Now in quantum mechanics it turns out thatmomentum is a different thing—it is no longer mv . It is hard to define exactly what is meant by the velocity of a particle,but momentum still exists. In quantum mechanics the difference is that when theparticles are represented as particles, the momentum is still mv , but when the particles are represented as waves, the momentum ismeasured by the number of waves per centimeter: the greater this number ofwaves, the greater the momentum. In spite of the differences, the law of conservationof momentum holds also in quantum mechanics. Even though the law F=mais false, and all the derivations of Newton were wrong for theconservation of momentum, in quantum mechanics, nevertheless, in the end, thatparticular law maintains itself!
现在，在量子力学中，结果就是，动量是一个不同的事物，而不再是mv了。一个粒子的矢速，究竟意味着什么，很难定义，但是，动量仍然存在。在量子力学中，区别就是，当粒子是被作为粒子来表现时，动量就仍是mv，但是，当粒子是作为波来表现时，动量就是通过每厘米波的数目，来测量的，这个波的数目越大，动量就越大。除了这点之外，动量守恒的规律，在量子力学中，照样成立。在量子力学中，尽管规律F=ma是错的，对于动量守恒来说，所有牛顿规律的推论都是错的，但是最终，粒子的规律，还能成立！
脚注
1. H. V. Neher and R. B. Leighton, Amer. Jour. of Phys. 31,255 (1963).
1、内赫，和雷顿，《物理学期刊》，31,255（1963）。

1 Chapter11.Vectors第11章 矢量

11–1Symmetry in physics 11-1 物理学中的对称
In this chapter we introduce a subject thatis technically known in physics as symmetry in physical law. The word“symmetry” is used here with a special meaning, and therefore needs to bedefined. When is a thing symmetrical—how can we define it? When we have a picturethat is symmetrical, one side is somehow the same as the other side. Professor HermannWeyl has given this definition of symmetry: a thing is symmetrical if one cansubject it to a certain operation and it appears exactly the same after theoperation. For instance, if we look at a silhouette of a vase that isleft-and-right symmetrical, then turn it 180∘ around the vertical axis, it looks the same. We shall adopt thedefinition of symmetry in Weyl’s more general form, and in that form we shalldiscuss symmetry of physical laws.
在这一章，我们要介绍的主题，是物理规律中的对称，它在物理学中有严格的意义。单词“对称”，在这里有着特殊的意义因此需要定义。一个事物，什么时候才是对称的--我们如何才能定义它呢？如果我们有一张图片，它是对称的，那么，它的一边，在某种意义上，与另一边是一样的。赫尔曼威尔教授，曾给出了这个对称的定义：一个事物，如果可以让它从属于某一操作，在此操作之后，它看上去，几乎一样，那么它就是对称的。例如，我们看一个左右对称的花瓶的轮廓像，然后，把它绕着垂直轴，转180度，它看起来是一样的。我们将在赫尔曼威尔的更广泛的形式上，采取这个关于对称的定义，我们将在这个形式上，讨论物理规律的对称。

Suppose we build a complex machine in a certainplace, with a lot of complicated interactions, and balls bouncing around withforces between them, and so on. Now suppose we build exactly the same kind ofequipment at some other place, matching part by part, with the same dimensionsand the same orientation, everything the same only displaced laterally by somedistance. Then, if we start the two machines in the same initial circumstances,in exact correspondence, we ask: will one machine behave exactly the same asthe other? Will it follow all the motions in exact parallelism? Of course theanswer may well be no, because if we choose the wrong place for ourmachine it might be inside a wall and interferences from the wall would make themachine not work.
假设我们在某地建，建立一种复杂的机器，有很多复杂的交互作用，球在它们之间，来回有力地反弹着，等等。现在，假设我们在另外一个地方，建立同样的设备，每一部分都匹配，有着同样的维度，和同样的方向，所有方面都一样，只是放在前者侧面，有些距离。因此，如果我们在相同的初始条件下，以精确的相应{每种条件都一样？}，启动这两台机器，我们问：一台机器的表现，会与另一台完全一样吗？它会以完全相似的方式，做所有的运动吗？当然，答案完全可能是否，因为，如果我们给我们的机器，选择了错误的地点，它可能在一堵墙里面，来自墙的干扰，可能会让机器无法工作。

All of our ideas in physics require a certainamount of common sense in their application; they are not purely mathematicalor abstract ideas. We have to understand what we mean when we say that thephenomena are the same when we move the apparatus to a new position. We meanthat we move everything that we believe is relevant; if the phenomenon is notthe same, we suggest that something relevant has not been moved, and we proceedto look for it. If we never find it, then we claim that the laws of physics donot have this symmetry. On the other hand, we may find it—we expect to find it—ifthe laws of physics do have this symmetry; looking around, we may discover, forinstance, that the wall is pushing on the apparatus. The basic question is, ifwe define things well enough, if all the essential forces are included insidethe apparatus, if all the relevant parts are moved from one place to another,will the laws be the same? Will the machinery work the same way?
我们关于物理学的所有想法，在它们的应用中，都要求一定量的常识；它们并不是纯粹数学的或抽象的想法。当我们说：‘当我们把仪器移到一个新的位置时，现象是一样的’，这种说法意味着什么，我们必须要理解。我们的意思是，所有我们相信是有关的东西，我们都移动了；如果现象并不一样，我们会认为，某个相关的东西，并未被移动，我们会去寻找它。如果我们永远都找不到它，那么，我们将会声明，物理学的规律，没有这种对称。另一方面，如果物理学的规律确实有这种对称，那么，我们可能会发现它，这也是我们的期待；例如，环顾四周，我们可能会发现，墙可能正在阻拦着仪器。基本的问题就是，如果我们把事物，定义地足够好，如果所有基本的力，都被包含在了仪器内，如果所有相关部分，都被从一个地方，移动到另一个地方，那么，规律还会一样吗？这个机器装置还会以同样的方式工作吗?

It is clear that what we want to do is tomove all the equipment and essential influences, but not everythingin the world—planets, stars, and all—for if we do that, we have the samephenomenon again for the trivial reason that we are right back where westarted. No, we cannot move everything. But it turns out in practicethat with a certain amount of intelligence about what to move, the machinery willwork. In other words, if we do not go inside a wall, if we know the origin ofthe outside forces, and arrange that those are moved too, then the machinery willwork the same in one location as in another.
有一点很清楚，我们想移动的，是所有的装备和基本影响，而不是世界上的所有事物—行星、恒星、及所有事物—因为，如果我们那样做，我们所拥有的现象，是会一样，原因很简单，我们只是回到了起点而已。不，我们不能移动所有事物。从实践上看，结果就是，对于什么应该移动，使用了一定的智慧后，此机械装置就将工作。换句话说，如果我们不走进墙内，如果我们知道外部力的起源，并且安排好那些被移动的东西{？}，那么，此机械装置，在一个位置上的工作运行，将与在另一位置上的一样。

11–2Translations 11-2 转换
We shall limit our analysis to just mechanics,for which we now have sufficient knowledge. In previous chapters we have seenthat the laws of mechanics can be summarized by a set of three equations foreach particle:
我们的分析，将只局限于力学，对它我们现在有充分的知识。在前面的章节中，我们已经看到，每个粒子的力学规律，可以被总结为一组方程，共三个：
m(d2x/dt2)=Fx, m(d2y/dt2)=Fy, m(d2z/dt2)=Fz. (11.1)
Now this means that there exists a way to measure x , y , and z on three perpendicular axes, and the forces along those directions,such that these laws are true. 现在，这就意味着，存在一种方法，可以测量在三个互相垂直的轴上的x , y ,和z，及沿着这些方向的力，这样，这些规律就是真的。

These must be measured from some origin,but where do we put the origin? All that Newton would tell us at firstis that there is some place that we can measure from, perhaps the centerof the universe, such that these laws are correct. But we can show immediatelythat we can never find the center, because if we use some other origin it wouldmake no difference. In other words, suppose that there are two people—Joe, whohas an origin in one place, and Moe, who has a parallel system whose origin is somewhereelse (Fig. 11–1). Nowwhen Joe measures the location of the point in space, he finds it at x , y , and z (we shall usually leave z out because it is too confusing to draw in a picture). Moe, on theother hand, when measuring the same point, will obtain a different x(in order to distinguish it, we will call it x′ ), and in principle a different y , although in our example they are numerically equal. So we have
x′=x−a, y′=y, z′=z. (11.2)
这些值，应该从某个原点开始测量，但是，原点放在什么地方呢？牛顿所能告诉我们的，首先就是，有某个地方，我们可以从它开始测量，或许是宇宙的中心，这样，这些规律就是正确的。但是，我们可以立即指出，我们永远也找不到那个中心，因为，如果我们使用某个其他的原点，不会有何不同。换句话说，假设有两个人，一个是Joe，他把一个地方当原点，一个是Moe，他有一个平行的系统，但原点在另一个地方（图11-1）。现在，当Joe测量空间中一个点的位置时，他发现，点在x , y , 和z（通常我们不画z轴，容易引起困惑）。另一方面，当Moe在测量同一个点时，他会得到不同的x (为了区别，我们将称之为 x′)，原则上也可得到一个不同的 y，虽然在我们的例子中，它们在数值上是相等的。于是我们有：
x′=x−a, y′=y, z′=z. (11.2)

Now in order to complete our analysis we must know what Moe would obtainfor the forces. The force is supposed to act along some line, and by the forcein the x -direction we mean the part of the total which is in the x -direction, which is the magnitude of the force times this cosine of itsangle with the x -axis. Now we see that Moe would use exactly the same projection as Joewould use, so we have a set of equations
Fx′=Fx, Fy′=Fy, Fz′=Fz. (11.3)
These would be the relationships between quantities as seen by Joe andMoe.
现在，为了完成我们的分析，我们应该知道，Moe能为力得到什么。力被假定是沿着某条线起作用，通过x方向的力，我们是指总的力在x方向的分量，它是力的大小，乘以，力与x轴的夹角的cosine。现在，我们看到，Moe使用的投影与Joe使用的投影一样，于是，我们就有一组方程：
Fx′=Fx, Fy′=Fy, Fz′=Fz. (11.3)
这就是Joe 和 Moe所看到的量之间的关系。

Fig. 11–1.Two parallel coordinate systems. 图11-1 两个平行的坐标系统。

The question is, if Joe knows Newton’slaws, and if Moe tries to write down Newton’s laws, will they also be correctfor him? Does it make any difference from which origin we measure the points?In other words, assuming that equations (11.1)are true, and the Eqs. (11.2)and (11.3)give the relationship of the measurements, is it or is it not true that
(a) m(d2x′/dt2)= Fx′
(b) m(d2y′/dt2)= Fy′ (11.4)
(c) m(d2z′/dt2)= Fz′
现在的问题是，如果Joe知道牛顿规律，并且，如果Moe想写出牛顿规律，那么，这些规律对他来说，也会是正确的吗？从不同原点出发，测量这些点，会有何不同？换句话说，假设方程(11.1)为真，方程（11.2）和（11.3）给出了测量的关系，那么，下面方程是否为真呢？
(a) m(d2x′/dt2)= Fx′
(b) m(d2y′/dt2)= Fy′ (11.4)
(c) m(d2z′/dt2)= Fz′
In order to test these equations we shall differentiate the formulafor x′ twice. First of all
dx′/dt=d(x−a)/dt=dx/dt−da/dt.
Now we shall assume that Moe’s origin is fixed (not moving) relativeto Joe’s; therefore a is a constant and da/dt=0 , so we find that
dx′/dt=dx/dt
and therefore
d2x′/dt2=d2x/dt2;
therefore we know that Eq. (11.4a)becomes
m(d2x/dt2)=Fx′.
(We also suppose that the masses measured by Joe and Moe are equal.)Thus the acceleration times the mass is the same as the other fellow’s. We havealso found the formula for Fx′ , for, substituting from Eq. (11.1),we find that
Fx′=Fx.
为了验证这些方程，我们将让方程对x′求微分两次。首先：
dx′/dt=d(x−a)/dt=dx/dt−da/dt.
现在，我们将假定Moe的原点，相对于Joe的原点，是固定的；因此，a是一个常数，且 da/dt=0，于是我们发现：
dx′/dt=dx/dt
从而：
d2x′/dt2=d2x/dt2;
因此，我们知道方程 (11.4a)就变为：
m(d2x/dt2)=Fx′.
(我们同样假设，Joe和Moe所测的质量，是相等的。)，这样，加速度乘以质量，就与另一伙计的一样。我们也为Fx′找到了公式，因为，为了替代公式（11.1），我们发现：
Fx′=Fx.
Therefore the laws as seen by Moe appear the same; he can writeNewton’s laws too, with different coordinates, and they will still be right.That means that there is no unique way to define the origin of the world,because the laws will appear the same, from whatever position they areobserved.
因此，Moe所看到的规律，是一样的；用不同的坐标，他也可以得出牛顿规律，且也是正确的。这就意味着，定义这个世界的原点，并非只有唯一的方法，因为，无论从哪个位置来观察规律，它们看上去都一样。
This is also true: if there is a piece ofequipment in one place with a certain kind of machinery in it, the same equipmentin another place will behave in the same way. Why? Because one machine, whenanalyzed by Moe, has exactly the same equations as the other one, analyzed byJoe. Since the equations are the same, the phenomena appear thesame. So the proof that an apparatus in a new position behaves the same as itdid in the old position is the same as the proof that the equations whendisplaced in space reproduce themselves. Therefore we say that the laws ofphysics are symmetrical for translational displacements, symmetrical in thesense that the laws do not change when we make a translation of our coordinates.Of course it is quite obvious intuitively that this is true, but it is interestingand entertaining to discuss the mathematics of it.
下面也是真的：如果某地有一个仪器，它有一定的机制，那么，在另外一个地方的同样的设备，将会有同样的表现。为什么？因为，一台机器，被Moe分析，能得到一组方程，另一台机器，被Joe分析，也能得到一组方程，两组方程一样。由于方程一样，所以，表现出来的现象也一样。于是，下面两个证据，就是一样的；一个证据用来证明：一台仪器放在新的地方的表现，与它在老地方的表现，是一样的；另一个证据用来证明，空间位置改变后，方程可以重复自己。因此，我们说，物理学的规律，对于位置转换来说，是对称的，对称的意义就是：当我们改变坐标系时，规律并不变化。当然，直观上显而易见，这是真的，然而，有趣且令人愉悦的则是，讨论其背后的数学。

11–3Rotations 11-3 旋转
The above is the first of a series of ever morecomplicated propositions concerning the symmetry of a physical law. The nextproposition is that it should make no difference in which direction wechoose the axes. In other words, if we build a piece of equipment in some placeand watch it operate, and nearby we build the same kind of apparatus but put itup on an angle, will it operate in the same way? Obviously it will not if it isa Grandfather clock, for example! If a pendulum clock stands upright, it worksfine, but if it is tilted the pendulum falls against the side of the case and nothinghappens. The theorem is then false in the case of the pendulum clock, unless weinclude the earth, which is pulling on the pendulum. Therefore we can make aprediction about pendulum clocks if we believe in the symmetry of physical lawfor rotation: something else is involved in the operation of a pendulum clockbesides the machinery of the clock, something outside it that we should lookfor. We may also predict that pendulum clocks will not work the same way whenlocated in different places relative to this mysterious source of asymmetry,perhaps the earth. Indeed, we know that a pendulum clock up in an artificial satellite,for example, would not tick either, because there is no effective force, and onMars it would go at a different rate. Pendulum clocks do involve somethingmore than just the machinery inside, they involve something on the outside.Once we recognize this factor, we see that we must turn the earth along withthe apparatus. Of course we do not have to worry about that, it is easy to do;one simply waits a moment or two and the earth turns; then the pendulum clockticks again in the new position the same as it did before. While we arerotating in space our angles are always changing, absolutely; this change doesnot seem to bother us very much, for in the new position we seem to be in thesame condition as in the old. This has a certain tendency to confuse one,because it is true that in the new turned position the laws are the same as in theunturned position, but it is not true that as we turn a thing itfollows the same laws as it does when we are not turning it. If we performsufficiently delicate experiments, we can tell that the earth is rotating,but not that it had rotated. In other words, we cannot locate itsangular position, but we can tell that it is changing.
还有很多更复杂的命题，牵扯到物理规律的对称性，上面就是其中之一。下一个命题则是，我们选择轴的方向是什么，应该没有什么不同。换句话说，如果我们把这个装备，在某地建立起来，并看着它运行，而在附近，我们建立同样的仪器，但是，让它转一个角度，它的运行，会一样吗？举个例子，如果它是老爷爷的钟，那么显然不行！如果一个单摆钟，站的笔直，那么，它会工作得很好，但是，如果它倾斜了，那么，单摆就会靠着一侧，什么就都不会发生了。因此，在单摆钟这种情况，那个定理就是错的，除非我们把地球也包括进来，它对单摆有吸引力。因此，如果对于旋转来说，我们相信物理规律的对称性：除了钟表的机制之外，还有其它的东西，卷入了单摆钟的运行，这种东西在它之外，我们应该去寻找。我们还可以预测，当单摆钟被放在不同的地方时，相对于这种神秘的对称源—或许是地球，单摆钟的工作，会不一样。确实，例如，我们知道，在人造地球卫星上，单摆钟也不会嘀嗒，因为没有有效的力，而在火星上，速率则会不同。单摆钟除了内部的机制外，确实还包含了更多的东西，它们包含着外面的一些东西。一旦我们认识到这个因素，我们就明白了，我们应该把地球与仪器一起转动。当然，我们无需担心这点，这很容易做到；简单地等待一会，地球就会转；然后，单摆钟就会在新的位置，开始嘀嗒，正如它以前所做那样。当我们在空间中旋转时，我们的角度总是在改变，这是绝对地；这种改变，似乎并不让我们感到很困惑，因为，在新的位置，我们的条件，与老的条件，似乎是一样的。这一点，似乎总容易迷惑人，因为，有一点是真的，即在转换后的新位置，规律与转换前的位置是一样的；但是，有一点并不为真，即当我们转了一个事物之后，它还遵循同样的规律，就像我们没有转它之前一样。如果我们执行足够精确的实验，我们可以告知，地球正在转动着，而不是它已经转动过了。换句话说，我们无法定位它的角度位置，但是，我们可以告知，它正在改变。

Fig. 11–2.Two coordinate systems havingdifferent angular orientations. 图11-2 两个坐标系，有不同的角度方位。
Now we may discuss the effects of angularorientation upon physical laws. Let us find out whether the same game with Joeand Moe works again. This time, to avoid needless complication, we shallsuppose that Joe and Moe use the same origin (we have already shown that the axescan be moved by translation to another place). Assume that Moe’s axes have rotatedrelative to Joe’s by an angle θ . The two coordinate systems are shown in Fig. 11–2, whichis restricted to two dimensions. Consider any point P having coordinates (x,y) in Joe’s system and (x′,y′) in Moe’s system. We shall begin, as in the previous case, byexpressing the coordinates x′ and y′ in terms of x , y , and θ . To do so, we first drop perpendiculars from P to all four axes and draw AB perpendicular to PQ .Inspection of the figure shows that x′ can be written as the sum of two lengths along the x′ -axis, and y′ as the difference of two lengths along AB . All these lengths are expressed in terms of x , y , and θ in equations (11.5),to which we have added an equation for the third dimension.
x′=xcosθ + y sinθ,
y′=ycosθ − xsinθ, (11.5)
z′=z
现在，我们可以讨论，角度方位对物理规律的影响。让我们看看，用Joe和 Moe来做这个游戏，是否可行。这次，为了避免不必要的复杂，我们将假设，Joe和Moe，用的是同一个原点（我们已经指出过，坐标轴可以通过转换，移到另一地方）。假设Moe的轴，相对于Joe的，转动了一个角度θ。图11-2所示的这两个坐标系，被限制在二维。考虑任一点P，在Joe的系统中，坐标为(x,y)，在Moe的系统中，坐标为 (x′,y′)。正如前一情况一样，我们将用x , y , 和 θ，来表示x′ 和 y′。要这样做，我们首先从点P，向四个轴做垂线，且做AB垂直于 PQ。从图中可看出，x′可被表示为沿着x′轴的两个长度之和， y′可被表示为沿着AB的两个长度之差。所有这些长度，都在方程组（11.5）中，用x , y , 和θ来表示了，我们还为它，增加了一个第三维的方程：
x′=xcosθ + y sinθ,
y′=ycosθ − xsinθ, (11.5)
z′=z

The next step is to analyze the relationship of forces as seen by thetwo observers, following the same general method as before. Let us assume thata force F , which has already been analyzed as having components Fxand Fy (as seen by Joe), is acting on a particle of mass m , located at point P in Fig. 11–2. Forsimplicity, let us move both sets of axes so that the origin is at P, as shown in Fig. 11–3. Moesees the components of F along his axes as Fx′ and Fy′ . Fx has components along both the x′ - and y′ -axes, and Fy likewise has components along both these axes. To express Fx′in terms of Fx and Fy , we sum these components along the x′ -axis, and in a like manner we can express Fy′in terms of Fx and Fy . The results are
Fx′=Fx cosθ + Fy sinθ,
Fy′=Fy cosθ − Fx sinθ, (11.6)
Fz′=Fz
下一步，与前面的步骤一样，就是分析两个观察者所看到的力之间的关系。我们假设，力F已经被分析为，具有分量Fx和Fy (由Joe所看到),且正作用于位于点P的质量 m ,见图 11–2。为了简单起见，我们移动两个轴，让原点在P，如图 11–3所示。 Moe看到，F沿着他的坐标轴的分量为Fx′ 和Fy′。 Fx在x′和y′ 轴上都有分量，Fy也是。要用Fx和 Fy来表示Fx′, 我们把它们沿着x′ 轴的分量加起来，同样，我们可以用Fx和 Fy来表示Fy′。结果就是：
Fx′=Fx cosθ + Fy sinθ,
Fy′=Fy cosθ − Fx sinθ, (11.6)
Fz′=Fz

It is interesting to note an accident of sorts, which is of extremeimportance: the formulas (11.5)and (11.6),for coordinates of P and components of F , respectively, are of identical form.
指出一个分类，非常有趣：公式(11.5)和 (11.6)分别是关于点P的坐标和 F的分量的，它们的形式相同。这点极为重要。

Fig. 11–3.Components of a force in the twosystems. 图11-3 一个力在两个坐标系中的分量。

As before, Newton’s laws are assumed to betrue in Joe’s system, and are expressed by equations (11.1).The question, again, is whether Moe can apply Newton’s laws—will the results becorrect for his system of rotated axes? In other words, if we assume that Eqs.(11.5)and (11.6)give the relationship of the measurements, is it true or not true that
m(d2x′m(d2y′m(d2z′/dt2)=Fx′,/dt2)=Fy′,/dt2)=Fz′? (11.7)
像以前一样，牛顿的规律，在Joe的坐标系中，被假定是真的，且通过公式（11.1）来表示。而问题则又是：Moe能应用牛顿规律吗？对于他的旋转了的坐标系，结果还是正确的吗？换句话说，如果我们假定公式(11.5)和(11.6)给出了测量的关系，那么，下面的是正确的吗？
(11.7)