8.3. Quantum Electrodynamics (QED) — Theoretical Physics Reference 0.5 documentation (2024)

8.3.1. Local Gauge Invariance¶

We use a metric with signature +2 in this section.

The Dirac equation for an electron is:

Physical quantities like a charge density () or a current(), are all invariant if we add a local phase to the field (this is called a local U(1) gauge transformation):

Where is a parameter that measures the strength of the phase transformation(this will be later interpreted as a charge, for example for electrons) and is the Planck constant. And so we require that theLagrangian is also invariant under the local gauge transformation, becausethere is no experiment that would change if this local gauge transformation isapplied on the wave functions. By putting this gauge transformation into theLagrangian density, we otain:

The reason the Lagrangian is not invariant is due to the derivative, which doesnot transform covariantly under a local gauge transformation:

In order to make the derivative transform covariantly (and thus the Lagrangiangauge invariant), we have to introduce a gauge field, in this case a vectorfield , as follows:

(8.3.1.1)¶

and the field must transform as . At this level, we are free to choose either plus orminus sign in (8.3.1.1), since the sign change can be absorbedin the definition of the field without loss of generality (if we changethe sign, the field transformation then changes to ). In the +2 metric signature we chose a minus sign, sothat coincides with the usual definition of the electromagnetic4-potential:

With signature -2, we must choose a plus sign and the identification goes asfollows:

And we obtain the same final equation. So the kinematic momentum is equal tocanonical momentum minus charge times the gauge field. The last expression isindependent of a metric signature, and that is what is e.g. in the kinetic termof a Schrödinger or Pauli equation (with the minus sign in ). We derive the non-relativistic limit rigorously later, but it givesthe same result. At this level we just have to make sure we choose the correctsign in (8.3.1.1), depending on the metric signature,otherwise we would get the electromagnetic 4-potential with the opposite sign(the sign of is ultimately just a convention, but later we want to getthe same equations as everybody else).

Another unrelated convention is in choosing the sign of the parameter . Wehave choosen it to coincide with an electric charge (negative for electrons).Some authors choose to be positive for electrons, then one must flip thesign in (8.3.1.1).

We will continue using the +2 signature in the rest of the section.

The operator is called a covariant derivative, because it does not change aform (is invariant) under a local gauge transformation:

Then the Lagrangian

(8.3.1.2)¶

is also gauge invariant:

The Lagrangian (8.3.1.2) can also be written as:

We can see that the condition of a local gauge invariance requires aninteraction with a vector field . Now we need to add the kinetic termfor the field :

The mass term is not gauge invariant, and so we have toset . Here is the full Lagrangian:

This is a Lagrangian for an electron and a massless vector boson (photon) ofspin 1. We can introduce a current ,then the Lagrangian density becomes:

For an electron, we can set , where is the elementary charge ( ispositive).

8.3.2. QED Lagrangian¶

We use a metric with signature -2 in this section.

The QED Lagrangian density is

where

and we must choose a plus sign in (8.3.1.1) since we use the-2 signature:

is the charge (negative for electrons ).

is the electromagnetic field tensor. It’s astonishing, that this simple Lagrangian can account for all phenomena from macroscopic scales down to something like . So it’s not a surprise that Feynman, Schwinger and Tomonaga received the 1965 Nobel Prize in Physics for such a fantastic achievement.

Plugging this Lagrangian into the Euler-Lagrange equation of motion for a field, we get:

The first equation is the Dirac equation in the electromagnetic field and thesecond equation is a set of Maxwell equations () with a source , which is a4-current comming from the Dirac equation.

8.3.3. Magnetic moment of an electron¶

In this section we derive the order- correction to the magnetic momentof an electron.

We start by computing the electron vertex function for the process:

where corresponds to some heavy target. If is a fixedclassical potential, we get:

Using general arguments (Lorentz invariance, parity-conservation, Wardidentity) we can always write as:

where and ar unknown functions of called form factors. As we will see below, in the lowest order we get and .

We can calculate the amplitude for elastic Coulomb scattering of anonrelativistic electron from a region of nonzero electrostatic potential bysetting , then:

If the electrostatic field is very slowly varying over a large (evenmacroscopic) region, will be concentrated about , then we can take the limit :

This corresponds to the Born approximation for scattering from a potential

Thus is the electric charge of the electron, in units of . Since already in the first order of perturbation theory, radiativecorrections to must vanish at .

Now we calculate the scattering from a static vector potential by setting, then:

In the limit this becomes:

where

is the Fourier transform of the magnetic field produced by .

This corresponds to the Born approximation for scattering from a potential

where

where

The coefficient is called the Landé g-factor, and since the leading orderof perturbation theory gives (and we know that to allorders), we get:

This is the standard prediction of the Dirac equation. The anomalous magneticmoment is then:

To calculate that, we need to evaluate the one-loop correction to the vertexfunction, so we start by deriving the appropriate Green function for theprocess :

where:

is the interacting Green function for the Lagrangian. In the first order:

so the amplitude is:

and we got , so and in the lowestorder. In the next order we get:

Now we can write:

where

So the expressions for the form factors are:

contains both ultraviolet and infrared divergencies. To cure the infrareddivergence, we add a term to . To cure the ultravioletdivergence, we make the substitution:

where is the first order (in ) correction to (i.e.):

so the corrected is:

Neither the ultraviolet nor the infrareddivergence affects , so we just set :

Thus we get the correction to the -factor of the electron:

Code:

>>> from math import pi>>> alpha = 1/137.035999049>>> a_e = alpha / (2*pi)>>> a_e0.0011614097331824923

Experiments give (arXiv:1412.8284, eq. (1)).

Higher order corrections from QED can also be calculated:

we already know that . See for example hep-ph/9410248 for the expression for :

Code:

>>> from sympy import zeta, S, log>>> A_2 = S(197)/144 + zeta(2)/2 + 3*zeta(3)/4 - 3*zeta(2) * log(2)>>> A_2.n()-0.328478965579194

See hep-ph/9602417 for the term:

Code:

>>> from sympy import pi, zeta, S, log, sum, var, oo>>> var("n")n>>> a4 = sum(1/(2**n * n**4), (n, 1, oo))>>> A_3 = 83*pi**2*zeta(3)/72 - 215*zeta(5)/24 + 100*(a4 + log(2)**4/24 - \...  pi**2*log(2)**2/24)/3 - \...  239*pi**4/2160 + 139*zeta(3)/18 - 298 * pi**2 * log(2)/9 + \...  17101 * pi**2 / 810 + S(28259)/5184>>> A_3.n()1.18124145658720

Higher terms are only known numerically. The and terms can be foundin arXiv:1412.8284:

We can now sum up to a given order by the following script:

from sympy import pi, zeta, S, log, summation, var, oovar("n")a4 = summation(1/(2**n * n**4), (n, 1, oo))A1 = S(1)/2A2 = S(197)/144 + zeta(2)/2 + 3*zeta(3)/4 - 3*zeta(2) * log(2)A3 = 83*pi**2*zeta(3)/72 - 215*zeta(5)/24 + 100*(a4 + log(2)**4/24 - \ pi**2*log(2)**2/24)/3 - \ 239*pi**4/2160 + 139*zeta(3)/18 - 298 * pi**2 * log(2)/9 + \ 17101 * pi**2 / 810 + S(28259)/5184A4 = -1.91298A5 = 7.795alpha = 1/137.035999049a_e_exp = 0.00115965218073a_e_exp_err = 0.00000000000028a_e_other = 0.00000000000448A = [A1, A2, A3, A4, A5]a_e= []for i in range(len(A)): a_e.append((A[i]*(alpha/pi)**(i+1)).n())print "========== ================"print "Order :math:`a_e`"print "========== ================"for i in range(len(A)): print "%d %16.14f" % (i+1, sum(a_e[:i+1]))print "Other %16.14f" % a_e_otherprint "Total %16.14f" % (sum(a_e) + a_e_other)print "Experiment %16.14f" % a_e_expprint "Difference %16.14f" % abs(sum(a_e) + a_e_other - a_e_exp)print "Exp. err %16.14f" % a_e_exp_errprint "========== ================"

and obtain the following table:

The “Other” line are contributions from the dependence on the muon and tauparticle masses, the hadronic vacuum-polarization, the hadroniclight-by-light-scattering and the electroweak contribution(see arXiv:1412.8284).The “Difference” line is the difference from the theory (the “Total” line) andexperiment. The “Exp. err” line is the experimental error.

At this level of accuracy, the uncertainty of the exact value of isthe primary cause of the difference from experiment, and one can use thisresult to predict a more accurate value for , assuming that QED and thestandard model are valid.