22 Going relativistic

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

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  • 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.

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lowest order scattering of electrons

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

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

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Some 4-vertex graphs

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

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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.