物理學 1949

量子電動力學的時空方法

理查·費曼

把圖畫出來，振幅便從圖上直接讀出。線、頂點，與那些圖。

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In depth · the introduction

要是那套關於光與物質如何相互作用、令人發狂的繁複算術，能被一幅塗鴉取代——就是你能在餐巾紙上隨手畫下的、寥寥幾條線和幾個點，會怎樣？

把這個想法拆開看

到了 1940 年代末，物理學家能寫下支配電子與光的定律，可真要拿它算出點什麼，卻意味著好幾頁磨人的代數，裡頭還遍布著無窮大，讓答案算出來淨是胡話。這套理論，在原理上行得通，一到實踐就卡住。

費曼的一招，是把圖畫出來。每個粒子化作一條線，每個兩粒子交互作用的瞬間化作一個線與線相交的點，整套糾纏的計算便化作一幅小小的圖。隨後，一套固定的規則把圖重新變回一個數——也就是這個過程的機率。不必再記帳，你只消畫出發生了什麼，再把答案讀出來。

它從哪裡來

戰後，實驗變得足夠精密，能看見原子裡那些舊理論無法解釋的微小移動。三位物理學家競相去修補它：朱利安·施溫格與朝永振一郎，造出了嚴謹、數學繁重的機器；而在康乃爾的費曼，造出了他的那些圖。這幾套辦法看上去如此不同，幾乎沒人相信它們竟是同一套理論——直到弗里曼·戴森證明，這三者給出的答案分毫不差。1965 年諾貝爾獎，由費曼、施溫格與朝永三人共享。

它為何重要

費曼的圖，不只與另兩套對上了——它們用起來還遠為輕便。一項專家要花上幾週的計算，如今只需畫出相關的幾幅圖，再把每幅翻譯成一個積分，便能架好。正是這份輕便，讓這些圖如野火般傳遍物理學，成了這一領域日常的語言。它們還捎帶著一個驚人的想法，借自斯蒂克爾貝格：一個反物質粒子——一個正電子——可以被當作一個逆著時間行進的、尋常的電子。

一種想像它的方式

把它想成一張電路接線圖。你並不去追蹤每一個電子；你只畫出元件與連接，幾條規則便告訴你電流是多少。費曼圖，就是粒子物理的接線圖：線是粒子，點是交互作用，而那些規則，把草圖換算成這個過程的強度。在下方玩一玩——滑動被交換的那個光子的能量，看看兩個電子彼此推搡得有多用力。

它落在何處

這是量子電動力學的實用引擎——那是人類檢驗得最精確的理論——而它，正從普朗克、愛因斯坦與狄拉克的量子革命中長出。這套「圖加規則」的方法，後來延續到了原子核內部的各種力；每當發現一種新粒子，物理學家最先畫下的，仍是這些圖。當你讀到大型強子對撞機裡的碰撞，那些正被檢驗的預言，正是以費曼圖架設出來的。

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