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Quantum field theory[edit]

Main article: Quantum field theory

Theory that brings quantum mechanics and special relativity together to account for subatomic theory.

Mathematics[edit]

Mathematically, QED is an abelian gauge theory with the symmetry group U(1). The gauge field, which mediates the interaction between the charged spin-1/2 fields, is the electromagnetic field. The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field is given by the real part of

where

are Dirac matrices;

a bispinor field of spin-1/2 particles (e.g. electron–positron field);

, called "psi-bar", is sometimes referred to as the Dirac adjoint;

is the gauge covariant derivative;

e is the coupling constant, equal to the electric charge of the bispinor field;

Aμ is the covariant four-potential of the electromagnetic field generated by the electron itself;

Bμ is the external field imposed by external source;

is the electromagnetic field tensor.

Equations of motion[edit]

To begin, substituting the definition of D into the Lagrangian gives us

Next, we can substitute this Lagrangian into the Euler–Lagrange equation of motion for a field:

 

(2)

to find the field equations for QED.

The two terms from this Lagrangian are then

Substituting these two back into the Euler–Lagrange equation (2) results in

with complex conjugate

Bringing the middle term to the right-hand side transforms this second equation into

 

The left-hand side is like the original Dirac equation and the right-hand side is the interaction with the electromagnetic field.

One further important equation can be found by substituting the Lagrangian into another Euler–Lagrange equation, this time for the field, Aμ:

 

(3)

The two terms this time are

and these two terms, when substituted back into (3) give us

 

Now, if we impose the Lorenz gauge condition, that the divergence of the four potential vanishes

then we get

which is a wave equation for the four potential, the QED version of the classical Maxwell equations in the Lorenz gauge. (In the above equation, the square represents the D'Alembert operator.)

Interaction picture[edit]

This theory can be straightforwardly quantized by treating bosonic and fermionic sectors as free. This permits us to build a set of asymptotic states which can be used to start a computation of the probability amplitudes for different processes. In order to do so, we have to compute an evolution operator that, for a given initial state, will give a final state in such a way to have

This technique is also known as the S-Matrix. The evolution operator is obtained in the interaction picture where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:

and so, one has

where T is the time ordering operator. This evolution operator only has meaning as a series, and what we get here is a perturbation series with the fine structure constant as the development parameter. This series is called the Dyson series.

Feynman diagrams[edit]

Despite the conceptual clarity of this Feynman approach to QED, almost no early textbooks follow him in their presentation. When performing calculations it is much easier to work with the Fourier transforms of the propagators. Quantum physics considers particle's momenta rather than their positions, and it is convenient to think of particles as being created or annihilated when they interact. Feynman diagrams then look the same, but the lines have different interpretations. The electron line represents an electron with a given energy and momentum, with a similar interpretation of the photon line. A vertex diagram represents the annihilation of one electron and the creation of another together with the absorption or creation of a photon, each having specified energies and momenta.

Using Wick theorem on the terms of the Dyson series, all the terms of the S-matrix for quantum electrodynamics can be computed through the technique of Feynman diagrams. In this case rules for drawing are the following

To these rules we must add a further one for closed loops that implies an integration on momenta, since these internal ("virtual") particles are not constrained to any specific energy–momentum – even that usually required by special relativity (see this article for details). From them, computations of probability amplitudes are straightforwardly given. An example is Compton scattering, with an electron and a photon undergoing elastic scattering. Feynman diagrams are in this case

and so we are able to get the corresponding amplitude at the first order of a perturbation series for the S-matrix:

from which we are able to compute the cross section for this scattering.

Renormalizability[edit]

Higher order terms can be straightforwardly computed for the evolution operator but these terms display diagrams containing the following simpler ones

 

One-loop contribution to the vacuum polarization function

 

One-loop contribution to the electron self-energy function

 

One-loop contribution to the vertex function

that, being closed loops, imply the presence of diverging integrals having no mathematical meaning. To overcome this difficulty, a technique called renormalization has been devised, producing finite results in very close agreement with experiments. It is important to note that a criterion for theory being meaningful after renormalization is that the number of diverging diagrams is finite. In this case the theory is said to be renormalizable. The reason for this is that to get observables renormalized one needs a finite number of constants to maintain the predictive value of the theory untouched. This is exactly the case of quantum electrodynamics displaying just three diverging diagrams. This procedure gives observables in very close agreement with experiment as seen e.g. for electron gyromagnetic ratio.

Renormalizability has become an essential criterion for a quantum field theory to be considered as a viable one. All the theories describing fundamental interactions, except gravitation whose quantum counterpart is presently under very active research, are renormalizable theories.

Nonconvergence of series[edit]

An argument by Freeman Dyson shows that the radius of convergence of the perturbation series in QED is zero.[24] The basic argument goes as follows: if the coupling constant were negative, this would be equivalent to the Coulomb force constant being negative. This would "reverse" the electromagnetic interaction so that like charges would attract and unlike charges would repel. This would render the vacuum unstable against decay into a cluster of electrons on one side of the universe and a cluster of positrons on the other side of the universe. Because the theory is 'sick' for any negative value of the coupling constant, the series do not converge, but are an asymptotic series.

From a modern perspective, we say that QED is not well defined as a QFT to arbitrarily high energy.[25] The coupling constant runs to infinity at finite energy, signalling a Landau pole. The problem is essentially that QED is not asymptotically free. This is one of the motivations for embedding QED within a Grand Unified Theory.

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