45 U(1) Gauge Symmetry and Electromagnetic Fields

We have now constructed both relativistic and non-relativistic quantum field theories for free scalar particles. In the non-relativistic case we had to use complex-valued fields, and found that the theory came with an action of a $U(1)$ group, the group of phase transformations on the fields. In the relativistic case real-valued fields could be used, but if we took complex-valued ones (or used pairs of real-valued fields), again there was an action of a $U(1)$ group of phase transformations. This is the simplest example of a so-called “internal symmetry” and it is reflected in the existence of an operator $\hat{Q}$ called the “charge”.

In this chapter we’ll see how to ${}_{\mathrm{go}}$ beyond the theory of free quantized charged particles, by introducing background electromagnetic fields that the charged particles will interact with. It turns out that this can be done using the $U(1)$ group action, but now acting independently at each point in spacetime, giving a large, infinite dimensional group called the “gauge group”. This requires introducing a new sort of space-time dependent field, called a “vector potential” by physicists, a “connection” by mathematicians. Use of this field allows the construction of a Hamiltonian dynamics invariant under the gauge group. This fixes the way charged particles interact with electromagnetic fields, which are described by the vector potential.

Most of our discussion will be for the case of the $U(1)$ group, but we will also indicate how this generalizes to the case of non-Abelian groups such as $SU(2)$ .

中文目录

Chapter contents

• 45.1 U(1) gauge symmetry
• 45.2 Curvature, electric and magnetic fields
• 45.3 Field equations with background electromagnetic fields
• 45.4 The geometric significance of the connection
• 45.5 The non-Abelian case
• 45.6 For further reading

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原书 PDF · 印刷页 484、485、486、487、488、489、490、491、492、493

来源版本：2025-10-20

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