# 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.