Why is a relativistic quantum theory of a finite number of particles impossible?

Question

In Dyson's book Advanced Quantum Mechanics , he said "These two examples (the discovery of antimatter and meson) are special cases of the general principle, which is the basic success of the relativistic quantum theory, that A Relativistic Quantum Theory of a Finite Number of Particles is Impossible." However, when we calculate the Feynman diagram of particles collision process, the number of particles should be finite.

So my question is why a relativistic quantum theory of a finite number of particles is impossible. What does it mean explicitly? Does it mean that we need to consider infinite number of harmonic oscillators when we quantize free field?

Answer 1

Because a pair of particle and anti-particle can be created from the vacuum, it means that infinite number of pairs of particle and anti-particle can be created from the vacuum. So when you consider relativistic quantum theory it's impossible to only consider a finite number of particles.

When you calculate Feynman diagram, you are actually only doing perturbation theory to some order. If you want to calculate the exact result by Feynman diagrams, then you need to consider infinite number of Feynman diagrams, because you can add loops to the diagrams. And it means that you need to consider infinite number of particles.

When quantize a field, we do have to consider an infinite number of harmonic oscillators. What do I mean by this is that if you look at the expression for the quantum field, say,a scalar field, you will find that it's a superposition of infinite creation operators and annihilation operators.