17 The principle of least action

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.)

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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:

    1. 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.

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almost straight chain of arrows

Figure 2.16.1 A large por­tion of the ampli­tudes (arrows) con­tributing to the sum (2.16.4) form an almost straight chain, causing the sum to be of con­sid­er­able magnitude.

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    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.

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coiled chain of arrows

Figure 2.16.2 The ampli­tudes (arrows) con­tributing to the sum (2.16.4) are more or less equally dis­trib­uted over all direc­tions in the com­plex plane. As a result, their sum — a tightly coiled chain of arrows — has a mag­ni­tude that is minute by comparison.

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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.

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