** Quantum Electrodynamics has Zero Radius of Convergence **
*Summarized from Toichiro Kinoshita* Quantum Electrodynamics is the best tested theory on earth (perhaps in the universe, depending on how many aliens there are). The practitioners (starting with Schwinger and

Feynman, and now being led by

Tom Kinoshita here at Cornell) typically calculate ``

**g-2**'' for the electron. They've calculated it as a series to

eighth order in the electron charge

*e*, or more properly to fourth order in the

fine-structure constant
which is about 1/137.03599993...

These calculations

aren't easy. The first term is represented by a Feynman diagram with one closed loop, and requires no more than a page or two of hand calculation. The second term represents seven diagrams, and took seven years. Seventy two Feynman diagrams are needed for the third term, and all of them have been evaluated exactly using symbolic manipulation programs on computers, after nearly thirty years of hard work. The fourth term requires the evaluation of 891 four-loop Feynman diagrams, and has been estimated numerically using large-scale computations on supercomputers. The last term (not involving the fine-structure constant) is a small correction caused by particles other than the electron, and interactions of a non-electronic type.

Extremely clever experiments have measured this same number. For the electron, it is measured to be

and for the positron (the anti-electron) it's

(If the electron and positron did not have the same values for

**g-2**, that would be an amazing surprise.)

Now, to compare experiment and theory, we need to know the value of the

fine-structure constant . The measurements of the value of this constant aren't as accurate as those of

**g-2**! One can measure

using the

quantum Hall effect, get 1/137.0360037(27) (an accuracy of 0.020ppm), and predict

**(g-2)qH** =1,159,652,156.4(22.9) x 10-12 One can measure

using the

ac Josephson effect to be 1/137.0359770(77) (0.056ppm), and predict

**(g-2)acJ** =1,159,652,378.0(65.3) x 10-12 Or, one can measure Planck's constant

*h* and the

mass of the neutron, and derive

to be 1/137.03601082(524) (0.039ppm) to predict

**(g-2)h/mn** =1,159,652,092.2(44.4) x 10-12 The numbers in parentheses are due to the uncertainty in the experimental value of

; the errors in the computer-measurement of the theoretical formula is much smaller (plus or minus 1.2). If you trust the theory, you can work backwards to an even better estimate of

: 1/137.03599993(52), with an estimated error of 0.0038ppm. This is undoubtedly the most accurate prediction ever made, and one of the most difficult. It's also one of the most accurate measurements ever made.

What's the point? Well, it turns out that this series does not converge!

Freeman Dyson gave a wonderful argument that the

radius of convergence of this function is zero. He noticed that the theory was sick (unstable) for negative

, (complex electron charge), no matter how small alpha was. Since any circle about

includes some of the sick region, the series could not converge! Just like

Stirling's formula, the theory is mighty useful even though it won't keep improving as we calculate more and more terms.

This was the inspiration for our work showing that

elastic theory has zero radius of convergence.
**Quantum Electrodynamics (QED)**
Quantum electrodynamics, commonly referred to as QED, is a quantum field theory of the

electromagnetic force. Taking the example of the force between two

electrons, the classical theory of electromagnetism would describe it as arising from the

electric field produced by each electron at the position of the other. The force can be calculated from

Coulomb's law.

The quantum field theory approach visualizes the force between the electrons as an

exchange force arising from the exchange of virtual

photons. It is represented by a series of

Feynman diagrams, the most basic of which is

With time proceeding upward in the diagram, this diagram describes the electron interaction in which two electrons enter, exchange a photon, and then emerge. Using a mathematical approach known as the Feynman calculus, the strength of the force can be calculated in a series of steps which assign contributions to each of the types of Feynman diagrams associated with the force.

QED applies to all electromagnetic phenomena associated with charged fundamental particles such as electrons and positrons, and the associated phenomena such as pair production, electron-positron annihilation, Compton scattering, etc. It was used to precisely model some quantum phenomena which had no classical analogs, such as the Lamb shift and the anomalous magnetic moment of the electron. QED was the first successful quantum field theory, incorporating such ideas as particle creation and annihilation into a self-consistent framework. The development of the theory was the basis of the 1965 Nobel Prize in physics, awarded to Richard Feynman, Julian Schwinger and Sin-itero Tomonaga.

**HOMEWORK FOR THURSDAY:**

Using between 500 and 700 words, tell the class what you think Quantum ElectroThermoDyanmics means to you

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