18 From quantum to classical (effects)

As you will remember, a gen­eral way of accom­mo­dating influ­ences on the behavior of a stable par­ticle is to modify the rate at which it “ticks.” We write the mod­i­fied ampli­tude asso­ci­ated with an infin­i­tes­imal path seg­ment dC in the form Z(dC) = [1:dS/], where the infin­i­tes­imal action incre­ment dS is a func­tion of x, y, z, t, dx, dy, dz, and dt, which sat­is­fies two con­di­tions: it is (i) invariant under Lorentz trans­for­ma­tions and (ii) homo­ge­neous.

There are exactly two ways of mod­i­fying the ampli­tude asso­ci­ated with dC without vio­lating these con­di­tions (and without turning the ampli­tude into a quan­tity that has mul­tiple com­po­nents like a vector). The first con­sists in adding to dS = −mc2ds (the action incre­ment asso­ci­ated with a freely prop­a­gating stable par­ticle) a sum of four terms each con­taining one of the coor­di­nate intervals:

(2.17.1)   dS = −mc2ds + (q/c)(Ax dx + Ay dy + Az dz − V dt).

The second con­sists in the substitution

(2.17.2)   c ds → √(Σjk gjk dxj dxk).

The so-​​called (scalar) poten­tial V and the three com­po­nents of the so-​​called vector poten­tial A are “fields,” in the sense (and only in the sense) that they are func­tions of x,y,z,t. The overall factor q — the particle’s charge — allows the effects on the behavior of a par­ticle encap­su­lated by V and A to differ between one par­ticle species and another. The indices j and k run from 0 to 3, so the root con­tains a sum of six­teen terms. The inter­vals dx0, dx1, dx2, and dx3 stand for c dt, dx, dy, and dz, respec­tively. (The super­scripts are indices, not powers.) The six­teen com­po­nents gjk of the so-​​called metric tensor g are also fields in the sense just spelled out. With g00 = 1, gjj = −1 for j=1,2,3, and gjk = 0 for all other index pairs, we recover the orig­inal expres­sion c ds from the right-​​hand side of the sub­sti­tu­tion (2.17.2).

As pre­vi­ously noted, dS defines a dif­fer­en­tial geom­etry, and clas­sical par­ti­cles follow the geo­desics of the dif­fer­en­tial geom­etry defined by the action incre­ment dS.

The geo­desics defined by (2.17.1) are char­ac­ter­ized by the so-​​called Lorentz force law, a com­pu­ta­tional tool that encap­su­lates all elec­tro­mag­netic effects on the motion of a clas­sical par­ticle. It takes as its input the particle’s charge q, the so-​​called elec­tric field E, the so-​​called mag­netic field B, and the coor­di­nate inter­vals dt,dx,dy,dz per­taining to an infin­i­tes­imal seg­ment dG of a geo­desic, and it yields as its output the change dpk of the particle’s kinetic momentum pk that takes place as the par­ticle travels dG. E and B are defined in terms of the poten­tials V and A in (2.17.1).

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Energy and momentum

What do we mean by the kinetic momentum of a clas­sical par­ticle? We begin by observing that space­time coor­di­nates are human inven­tions. They belong to the lan­guage we use to describe a phys­ical sit­u­a­tion rather than to the phys­ical sit­u­a­tion itself. To make the laws of physics as simple as pos­sible, we require of our coor­di­nates that equal coor­di­nate inter­vals be phys­i­cally equiv­a­lent. This implies, among other things, that freely moving clas­sical par­ti­cles travel equal space inter­vals Δx,Δy,Δz in equal time inter­vals Δt; the ratios formed of Δx, Δy, Δz, and Δt are con­stants. In con­se­quence of the phys­ical equiv­a­lence of equal time inter­vals there is a con­served phys­ical quan­tity — the particle’s energy E. And in con­se­quence of the phys­ical equiv­a­lence of equal inter­vals of the space coor­di­nates there is another con­served phys­ical quan­tity — the particle’s momentum p. Because space has three dimen­sions, p has three com­po­nents (px, py, and pz). (If a phys­ical system is closed, meaning not sub­ject to any external influ­ences, then the values of its con­served quan­ti­ties do not change as time passes.)

The def­i­n­i­tions of E and p depend on the form of dS. If dS is given by (2.17.1), then

(2.17.3)   E = mc²/√1 − v²/c² + qV   and   p = mv/​√1 − v²/c² + (q/​c)A.

The terms con­taining the root are the particle’s kinetic energy and momentum, respec­tively, while qV and (q/​c)A are its poten­tial energy and momentum, respec­tively. v is the particle’s velocity, and v is the cor­re­sponding mag­ni­tude, the particle’s speed.

Let us note in passing that if v is so small com­pared to c that all powers of v2/​c2 but the first can be ignored, we obtain the fol­lowing expres­sions for the non-​​relativistic (New­tonian) theory:

E = mc2 + (m/2)v2 + qV,   pk = mv.

Since the rest energy mc2 is now equiv­a­lent to a con­stant poten­tial energy, and since only the dif­fer­ences between poten­tial ener­gies at dif­ferent places and/​or times are phys­i­cally rel­e­vant, the rest energy term is usu­ally dropped, so that the kinetic energy is given by (m/2)v2 = pk2/​2m.

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Cur­va­ture

Influ­ences or effects on the motion of clas­sical par­ti­cles are thus math­e­mat­i­cally rep­re­sented by a dif­fer­en­tial geom­etry. Par­ti­cles affected or acted on follow the geo­desics of that geom­etry. Figure 2.17.1 illus­trates the effects rep­re­sented by E and B.

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curvature due to E and B

Figure 2.17.1

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Con­sider first the rec­tangle at the right, which lies in a spa­tial plane. A math­e­mat­ical expres­sion called the flux of the mag­netic field through this rec­tangle deter­mines the dif­fer­ence between the actions of the paths A→B→C and A→D→C. If B van­ishes, the actions of the two paths are equal, and the geo­desic from A to C — the path with the least action — is the straight line from A to C. If on the other hand B does not vanish, then the path via B (say) is shorter than the path via D — shorter according to the space­time geom­etry defined by (2.17.1). As a result, the geo­desic from A to C is curved as indi­cated. According to a story pop­ular among teachers of clas­sical physics (and hence, unfor­tu­nately, among their stu­dents), the cur­va­ture is due to a mag­netic “force” that pulls (or pushes) in a direc­tion per­pen­dic­ular to the particle’s motion.

Con­sider now the rec­tangle at the left, which lies in a space­time plane con­taining (or par­allel to) the time axis. In this case the dif­fer­ence between the actions of the paths A→B→C and A→D→C. is deter­mined by the flux of the elec­tric field through the rec­tangle. If E van­ishes, the actions of the two paths are again equal, and the geo­desic from A to C is again the straight line from A to C. If B does not vanish, then the path via B (say) is again shorter than the path via D. As a result, the geo­desic from A to C is curved as indi­cated. But cur­va­ture in a space­time plane con­taining the time axis means that the par­ticle is accel­er­ating. According to the story just men­tioned, the accel­er­a­tion is due to an elec­tric “force” that pulls (or pushes) in the direc­tion in which the par­ticle is moving.

While the geo­desics defined by the action incre­ment (2.17.1) are particle-​​specific — they depend on the masses and the charges of the par­ti­cles acted on — the geom­etry defined by the sub­sti­tu­tion (2.17.2) is inde­pen­dent of particle-​​specific para­me­ters. For this reason it has become cus­tomary to attribute this geom­etry to “space­time itself.” The attri­bu­tion of a geom­etry to “space­time itself,” how­ever, is as uncalled for as the notion of forces bending the paths of moving objects. Both kinds of geom­etry — the one that rep­re­sents influ­ences on the motion of a par­ticle in a particle-​​specific manner (tech­ni­cally known as a Finsler geom­etry) and the one that rep­re­sents influ­ences on the motion of a par­ticle in a non­spe­cific manner (tech­ni­cally known as a pseudo-​​Riemannian geom­etry) — are math­e­mat­ical tools used for rep­re­senting influ­ences on the motion of par­ti­cles. The fact that one of these tools is non­spe­cific as far as the affected par­ti­cles are con­cerned, by no means war­rants thinking of it as a prop­erty of “space­time itself.”

It is, how­ever, worth an illus­tra­tion that the influ­ences which Newton attrib­uted to the force of gravity are indeed encap­su­lated by a uni­versal space­time geom­etry. If a ball is thrown so that it rises to a height of about 5 m and covers a dis­tance of 10 m, it hits the ground after 2 sec­onds. If a bullet is fired so that it rises to a height of about 0.5 mm and covers the same dis­tance, it hits the ground after 20 mil­lisec­onds. The cur­va­tures of their tra­jec­to­ries in space are obvi­ously very dif­ferent, but not the cur­va­tures of their paths in space­time, as one gathers from Fig. 2.17.2.

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A bullet and a ball

Figure 2.17.2 In space­time, the paths of a ball and of a bullet near the sur­face of the Earth have the same cur­va­ture (not to scale).

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