费曼，鼓
Feynman, The Drum
一、所有的路
粒子从A到B怎么走？
在一张纸上点出两个点，很容易顺手画一条线。倘若画的是一颗抛出去的球，线也许会弯起来。经典力学要我们说明起初的状态、受到的力，再算它怎样运动。路不是因为看上去最短，就被自然选中的。
费曼换了一种算法。
先看非相对论量子力学中的简单情形。给定起点、终点和相应的时刻，把连接它们的各种路径放进计算。每条路径贡献一个概率振幅。可以暂且把振幅想成有方向的小箭头：有些相加，有些抵消。先合成振幅，再把合成结果的大小平方，用来计算探测概率。把各条路的概率直接相加，会漏掉干涉这件事。
纸上有很多路。这里的热闹发生在数学的求和中。我们没有给粒子装上一部相机，看着它逐条游历宇宙，再赶回B点。
那条熟悉的经典路径怎样回来？在适当的经典极限下，许多贡献因相位变化而抵消，作用量驻定的路径附近可以留下集中的贡献。“驻定”说的是稍稍改变路径时，作用量的一阶变化为零；它不保证路程最短，甚至不总是作用量最小。
这仍然是一件令人高兴的事。同一个问题，不必永远沿着最初学会的那套写法往下做。1942年的博士研究，后来展开成1948年发表的时空表述；旧方程所能算出的东西，在另一种组织里显出了新的入口。
薛定谔已经在本系列中留下了一只猫和一张纸。费曼没有把那张纸撕掉。他让另一套写法来到旁边。不同的入口，可以接受同一套实验的追问。
人的生活却没有一项相应的求和，能替他把没有走的路也全部活过。选了一件事，时间就花在这里；答应了一个人，那个人会在具体的一天等他。
数学中的“所有”，到了生活里，要重新学习有限。
他后来拿起鼓，也没有因此不必再放下它。
二、他
理查德·费曼。1918年，生于纽约。
他回忆父亲时，常讲起制服。父亲做过制服生意，知道衣服里面仍然是一个要吃饭、要过日子的人。头衔可以告诉你他在什么位置，却不能替你检查他说的话。
家里还有妹妹琼。她后来也成为物理学家，研究太阳风与地球周围的空间。一个姓氏，通向了不同的工作。
少年费曼修收音机，拆开机壳，找究竟哪里出了问题。后来是麻省理工、普林斯顿，还有导师约翰·惠勒。1942年，他取得博士学位。一个问题会跟着他从一张桌子走到另一张，不因为换了校门就自动解决。
1943年，他来到洛斯阿拉莫斯，在贝特领导的理论部门工作。计算也需要机器和操作机器的人。娜奥米·利夫西等人的穿孔卡片工作，与理论人员的推算接在一起。后来故事里那个机灵的年轻人，并不是房间里唯一一双手。
他还研究锁，喜欢弄清楚一些看似严密的东西究竟怎样工作。柜门一开，很适合讲成笑话。柜子里放着的工作，却不能因此也变成一个没有后果的游戏。
工作之外，还有另一条路。
去阿尔伯克基的路。阿琳在那里疗养。洛斯阿拉莫斯的计算、借来的车、病房里的等待，并不占据同一种时间。再快的脑子，也不能把这一段路省掉。
在这段生活里，他有时是计算的人，有时是来探望的人。后一个身份，没有因为前一个身份后来得了奖，就变得次要。
三、阿琳
阿琳·格林鲍姆。费曼在学生时代就认识的女孩。
1942年6月29日，两人在纽约斯塔滕岛结婚。她已患上结核病。1943年春，他们来到新墨西哥，她住进阿尔伯克基的西南长老会疗养院。婚礼、远行与疗养，是一段日子里的不同页。
他们写信，也开玩笑。阿琳寄过写在拼图上的信，还为他的生日设计过一份夸张的报纸。她会出主意，会让通信里多出一点意料之外的东西。病人的地址没有把她变成一个只会等候探望的人。
两个人还想一起学做衣服，学中文，把一部电影放映机弄到手。后来读到这些计划，比读到“他爱她”三个字更慢一些：衣服要裁，字要学，机器要找地方放。生活曾经往前铺开，铺到了这些小事上。
1945年6月16日，阿琳去世。费曼二十七岁。
他回到工作中。日历继续往下翻。她没有回来。
1946年10月17日，他又给她写信。离她去世已经十六个月。他知道信不会送到，仍然想同她说话。那些共同的计划也回到纸上：他想照顾她，仍想听她的意见。
附言写着：“请原谅我没有把这封信寄出——我不知道你的新地址。”
这句话里有费曼式的轻轻一转，也有那个转不过去的地方。信能够写完，地址不能补上。
在这个系列里，艾米莉让感情成为荒原上的风，夏洛蒂让一个人站到另一个人面前说话。到阿琳这里，留下的是小得多的物件：拼图，报纸，一张未寄出的信。声音各不相同，不必排出哪一种更深。
阿琳也不必因为后来被一位物理学家如此怀念，才成为值得我们读到的人。她曾参与那段生活，给它添过自己的想法。她留下的，不只是他的悲伤。
纸能把一句话留住。不能把写信的人和收信的人，重新放回同一个房间。
四、在碎片上跳舞
在这座系列的桥上，亚里士多德铺地板，法拉第掀起木板，后面又有人带来场、光和分子的运动。费曼来到这里，也许会先敲一敲：这块木板发出什么声音？
他并没有等全部问题解决了，才允许自己做一点别的事。
鼓，巴西的桑巴，画纸，剧场。它们不只是“物理学家原来也会玩”的花边。要加入一段音乐，得听旁边的人什么时候进来；要画一个人，得让眼睛多停一会儿。
在他的巴西回忆里，参加桑巴队时，他敲的是一种像小煎锅的金属打击乐器。排练曾停下来，指出他的节奏出了问题。于是他拿着小棍，沿海边练手腕。后来学画，艺术家吉雷尔·佐西安先让他试画鞋与花盆。新东西来到手里，也会笨拙。
这使“玩”有了重量。不是随便碰一下就宣布自己什么都懂，而是愿意有一段时间不会。
战后在康奈尔，他曾经觉得自己做不出重要的东西了。按他后来的回忆，食堂里有人抛起一只盘子，红色校徽在转，盘面在晃。他看着看着，想算清这两种运动的关系。
他跑去同贝特讲。贝特问这有什么重要性。他说自己只是觉得好玩。那只盘子，在他的讲述里，成了重新愿意做物理的入口。
我们容易从后来回望，把盘子看成诺贝尔奖的预告。可当时吸引他的，是眼前这一点晃动。若每一次好奇都必须先填上将来的用途，它还没开始，就已经在替一个结果干活。
后来的研究当然还要面对同行、计算和实验。盘子让他回到桌边，不能代替桌上剩下的工作。
苏格拉底的追问不要求在散步前全部完成。费曼的鼓也不必等到心里再没有缺口才响。只是跳舞的人仍要知道，脚落在哪里，旁边是谁。
碎片上也能有节奏。碎片并不因此失去边角。
五、费曼图
一条直线，一条波浪线，几处相接的地方。
在量子电动力学的常见画法里，带箭头的实线与电子有关，波浪线与光子有关，连接处记下相互作用。纸面很省。笔划背后却各有运算规则。
1949年的论文里，费曼把一幅两电子相互作用的图放在公式旁边。图中的一处连接对应一个因子，内部发生相互作用的位置还要纳入求和或积分。它记录怎样组织一项计算，不是高速相机拍到的电子路线。波浪线也没有告诉我们，光子在空间里就那样一上一下地游过去。
困难没有被画没。它被安排得更容易找到。
少了一项，接错了一处，别人可以沿着这些线回去检查。这是一种能交到别人手里的清楚。画得轻快，不等于结果只凭画者的感觉成立。
1949年，戴森也发表论文，把费曼的方法与朝永—施温格一系的表述联系起来，说明它们之间的等价关系，并改进实际应用的组织。1965年，费曼与朝永振一郎、朱利安·施温格共同获得诺贝尔物理学奖。那条研究道路上，有彼此独立的工作，也有把不同工作接起来的人。
这个系列里的梵高，把他看见的世界留在颜色里。费曼把纷繁的计算画成了线。两张纸摆在面前，我们都可能忍不住凑近一点：原来还可以这样看。画前的人慢慢看；算式旁的人接着算，再把答案带回实验。那一点好奇，却很相像。
把图叫作舞谱，是一个读者的比喻。真正的舞者可以走错一步，再笑着回来；漏算的项，却不会因为作者笑了就自行补齐。
费曼留下的线，后来仍由别人使用。它们已经不必等画出它们的人坐到桌旁。
六、O型环
1986年1月28日。挑战者号发射。
七十三秒后，航天飞机结构解体。七名机组成员全部遇难。
费曼参加了事故调查。2月11日的公开听证会上，他把受压的密封材料浸在冰水里，再解除压力。材料没有立刻恢复原状。他请在场的人看这几秒钟的迟滞。
一小段材料。一个杯子。冷在这里不再只是报告里的温度数字。接缝需要及时密封时，这样的迟滞会意味着什么？
在那杯水之前，工程师已经提出过问题。
调查报告记下，发射前一晚，阿诺德·汤普森拿纸画出接缝，罗杰·博伊斯乔利拿出密封环受损的照片，继续反对在低温下发射。制造商管理层在内部讨论后改变了原先不建议发射的意见。警告、争执和改变，发生在火箭升空之前。
因而，后来被反复播放的演示旁边，还应放着那张画过接缝的纸。镜头里的清楚，不该使先前已经说过话的人再次消失。
冰水没有独自完成整场调查。接缝的设计、以往的损伤、温度资料、信息怎样向上传递，都要接进来。费曼在报告附录中也追问：过去出现过异常却仍安全返回，为什么会被逐渐当成以后也能接受的理由？
一次没有出事，不是下一次的保证。对坐进航天飞机的人，这不是一句关于概率的机智话。
附录最后留下那句：“自然是不能被愚弄的。”
那天演示结束，材料会慢慢回暖。需要重新检查的东西，却不只有杯子里的那一小段。
七、他和奥本海默
奥本海默在这座桥上背着灰。费曼带来鼓。
这很容易被看成两种已经解决的人生：一个忧虑，一个轻松。但鼓声里也有一段费曼自己不肯略去的记录。
1981年的访谈中，他回到参加原子弹工作的起点。他说，最初推动他的，是德国可能造出这种武器的危险。德国战败之后，他却没有重新问自己，为何仍在继续。这是他对自己当年行为的追问。
他还记得广岛消息传来后的庆祝：自己坐在吉普车前盖上敲鼓。讲述那一幕时，他把洛斯阿拉莫斯的兴奋同广岛正在发生的死亡并置。后来在纽约，看见有人造桥，他又觉得那些建设很快都会被毁去。鼓后的日子，没有始终保持同一种轻快。
承认这一段，并没有把那一天取消。持续忧虑，也不会使已经发生的工作从记录里退出。奥本海默与费曼各自留下的文字，需要分别读；灰和鼓不能替他们把责任分配完毕。
这里的难处，不是要求一个人从此再也不能笑。把所有活着的日子都变成惩罚，仍然不能替别人多活一天。难处是笑起来以后，是否还容得下关于过去的提问。
阿琳的死亡是他切身的失去。远处那些人的死亡，同样各有不能替代的生活。一个人知道自己的痛，并不因此已经知道所有人的痛；愿意继续活，也不意味着可以把别人的痛关在故事外面。
这段鼓声，令本篇的鼓变得不那么容易听。它可以是探索的快乐，也可能被卷进一次尚未认真追问的庆祝。乐器没有替拿着它的人作选择。
后来有人仍愿意学他的物理，听他讲笑话。读到令人愉快的地方，也可以把书翻回这一页。
灰还在。鼓也还在。它们在这里相遇，并没有互相抵消。
八、他说的话
费曼谈量子力学时，会先提醒听者：那些熟悉的小球和水波，都不能完整承担我们在这里要描述的行为。习惯的画面不够用了，问题却仍然可以计算，可以拿实验来问。
这样的提醒，并不是把门关上。更像是在门口说：带着旧地图进来可以，但别因为某条路没有画在上面，就拒绝看脚下。
还有一句，来自1974年的毕业演说：“首要原则是别欺骗自己，而你是最容易被自己骗过的人。”
他说的是研究中的具体诚实。那些不利于结论的结果、可能使解释失效的条件，也应当交出来。把最顺手的证据排成一列，很容易觉得自己已经把事情做完；别人却需要看见没有站进这一列的东西。
这句话同样可以放在讲故事的费曼旁边。童年的聪明、食堂的盘子、听证会的杯子，都很容易被编成一条毫无停顿的直线。细读时，旁边还有教过他的人、先前的工作、未作出的追问。
“不知道”因此也有几种不同的分量。有些不知道，让人继续查；有些不知道，要求行动之前更谨慎。一个人可以容许宇宙还有谜，却不能把一句“谁知道呢”交给将要承担风险的人，当作已经尽到说明。
在1981年的谈话末尾，他谈到自己能够与疑惑和不确定一起生活，对不同的事持有不同程度的把握。未知没有使他失去探索的兴致。
这是可以读出的活法，不是一张要求别人同样轻松的处方。有人此刻没有力气敲鼓，仍然可以坐在旁边。好奇不该变成另一种必须表现给人看的精神饱满。
那封没有地址的信，已经把这一点说得很轻。问不出的去处，仍然会让人想念。手可以停一会儿，再去做今天的一件事。
九、鼓
1988年2月15日，费曼在洛杉矶去世。六十九岁。
他的线仍在论文里。他的声音仍在录音里。那封没有寄出的信，也留在纸上。
现在，让他走到这座想象的桥头。
鼓挂在身前。他的手落上去，敲出两下，又添了几下。声音先到了，人才从桥边走出来。头发有些乱，身上没有一套能替他说明一切的制服。
苏格拉底站在空地上。柏拉图蹲着画图纸。休谟守着双陆棋盘。叔本华看桥底下。克尔凯郭尔跳了。图灵看着手里的苹果。契诃夫靠着栏杆。康托尔看天上。哥白尼放下书走了。萨特拿着烟斗踱步。波伏瓦举着镜子。蒯因轻轻说了一句话。
特斯拉听嗡嗡声。爱迪生拿着没有亮的灯泡。海森堡的位置不确定。玻尔拿着没寄出的信。托尔斯泰拿着药方，面对契诃夫。莎士比亚不在岸上——他是桥下面的水。斯宾诺莎手上有玻璃粉。亚里士多德蹲着铺地板。法拉第掀起一块木板。麦克斯韦站着写方程。
贞德带着火在桥的上方。王尔德站得很好看，手里拿着那句话。拉马努金从缝隙里探出半个身子。奥本海默背着灰。夏洛蒂拿着笔。艾米莉在桥外的荒原上，风从她那里吹来。玻尔兹曼抱着刻有方程的石头。梵高的画笔还在滴颜色。狄更斯站在裂缝上，笔和火柴盒都在手里。丘吉尔扶着肩上的旗。罗斯福的轮椅停在缓坡旁。薛定谔抱着猫。
费曼走到奥本海默身边，鼓声低了一点。他看见衣服上的灰，没有伸手替他拍掉。两个人站了一会儿，谁也没有用一句笑话把那一会儿省掉。
玻尔兹曼挪动怀里的石头。费曼让开一步，等他站稳，再敲一小段节奏。石头很沉，脚步也沉。鼓声跟着那脚步走了几下，轻轻转了一个弯。
薛定谔怀里的猫抬起头。费曼停住手，伸出一根手指。猫闻了闻，转过脸去，仍然舒服地窝着。他收回手，笑了一下，又低低敲了两声。猫没有再抬头。
鼓声又响起来，混进水声里。丘吉尔那段话的回声还在桥远处。这里不必决定谁的声音更能传得远。休谟移动一颗棋子，狄更斯脚下的纸也轻轻响了一声。
费曼的衣袋里放着一封信。走动的时候，纸边碰到衣服。他把一只手从鼓上挪开，隔着布按了一下。信没有地址，鼓也不能把它送走。
远处，康德已经站在那里。鼓声传过去，他转过脸，等费曼走近一点。
费曼向他走去，经过别人修过的木板和仍待修整的接缝。他有时敲鼓，有时停下来听；停下的时候，脚步还在继续。
亚里士多德抬头看了一眼，手里仍托着那块木板。费曼经过他身旁，把声音放轻，等他把板子放好。桥并不因为有人跳舞，就不再需要有人照料。
风从荒原吹来。费曼嘴角动了一下。
也许是笑。鼓暂时停了。信还在。
注释
[1] 路径积分及经典极限参《费曼物理学讲义》II卷第19章；1949年论文的图与运算见772–773页。盘子、桑巴及学画来自事后回忆，不能代替完整的科学发现史。
[2] 阿琳的经历参洛斯阿拉莫斯档案研究介绍；1946年10月17日未寄信使用经许可重刊的文本。挑战者号冰水演示见1986年2月11日听证记录679–680页，工程师预警见调查报告第五章；战争反思来自1981年访谈。
[3] 第九节为文学想象。鼓、信与猫延续本系列的群像；快乐不替战争责任免责，科学比喻也不替一个人作出生活的选择。
I. All Paths
How does a particle get from A to B?
Put two dots on a sheet of paper and the hand almost draws the line for you. If the picture is of a ball being thrown, the line may curve. Classical mechanics asks for the initial conditions and the forces, then calculates the motion. Nature does not choose a route simply because it looks shortest on the page.
Feynman found another way to calculate.
Begin with a simple case in nonrelativistic quantum mechanics. Fix the starting and finishing points, and the times at which the particle is to be there. Bring the paths connecting them into the calculation. Each contributes a probability amplitude. For the moment, think of amplitudes as little arrows: some reinforce one another, some cancel. Add the amplitudes first, then use the squared magnitude of the result to calculate a detection probability. Adding probabilities for the individual paths instead would leave interference out of the story.
There are many paths on the paper. This bustle belongs to a mathematical sum. We have not fitted a camera to the particle and watched it tour the universe, route by route, before hurrying back to B.
How does the familiar classical trajectory return? In an appropriate classical limit, rapidly changing phases cancel many contributions, while contributions near a path of stationary action can accumulate. “Stationary” means that a small change in the path produces no first-order change in the action. It promises neither the shortest distance nor, in every case, the least action.
There is pleasure in this nonetheless. A question need not always be worked through in the first notation one learned. The doctoral work of 1942 developed into the space-time formulation published in 1948. Results accessible through older equations acquired another entrance.
Schrödinger has already left a cat and a sheet of paper in this series. Feynman does not tear the paper up. He brings another formulation alongside it. Both entrances can be questioned by the same experiments.
A human life offers no corresponding sum through which we can also live every road we did not take. Choose one thing, and time is spent on it. Make a promise to someone, and that person will wait on a particular day.
Mathematics has its “all.” In life we must learn again how much can fit into a day.
Taking up a drum would not release him from the need to put it down.
II. The Man
Richard Feynman. Born in New York in 1918.
When he recalled his father, he often spoke about uniforms. His father had worked in the uniform business. He knew that inside the clothes was still someone who needed to eat and get through a day. A title could tell you the position a person occupied. It could not check what the person said.
There was also his sister Joan. She became a physicist too, studying the solar wind and the space around Earth. One family name led to different work.
As a boy, Richard repaired radios. He opened the casing and tried to find out what had gone wrong. Later came MIT, Princeton, and his doctoral supervisor, John Wheeler. He received his doctorate in 1942. A problem could follow him from one desk to the next; passing through a different university gate did not solve it.
In 1943 he arrived at Los Alamos, joining the theoretical division led by Hans Bethe. Calculation also required machines and people to operate them. Naomi Livesay and others worked with punched cards, connecting their labor with the theorists' calculations. The quick-witted young man in the later stories was not the only pair of hands in the room.
He investigated locks as well. He wanted to know how something apparently secure actually worked. An open cabinet door makes a good ending to a joke. The work inside the cabinet does not thereby become a game without consequences.
Outside the work, another road awaited him.
The road to Albuquerque, where Arline was in a sanatorium. Calculations at Los Alamos, a borrowed car, waiting in a sickroom: these made different demands on time. Even the quickest mind could not make the journey unnecessary.
In this part of his life he was sometimes the man doing the calculations, sometimes the man coming to visit. The second role did not become less important because the first would one day bring a prize.
III. Arline
Arline Greenbaum. The girl Feynman had known since his school days.
They married on Staten Island, New York, on June 29, 1942. She already had tuberculosis. In the spring of 1943 they went to New Mexico, where she entered the Southwest Presbyterian Sanatorium in Albuquerque. The wedding, the journey, and the stay in the sanatorium occupied different pages of their lives.
They wrote letters and played jokes. Arline sent a letter written on a jigsaw puzzle and designed an extravagant newspaper for his birthday. She had ideas of her own. She could make something unexpected happen in the post. A patient's address had not reduced her to someone who only waited for a visitor.
They also wanted to learn to make clothes together, learn Chinese, get hold of a movie projector. Reading those plans takes longer than reading “he loved her.” Cloth needs cutting. Characters need learning. A machine needs somewhere to stand. Their life had opened ahead of them as far as these small things.
Arline died on June 16, 1945. Feynman was twenty-seven.
He returned to work. The calendar went on turning. She did not return.
On October 17, 1946, sixteen months after her death, he wrote to her again. He knew the letter could not arrive. He still wanted to speak to her. The plans they had made returned to the page: he wanted to care for her, still wanted to hear her ideas.
The postscript reads: “Please excuse my not mailing this—but I don't know your new address.”
There is the small Feynman turn in the sentence, and the place where no turn is possible. He could finish the letter. He could not fill in the address.
In this series, Emily has made feeling into the wind over a moor. Charlotte has let one person stand before another and speak. Around Arline, smaller things remain: a puzzle, a newspaper, an unmailed sheet of paper. These are different voices. They do not have to compete over which runs deepest.
Nor does Arline become worth our attention only because a physicist later missed her this much. She had helped make that life, adding ideas of her own. What she left was more than his grief.
Paper can keep a sentence. It cannot put its writer and its reader back in the same room.
IV. Dancing on the Fragments
On this series' bridge, Aristotle lays the floor and Faraday lifts a plank. Others arrive with fields, light, and the motion of molecules. Feynman might begin by tapping the wood. What sound does this piece make?
He did not wait for every problem to be solved before allowing himself something else to do.
Drums, samba in Brazil, drawing paper, the theater. More is at stake here than the novelty of a physicist who also has hobbies. To join a piece of music, you must listen for the person beside you. To draw someone, you must let your eyes stay a little longer.
In his recollections of Brazil, the instrument he played in the samba group was a metal percussion instrument resembling a little frying pan. A rehearsal stopped because something was wrong with his rhythm. He went along the beach with his stick, practicing the movement of his wrist. When he later learned to draw, the artist Jirayr Zorthian began by having him try a shoe and a flowerpot. Something new in the hands can make them clumsy.
That gives play a little weight. It involves being willing not to know how, for a while, rather than touching something once and claiming mastery.
After the war, at Cornell, he had felt unable to do important work. In his later account, someone in the cafeteria tossed a plate into the air. The red university emblem turned while the plate wobbled. He watched and wanted to calculate how the two motions were related.
He took the problem to Bethe. Asked what made it important, he said he was doing it for the fun of it. In the story he told afterward, the plate became a way back into wanting to do physics.
From the far end of the story, it is tempting to see the plate as an announcement of the Nobel Prize. What caught his eye at the time was that small wobble. Require curiosity to name its eventual use before it can begin, and it is already working for an outcome someone has assigned it.
The research that followed still had to meet colleagues, calculations, and experiments. The plate brought him back to the desk. It could not do the work waiting there.
Socrates need not finish every question before going for a walk. Feynman's drum need not wait until nothing is missing from his life. The dancer must still notice where his foot will land, and who is beside him.
There can be rhythm on fragments. They do not lose their edges.
V. Feynman Diagrams
A straight line, a wavy line, a few places where they meet.
In a familiar convention of quantum electrodynamics, solid lines with arrows concern electrons; wavy lines concern photons; junctions mark interactions. The drawing is economical. Every stroke carries rules for a calculation.
In his 1949 paper, Feynman placed a diagram of two electrons interacting beside an equation. A junction in the diagram corresponded to a factor. The internal locations of interactions still had to be included in sums or integrals. The diagram organized a calculation; it was not an electron's route captured by a high-speed camera. The wavy line did not report a photon swimming up and down through space.
The difficulty had not been drawn away. It had been arranged so that one could find it.
Leave something out, connect something incorrectly, and another person can go back along the lines and check. Here was a clarity that could pass into someone else's hands. The ease of the drawing did not make its maker's feeling the test of the result.
In 1949 Dyson also published a paper connecting Feynman's method with the Tomonaga–Schwinger formulations, showing their equivalence and improving the organization of their practical application. In 1965 Feynman shared the Nobel Prize in Physics with Shin'ichirō Tomonaga and Julian Schwinger. There were independent lines of work, and people who brought different lines together.
Van Gogh, in this series, leaves the world he has seen in color. Feynman turns a thicket of calculation into lines. Either sheet can draw us closer: so there is another way to look. Before the painting we linger; beside the diagram we calculate, then take the answer back to experiment. The work differs. That first flicker of curiosity feels familiar.
To call the diagram a dance score is a reader's metaphor. A dancer can miss a step and laugh on the way back. Laughter will not restore a missing term to a calculation.
Other people went on using Feynman's lines. They no longer needed the man who drew them to sit beside the page.
VI. The O-Ring
January 28, 1986. The launch of Challenger.
Seventy-three seconds later, the shuttle broke apart. All seven crew members died.
Feynman joined the investigation. At the public hearing on February 11, he immersed a compressed piece of sealing material in ice water, then released the pressure. It did not immediately recover its shape. He asked those present to look at those seconds of delay.
A small piece of material. A cup. Cold was no longer only a number in a report. What could such a delay mean when a joint needed to seal in time?
Before the cup of water, engineers had already raised the problem.
The commission's report records Arnold Thompson drawing the joint on paper the night before launch. Roger Boisjoly produced photographs of damage to the seals. They continued to oppose launching in the cold. After an internal discussion, the manufacturer's managers reversed their earlier recommendation against launch. The warnings, the argument, and the change of recommendation all preceded liftoff.
Beside the demonstration that would be played over and over, there should also be the sheet on which an engineer drew the joint. The clarity of the camera's image should not make those who had already spoken disappear again.
Ice water did not complete the investigation by itself. Joint design, earlier damage, temperature records, and the way information traveled upward all had to be brought together. In his appendix to the report, Feynman also asked why a flight that returned safely despite an anomaly gradually became a reason to accept that anomaly again.
Getting through once is no guarantee of getting through the next time. For the people boarding the shuttle, this is more than a clever remark about probability.
At the end of the appendix are the words “nature cannot be fooled.”
After the demonstration, the material would slowly warm. It was not the only thing that needed another look.
VII. Feynman and Oppenheimer
Oppenheimer carries ash on this bridge. Feynman brings a drum.
It is easy to turn them into two settled ways of living: one anxious, one lighthearted. Yet the drum contains a memory Feynman himself did not wish to leave out.
In an interview in 1981, he returned to his reasons for joining the atomic-bomb project. He recalled the danger that Germany might build the weapon. After Germany's defeat, however, he had failed to ask himself again why he was continuing. Many years later, he was questioning what he had done then.
He also remembered the celebration when the news from Hiroshima arrived: sitting on the hood of a jeep, beating a drum. In telling it, he set the excitement at Los Alamos beside the deaths in Hiroshima. Later, in New York, he saw people building bridges and felt that their work would soon be destroyed. The days after the drumming did not keep a single lighthearted rhythm.
Acknowledging that memory did not cancel the day. Continuing to worry would not remove the work from the record either. Oppenheimer's writings and Feynman's must each be read. Ash and drum cannot finish assigning responsibility for them.
The difficulty is not that a person must never laugh again. Making every remaining day a punishment would not give someone else another day to live. The difficulty is whether, once laughter returns, questions about the past are still allowed into the room.
Arline's death was his own intimate loss. The people who died far away each had irreplaceable lives as well. Knowing one's own pain does not supply knowledge of everyone else's. Wanting to go on living does not authorize a story to shut their pain outside.
This remembered beat makes the drum in our essay harder to hear simply. It can carry the pleasure of discovery. It can also become part of a celebration whose reasons have not been examined. The instrument has made no choice on its player's behalf.
Later readers may still want to learn his physics and listen to his jokes. At a passage that gives them pleasure, they can also turn back to this page.
The ash remains. So does the drum. Meeting here does not make either cancel the other.
VIII. What He Said
When Feynman spoke about quantum mechanics, he warned his listeners that familiar little balls and water waves could not fully picture the behavior being described. The habitual image was insufficient. The problem could still be calculated, still put to an experiment.
Such a warning need not close a door. It is more like someone at the entrance saying: bring the old map, but do not refuse to look at the ground because a road is missing from it.
Another sentence comes from his commencement address of 1974: “The first principle is that you must not fool yourself—and you are the easiest person to fool.”
He was speaking of a particular honesty in research. Results that count against a conclusion, conditions under which an explanation might fail: these must be disclosed too. Line up the evidence that comes most readily to hand and it is easy to feel the job is done. Another person needs to see what has not been allowed into the line.
We can place this sentence beside Feynman the storyteller as well. The clever child, the plate in the cafeteria, the cup at the hearing can readily become one unbroken line. Read more closely, and other people appear beside it: teachers, earlier workers, questions he did not ask.
Not knowing therefore has different weights. Sometimes it sends a person back to investigate. Sometimes it requires greater caution before acting. One can live with mysteries in the universe without handing “Who knows?” to someone about to bear a risk as though an explanation had been provided.
Near the end of the 1981 conversation, Feynman spoke of living with doubt and uncertainty, holding different degrees of confidence about different things. The unknown had not taken away his appetite for discovery.
A reader can find a way of living in this. It is not a prescription requiring everyone else to feel equally light. Someone without the strength to play a drum today may still sit beside it. Curiosity must not become another demand to look full of life for an audience.
The letter without an address has already said this very quietly. The absence of an address does not put an end to missing someone. The hands can rest before turning to one thing that needs doing today.
IX. The Drum
Feynman died in Los Angeles on February 15, 1988. He was sixty-nine.
His lines remain in papers. His voice remains in recordings. The letter he did not send remains on paper too.
Now let him come to this imagined bridge.
The drum hangs in front of him. His hands fall on it twice, then add a few more beats. The sound arrives before the man steps out from the end of the bridge. His hair is untidy. No uniform he could wear would explain everything about him.
Socrates stands in the clearing. Plato crouches over his blueprints. Hume is at his backgammon board. Schopenhauer looks beneath the bridge. Kierkegaard has made his leap. Turing looks at the apple in his hand. Chekhov leans against the railing. Cantor looks up at the sky. Copernicus has put down his book and walked away. Sartre paces with his pipe. Beauvoir holds a mirror. Quine has said something quietly.
Tesla listens to a hum. Edison holds an unlit bulb. Heisenberg's position is uncertain. Bohr holds an unsent letter. Tolstoy faces Chekhov with a prescription. Shakespeare is not on the bank: he is the water below the bridge. Spinoza has glass dust on his hands. Aristotle crouches to lay the floor. Faraday lifts a plank. Maxwell stands writing equations.
Joan of Arc carries fire above the bridge. Wilde stands with easy elegance, holding that sentence. Ramanujan has emerged halfway through a gap. Oppenheimer carries ash. Charlotte holds a pen. Emily is on the moor beyond the bridge; the wind comes from where she is. Boltzmann cradles a stone engraved with an equation. Van Gogh's brushes are still dripping color. Dickens stands over a crack with pen and matchbox in his hands. Churchill steadies the flag on his shoulder. Roosevelt's wheelchair is beside the gentle ramp. Schrödinger holds a cat.
Feynman goes up to Oppenheimer. The drum grows quieter. He sees the ash on the clothes and does not reach out to brush it off. The men stand together for a while. Neither saves himself that interval with a joke.
Boltzmann shifts the stone in his arms. Feynman steps aside and waits until he has his balance, then plays a short rhythm. The stone is heavy; so are the footsteps. For a few beats the drum follows them, then takes a small, light turn.
The cat in Schrödinger’s arms looks up. Feynman stops playing and offers a finger. The cat sniffs, turns away, and stays comfortably curled up. He withdraws his hand, smiles, and taps the drum twice, softly. The cat does not look up again.
The drum starts again, mixing with the water. An echo of Churchill's speech remains at the far end of the bridge. There is no need to decide whose sound carries farther. Hume moves a piece. The paper beneath Dickens's feet makes a small noise of its own.
A letter lies in Feynman's pocket. As he walks, the edge of the paper brushes against the cloth. He lifts one hand from the drum and presses it there. The letter has no address. The drum cannot send it.
In the distance, Kant is already standing. He hears the drum, turns his face, and waits for Feynman to come a little closer.
Feynman walks toward him, over planks other people have repaired and joints still needing work. Sometimes he plays; sometimes he stops to listen. When his hands stop, his feet go on.
Aristotle looks up, still supporting a plank. Passing him, Feynman quiets the drum and waits for the wood to be set in place. Someone dancing on the bridge does not remove its need for care.
The wind comes from the moor. The corner of Feynman's mouth moves.
Perhaps a smile. The drum rests for a moment. The letter is still there.
Notes
[1] For path integrals and the classical limit, see The Feynman Lectures, II.19; for diagrams and calculations, the 1949 paper, pp. 772–773. The plate, samba and drawing stories are later recollections, not a complete history of discovery.
[2] Arline’s life is documented in the Los Alamos archival accounts; the letter dated October 17, 1946 is available in an authorized reprint. The ice-water demonstration appears in the February 11, 1986 hearing, pp. 679–680; earlier engineering warnings appear in Chapter V of the report. The wartime reflections come from the 1981 interview.
[3] Section IX is literary imagination. The drum, letter and cat continue the series’ ensemble. Pleasure does not cancel responsibility, and a scientific metaphor cannot make a life’s choices.