For a freely propagating stable particle, the amplitude associated with an infinitesimal path segment dC — its end points given by the coordinates (x,y,z,t) and (x+dx,y+dy,z+dz,t+dt) — was found to be Z(dC) = [1: −b ds], with ds = √dt² − (dx² + dy² + dz²)/c². We know now that ds is an interval of (proper) time, so the physical dimension of the particle-specific constant b is that of a frequency. To convert it to energy units we have to divide it by ℏ, and to further convert it to its conventional mass units, we have to multiply it by c2. Once this is done, we use the letter m and write
(2.16.1) Z(dC) = [1: −(mc2/ℏ) ds].
(The famous formula E=mc2 is as unexciting as any conversion formula, for E and m are the same physical quantity, except that once it is measured in conventional units of energy and once in conventional units of mass.)
From particles as clocks to the principle of least action
Let us follow a particle as it travels from (A,tA) to (B,tB) along C, and let s stand for the proper time that has passed since its departure at (A,tA). As s increases, the complex number [1: −(mc2/ℏ) s] — an arrow of unit magnitude — rotates in the plane of complex number (the “complex plane,” for short). As suggested by Feynman, we may think of this rotating arrow as a clock. Though a feature of the quantum-mechanical probability calculus rather than a feature of the physical world, this clock reveals a deep connection between the quantum-mechanical probability calculus and the metric properties of the physical world. For what makes it possible to use a global system of spacetime units is the fact that the rates at which free particles “tick” in their momentary rest frames — which is to say, their masses — are constant. This is the reason why we can use meters and seconds everywhere and “everywhen,” secure in the knowledge that a second or meter here and now is equal to second or meter there and then.
A general way of accommodating influences on the behavior of a stable particle (albeit not the most general, as we shall see) is to modify the rate at which it “ticks.” It is customary to write the modified amplitude associated with an infinitesimal path segment dC in the form
(2.16.2) Z(dC) = [1:dS/ℏ],
where the infinitesimal increment dS of the so-called action S is a function of x, y, z, t, dx, dy, dz, and dt, which has to satisfy the following conditions:
- Like ds, dS must be invariant under Lorentz transformations. Otherwise it would not be the case that all inertial coordinate systems are “created equal”: the behavior of a particle (encapsulated by dS) would depend on the language (the inertial frame) we use to describe it.
- dS must be “homogeneous” in the following sense: multiplying by a factor u each of the infinitesimal coordinate intervals on which dS depends is equivalent to multiplying dS itself by u. This is a consequence of two facts: the multiplication of two complex numbers of unit magnitude is equivalent to the addition of their phases, and the amplitude associated with consecutive path segments is the product of the amplitudes of the individual segments.
With (2.16.2), the particle propagator takes the form
(2.16.3) <B,tB|A,tA> = ∫DC [1:∫C dS/ℏ].
One may be tempted to ask: what is the probability of finding that a particle with this propagator has traveled from (A,tA) to (B,tB) via a specific path C? Since the magnitude of the amplitude [1:∫C dS/ℏ] is the same for all paths (namely, 1), that probability seems to be the same for all paths. It is, however, strictly impossible to ascertain that a particle has traveled via a precise path. We obtain a more useful answer if we make the half realistic assumption that what can be ascertained is whether a particle has traveled from (A,tA) to (B,tB) inside a narrow bundle of paths.
Imagine a narrow “tube” T “filled” with paths whose endpoints are (A,tA) and (B,tB). The probability of finding that the particle has traveled inside T is (formally) given by the squared magnitude of
(2.16.4) ∫T DC [1:∫C dS/ℏ],
which sums the amplitudes of all paths contained in T. To make sure that a single path is obtained in the so-called classical limit, in which quantum physics degenerates into classical physics, we assume that there is a unique path G from (A,tA) to (B,tB) for which the action S(C) = ∫C dS is stationary: it does not change under infinitesimal deformations (“variations”) of G. We will consider two cases:
- G is contained in T. Because the action is stationary for G, T contains a large number of paths with almost equal amplitudes (think arrows in the complex plane). As a consequence, the magnitude of the sum of these arrows is almost as large as the sum of their magnitudes. This is illustrated by the almost straight chain of arrows in Fig. 2.16.1.
- G is not contained in T. In this case the arrows contributed by the paths in T are more or less equally distributed over all directions in the complex plane. As a result, the magnitude of the sum (2.16.4) is small by comparison. This is illustrated by the coiled chain of arrows in Fig. 2.16.2.
If we let ℏ go to zero (ℏ→0), the probability of finding that the particle has traveled inside T goes to 1 if G is contained in T, and it goes to 0 if G is not contained in T. Since we can make T as narrow as we wish, it is safe to assert that in this limit a particle traveling from (A,tA) to (B,tB) travels along G, as classical physics has it.
Because dS is homogeneous in the sense spelled out above, it defines a differential geometry — the kind of geometry appropriate for assigning lengths to paths on a warped surface, in a warped space, or in a warped spacetime. The next best thing to a straight line (which does not exist in a warped spacetime) is a geodesic: a path G that is either longer or shorter than all paths that have the same endpoints as G and lie in a sufficiently small neighborhood of G. What we just found is that classical particles follow the geodesics of the spacetime geometry defined by the action increment dS.
If we are dealing with a classical physical system that has N degrees of freedom, rather than just three (like a single particle), we can specify their values at any given time by a point in an N-dimensional “configuration space.” This system point, too, follows the geodesics of the geometry defined by the system’s action increment dS. Classical physics can therefore be encapsulated by the so-called principle of least action, although this is something of a misnomer, since what is actually required of a geodesic is that it either minimizes or maximizes the action when compared to the other paths in its neighborhood.