Quantum Field Theory

Quantum Electrodynamics (QED)

Quantum electrodynamics, or QED, is a quantum theory of the interactions of charged particles with the electromagnetic field. It describes mathematically not only all interactions of light with matter but also those of charged particles with one another. QED is a relativistic theory in that Albert Einstein's theory of special relativity is built into each of its equations. That is, the equations are invariant under a transformation of space-time. The QED theory was refined and fully developed in the late 1940s by Richard P. Feynman, Julian S. Schwinger, and Shin'ichiro Tomonaga, independently of one another. Because the behavior of atoms and molecules is primarily electromagnetic in nature, all of atomic physics can be considered a test laboratory for the theory. Agreement of very high accuracy makes QED one of the most successful physical theories so far devised.

The formulism for QED is very similar to the nucleon-pion system in Eqs.(12) and (13). While Eq.(13) for fermion is readily applicable (with appropriate value for ko, which is proportional to the mass of the fermion); Eq.(12) is replaced by the Maxwell equations:

where E, B are the electric and magnetic field respectively, j is the current density, is the charge density, and c is the velocity of light.

By virtue of the anti-symmetric property of Eq.(21), the continuity equation for the charge-current density is automatically satisfied, i.e., where = /x. The energy-momentum tensor of the electromagnetic field can be expressed as:
, where .
In curved space-time Eq.(23b) remains unchanged, while Eq.(23a) becomes , where g is the determinant of the metric tensor ( g = g11g22g33g44 in isotropic space-time). In general, the field component and the source are multiplied by a factor of (-g)½ with the presence of gravity. The energy-momentum tensor is in a more complicated form: . See "temporal waves" from early universe as dark energy.

Another form of the Maxwell equations is in terms of the four-vector potentials A defined by:
 ---------- (24)
This definition is used to simplify computations. It has a redundancy in the degree of freedom for the four-vector potentials A when Eq.(23a) is expressed in this form. By imposing the Lorenz condition to the arbitrariness,

By the way, Eq.(23b) is automatically satisfied by the relationship between the anti-symmetric tensor fields and the four-vector potentials in Eq.(24).

The 3-vector potential A can always be decomposed into a transverse component and a longitudinal component (parallel to the direction of motion) as shown in Eq.(29) such that the transversality condition Eq.(30) (which dictates that the transverse components are perpendicular to the direction of motion) and the irrotational condition of the longitudinal component Eq.(31) are satisfied:

It can be shown that and A0 together give rise to the instantaneous static Coulomb interactions between charged particles with the total Hamiltonian in the form (the Hamiltonian H is related to the Lagrangian L by the formula H = p - L):
 ---------- (32)
where and A0 have been completely eliminated in favor of the instantaneous Coulomb interaction in the last term. It is instantaneous in the sense that its value at t is determined by the charge distribution at the same instant of time. This formalism was derived by E. Fermi in 1930. Since the separation into different terms is not relativistically covariant, nor is the transversality condition itself, the whole formalism appears noncovariant; each time we perform a Lorentz transformation, we must simultaneously make a gauge transformation to obtain a new set of A and A0. It had been shown that it is possible to develop manifestly covariant calculational techniques starting with this Hamiltonian. It is also possible to construct a formalism, which preserves relativistic covariance at every stage.

Note that the long range instantaneous Coulomb interaction does not imply a force travelling faster than the speed of light. Although the intereaction is instantaneous, it can be considered as the interaction between two overlapping Coulomb tails (clouds of virtual photons) of the two charged particles, so there is no need for the interaction to travel from one point to another in zero time.

See a slightly different treatment for derivation of the free field equations in "Quantization and Field Equations".

QED concerns mainly with the transverse components, which account for the electromagnetic radiation of accelerating charged particles. The transverse electromagnetic fields provide a simple and physically transparent description of a variety of processes in which real photons are emitted, absorbed, or scattered. The three basic equations for the free-field case are:

where A satisfies the transversality condition in Eq.(30).

Eq.(33b) is in a form very similar to the Klein-Gordon Equation Eq.(1l) or Eq.(12) except that the mass term vanishes (because the photon has no rest mass) and the field is a vector (instead of scalar) with two transverse components (polarization) perpendicular to each other. Thus A can be expressed in Fourier series similar to Eq.(2):
The quantization rules for the electromagnetic field is very similar to that in Eq.(4):
 ---------- (36)
where the ak's are related to the ck's by:
 ---------- (37)
Construction of the eigenvectors follows exactly the same way as in Eqs.(5) and (6) with an additional index for polarization.

Interaction between photon and fermion, e.g., electron can be introduced by demanding local gauge invariance for the formulism. With this constraint on the quantum field theory, the ordinary derivative in Eq.(13) becomes the covariant derivative:

and the interaction takes the form:

where e is the coupling constant. (See appendix on "Abelion/non-Abelion Groups and U(1), SU(2), SU(3)" for a discussion about the concept of gauge or phase transformation.)

The photon-electron processes are described by substituting Eq.(38b) to HI in the S matrix expansion Eq.(15):

• The first order graphs -

where the electron field has been decomposed to:

and the symbol X may stand for an interaction with an external classical potential, the emission of a photon, or the absorption of a photon. The realization of a particular process depends entirely on the initial and final states.

• The second order graphs (tree diagrams) -

• Two-photon annihilation - Pair of an electron and positron into two gamma rays.
• Compton scattering - It occurs when an electron and a photon collide and scatter elastically.
• Moller scattering - It is the scattering of two relativistic electrons.
• Bremsstrahlung - It is the process by which radiation is emitted from an electron as it moves past a nucleus.

• Second order graphs (loop diagrams) -

• Vacuum polarization - It produces a correction to the coupling constant (electric charge) and contributes to the Lamb shift, which is a small splitting of hydrogen atom energy levels caused by the interaction with the virtual pair (of electron and positron).
• Vertex correction - A correction to the electron vertex function, which contributes to the anomalous electron magnetic moment.
• Self-energy - The interaction of a charged particle with its own field and it usually gives rise to infinite self-energy and infinite mass.
• Mass renormalization - The observed mass is generated by combining the bare mass and the (calculated) divergent mass.
In summary QED rests on the idea that charged particles (e.g., electrons and positrons) interact by emitting and absorbing photons, the particles of light that transmit electromagnetic forces. These photons are virtual; that is, they cannot be seen or detected in any way because their existence violates the conservation of energy and momentum. Interaction also occurs by the exchange of virtual electron/positron. The exchange of virtual photon is manifested as the "force" in the electromagnetic interaction, because the interacting particles change their speed and direction of travel as they release or absorb the energy of a photon. Photons also can be emitted in a free state, in which case they may be observed. The interaction of real particles occurs in a series of graphs of increasing complexity. In the first order graph, no virtual photon or virtual electron/positron is involved; in the second-order graph, there are either one virtual photon or one virtual electron/positron; and so forth. The graphs correspond to all the possible ways in which the particles can interact by the exchange of virtual photons and virtual electrons/positrons, and each of them can be represented graphically by means of the Feynman diagrams. Besides furnishing an intuitive picture of the process being considered, this type of diagram prescribes precisely how to calculate the observable quantity involved.

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