Quantum electrodynamics

In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics.

In technical terms, QED can be described as a perturbation theory of the electromagnetic quantum vacuum. Richard Feynman called it "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the electron and the Lamb shift of the energy levels of hydrogen. It is the most precise and stringently tested theory in physics.

History

upright|thumb|right|[[Paul Dirac]]

The first formulation of a quantum theory describing radiation and matter interaction is attributed to Paul Dirac, who during the 1920s computed the coefficient of spontaneous emission of an atom. He is credited with coining the term "quantum electrodynamics".

Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. In the following years, with contributions from Wolfgang Pauli, Eugene Wigner, Pascual Jordan, Werner Heisenberg and Enrico Fermi, physicists came to believe that, in principle, it was possible to perform any computation for any physical process involving photons and charged particles. However, further studies by Felix Bloch with Arnold Nordsieck, and Victor Weisskopf, in 1937 and 1939, revealed that such computations were reliable only at a first order of perturbation theory, a problem already pointed out by Robert Oppenheimer. At higher orders in the series infinities emerged, making such computations meaningless and casting doubt on the theory's internal consistency. This suggested that special relativity and quantum mechanics were fundamentally incompatible.

upright|thumb|[[Hans Bethe ]]

Difficulties increased through the end of the 1940s. Improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom, later known as the Lamb shift and magnetic moment of the electron. These experiments exposed discrepancies that the theory was unable to explain.

A first indication of a possible solution was given by Hans Bethe in 1947. He made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by Willis Lamb and Robert Retherford. Despite limitations of the computation, agreement was excellent. The idea was simply to attach infinities to corrections of mass and charge that were actually fixed to a finite value by experiments. In this way, the infinities get absorbed in those constants and yield a finite result with good experimental agreement. This procedure was named renormalization.

thumb|right|[[Richard Feynman|Feynman (center) and Oppenheimer (right) at Los Alamos.]]

Based on Bethe's intuition and fundamental papers on the subject by Shin'ichirō Tomonaga, Julian Schwinger, Richard Feynman and Freeman Dyson, it was finally possible to produce fully covariant formulations that were finite at any order in a perturbation series of quantum electrodynamics. Tomonaga, Schwinger, and Feynman were jointly awarded the 1965 Nobel Prize in Physics for their work in this area. Their contributions, and Dyson's, were about covariant and gauge-invariant formulations of quantum electrodynamics that allow computations of observables at any order of perturbation theory. Feynman's mathematical technique, based on his diagrams, initially seemed unlike the field-theoretic, operator-based approach of Schwinger and Tomonaga, but Dyson later showed that the two approaches were equivalent.

QED is the model and template for all subsequent quantum field theories. One such subsequent theory is quantum chromodynamics, which began in the early 1960s and attained its present form in the 1970s, developed by H. David Politzer, Sidney Coleman, David Gross and Frank Wilczek. Building on Schwinger's pioneering work, Gerald Guralnik, Dick Hagen, and Tom Kibble, Peter Higgs, Jeffrey Goldstone, and others, Sheldon Glashow, Steven Weinberg and Abdus Salam independently showed how the weak nuclear force and quantum electrodynamics could be merged into a single electroweak force.

Feynman's view of quantum electrodynamics

Introduction

Near the end of his life, Richard Feynman gave a series of lectures on QED intended for the lay public. These lectures were transcribed and published as Feynman (1985), QED: The Strange Theory of Light and Matter,

= \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \bar\psi\,(i\gamma^\mu D_\mu - m)\,\psi\right]</math>

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where

<math> \gamma^\mu </math> are Dirac matrices.
<math>\psi</math> a Dirac spinor field of spin-1/2 particles (e.g. electron–positron field).
<math>\bar\psi\equiv\psi^\dagger\gamma^0</math>, called "psi-bar", is sometimes referred to as the Dirac adjoint.
<math>D_\mu \equiv \partial_\mu+ieA_\mu+ieB_\mu </math> is the gauge covariant derivative.
e is the coupling constant, equal to the electric charge of the Dirac spinor field.
<math>A_\mu</math> is the covariant four-potential of the electromagnetic field generated by the electron itself. It is also known as a gauge field or a <math>\text{U}(1)</math> connection.
<math>B_\mu</math> is the external field imposed by external source.
m is the mass of the electron or positron.
<math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu </math> is the electromagnetic field tensor. This is also known as the curvature of the gauge field.

Expanding the covariant derivative reveals a second useful form of the Lagrangian (external field <math>B_\mu</math> set to zero for simplicity)

:<math>\mathcal{L} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \bar\psi(i\gamma^\mu \partial_\mu - m)\psi - ej^\mu A_\mu</math>

where <math>j^\mu</math> is the conserved <math>\text{U}(1)</math> current arising from Noether's theorem. It is written

:<math>j^\mu = \bar\psi\gamma^\mu\psi.</math>

Equations of motion

Expanding the covariant derivative in the Lagrangian gives

:<math>\mathcal{L} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i \bar\psi \gamma^\mu \partial_\mu \psi - e\bar{\psi}\gamma^\mu A_\mu \psi -m \bar{\psi} \psi </math>

:<math> = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i \bar\psi \gamma^\mu \partial_\mu \psi -m \bar{\psi} \psi - ej^\mu A_\mu .</math>

For simplicity, <math>B_\mu</math> has been set to zero, with no loss of generality. Alternatively, we can absorb <math>B_\mu</math> into a new gauge field <math>A'_\mu = A_\mu + B_\mu</math> and relabel the new field as <math>A_\mu.</math>

From this Lagrangian, the equations of motion for the <math>\psi</math> and <math>A_\mu</math> fields can be obtained.

Equation of motion for ψ

These arise most straightforwardly by considering the Euler-Lagrange equation for <math>\bar\psi</math>. Since the Lagrangian contains no <math>\partial_\mu\bar\psi</math> terms, we immediately get

:<math>\frac{\partial \mathcal{L{\partial(\partial_\mu \bar\psi)} = 0</math>

so the equation of motion can be written

<math>(i\gamma^\mu\partial_\mu-m)\psi = e\gamma^\mu A_\mu\psi.</math>

Equation of motion for A<sub>μ</sub>

Using the Euler–Lagrange equation for the <math>A_\mu</math> field,

{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L{\partial A_\mu} = 0,</math>|3

the derivatives this time are

<math display="block">\partial_\nu \left( \frac{\partial \mathcal{L{\partial ( \partial_\nu A_\mu )} \right) = \partial_\nu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right),</math>

<math display="block">\frac{\partial \mathcal{L{\partial A_\mu} = -e\bar{\psi} \gamma^\mu \psi.</math>

Substituting back into () leads to

:<math>\partial_\mu F^{\mu\nu} = e\bar\psi \gamma^\nu \psi</math>

which can be written in terms of the <math>\text{U}(1)</math> current <math>j^\mu</math> as

Now, if we impose the Lorenz gauge condition

<math display="block">\partial_\mu A^\mu = 0,</math>

the equations reduce to

which is a wave equation for the four-potential, the QED version of the classical Maxwell equations in the Lorenz gauge. (The square represents the wave operator, <math>\Box = \partial_\mu \partial^\mu</math>.)

Interaction picture

This theory can be straightforwardly quantized by treating bosonic and fermionic sectors as free. This permits us to build a set of asymptotic states that can be used to start computation of the probability amplitudes for different processes. In order to do so, we have to compute an evolution operator, which for a given initial state <math>|i\rangle</math> will give a final state <math>\langle f|</math> in such a way to have cannot be understood in terms of any finite number of Feynman diagrams and hence is described as nonperturbative. Mathematically, it can be derived by a semiclassical approximation to the path integral of quantum electrodynamics.

Renormalizability

Higher-order terms can be straightforwardly computed for the evolution operator, but these terms display diagrams containing the following simpler ones 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 does not converge but is at best an asymptotic series.

From a modern perspective, we say that QED is not well defined as a quantum field theory to arbitrarily high energy. The coupling constant runs to infinity at finite energy, signalling a Landau pole. The problem is essentially that QED appears to suffer from quantum triviality issues. This is one of the motivations for embedding QED within a Grand Unified Theory.

Electrodynamics in curved spacetime

This theory can be extended, at least as a classical field theory, to curved spacetime. This arises similarly to the flat spacetime case, from coupling a free electromagnetic theory to a free fermion theory and including an interaction which promotes the partial derivative in the fermion theory to a gauge-covariant derivative.

References

External links

Feynman's Nobel Prize lecture describing the evolution of QED and his role in it
Feynman's New Zealand lectures on QED for non-physicists
The Strange Theory of Light | Animation of Feynman pictures light by QED – Animations demonstrating QED