Propagator derivations

Peter Woit is writing a book, Quantum Theory, Groups and Representations: An Introduction, and has a PDF of the draft version linked here.  He has now come up with the slogan “Quantum Theory is Representation Theory”, after postulating “What’s Hard to Understand is Classical Mechanics, Not Quantum Mechanics“.

I’ve recently become interested in the mathematics of QFT, so I’ll just make a suggestion for Dr Woit regarding his section “42.4 The propagator” which is incomplete (he has only the heading there on page 404 of the 11 August 2014 revision, with no test under it at all).

The Propagator is the greatest part of QFT from the perspective of Feynman’s 1985 book QED: you evaluate the propagator from either the Lagrangian or Hamiltonian, since the Propagator is simply the Fourier transform of the potential energy (the interaction part of the lagrangian provides the couplings for Feynman’s rules, not the propagator).  Fourier transforms are simply Laplace transforms with a complex number in the exponent.  The Laplace and Fourier transforms are used extensively in analogue electronics for transforming waveforms (amplitudes as a function of time) into frequency spectra (amplitudes as a function of frequency).  Taking the concept at it’s simplest, the Laplace transform of a constant amplitude is just the reciprocal (inverse), e.g. an amplitude pulse lasting 0.1 second has a frequency of 1/0.1 = 10 Hertz.  You can verify that from dimensional analysis.  For integration between zero and infinity, with F(f) = 1 we have:

Laplace transform, F(t) = Integral [F(f) * exp(-ft)] df

= Integral [exp(-ft)] df

= 1/t.

If we change from F(f) = 1 to F(f) = f, we now get:

Frequency, F(t) = Integral [f * exp(-ft)] df = 1/(t squared).

The trick of the Laplace transform is the integration property of the exponential function by itself, i.e. it’s unique property of remaining unchanged by integration (because e is the base of natural logarithms), apart from multiplication by the constant (the factor which is not a function of factor you’re integrating over) in its power.  The Fourier transform is the same as the Laplace transform, but with a factor of “i” included in the exponential power:

Fourier transform, F(t) = Integral [F(f) * exp(-ift)] df

In quantum field theory, instead of inversely linked frequency f and time t, you have inversely linked variables like momentum p and distance x.   This comes from Heisenberg’s ubiquitous relationship, p*x = h-bar.  Thus, p ~ 1/x.  Suppose that the potential energy of a force field is given by V = 1/x.  Note that field potential energy V is part of the Hamiltonian, and also part of the Lagrangian, when given a minus sign where appropriate.  You want to convert V from position space, V = 1/x, into momentum space, i.e. to make V a function of momentum p.  The Fourier transform of the potential energy over 3-d space shows that V ~ 1/p squared.  (Since this blog isn’t very suitable for lengthy mathematics, I’ll write up a detailed discussion of this in a vixra paper soon to accompany the one on renormalization and mass.)

What’s interesting here is that this shows that the propagator terms in Feynman’s diagrams, which, integrated-over produce the running couplings and thus renormalization, are simply dependent on the field potential, which can be written in terms of classical Coulomb field potentials or quantum Yukawa type potentials (Coulomb field potentials with an exponential decrease included.  There are of course two types of propagator: bosonic (integer spin) and fermionic (half integer spin).  It turns out that the classical Coulomb field law gives a potential of V = 1/x which, when Fourier transformed, gives you V ~ 1/p squared, and when you include a Yukawa exp(-mx) short-range attenuation or decay term, i.e. V = (1/x)exp(-mx), you get a Fourier transform of 1/[(p squared) – (m squared)] which is the same result that a Fourier transform of the spin-1 field propagator (boson propagators) give using a Klein-Gordon lagrangian.
However, using the Dirac lagrangian, which is basically a square-root version of the Klein-Gordon equation with Dirac’s gamma matrices to avoid losing solutions due to the problem that minus signs and complex numbers tend to disappear when you square them, you get a quite different propagator: 1 /(p – m).  The squares on p and m which occur for spin-1 Klein-Gordon equation propagators, disappear for Dirac’s fermion (half integer spin) propagator!
So what does this tell us about the physical meaning of Dirac’s equation, or put another way, we know that Coulomb’s law in QFT (QED more specifically) physically involves field potentials consisting of exchanged spin-1 virtual photons which is why the Fourier transform of Coulomb’s law gives the same result as the propagator from the Klein-Gordon equation but without a mass term (Coulomb’s virtual photons are non-massive, so the electromagnetic force is infinite ranged), but what is the equivalent for Coulomb’s law for Dirac’s spin-1/2 fermion fields?  Doing the Fourier transform in the same way but ending up with Dirac’s 1 /(p – m) fermion propagator gives an interesting answer which I’ll discuss in my forthcoming vixra paper.
Another question is this: the Higgs field and the renormalization mechanism only deal with problems of mass at high energy, i.e. UV cutoffs as discussed in detail in my previous paper.  What about the loss of mass at low energy, the IR cutoff, to prevent the coupling from running due to the presence of a mass term in the propagator?
In other words, in QED we have the issue that the running coupling polarizable pair production only kicks in at 1.02 MeV (the energy needed to briefly form an electron-positron pair).  Below that energy, or in weak fields beyond the classical electron radius, the coupling stops running, so the effective electronic charge is constant.  This is why there is a standard low energy electronic charge that was measured by Millikan.  Below the IR cutoff, or at distances larger than the classical electron radius, the charge of an electron is constant and the force merely varies with the Coulomb geometric law (the spreading or divergence of field lines or field quanta over an increasing space, diluting the force, but with no additional vacuum polarization screening of charge, since this screening is limited to distances shorter than the classical electron radius or energies beyonf about 1 MeV).
So how and why does the Coulomb potential suddenly change from V = 1/x beyond a classical electron radius, to V = (1/x)exp(-mx) within a classical electron radius? Consider the extract below from page 3 of
Integrating using a massless Coulomb propagator to obtain correct low energy mass
The key problem for the existing theory is very clear when looking at the integrals in Fig. 1.  Although we have an upper case Lambda symbol included for an upper limit (or high energy, i.e. UV cutoff) on the integral which includes an electron mass term, we have not included a lower integration limit (IR cutoff): this is in keeping with the shoddy mathematics of most (all?) quantum field theory textbooks, which either deliberately or maliciously cover-up the key (and really interesting or enlightening) problems in the physics by obfuscating or by getting bogged down in mathematical trivia, like a clutter of technical symbolism.  What we’re suggesting is that there is a big problem with the concept that the running coupling merely increases the “bare core” mass of a particle: this standard procedure conflates and confuses the high energy bare core mass that isn’t seen at low energy, with the standard value of electron mass which is what you observe at low energy.
In other words, we’re arguing for a significant re-interpretation of physical dogmas in the existing mathematical structure of QFT, in order to get useful predictions, nothing useful is lost by our approach while there is everything to be gained from it.  Unfortunately, physics is now a big money making industry in which journals and editors are used as a professional status-political-financial-fame-fashion-bigotry-enforcing-consensus-enforcing-power grasping tool, rather than informative tool designed solely and exclusively to speed up the flow of information that is helpful to those people focused merely upon making advances in the basic science.  But that’s nothing new.  [When Mendel’s genetics were finally published after decades of censorship, his ideas had been (allegedly) plagiarized by two other sets of bigwig researchers whose papers the journal editors had more from gain by publishing, than they had to gain from publishing the original research of someone then obscure and dead!  Neat logic, don’t you agree?  Note that is statement of fact is not “bitterness”, it is just fact.  A lot of the bitterness that does arise in science comes not from the hypocrisy of journals and groupthink, but because these are censored out from discussion.  (Similarly the Oscars are designed to bring the attention to the Oscars, since the prize recipients are already famous anyway.  There is no way to escape the fact that the media in any subject, be it science or politics, deems one celebrity more worthy of publicity than the diabolical murder of millions by left wing dictators.  The reason is simply that the more “interesting” news sells more journals than the more difficult to understand problems.)]
 17 August 2014 update:

(1) The Fourier transform of the Coulomb potential (or the Fourier transform of the potential energy term in the Lagrangian or Hamiltonian) gives rest mass.

(2) Please note in particular the observation that since the Coulomb (low energy, below IR cutoff) potential’s Fourier transform gives a propagator omitting a mass term, this propagator does not contribute a logarithmic running coupling. This lack of a running coupling at low energy is observed in classical physics for energy below about 1 Mev where no vacuum polarization or pair production occurs because pair production requires at least the mass of the electron and positron pair, 1.02 MeV. The Coulomb non-mass term propagator contribution at one-loop to electron mass is then non-logarithmic and simply equal to a factor like alpha times the integral (between 0 and A) of (1/k3)d4k = alpha * A. As shown in the diagram we identify this “contribution” from the Coulomb low energy propagator without a mass term to be the actual ground state mass of the electron, with the cutoff A corresponding to the neutral currents that mire down the electron charge core, causing mass, i.e. A is the mass of the uncharged Z boson of the electroweak scale (91 GeV). If you have two one loop diagrams, this integral becomes alpha * A squared.

(3) The one loop corrections shown on page 3 to electron mass for the non-Coulomb potentials (i.e. including mass terms in the propagator integrals) can be found in many textbooks, for example equations 1.2 and 1.3 on page 8 of “Supersymmetry Demystified”. As stated in the blog post, I’m writing a further paper about propagator derivations and their importance.

If you read Feynman’s 1985 QED (not his 1965 book with Hibbs, which misleads most people about path integrals and is preferred to the 1985 book by Zee and Distler), the propagator is the brains of QFT. You can’t directly do a path integral over spacetime with a lagrangian integrated to give action S and then re-integrated in the path integral, the integral of amplitude exp(iS) taken over all possible geometric paths, where S is the lagrangian integral. So, as Feynman argues, you have to construct a perturbative expansion, each term becoming more complex and representing pictorially the physical interactions between particles. Feynman’s point in his 1985 book is that this process essentially turns QFT simple. The contribution from each diagram involves multiplying the charge by a propagator for an internal line and ensuring that momentum is conserved at verticles.