物理学 1949

量子电动力学的时空方法

理查德·费曼

把图画出来，振幅便从图上直接读出。线、顶点，与那些图。

Choose your version

In depth · the introduction

要是那套关于光与物质如何相互作用、令人发狂的繁复算术，能被一幅涂鸦取代——就是你能在餐巾纸上随手画下的、寥寥几条线和几个点，会怎样？

把这个想法拆开看

到了 1940 年代末，物理学家能写下支配电子与光的定律，可真要拿它算出点什么，却意味着好几页磨人的代数，里头还遍布着无穷大，让答案算出来净是胡话。这套理论，在原理上行得通，一到实践就卡住。

费曼的一招，是把图画出来。每个粒子化作一条线，每个两粒子相互作用的瞬间化作一个线与线相交的点，整套纠缠的计算便化作一幅小小的图。随后，一套固定的规则把图重新变回一个数——也就是这个过程的概率。不必再记账，你只消画出发生了什么，再把答案读出来。

它从哪里来

战后，实验变得足够精密，能看见原子里那些旧理论无法解释的微小移动。三位物理学家竞相去修补它：朱利安·施温格与朝永振一郎，造出了严谨、数学繁重的机器；而在康奈尔的费曼，造出了他的那些图。这几套办法看上去如此不同，几乎没人相信它们竟是同一套理论——直到弗里曼·戴森证明，这三者给出的答案分毫不差。1965 年诺贝尔奖，由费曼、施温格与朝永三人共享。

它为何重要

费曼的图，不只与另两套对上了——它们用起来还远为轻便。一项专家要花上几周的计算，如今只需画出相关的几幅图，再把每幅翻译成一个积分，便能架好。正是这份轻便，让这些图如野火般传遍物理学，成了这一领域日常的语言。它们还捎带着一个惊人的想法，借自斯蒂克尔贝格：一个反物质粒子——一个正电子——可以被当作一个逆着时间行进的、寻常的电子。

一种想象它的方式

把它想成一张电路接线图。你并不去追踪每一个电子；你只画出元件与连接，几条规则便告诉你电流是多少。费曼图，就是粒子物理的接线图：线是粒子，点是相互作用，而那些规则，把草图换算成这个过程的强度。在下方玩一玩——滑动被交换的那个光子的能量，看看两个电子彼此推搡得有多用力。

它落在何处

这是量子电动力学的实用引擎——那是人类检验得最精确的理论——而它，正从普朗克、爱因斯坦与狄拉克的量子革命中长出。这套「图加规则」的方法，后来延续到了原子核内部的各种力；每当发现一种新粒子，物理学家最先画下的，仍是这些图。当你读到大型强子对撞机里的碰撞，那些正被检验的预言，正是以费曼图架设出来的。

The original document

Original source text

R. P. Feynman · Department of Physics, Cornell University · Physical Review 76, 769–789 (received May 9, 1949; published September 15, 1949)

Abstract — what the paper does

In this paper two things are done. (1) It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem.

The abstract then turns to the second task — modifying the interaction of electrons at short distances so that the matrix elements become finite — and notes that for directly observable phenomena the results agree with those of Schwinger.

A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons.

The space-time viewpoint

The simplification in writing the expressions results from an emphasis on the over-all space-time view resulting from a study of the solution of the equations of electrodynamics.

Feynman flags that this paper is a direct continuation of his companion paper, "The Theory of Positrons" (Phys. Rev. 76, 749, 1949), referred to throughout simply as "I."

Why processes can be combined

For example, in the exchange of a quantum between two electrons there were two terms depending on which electron emitted and which absorbed the quantum. Yet, in the virtual states considered, timing relations are not significant.

Because the timing is not significant — only the order of operators in the matrix must be kept — separate terms that older methods wrote apart can be combined into one, and the relativistic invariance of the result becomes self-evident.

[ … ]

The cost, stated honestly

Unfortunately, the modification proposed is not completely satisfactory theoretically (it leads to some difficulties of conservation of energy). It does, however, seem consistent and satisfactory to define the matrix element for all real processes as the limit of that computed here as the cut-off width goes to zero.

The body of the paper writes down the self-energy expression, applies the cut-off to make it finite, treats vacuum polarization in a manner suggested by Pauli and Bethe, and works illustrative examples; the integrals are evaluated by methods collected in the appendix. The complete paper, with its diagrams, runs from page 769 to 789 and is available in full at the source below.

Cornell University · received May 9, 1949