22 Going relativistic

Formally, the Schrödinger equation can be obtained via the following steps:

  • Start with the approximate, non-relativistic relation (2.3.2) between the kinetic energy and the kinetic momentum associated with a particle of mass m.
  • Write this in terms of the total energy E, the total momentum p, the potential energy qV, and the potential momentum (q/c)A (the latter introduced on this page),

E − qV = (1/2m)[p − (q/c)A]2

or (using units in which c=1 to unclutter the notation somewhat)

E − qV = (1/2m)[(px − qAx)2 + (py − qAy)2 + (pz − qAz)2],

  • replace the total energy E by the operator iℏ∂t and the components px, py, pz of the total momentum by the operators (ℏ/i)∂x, (ℏ/i)∂y, and (ℏ/i)∂z from Eq. (2.3.1–7),
  • multiply both sides from the right by ψ:

(2.21.1)xxx(iℏ∂t − qV)ψ = (1/2m)[((ℏ/i)∂x − qAx)2 + ((ℏ/i)∂y − qAy)2 + ((ℏ/i)∂z − qAz)2]ψ.

If we dispense with the potential A, this boils down to Eq. (2.3.7.).

The correct, relativistic relation between the kinetic energy and the kinetic momentum associated with a particle of mass m, already written in terms of the total and potential energies and momenta, follows from Equations (2.17.3):

(2.21.2)xxx[E − qV]2 − [p − (q/c)A]2 c2 = m2c4.

If we start with this and carry out the remaining steps (again using units in which c=1), the result is the Klein–Gorden equation:

(2.21.3)xxx[(iℏ∂t − qV)2 − ((ℏ/i)∂x − qAx)2 − ((ℏ/i)∂y − qAy)2 − ((ℏ/i)∂z − qAz)2]ψ = m2 ψ.

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Antiparticles

Because the Klein–Gordon equation, unlike the Schrödinger equation, is quadratic in the energy term, it appears to have among its solutions particle states whose energies are negative. But appearances are deceptive, as they say. In actual fact, the Klein-Gordon equation describes the propagation — in terms of correlations between detection events — of two types of particles: “ordinary” ones as well as their antiparticles, which have the same properties as the “ordinary” ones except for their opposite charges. (The only antiparticle that has its own name is the positron, which has the same properties as the electron except for its positive electric charge.)

The big news here is that the marriage of quantum mechanics with special relativity predicts not only that for every particle that “carries” a charge of some kind there is an antiparticle, but also that individual particles are no longer conserved. What is conserved is the total charge. As a consequence, particles can be created or annihilated in pairs that carry opposite charges, or a group of particles can metamorphose into a different group of particles as long as for every type of charge the total charge remains the same.

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The Dirac equation

Like the utility of the Schrödinger equation, that of the Klein–Gordon equation is restricted to particles without spin. To arrive at the appropriate equation for a particle with spin, one needs to let the wave function have more than one component. This equation will therefore contain operators with one input slot per component and as many output slots. The simplest ansatz is linear in iℏ∂t, (ℏ/i)∂x, (ℏ/i)∂y, and (ℏ/i)∂z. If one requires in addition that each component of ψ should satisfy the Klein–Gordon equation, one finds that the lowest possible number of components is four, and that this number yields the appropriate equation for a spin-1/2 particle like the electron — the Dirac equation. The four components of ψ correspond to the two components each of the particle (the electron) and its antiparticle (the positron).

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More fuzziness

As said, a group of particles can metamorphose into a different group of particles as long as the total of every type of charge stays the same. (In addition the total energy and the total momentum associated with the incoming particles has to be the same as the total energy and the total momentum, respectively, associated with the outgoing particles.) This means that we have to come to grips with a new kind of fuzziness — the fuzziness of particle numbers. To this end particle physicists study inelastic scattering events. (During an elastic scattering event, you will remember, no particles are created or annihilated.) To produce inelastic scattering events, powerful particle accelerators are used, for the higher the energies of the incoming particles are, the greater will be the variety of possible sets of outgoing particles.

The first thing we know is the set of incoming particles. The next thing we know is the set of outgoing particles. What happens between the “preparation” of the incoming set and the detection of the outgoing set is anybody’s guess. Hence when we calculate the amplitude <out|S|in>associated with the possibility that a given incoming set metamorphoses into a specific outgoing set, we must “sum over” the possible ways in which the incoming particles can metamorphose into the outgoing ones — that is, we must add the corresponding amplitudes.

The resulting perturbation series can be obtained by expanding a summation of the form

<out|S|in> = ∫DC [1:(1/ℏ)∫L d4x].

In compliance with the requirement of relativistic invariance, the action now has the form ∫L d4x — it sums contributions from infinitesimal spacetime volumes d4x = dt dx dy dz. The so-called Lagrangian L is a function of the wave functions (or fields) ψa, ψb, ψc,… associated with the particles the theory is dealing with and of their partial derivatives ∂tψa, ∂xψa, etc. By letting the Lagrangian depend on wave functions and by giving its free part the form that yields the corresponding wave equations (for example, the free Dirac equation), the fuzziness associated with freely propagating particles has been taken care of. The amplitudes being summed therefore correspond to alternatives that only differ in the number of particles that are created and/or annihilated, the spacetime locations at which particles are created and/or annihilated, and the different ways in which the vertices can be connected by free-particle propagators. In the case of quantum electrodynamics (QED), the remaining part of the Lagrangian once again leaves nothing to the physicist’s discretion.

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Feynman diagrams

The calculation of the perturbation expansion is greatly facilitated if each term is represented by a Feynman diagram. Feynman diagrams are made up of (i) lines and (ii) vertices at which lines meet. Internal lines — those corresponding to neither incoming nor outgoing particles — are generally referred to as “virtual” particles. The Feynman diagrams for QED contain straight lines representing electrons or positrons, wiggly lines representing photons, and vertices at which two electron/positron lines and one photon line meet. The following diagram is the lowest-order graph that contributes to the amplitude for electron–electron scattering in QED. (“Lowest order” means “containing the smallest number of vertices.”) The upward direction of the arrows tells us that the interacting particles are electrons rather than positrons.

qed scattering
Figure 2.17.1 Lowest-order graph for the scattering of two electrons in QED.

And here are some 4-vertex graphs that contribute to to the amplitude for electron–electron scattering in QED:

qed vertices
Figure 2.17.2 Some 4-vertex graphs for the scattering of two electrons in QED.

How should we think about these diagrams? According to Zee,[1]

Spacetime Feynman diagrams are literally pictures of what happened. … Feynman diagrams can be thought of simply as pictures in spacetime of the antics of particles, coming together, colliding and producing other particles, and so on.

Mattuck[2] is more cautious:

Because of the unphysical properties of Feynman diagrams, many writers do not give them any physical interpretation at all, but simply regard them as a mnemonic device for writing down any term in the perturbation expansion. However, the diagrams are so vividly “physical looking,” that it seems a bit extreme to completely reject any sort of physical interpretation whatsoever. … Therefore, we shall here adopt a compromise attitude, i.e., we will “talk about” the diagrams as if they were physical, but remember that in reality they are only “apparently physical” or “quasi-physical.”

Consider a two-electron scattering event. Whereas the momenta of the incoming particles are known, and those of the outgoing particles are stipulated (inasmuch as we want to calculate the probability of the incoming particles metamorphosing in the outgoing ones), whatever happened in the meantime is anybody’s guess. One thing is sure, though: whenever quantum mechanics requires us to sum alternatives (Rule A), no single alternative can be a picture of what actually happened. And if there is a sense in which they all happened, Zee fails to spell it out.

Most Feynman diagrams contain so-called “vacuum parts,” which are not connected by any line or sequence of lines to an incoming or outgoing particle. These are artifacts of the methods employed in generating the perturbation series. They are systematically canceled out in every actual calculation of a scattering amplitude. They certainly do not warrant Zee’s[1] claim that “the vacuum in quantum field theory is a stormy sea of quantum fluctuations.”

Mattuck’s plea for cognitive dissonance, on the other hand, is a recipe for philosophical disaster. Who has not heard the song and dance about a cloud of virtual photons and virtual particle-antiparticle pairs surrounding every real particle, which is routinely invoked in explanations of the dependence of physical masses and charges on the momentum scale at which they are measured? As long as this naive reification of computational tools is the standard of philosophical rigor in theoretical physics, it is not surprising that quantum mechanics keeps being vilified as unintelligible, absurd, or plain silly.

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1. [↑] Zee, A. (2003). Quantum Field Theory in a Nutshell, Princeton University Press, pp. 53–57, 19.

2. [↑] Mattuck, R.D. (1976). A Guide to Feynman Diagrams in the Many-Body Problem, McGraw-Hill.