In Coleman's lecture on quantum field theory he says that when a particle is confined in in a region shorter than its Compton wavelength, very many particles can be produced. My question is whether this will happen even in vacuum and, if so, where does the energy required for their production come from?
People explain these using uncertainty principle in the form of the energy-time uncertainty relation. However, in quantum field theory, the only uncertainty principle is the uncertainty between a field $\phi(x)$ and the conjugate momentum $\pi(x)$.
EDIT: Page 16 of this note by Coleman.
Answer
Yes, this will happen. But you cannot confine particle in the vacuum. To confine a particle, you must have some potential. The energy to produce pairs must come exactly from this binding potential. For example, you can confine electron using a very strong electric field. To confine an electron in a region smaller than its Compton wavelength you need a field with enough energy to create electron position pairs. Particle in a vacuum will never be confined.
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