Cambridge physicists: Gauge theory 'more than normally recognized’

A recent paper by two Cambridge University scientists shows how a concept important in physics as a whole, called “gauge,” plays out in the specific theory of electromagnetism. 

“The moral of the paper,” says coauthor Jeremy Butterfield, “is to correct a widespread view of gauge. There is more to it than is normally recognized.”

The paper, titled “How to Choose a Gauge? The Case of Hamiltonian Electromagnetism,” appeared as a preprint Oct. 25, 2021, on arxiv.org. 

A gauge theory is a type of field theory, a type of mathematical formalism that can be applied to many branches of physics, from gravitation to quantum mechanics. The “gauge” is a specific formulation chosen to specify the degrees of freedom of the system being described.

“Gauge is normally considered to be a matter of choice that is guided only by convenience for solving the physical problem at hand,” Butterfield said. “But we show that in the theory of electromagnetism there is a systematic reason to make one choice.”

That choice, Butterfield says, “is called the Coulomb gauge,” named after the French physicist who showed that an electrically charged body obeys the same inverse square law of attraction as is seen in gravitation. 

The gauge in electromagnetism

The behavior of electricity and magnetism was the subject of much study in the 19th Century. The British physicists Michael Faraday and James Clerk Maxwell adopted the view that electricity and magnetism were two aspects of the same natural phenomenon, now called electromagnetism. Maxwell wrote down equations describing electromagnetism. 

“Nowadays we write these equations in very different forms from the one adopted by Maxwell,” said Butterfield, pointing out that “a great deal of theoretical science involves, not only applying given equations, but systematically reformulating them, i.e. casting them in terms of different concepts than the given ones.” 

“One of these current forms is called ‘Hamiltonian,’ named after the 19th Century Irish physicist William Rowan Hamilton,” Butterfield said.

Advantages of Hamiltonian form

“Writing physical theories in Hamiltonian form has many advantages,” Butterfield said. “When cast in this form one can see in the theory how notions that are like position get paired one-to-one with notions that are like momentum [i.e. mass times velocity]. Another main one is that the treatment of symmetries becomes clearer and more systematic. 

“In effect there is a systematic procedure, an algorithm, for deriving from the theory’s equations its symmetries," he added. (This is often called the ‘Dirac algorithm,’ after the 20th Century British physicist Paul Dirac.) “In our paper Henrique Gomes and I show how these two advantages, applied to Maxwell’s equations for electromagnetism, imply a special and deep role for the Coulomb gauge, independent of what problem you are trying to solve.”

Summing up the argument, Butterfield said, “In physics gauge is generally regarded as a matter of re-description. This is taken to mean that many choices of gauge are legitimate, and which is the sensible choice depends on the context, in particular the problem you are trying to solve. But there is more to it than that. Although you can freely choose, there is a general and systematic reason for one choice.”

Henrique Gomes and Jeremy Butterfield are at Trinity College, University of Cambridge in the United Kingdom. 

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H. Gomes and J. Butterfield. How to Choose a Gauge? The case of Hamiltonian Electromagnetism, arxiv.org(Oct. 25, 2021). https://arxiv.org/pdf/2110.11682.pdf