then has two poles at

so different choices of how to avoid these lead to different propagators.
Causal propagator[edit]
Retarded propagator[edit]

A contour going clockwise over both poles gives the causal retarded propagator. This is zero if x and y are spacelike or if x ⁰< y ⁰ (i.e. if y is to the future of x).
This choice of contour is equivalent to calculating the limit,

Here

is the proper time from x to y and  is a Bessel function of the first kind. The expression  means y causally precedes x which, for Minkowski spacetime, means
 and 
This expression can also be expressed in terms of the vacuum expectation value of the commutator of the free scalar field operator,

where

is the Heaviside step function and

is the commutator.
Advanced propagator[edit]

A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if x and y are spacelike or if x ⁰> y ⁰ (i.e. if y is to the past of x).
This choice of contour is equivalent to calculating the limit

This expression can also be expressed in terms of the vacuum expectation value of the commutator of the free scalar field. In this case,

Feynman propagator[edit]

A contour going under the left pole and over the right pole gives the Feynman propagator.
This choice of contour is equivalent to calculating the limit (see Huang p. 30)

Here

where x and y are two points in Minkowski spacetime, and the dot in the exponent is a four-vector inner product. H₁⁽¹⁾ is a Hankel function and K₁ is a modified Bessel function.
This expression can be derived directly from the field theory as the vacuum expectation value of the time-ordered product of the free scalar field, that is, the product always taken such that the time ordering of the spacetime points is the same,

This expression is Lorentz invariant, as long as the field operators commute with one another when the points x and y are separated by a spacelike interval.
The usual derivation is to insert a complete set of single-particle momentum states between the fields with Lorentz covariant normalization, then show that the Θ functions providing the causal time ordering may be obtained by a contour integral along the energy axis if the integrand is as above (hence the infinitesimal imaginary part, to move the pole off the real line).
The propagator may also be derived using the path integral formulation of quantum theory.
Momentum space propagator[edit]
The Fourier transform of the position space propagators can be thought of as propagators in momentum space. These take a much simpler form than the position space propagators.
They are often written with an explicit ε term although this is understood to be a reminder about which integration contour is appropriate (see above). This ε term is included to incorporate boundary conditions and causality (see below).
For a 4-momentum p the causal and Feynman propagators in momentum space are:

For purposes of Feynman diagram calculations it is usually convenient to write these with an additional overall factor of −i (conventions vary).
Faster than light?[edit]
The Feynman propagator has some properties that seem baffling at first. In particular, unlike the commutator, the propagator is nonzero outside of the light cone, though it falls off rapidly for spacelike intervals. Interpreted as an amplitude for particle motion, this translates to the virtual particle traveling faster than light. It is not immediately obvious how this can be reconciled with causality: can we use faster-than-light virtual particles to send faster-than-light messages?
The answer is no: while in classical mechanics the intervals along which particles and causal effects can travel are the same, this is no longer true in quantum field theory, where it is commutators that determine which operators can affect one another.
So what does the spacelike part of the propagator represent? In QFT the vacuum is an active participant, and particle numbers and field values are related by an uncertainty principle; field values are uncertain even for particle number zero. There is a nonzero probability amplitude to find a significant fluctuation in the vacuum value of the field Φ(x) if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially correlated fluctuations to some extent. The nonzero time-ordered product for spacelike-separated fields then just measures the amplitude for a nonlocal correlation in these vacuum fluctuations, analogous to an EPR correlation. Indeed, the propagator is often called a two-point correlation function for the free field.
Since, by the postulates of quantum field theory, all observable operators commute with each other at spacelike separation, messages can no more be sent through these correlations than they can through any other EPR correlations; the correlations are in random variables.
In terms of virtual particles, the propagator at spacelike separation can be thought of as a means of calculating the amplitude for creating a virtual particle-antiparticle pair that eventually disappear into the vacuum, or for detecting a virtual pair emerging from the vacuum. In Feynman's language, such creation and annihilation processes are equivalent to a virtual particle wandering backward and forward through time, which can take it outside of the light cone. However, no causality violation is involved.
Propagators in Feynman diagrams[edit]
The most common use of the propagator is in calculating probability amplitudes for particle interactions using Feynman diagrams. These calculations are usually carried out in momentum space. In general, the amplitude gets a factor of the propagator for every internal line, that is, every line that does not represent an incoming or outgoing particle in the initial or final state. It will also get a factor proportional to, and similar in form to, an interaction term in the theory's Lagrangian for every internal vertex where lines meet. These prescriptions are known as Feynman rules.
Internal lines correspond to virtual particles. Since the propagator does not vanish for combinations of energy and momentum disallowed by the classical equations of motion, we say that the virtual particles are allowed to be off shell. In fact, since the propagator is obtained by inverting the wave equation, in general it will have singularities on shell.
The energy carried by the particle in the propagator can even be negative. This can be interpreted simply as the case in which, instead of a particle going one way, its antiparticleis going the other way, and therefore carrying an opposing flow of positive energy. The propagator encompasses both possibilities. It does mean that one has to be careful about minus signs for the case of fermions, whose propagators are not even functions in the energy and momentum (see below).
Virtual particles conserve energy and momentum. However, since they can be off shell, wherever the diagram contains a closed loop, the energies and momenta of the virtual particles participating in the loop will be partly unconstrained, since a change in a quantity for one particle in the loop can be balanced by an equal and opposite change in another. Therefore, every loop in a Feynman diagram requires an integral over a continuum of possible energies and momenta. In general, these integrals of products of propagators can diverge, a situation that must be handled by the process of renormalization.
Other theories[edit]
If the particle possesses spin then its propagator is in general somewhat more complicated, as it will involve the particle's spin or polarization indices. The momentum-space propagator used in Feynman diagrams for a Dirac field representing the electron in quantum electrodynamics has the form

where the  are the gamma matrices appearing in the covariant formulation of the Dirac equation. It is sometimes written, using Feynman slash notation,

for short. In position space we have:

This is related to the Feynman propagator by

where .
The propagator for a gauge boson in a gauge theory depends on the choice of convention to fix the gauge. For the gauge used by Feynman and Stueckelberg, the propagator for a photon is

The propagator for a massive vector field can be derived from the Stueckelberg Lagrangian. The general form with gauge parameter λ reads

With this general form one obtains the propagator in unitary gauge for λ = 0, the propagator in Feynman or 't Hooft gauge for λ = 1 and in Landau or Lorenz gauge for λ = ∞. There are also other notations where the gauge parameter is the inverse of λ. The name of the propagator however refers to its final form and not necessarily to the value of the gauge parameter.
Unitary gauge:

Feynman ('t Hooft) gauge:

Landau (Lorenz) gauge:

Related singular functions[edit]
The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important in quantum field theory. We follow the notation in Bjorken and Drell.^[2] See also Bogolyubov and Shirkov (Appendix A). These function are most simply defined in terms of the vacuum expectation value of products of field operators.
Solutions to the Klein–Gordon equation[edit]
Pauli–Jordan function[edit]
The commutator of two scalar field operators defines the Pauli–Jordan function  by^[2]

with

This satisfies  and is zero if .
Positive and negative frequency parts (cut propagators)[edit]
We can define the positive and negative frequency parts of , sometimes called cut propagators, in a relativistically invariant way.
This allows us to define the positive frequency part:
,
and the negative frequency part:
.
These satisfy^[2]

and

Auxiliary function[edit]
The anti-commutator of two scalar field operators defines  function by

with

This satisfies 
Green's functions for the Klein-Gordon equation[edit]
The retarded, advanced and Feynman propagators defined above are all Green's functions for the Klein-Gordon equation. They are related to the singular functions by^[2]

where 
References[edit]
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1. Jump up^ Saddle point approximation, planetmath.org
   
2. ^ Jump up to:^a b c d Bjorken and Drell, Appendix C
   

• Bjorken, J.D., Drell, S.D., Relativistic Quantum Fields (Appendix C.), New York: McGraw-Hill 1965, ISBN 0-07-005494-0.
  
• N. N. Bogoliubov, D. V. Shirkov, Introduction to the theory of quantized fields, Wiley-Interscience, ISBN 0-470-08613-0 (Especially pp. 136–156 and Appendix A)
  
• Edited by DeWitt, Cécile and DeWitt, Bryce, Relativity, Groups and Topology, section Dynamical Theory of Groups & Fields, (Blackie and Son Ltd, Glasgow), Especially p615-624, ISBN 0-444-86858-5
  
• Griffiths, David J., Introduction to Elementary Particles, New York: John Wiley & Sons, 1987. ISBN 0-471-60386-4
  
• Griffiths, David J., Introduction to Quantum Mechanics, Upper Saddle River: Prentice Hall, 2004. ISBN 0-131-11892-7
  
• Halliwell, J.J., Orwitz, M. Sum-over-histories origin of the composition laws of relativistic quantum mechanics and quantum cosmology, arXiv:gr-qc/9211004
  
• Huang, Kerson, Quantum Field Theory: From Operators to Path Integrals (New York: J. Wiley & Sons, 1998), ISBN 0-471-14120-8
  
• Itzykson, Claude, Zuber, Jean-Bernard Quantum Field Theory, New York: McGraw-Hill, 1980. ISBN 0-07-032071-3
  
• Pokorski, Stefan, Gauge Field Theories, Cambridge: Cambridge University Press, 1987. ISBN 0-521-36846-4 (Has useful appendices of Feynman diagram rules, including propagators, in the back.)
  
• Schulman, Larry S., Techniques & Applications of Path Integration, Jonh Wiley & Sons (New York-1981) ISBN 0-471-76450-7
  

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External links[edit]
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• Three Methods for Computing the Feynman Propagator
  

Formal motivation of the magic formulas. In rigorous terms, we have
shown in Sect. 7.25 on page 459 that finite-dimensional functional integrals
can be used in order to justify both
• the magic quantum action reduction formula and
• the magic LSZ reduction formula.
In the case of the present continuum ϕ4-model, the same arguments can
be applied formally to (14.44) in order to get the magic formulas (14.38)
and (14.41) from Sects. 14.2.4 and 14.2.5, respectively. Observe that this
represents only a formal approach, since the infinite-dimensional functional
integral (14.44) is not a well-defined mathematical object.

Bose–Einstein Condensation of Dilute Gases
The ϕ4-model as a toy model in particle physics. From the modern
point of view of elementary particle physics, the ϕ4-model is only a toy model.
It describes the interactions of elementary particles in the wrong way. It was
discovered by physicists in the 1960s and in the early 1970s that
The Standard Model in particle physics has to be based on the idea
of gauge field theory.
Here, in contrast to the ϕ4-model, the interactions between elementary particles
are not described by self-interactions of the quantum field ϕ itself, but
by additional gauge fields which correspond to particles called gauge bosons
(photons, vector bosons W+,W−, Z0, and eight gluons).
• The prototype of the Standard Model is quantum electrodynamics. In Sect.
14.3, we will sketch the basic ideas of quantum electrodynamics. In terms
of gauge field theory, quantum electrodynamics refers to the commutative
gauge group U(1). A detailed study of quantum electrodynamics will be
carried out in Volume II.
• The Standard Model in particle physics refers to the product gauge group
U(1)× SU(2)× SU(3). This has to be supplemented by the Higgs mechanism
which equips the vector bosons W+,W−, Z0 with large masses compared
with the proton mass. The point is that the gauge groups SU(2)
and SU(3) are not commutative, in contrast to U(1). The prototype of the
Standard Model in the form of a SU(N)-gauge field theory will be studied
in Sect. 16.2 on page 878. In particular, in the case where N = 3, we
get quantum chromodynamics which describes strong interactions based
on quarks and gluons. A detailed study of the Standard Model in particle
physics can be found in Volume III (gauge field theory) and Volume V
(physics of the Standard Model).
Many physicists believe that the ϕ4-model in 4-dimensional space-time M4
describes only a trivial free quantum field after renormalization. This means
that the influence of the interaction term κLint vanishes after renormalization.
In terms of physics, quantum fluctuations destroy the interaction. Recommendations
for further reading can be found on page 871 (triviality of
the ϕ4-model). It is thinkable that a variant of the ϕ4-model on a modified
4-dimensional space-time (in the setting of noncommutative geometry) corresponds
to a nontrivial quantum field. This is the subject of recent research.
The ϕ4-model as an effective theory for Bose–Einstein condensation.
It was conjectured by Gross and Pitaevskii that the ϕ4-model can
be used in order to describe Bose–Einstein condensation of dilute gases, in
the sense of an effective potential. This conjecture was rigorously proven in
the fundamental paper by
E. Lieb and R. Seiringer, Proof of Bose–Einstein condensation for dilute
trapped gases, Phys. Rev. Lett. 88 (2002), No. 170409.
Internet: http://www.arXiv math-ph/0112032

15. The Operator Approach

Dyson’s magic formula for the S-matrix represents a far-reaching generalization
of Lagrange’s variation-of-the-parameter method in celestial mechanics.
Folklore
In Chap. 14, we have described the approach to quantum field theory which
can be traced back to Feynman’s approach in the 1940s based on the Feynman
rules for Feynman diagrams and the representation of propagators by functional
integrals. Typically, this approach does not use operators in Hilbert
spaces, that is, the methods of functional analysis do not play any role. Historically,
in the 1920s quantum mechanics was first based on operator theory
by Heisenberg, Born, Jordan, Dirac, Pauli, and von Neumann. In order to
understand Feynman’s very effective approach, Dyson related this to operator
theory via the magic Dyson formula for the S=matrix.Conceptually, the
advantage of operator theory is that the duality between particles and waves
is formulated in a very transparent manner.
• The waves appear as solutions of classical wave equations. These equations
arise as equations of motion from the classical principle of critical action.
• The particles appear after introducing creation and annihilation operators.
• The free quantum field is a linear combination of creation and annihilation
operators where the coefficients are classical wave functions (that is,
solutions of the free equations of motion).
The disadvantage of operator theory is the fact that there arise serious mathematical
difficulties in applying the rigorous theory of functional analysis to
quantum electrodynamics and the Standard Model in particle physics. These
difficulties are caused by the interactions which are related to nonlinearities.
Basic strategy. Let us describe the main steps of the approach in this
chapter. In what follows, we will use
• a finite number of creation and annihilation operators,
• the quantized finite Fourier series for the free quantum field (Fourier quantization),
• the computation of the 2-point Green’s function of the free quantum field
via Cauchy’s residue theorem,
• the magic Dyson formula for the S-matrix as an axiom,
• the Wick theorem in order to compute S-matrix elements and to represent
them as Feynman diagrams which are very close to physical intuition.
Finally, we will show how one can compute cross sections of scattering processes
by means of S-matrix elements. In order to get rigorous formulas in
each order of perturbation theory, we put the system in a box of finite volume,
we consider a finite time interval, and we use a finite lattice in 4-dimensional
space (i.e., energy space and 3-dimensional momentum space).
This is the best approach to scattering processes in quantum field
theory from the point of view of physical intuition.
This approach has to be complemented by the method of renormalization.
This means that we have to study the limit where
• the box goes to R3,
• the finite time interval goes to R,
• the finite energy interval goes to R, and
• the finite lattice in 4-dimensional momentum space goes to R4.
Explicitly, we have to add counterterms in order force the existence of this
limit.
System of units. In this chapter, we will use the energetic system of
units with
h = c = 1.

Fourier Quantization
The idea of Fourier quantization is to consider solutions of the Klein–Gordon
equation in the form of finite Fourier series and to replace the Fourier coefficients
by creation and annihilation operators. This way we obtain free quantum
fields. Later on, we will use the free quantum field in order to construct
the S-matrix operator S(T ) by using Dyson’s magic formula. The operator S
allows us to compute transition probabilities for scattering processes.

The S-matrix operator describes interactions of the quantum field
in powers of the coupling constant by using nonlinear terms which
depend on the known free quantum field.
The S-matrix knows all about scattering processes of elementary particles.

The differential cross section is crucial in physics, since it does not
depend on the volume V of the box, the time interval [−T/2 , T/2 ], and
the relative velocity vrel.

A Glance at Quantum Electrodynamics
The application of the operator approach to quantum electrodynamics including
Feynman rules and renormalization will be studied in Volume II.
We will use creation and annihilation operators for electrons,
positrons, and photons.
Let us only mention the following points concerning renormalization. In quantum
electrodynamics, the original bare mass me and the bare charge −e of
the electron are replaced by the renormalized electron mass mren and the
renormalized electron charge −eren which coincide with the values measured
in physical experiments. The bare quantities me and −e only appear at the
very beginning of the renormalization process, but they are eliminated after
renormalization. We call this mass and charge renormalization. The physical
philosophy behind this reads as follows:
• The original Lagrangian density does not include quantum fluctuations.
• In contrast to the classical theory, mass and electric charge of the electron
are the result of complicated quantum interaction processes. This yields
the values mren and eren which are observed in physical experiments
• The method of renormalization allows us to introduce renormalized parameters
mren and −eren along with so-called radiative corrections to Feynman
diagrams. This way, we get close agreement with physical experiments. For
example, this allows us to compute the fine structure of the spectrum of
the hydrogen atom (the Lamb shift) and the anomalous magnetic moment
of both the electron and the muon.
The renormalized parameters mren and −eren are also called effective parameters.

A Glance at Renormalization
The subject of interacting quantum fields is full of nonsense.
Paul Dirac, 1981
Renormalization theory is a notoriously complicated and technical subject.
. . I want to tell stories with a moral for the earnest student: Renormalization
theory has a history of egregious errors by distinguished savants.
It has a justified reputation for perversity; a method that works up
to 13th order in the perturbation theory fails in the 14th order. Arguments
that sound plausible often dissolve into mush when examined closely. The
worst that can happen often happens. The prudent student would do well
to distinguish sharply between what has been proved and what has been
plausible, and in general he should watch out!
Artur Wightman, 1976
In 1951 Matthews and Salam11 formulated a requirement for renormalization
procedures that has become popularly known as the Salam criterion:
“The difficulty, as in all this work, is to find a notation which is both concise
and intelligible to at least two persons, of whom one may be an author.”
Possibly there are many proofs of the renormalizability of quantum electrodynamics
which satisfy the Salam criterion. But we must confess that
none of us has yet qualified as that other person who is the guarantor of
the criterion. While there are today many standard texts which discuss
the renormalizability of quantum electrodynamics, we are not aware of
any which represents a complete proof and in particular justifies the claim
that only gauge invariant counterterms are required. We here submit to
you a direct and complete proof and we invite you to judge whether you
can vouch for the Salam criterion.
Joel Feldman, Thomas Hurd, Lon Rosen, and Jill Wright, 1988
I shall roughly divide the history of renormalization theory into two main
chapters.
First the structure of the infinities or “divergences” in physical quantum
field theory such as electrodynamics was elucidated. A recursive process,
due to Bogoliubov and his followers, was found to hide these infinities
into unobservable “bare ” parameters that describe the fundamental laws
of physics at experimentally inaccessible extremely short distances. Although
technically very ingenious, this solution left many physicists and
probably most mathematicians under the impression that a real difficulty
had been “pulled under the rug”.
It would be unfortunate however to remain under this impression. Indeed
the second chapter of the story, known under the curious and slightly
inaccurate name of the “renormalization group”, truly solved the difficulty.
It was correctly recognized by Wilson and followers that in a quantum
theory with many scales involved, the change of parameters from bare to
renormalized values is a phenomenon too complex to be described in a
single step.
Just like the trajectory of a complicated dynamical system, it must be
instead studied step by step through a local evolution rule. The change of
scale in the renormalization group plays the role of time in dynamical systems.
This analogy is deep. There is a natural arrow of time, related to the
second principle of thermodynamics, and there is similarly a natural arrow
for the renormalization group evolution: microscopic laws are expected to
determine macroscopic laws, not the converse. The renormalization group
erases unnecessary detailed short scale information. . .
If we consider the universal character of the action principle both at the
classical and quantum level, and observe that the relation between microscopic
and macroscopic laws is perhaps the most central of all physical
questions, it is probably not an exaggeration to conclude that the renormalization
group is in some deep sense the “soul” of physics.
Vincent Rivasseau, 2002
Renormalization theory will play a crucial role in the following volumes.
At this point, we only want to discuss a few basic ideas. As an introduction
into renormalization theory formulated in the language of physicists, we
recommend the textbooks by Nash (1978), Collins (1984), Veltman (1995),
Kugo (1997), Ryder (1999), and Zinn–Justin (2004). Renormalization theory
in terms of mathematics can be found in Manoukian (1983) and Rivasseau
(1991).

The Trouble with the Continuum Limit
To illustrate the typical situation, let us consider the lattice ϕ4-model from
Sect. 15.1.2 on page 817. In order to get the continuum model, we have to
carry out the following limits.15
(i) High energy limit: N → +∞. This implies Emax → +∞.
(ii) Low-energy limit: V →+∞. This implies Δp → +0.
(iii) Large-time limit: T → +∞.
(iv) Regularization limit: ε → +0.
This corresponds to a passage from a finite number to an infinite number
of degrees of freedom. Since light of low (resp. high) energy is violet (resp.
red), the high-energy (resp. the low-energy) limit is also called the ultraviolet
(resp. infrared) limit by physicists. The trouble is that, as a rule, these limits
do not exist.
To overcome this trouble, the main idea is to change the classical
Lagrangian by adding counterterms.
The mathematical prototypes of this technique are the Weierstrass theorem
and the Mittag–Leffler theorem considered in Sect. 8.5.1 on page 509. From
the physical point of view, the philosophy is that there arise additional quantum
fluctuations in a quantum field theory. Such quantum fluctuations can
be described by the counterterms of the classical Lagrangian density.

Basic Ideas of Renormalization
The crucial point is that from the physical point of view, renormalization theory
allows us to pass from microscopic quantities to macroscopic quantities.
In contrast to the microscopic quantities, the macroscopic quantities can be
measured in physical experiments which depend on the available scale (e.g.,
the energy scale). In this context, the following two ideas play the decisive
role:
(i) The idea of counterterms.
(ii) The idea of essential and inessential scales.

Essential and inessential scales. If one wants to pass from microscopic
quantities to macroscopic quantities, one has to distinguish between the essential
scale and the inessential scale. The behavior of the physical system on
the inessential scale can be replaced by suitable averages. This idea is used
systematically in the method of the renormalization group.18
Methods of renormalization theory. The most important methods
in renormalization theory read as follows:
(a) BPHZ renormalization and the Weinberg power-counting theorem.
(b) Pauli–Villars regularization by introducing fictitious masses.
(c) Dimensional regularization.
(d) The BRST symmetry and algebraic renormalization.
(e) The renormalization group approach.
(f) The Epstein–Glaser approach.
(g) The Zimmermann forest formula and the importance of Hopf algebras.
(h) Gauge symmetries of functional integrals and the Ward–Takehashi identities
and the Taylor–Slavnov identities for Green’s functions.


正规化[编辑]





[ltr]物理学中，尤其是量子场论，正规化(regularization)是一项处理无限大、发散以及一些不合理表示式的方法，其方法透过引入一项辅助性的概念——正规子(regulator)。举例来说，若短距离物理效应出现发散，则设定一项空间中最小距离来解决这情形。正确的物理结果是让正规子消失(此例是)的极限情形，不过正规子的用意就在于当它是有限值，理论结果也是有限值的。正规化是将数学中的发散级数的可和性方法(summability methods)用在物理学问题上。
然而，理论结果通常包含了一些项，是正比于例如的式子，若取极限则会没有良好定义。正规化是获得一个完整、有限且有意义的结果的第一步；在量子场论，通常会接着一个相关但是独立的技术方法称作重整化。重整化则是基于对一些有着类似表示式的物理量的要求，要求其应该等于观测值。如此的约束条件则允许我们计算一些看似发散的物理量的有限值。
特定例子[size=13][编辑]
正规化的特定例子有：[/ltr][/size]

• 维度正规化(Dimensional regularization)
  
• 泡立-维拉斯正规化(Pauli-Villars regularization)
  
• 晶格正规化(Lattice regularization)
  
• ζ函数正规化(Zeta function regularization)
  
• 哈达玛正规化(Hadamard regularization)
  
• 点分裂正规化(Point-splitting regularization)
  

[ltr]
相关条目[编辑][/ltr]

• 重整化
  
• 量子场论
  

重整化（Renormalization）是量子场论、场的统计力学和自相似几何结构中解决计算过程中出现无穷大的一系列方法。

在量子场论发展的早期，人们发现许多圈图（即微扰展开的高阶项）的计算结果含有发散（即无穷大）项。重整化是解决这个困难的一个方案。一个理论如果只有有限种发散项，则可以在拉氏量中引进有限数目的项来抵消这些无穷大项，这种情形被称为可重整。反之，如果理论中有无限种发散项，则称为不可重整。

可重整化曾被认为一个场论所必需满足的自洽性要求。它在量子电动力学和量子规范场论的发展过程中起过重要的作用。粒子物理的标准模型也是可重整的。

现代场论的观点认为所有理论都只是有效理论，它们都有它们的适用范围。除了所谓的终极理论，所有理论在原则上都是不可重整的。在这种观点下，重整化只是联系不同能标下理论的一种方法。

例如:  的后两项发散.

为了消除发散，把积分下限分别改为无穷小的和，这样积分就变成了

如果能保证,那么就可以得到.

自发对称性破缺[编辑]





[ltr]
自发对称性破缺（spontaneous symmetry breaking）是某些物理系统实现对称性破缺的模式。当物理系统所遵守的自然定律具有某种对称性，而物理系统本身并不具有这种对称性，则称此现象为自发对称性破缺。^{[1]:141[2]:125}这是一种自发性过程（spontaneous process），由于这过程，本来具有这种对称性的物理系统，最终变得不再具有这种对称性，或不再表现出这种对称性，因此这种对称性被隐藏。因为自发对称性破缺，有些物理系统的运动方程或拉格朗日量遵守这种对称性，但是最低能量解答不具有这种对称性。从描述物理现象的拉格朗日量或运动方程，可以对于这现象做分析研究。
对称性破缺主要分为自发对称性破缺与明显对称性破缺两种。假若在物理系统的拉格朗日量里存在着一个或多个违反某种对称性的项目，因此导致系统的物理行为不具备这种对称性，则称此为明显对称性破缺。
如右图所示，假设在墨西哥帽（sombrero）的帽顶有一个圆球。这个圆球是处于旋转对称性状态，对于绕着帽子中心轴的旋转，圆球的位置不变。这圆球也处于局部最大引力势的状态，极不稳定，稍加微扰，就可以促使圆球滚落至帽子谷底的任意位置，因此降低至最小引力势位置，使得旋转对称性被打破。尽管这圆球在帽子谷底的所有可能位置因旋转对称性而相互关联，圆球实际实现的帽子谷底位置不具有旋转对称性──对于绕着帽子中心轴的旋转，圆球的位置会改变。^[3]:203
大多数物质的简单相态或相变，例如晶体、磁铁、一般超导体等等，可以从自发对称性破缺的观点来了解。像分数量子霍尔效应（fractional quantum Hall effect）一类的拓扑相（topological phase）物质是值得注意的例外。[/ltr]

• 1 概述
  
• 2 凝聚体物理学
  
• 3 粒子物理学
  
  
  
  • 3.1 手征对称性破缺
    
  • 3.2 希格斯机制
    
    
    
    • 3.2.1 外显的对称性案例
      
    • 3.2.2 自发对称性破缺案例
      
    
    
    
  
  
  
• 4 实例
  
• 5 诺贝尔奖
  
• 6 数学范例：墨西哥帽势能
  
• 7 参见
  
• 8 注释
  
• 9 参考文献
  
• 10 外部链接
  

[ltr]

概述[编辑]
量子力学的真空与一般认知的真空不同。在量子力学里，真空并不是全无一物的空间，虚粒子会持续地随机生成或湮灭于空间的任意位置，这会造成奥妙的量子效应。将这些量子效应纳入考量之后，空间的最低能量态，是在所有能量态之中，能量最低的能量态，不具有额外能量来制造粒子，又称为基态或“真空态”。最低能量态的空间才是量子力学的真空。^[4]
设想某种对称群变换，只能将最低能量态变换为自己，则称最低能量态对于这种变换具有“不变性”，即最低能量态具有这种对称性。尽管一个物理系统的拉格朗日量对于某种对称群变换具有不变性，并不意味着它的最低能量态对于这种对称群变换也具有不变性。假若拉格朗日量与最低能量态都具有同样的不变性，则称这物理系统对于这种变换具有“外显的对称性”；假若只有拉格朗日量具有不变性，而最低能量态不具有不变性，则称这物理系统的对称性被自发打破，或者称这物理系统的对称性被隐藏，这现象称为“自发对称性破缺”。^[5]:116-117
回想先前提到的墨西哥帽问题，在帽子谷底有无穷多个不同、简并的最低能量态，都具有同样的最低能量。对于绕着帽子中心轴的旋转，会将圆球所处的最低能量态变换至另一个不同的最低能量态，除非旋转角度为360°的整数倍数，所以，圆球的最低能量态对于旋转变换不具有不变性，即不具有旋转对称性。总结，这物理系统的拉格朗日量具有旋转对称性，但最低能量态不具有旋转对称性，因此出现自发对称性破缺现象。^[3]:203
凝聚体物理学[编辑]
大多数物质的相态可以通过自发对称性破缺的透镜来理解。例如，晶体是由原子以周期性矩阵排列形成，这排列并不是对于所有平移变换都具有不变性，而只是对于一些以晶格矢量为间隔的平移变换具有不变性。磁铁的磁北极与磁南极会指向某特定方向，打破旋转对称性。除了这两个常见例子以外，还有很多种对称性破缺的物质相态，包括液晶的向列相（nematic phase）、超流体等等。
类似的希格斯机制应用于凝聚态物质会造成金属的超导体效应。在金属里，电子库柏对的凝聚态自发打破了电磁相互作用的U(1)规范对称性，造成了超导体效应。更详尽细节，请参阅条目BCS理论。
有些物质的相态不能够用自发对称性破缺来解释。例如，分数量子霍尔液体（fractional quantum Hall liquid）、旋液体（spin liquid）这一类物质的托普有序相态。这些相态不会打破任何对称性，是不同种类的相态，没有比较通用的理论论述来描述这些相态。
粒子物理学[编辑]
在粒子物理学里，描述基本粒子的方程可能遵守某种对称性，可是方程的解并不能满足这对称性，例如，假设某种场方程可以用来估算两种夸克A、B的质量，并且对于这两种夸克具有对称性，解析这场方程或许给出了两个解，在第一个解里，夸克A比夸克B沉重，而在第二个解里，以同样的重量差，夸克B比夸克A沉重。对于这案例，场方程的对称性并没有被场方程的每一个单独解反映出来，而是被所有解共同一起反应出来。由于每一次做实际测量只能得到其中一个解，这表征了所倚赖理论的对称性被打破。对于这案例，使用术语“隐藏”可能会比术语“打破”更为恰当，因为对称性已永远嵌入在场方程里。由于物理学者并未找到任何外在因素涉及到场方程的对称性破缺，这现象称为“自发”对称性破缺。^[6]:194-195
手征对称性破缺[编辑]
主条目：手征对称性破缺
在粒子物理学里，手征对称性破缺指的是强相互作用的手征对称性被自发打破，是一种自发对称性破缺。假若夸克的质量为零（这是手征性（chirality）极限），则手征对称性成立。但是，夸克的实际质量不为零，尽管如此，跟强子的质量相比较，上夸克与下夸克的质量很小，因此可以视手征对称性为一种“近似对称性”。
在量子色动力学的真空里，夸克与反夸克彼此会强烈吸引对方，并且它们的质量很微小，生成夸克-反夸克对不需要用到很多能量，因此，会出现夸克-反夸克对的夸克-反夸克凝聚态，就如同在金属超导体里电子库柏对的凝聚态一般。夸克-反夸克对的总动量与总角动量都等于零，总手征荷不等于零，所以，夸克-反夸克凝聚的真空期望值（vacuum expectation value）不等于零，促使物理系统原本具有的手征对称性被自发打破，这也意味着量子色动力学的真空会将夸克的两个手征态混合，促使夸克在真空里获得有效质量。^[7]:669-672
根据戈德斯通定理，当连续对称性被自发打破后必会生成一种零质量玻色子，称为戈德斯通玻色子。手征对称性也具有连续性，它的戈德斯通玻色子是π介子。假若手征对称性是完全对称性，则π介子的质量为零；但由于手征对称性为近似对称性，π介子具有很小的质量，比一般强子的质量小一个数量级。这理论成为后来电弱对称性破缺的希格斯机制的初型与要素。^[7]:669-672
根据宇宙学论述，在大爆炸发生10^-6秒之后，开始强子时期，由于宇宙的持续冷却，当温度下降到低于临界温度KT_c≈173MeV之时 ，会发生手征性相变（chiral phase transition），原本具有的手征对称性的物理系统不再具有这性质，手征对称性被自发性打破，这时刻是手征对称性的分水岭，在这时刻之前，夸克无法形成强子束缚态，物理系统的有序参数反夸克-夸克凝聚的真空期望值等于零，物理系统遵守手征对称性；在这时刻之后，夸克能够形成强子束缚态，反夸克-夸克凝聚的真空期望值不等于零，手征对称性被自发性打破。^[8] [9]
希格斯机制[编辑][/ltr]


[ltr]
主条目：希格斯机制
在标准模型里，希格斯机制是一种生成质量的机制，能够使基本粒子获得质量。为什么费米子、W玻色子、Z玻色子具有质量，而光子、胶子的质量为零？^[10]:361-368希格斯机制可以解释这问题。希格斯机制应用自发对称性破缺来赋予粒子质量。在所有可以赋予规范玻色子质量，而同时又遵守规范理论的可能机制中，这是最简单的机制。^[10]:378-381根据希格斯机制，希格斯场遍布于宇宙，有些基本粒子因为与希格斯场之间相互作用而获得质量。
更仔细地解释，在规范场论里，为了满足局域规范不变性，必须设定规范玻色子的质量为零。由于希格斯场的真空期望值不等于零，^{[注 1]}造成自发对称性破缺，因此规范玻色子会获得质量，同时生成一种零质量玻色子，称为戈德斯通玻色子，而希格斯玻色子则是伴随着希格斯场的粒子，是希格斯场的振动。通过选择适当的规范，戈德斯通玻色子会被抵销，只存留带质量希格斯玻色子与带质量规范矢量场。^{[注 2][10]:378-381}
费米子也是因为与希格斯场相互作用而获得质量，但它们获得质量的方式不同于W玻色子、Z玻色子的方式。在规范场论里，为了满足局域规范不变性，必须设定费米子的质量为零。通过汤川耦合，费米子也可以因为自发对称性破缺而获得质量。^[7]:689ff
外显的对称性案例[编辑]
假定遍布于宇宙的希格斯场是由两个实函数  、 组成的复值标量场 ：
 ；
其中， 是四维坐标。
假定希格斯势的形式为
 ；
其中， 、 都是正值常数。
则这物理系统只有一个最低能量态，其希格斯场为零（）
对于这自旋为零、质量为零、势能为  的标量场  ，克莱因-戈尔登拉格朗日量  为^[7]:16-17
 。
注意到这拉格朗日量的第一个项目是动能项目。
由于拉格朗日量对于全域相位变换  具有不变性，而最低能量态对于全域相位变换也具有不变性：
 ，
所以，这物理系统对于全域相位变换具有外显的对称性。
自发对称性破缺案例[编辑]
假定遍布于宇宙的希格斯场是由两个实函数  、 组成的复值标量场 ：
 ；
其中， 是四维坐标。
假定希格斯势的形式为
 ；
其中， 、 都是正值常数。
对于这自旋为零、质量为零、势能为  的标量场  ，克莱因-戈尔登拉格朗日量  为^[7]:16-17
 。
如墨西哥帽绘图所示，这势能的猜想形状好似一顶墨西哥帽。希格斯势与拉格朗日量在  、 空间具有旋转对称性。位于z-坐标轴的帽顶为希格斯势的局域最大值，其复值希格斯场为零（），但这不是最低能量态；在帽子的谷底有无穷多个简并的最低能量态。从无穷多个简并的最低能量态中，物理系统只能实现出一个最低能量态，标记这最低能量态为 。这物理系统的拉格朗日量对于全域相位变换  具有不变性，即在  、 空间具有旋转对称性，而最低能量态  对于全域相位变换不具有不变性：
 ，
通常，  不等于  ，除非角弧  是  的整数倍数。所以，这物理系统对于全域相位变换的对称性被自发打破。这物理系统对于更严格的局域相位变换的对称性也应该会被自发打破。
实例[编辑][/ltr]

• 铁磁性物质对于空间旋转的不变性与居里温度有关。这物理系统的有序参数（order parameter）是量度磁偶极矩的磁化强度。假设温度高过居里温度，则自旋的取向是随机的，无法形成磁偶极矩，有序参数为零，基态对于空间旋转具有不变性，不存在对称性破缺。假设将系统冷却至温度低于居里温度，则自旋的取向会指向某特定方向，磁化强度不等于零，方向与自旋相互平行，基态不再具有旋转对称性，物理系统的旋转对称性被打破，产生自发对称性破缺现象，只剩下对于磁化强度所指方向的圆柱对称性。^[12]:283-284
  
• 描述固体的定律在整个欧几里德群（Euclidean group）之下具有不变性，但是固体自己将这欧几里德群打破为空间群（space group）。位移与取向是有序参数。
  
• 广义相对论具有洛伦兹对称性，但是在弗里德曼-罗伯逊-沃尔克模型里，将星系速度（在宇宙学尺寸，星系可以视为气体粒子）做平均而得到的平均四维速度场，变成打破这对称性的有序参数。关于宇宙微波背景也可以做类似论述。
  
• 在弱电相互作用模型里，希格斯场的真空期望值（vacuum expection value）是将电弱规范对称性打破成为电磁规范对称性的有序参数。如同铁磁性物质实例，这里也存在有电弱临界温度，在这临界温度会发生相变。
  
• 设想一根圆柱形细棒的两端被施加轴向应力，在发生屈曲（buckling）之前的状态S₀，整个系统对于以细棒为旋转轴的二维旋转变换具有对称性，因此可以观察到这系统的旋转对称性，可是这状态不是最低能量态，因为有应力能量储存于细棒的微观结构内，这状态极不稳定，稍有微扰就可以促使发生屈曲，释出应力能量，跃迁至最低能量态。注意到细棒有无穷多个最低能量态做选择，这些最低能量态之间因旋转对称性关联在一起，细棒可以选择跃迁至其中任意一个最低能量态，在发生屈曲之后的状态，完全改观为非对称性。尽管如此，仍旧存了旋转对称性的一些特征：假若忽略阻力，则不需施加任何作用力就可以自由地将细棒旋转，变换到另外一个最低能量态，这旋转模态实际就是不可避免的戈德斯通玻色子。^[12]:282-283
  
• 设想在无限宽长的水平平板上，有一层均匀厚度的液体。这物理系统具有欧几里德平面的所有对称性。现在从底部将平板均匀加热，使得液体的底部温度大于顶部温度很多。当温度梯度变得足够大的时候，会出现对流胞（convection cell），打破欧几里德对称性。
  

[ltr]
诺贝尔奖[编辑]
2008年10月7日，瑞典皇家科学院颁发诺贝尔物理学奖给三位日裔物理学者，赞赏他们在亚原子物理领域对于对称性破缺的研究成果。这三位物理学者分别为芝加哥大学的南部阳一郎、高能加速器研究机构的小林诚、京都大学基础物理学研究所的益川敏英。由于发现在强相互作用里自发对称性破缺的机制，特别是手征对称性破缺，南部阳一郎获得一半奖金，小林诚与益川敏英分享另外一半奖金，嘉勉他们在弱相互作用里CP对称被明显打破的原由。^[13]这原由最终是倚赖希格斯机制，但至今为止，被认知为只是希格斯耦合的一个特色，而不是一个自发对称性破缺现象。
数学范例：墨西哥帽势能[编辑]
在最简单的理想相对论性模型里，自发对称性破缺可以由标量场理论（scalar field theory）来概述。理论而言，自发对称破缺一般是从拉格朗日量来探讨。拉格朗日量可拆作动能部分和势能部分。
 。
对称性破缺来自于其势能部分  。如墨西哥帽绘图所示，
 。
这个势能有无限多个可能的势能最低点（真空态）：
 ；
其中， 值介于  到  之间。
参见[编辑][/ltr]

• CP破坏
  
• 动力学对称性破缺
  
• 明显对称性破缺
  
• 南部-戈德斯通定理
  
• 大统一理论
  
• 量子涨落
  
• 对称性破缺
  
• 迅子
  

[th] [隐藏] 查  论  编 量子场论[/th][th]背景[/th][th]对称性[/th][th]工具[/th][th]方程[/th][th]标准模型[/th][th]未完成理论[/th][th]科学史[/th] 场   规范场论   经典场论   庞加莱对称   量子力学   自发对称性破缺 交叉   电荷共轭   宇称   时间反演 反常   有效场论   真空期望值   法捷耶夫波波夫鬼态   费曼图 LSZ约化公式   配分函数   传播函数   量子化   重整化   真空态 维克定理   怀特曼公理体系 狄拉克方程   克莱因-戈尔登方程   普洛卡方程   惠勒－德威特方程 电弱相互作用   希格斯机制   量子色动力学   量子电动力学 杨-米尔斯存在性与质量间隙 量子引力   弦理论   超对称   人工色   万有理论 阿德勒   贝特   波古留波夫   卡伦   科尔曼   德维特   狄拉克   戴森 费米   费曼   菲尔茨   弗罗利希   盖尔曼   戈德斯通   格娄斯   胡夫特 贾基夫   克莱因   朗道   李政道   雷曼   马约拉纳   南部阳一郎 巴雷西   泊里雅科夫   萨拉姆   施温格   斯卡姆   斯塔克伯格 西蒙泽克   朝永振一郎   韦尔特曼   史蒂文·温伯格   韦斯柯夫 威耳逊   威滕   杨振宁   汤川秀树   齐默尔曼   金恩-贾斯廷

[ltr]
注释[编辑][/ltr]

1. ^ 希格斯场在最低能量态的平均值，就是“希格斯场的真空期望值”。费曼微积分（Feymann calculus）用来计算的是希格斯场在最低能量态的振动，即希格斯玻色子。
   
2. ^ 根据量子场论，所有万物都是由量子场形成或组成，而每一种基本粒子则是其对应量子场的微小振动，就如同光子是电磁场的微小振动，夸克是夸克场的微小振动，电子是电子场的微小振动，引力子是引力场的微小振动等等。^[11]:32-33
   

拓扑绝缘体[编辑]

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[size][ltr]
拓扑绝缘体是一种内部绝缘，界面允许电荷移动的材料。
在拓扑绝缘体的内部，电子能带结构和常规的绝缘体相似，其费米能级位于导带和价带之间。在拓扑绝缘体的表面存在一些特殊的量子态，这些量子态位于块体能带结构的带隙之中，从而允许导电。这些量子态可以用类似拓扑学中的亏格的整数表征，是拓扑有序的一个特例。^[1]
拓扑保护的边缘状态（一维）在碲化汞/碲化镉量子阱中被预言，^[2]随后由实验观测证实。^[3]很快拓扑绝缘体又被预言存在于含铋的二元化合物三维固体中。^[4]第一个实验实现的三维拓扑绝缘体在锑化铋中被观察到，^[5]随后不同实验组又通过角分辨光电子谱的方法，在锑，碲化铋，硒化铋，碲化锑中观察到了拓扑保护的表面量子态。^[6] 现在人们相信，在其他一些材料体系中，也存在拓扑绝缘态。^[7]在这些材料中，由于自然存在的缺陷，费米能级实际上或是位于导带或是位于价带，必须通过掺杂或者通过改变其电势将费米能级调节到禁带之中。^[8][9]
类似的边缘效应同样出现于量子霍尔效应之中，但仅在强垂直磁场，低温的二维系统中出现。
参见[编辑]
[/ltr][/size]

• Moore, Joel. The Birth of Topological Insulators. Nature. 2010, 464(7286): 194. doi:10.1038/nature08916. PMID 20220837.
  
• Kane, C. L.; Mele, E. J. A New Spin on the Insulating State. Science. 2006, 314 (5806): 1692. doi:10.1126/science.1136573.PMID 17170283.
  
• Kane, C.L. Topological Insulator: An Insulator with a Twist. Nature. 2008, 4 (5): 348. doi:10.1038/nphys955.
  
• Witze, Alexandra. Topological Insulators: Physics On the Edge.Science News. 2010.
  
• Brumfield, Geoff. Topological insulators: Star material : Nature News. Nature. 2010, 466 (7304): 310–311 [2010-08-06].doi:10.1038/466310a. PMID 20631773.
  
• Murakami, Shuichi. [url=http://iopscience.iop.org/1367-2630/focus/Focus on Topological Insulators]Focus on Topological Insulators[/url]. New Journal of Physics. 2010.
  

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参考文献[编辑]
[/ltr][/size]

1. ^ Kane, C. L.; Mele, E. J. Z₂ Topological Order and the Quantum Spin Hall Effect. Physical Review Letters. 30. September 2005, 95 (14): 146802.doi:10.1103/PhysRevLett.95.146802.
   
   
2. ^ Bernevig, B. Andrei; Taylor L. Hughes, Shou-Cheng Zhang. Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells. Science. 2006-12-15, 314 (5806): 1757–1761 [2010-03-25].doi:10.1126/science.1133734. PMID 17170299.
   
   
3. ^ Konig, Markus; Steffen Wiedmann, Christoph Brune, Andreas Roth, Hartmut Buhmann, Laurens W. Molenkamp, Xiao-Liang Qi, Shou-Cheng Zhang. Quantum Spin Hall Insulator State in HgTe Quantum Wells. Science. 2007-11-02, 318 (5851): 766–770 [2010-03-25].doi:10.1126/science.1148047. PMID 17885096.
   
   
4. ^ Fu, Liang; C. L. Kane. Topological insulators with inversion symmetry. Physical Review B. 2007-07-02, 76 (4): 045302 [2010-03-26].doi:10.1103/PhysRevB.76.045302. Shuichi Murakami. Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase. New Journal of Physics. 2007, 9 (9): 356–356 [2010-03-26]. doi:10.1088/1367-2630/9/9/356. ISSN 1367-2630.
   
   
5. ^ Hsieh, D.; D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava & M. Z. Hasan. A Topological Dirac insulator in a 3D quantum spin Hall phase. Nature. 2008, 452 (9): 970–974 [2010]. doi:10.1038/nature06843.PMID 18432240.
   
   
6. ^ Hasan, M. Z; C. L Kane. Topological Insulators. 1002.3895. 2010-02-20 [2010-04-27].
   
   
7. ^ Lin, Hsin; L. Andrew Wray, Yuqi Xia, Suyang Xu, Shuang Jia, Robert J. Cava, Arun Bansil, M. Zahid Hasan. Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena. Nat Mater. 2010-07, 9 (7): 546–549 [2010-08-05].doi:10.1038/nmat2771. ISSN 1476-1122. PMID 20512153.
   
   
8. ^ Hsieh, D.; Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, M. Z. Hasan. Observation of Time-Reversal-Protected Single-Dirac-Cone Topological-Insulator States in Bi2Te3 and Sb2Te3. Physical Review Letters. 2009, 103 (14): 146401 [2010-03-25].doi:10.1103/PhysRevLett.103.146401. PMID 19905585.
   
   
9. ^ Noh, H.-J.; H. Koh, S.-J. Oh, J.-H. Park, H.-D. Kim, J. D. Rameau, T. Valla, T. E. Kidd, P. D. Johnson, Y. Hu and Q. Li. Spin-orbit interaction effect in the electronic structure of Bi2Te3 observed by angle-resolved photoemission spectroscopy. EPL Europhysics Letters. 2008, 81 (5): 57006 [2010-04-25]. doi:10.1209/0295-5075/81/57006.
   
   

截面 (物理)[编辑]





[ltr]在原子核物理学和粒子物理学中，截面是一个用于表达粒子间发生相互作用可能性的术语。[/ltr]

• 1 概念
  
• 2 散射截面
  
• 3 原子核物理图
  
• 4 参考文献
  
• 5 外部链接
  

[ltr]

概念[编辑]
假设有一束射出的粒子，另有一个由某种材料制成的平面状箔，粒子以垂直于这个平面的方向射来。平面箔上被击中的微粒在平面上代表的面积，即为所说的“截面”，^[1]用希腊字母表示。这些粒子在接近目标粒子时组成的薄面时，会发生一些相互作用。
这个术语起源于点粒子被射向固体目标的经典物理图景。假设发射粒子与目标粒子一旦靠近就一定会发生相互作用，如果错失了就完全不发生相互作用，则总的相互作用可能性就等于截面与整个靶面积的比值。
上述的基本概念可以引申到其他的情况，例如靶区域呈现介于0至100%反应几率的情况——因为目标中的物质不完全相同，或者因为不均匀的场使之减弱。一个特殊的例子就是散射。
散射截面[编辑]
参见：散射
散射截面是用于描述光（或者其他形式的辐射）被粒子散射的可能性。一般地说，散射截面里所指的“截面”与几何上的“截面”不同，它与入射粒子的波长和靶粒子的电容率、形状、大小有关。稀疏介质的散射总数决定于散射截面和粒子个数的乘积。考虑到吸收、散射和发光，总散射截面（）可以用面积表示为下面的式子：

根据比尔－朗伯定律，吸光度与浓度成比例，即，其中是浓度，为给定波长的吸光度，为路径长度，由此总散射面积还与光的吸光度有关。入射辐射的吸光度是透光率倒数的对数^[2]：

在考虑粒子的散射时，通常引入另一个物理量微分散射截面，而将称作总散射截面。微分散射截面表达为：

其中为出射粒子的空间角。这个微分表示每单位空间角的出射粒子对应的入射区域，因此对这个量在一个完整的空间角中积分即可获得总截面。微分散射截面在量子力学中可方便地由求得；而由量子力学中散射结果，进行渐近分析分解为入射波与散射波后（如用分波方法分解为球谐函数，或玻恩近似），设定入射项的系数为1，出射项系数即为。
原子核物理图[编辑][/ltr]


[ltr]
在原子核物理学中，截面的概念可以很方便地表达特定事件发生的可能性。在统计上，薄膜上原子的中心可以被看做均匀分布在一个平面上的点的集合。参与撞击原子的中心与其他原子以为距离通过的概率是确定的。事实上，如果在平面上区域有个原子中心，那么这个概率为，这仅仅是所有原子中心以为半径的圆的总面积与整个平面的比值。如果我们将原子考虑成不可穿透的钢制盘，将与之相互作用的粒子看做直径可以忽略的子弹，这个比值就是“子弹”被“钢质盘”截止的可能性，也就是说入射粒子被被射原子平面阻挡的的可能性。如果计算通过的原子，那么结果则可以被表达为原子的等效截止截面。这一概念可以延伸到到任何有关入射粒子与靶粒子间的相互作用。例如入射粒子阿尔法粒子轰击靶粒子铍会产生中子的可能性可以表示为这种原子核反应的反应截面。
参考文献[编辑][/ltr]

1. ^ 褚圣麟. 《原子物理学》. 高等教育出版社. ISBN 978-7-04-001312-2.
   
2. ^ Bajpai, P.K. 2. Spectrophotometry//Biological Instrumentation and Biology. ISBN 8121926335.
   

• J.D.Bjorken, S.D.Drell, Relativistic Quantum Mechanics, 1964
  
• P.Roman, Introduction to Quantum Theory, 1969
  
• W.Greiner, J.Reinhardt, Quantum Electrodinamics, 1994
  
• R.G. Newton. Scattering Theory of Waves and Particles. McGraw Hill, 1966.
  

[ltr]
外部链接[编辑][/ltr]

• Nuclear Cross Section
  
• Scattering Cross Section
  
• IAEA - Nuclear Data Services
  
• BNL - National Nuclear Data Center
  
• Particle Data Group - The Review of Particle Physics
  
• IUPAC Goldbook - Definition: Reaction Cross Section
  
• IUPAC Goldbook - Definition: Collision Cross Section
  

中子截面[编辑]



[ltr]中子截面（英语：Neutron cross-section）常用于核物理学与粒子物理学中，表示入射中子与靶核交互作用的一种带有机率意义的常数。单位以barn表示，等于10⁻²⁴cm²。中子截面与中子通量、核反应速率计算有关，例如：计算一座核电厂的功率。

[/ltr]

1 决定中子截面的参数



• 1.1 核种
  
• 1.2 交互作用方式
  
  
  
  • 1.2.1 中子吸收
    
  • 1.2.2 中子散射
    
  
  
  
• 1.3 入射中子能量
  
• 1.4 靶材温度
  



2 与反应速率的关系

3 连续与平均截面

4 巨观与微观截面

5 平均自由径

6 常见中子截面数据

7 外部链接

8 参考资料

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决定中子截面的参数[编辑]
中子截面与下列几个参数有关：
[/ltr][/size]

• 靶材核种
  
• 交互作用方式
  
• 入射中子能量或强度
  
• 靶材温度
  

[size][ltr]
核种[编辑]
核种与中子截面有关，例如：¹H与同位素²H的中子吸收截面并不一样，后者较小。这就是为何重水作为中子减速剂的效果较轻水佳，前者吸收中子较后者少，因而使用天然铀即可达到临界，减少使用浓缩铀的成本。
交互作用方式[编辑]
若我们只考虑总反应截面σ_T，则与个别作用方式无关。然而，σ_T可由不同交互作用方式的反应截面加总得到：^[1]

σ_S是总中子散射截面，σ_A是总中子吸收截面。
中子吸收[编辑]
核子吸收中子后，会成为其核种的同位素。以U-235为例，其吸收中子成为U-236*（星号代表能量较高）。
不稳定的原子核会透过不同的方式将能量释放出来：
[/ltr][/size]

• 释出一个中子（与散射情况类似）。
  

[size][ltr]

[/ltr][/size]

• 放出γ射线（不会改变质子数）。
  
• β^-衰变。使中子变换成质子、电子和反电子微中子（会改变质子数）。以此作为主要释放能量方法的核种：²³²Th。
  

[size][ltr]

[/ltr][/size]

• 约81%的U-236*能量较高，会直接产生核分裂，其能量会以分裂产物的动能形式表现出来，并释放出5个自由中子。以此作为主要释放能量方法的核种：²³³U、²³⁵U、²³⁷U、²³⁹Pu、²⁴¹Pu。
  

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中子散射[编辑]
散射可分为相干散射和非相干散射。因为中子极为靠近原子核时会产生核力作用，且不同的同位素有不同的截面变化。一个明显的例证是¹H和²H，前者的总截面是后者的10倍，这是因为氢的非相干散射长度较大所造成的。铝和锆也有类似的情况。
入射中子能量[编辑]
主条目：中子温度
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当确定了反应方式与核种后，中子截面大小明显地会与入射中子速率有关。在极端情况下，若入射中子速率过低，无法使核子超过阈能，则无法启动核反应。因此，中子截面的数值取决于特定能量或某个能量区间内。
举例来说，右方的U-235核分裂中子截面随能量变化图中，随着能量增高，反应截面下降。所以在核反应炉中，我们会使用中子减速剂来降低中子能量，便于促使核分裂连锁反应发生。
一个简单估计能量与中子截面关系的模型——拉姆绍尔模型。^[2]是以中子热德布罗意波长作为核反应的有效体积大小：

为中子有效半径，为圆形截面面积，为原子核半径，它们有以下关系：

若中子有效半径远大于原子核半径（1–10fm，E = 10–1000keV），则原子核半径可忽略。对低能量中子来说（如热中子），与中子能量成反比关系，这可用来解释在核反应器内中子减速剂的使用。另一方面，高能中子（1MeV以上）的可忽略，中子截面约为常数，只与原子核有关。
然而，这个模型无法解释中子共振区（1eV–10keV）和一些核反应的阈能大小的影响。
靶材温度[编辑]
目前中子截面的数据大多是20°C的测量值，为了计算中子截面随靶材温度的变化，可利用下列公式：^[1]:

σ是在温度T下的中子截面，σ₀则是在温度T₀下的中子截面，温度单位为K。
与反应速率的关系[编辑]
[/ltr][/size][size][ltr]
让我们想像一个静置不动的球形靶（右图黑色圆形），和一群以速率v向右移动的入射粒子（右图蓝色圆形）。假设一个入射粒子在dt单位时间和σ单位截面内，以速率v移动所形成的体积（右图黑色圆柱）：

若有每单位体积有n个粒子使靶材以r的反应速率进行反应：

代入中子通量Φ = n v^[1]：

若每单位体积有N个靶材粒子以每单位体积R的反应速率进行反应：

一个典型原子核半径r约为10⁻¹²厘米，其截面π r²约为10⁻²⁴平方厘米（这也是使用靶恩作为单位的原因），但是不同的截面有较大的数量级变化。例如，慢中子的（n,γ）反应截面约等于1,000 b，但伽玛射线的反应吸收截面就只有0.001 b。
连续与平均截面[编辑]
但是一群粒子通常具有不同的入射速率，所对应的反应速率R可由积分式得出：

σ(E)是随能量变化的连续截面，Φ(E)是随能量变化的粒子通量，N是靶材原子密度。 平均截面定义为：

Φ= Φ(E) dE是整个能量范围的粒子通量积分值。
利用Φ和σ可得出：

巨观与微观截面[编辑]
从上可知，前面的截面都是指微观截面σ。然而，我们可以定义巨观截面Σ：^[1]

N是原子密度，单位cm⁻³。
因此，微观截面的单位是cm²，巨观截面单位是cm⁻¹。所以反应速率R可表示成：

平均自由径[编辑]
平均自由径λ是任一入射粒子在两次与靶核交互作用之间所能移动的平均距离。L是在单位时间dt、单位体积dV内所有未碰撞粒子移动的总距离，可用个别粒子所走距离l与总粒子数N的乘积表示：

l与N又可以用粒子速率v和单位体积粒子数n表示：

代入上式可得：

利用中子通量Φ的定义：^[1]

得到：

在这我们引入平均自由径λ，用未碰撞粒子移动的总距离L与发生的反应数目R来表示：

且：

导出：

在此，λ是微观平均自由径，Σ是巨观平均自由径。[/ltr][/size]