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Quantum Field Theory

Quantum Field Theory (QFT) is a happy marriage of Quantum Mechanics and Special Relativity. Quantum Electrodynamics, the theory of electromagnetic interactions, is an example of QFT. Its agreement with experiments is phenomenal—it's the best theory in Physics, bar none. So if we are serious about studying the structure of our world, we have to take into account all the interesting things that QFT has to tell us.

To begin with, since QFT is relativistic, all events described by it take place in four-dimensional space-time. (For those familiar with superstring theory, this becomes a 10-dimensional space-time plus some fermionic dimensions.) We already know what it means—the events, both past and future, must co-exist trapped in a preordained universe. On the other hand, since QFT is an extension of Quantum Mechanics, it has the same probabilistic interpretation. In QFT, instead of calculating a wave function, we calculate the amplitude. Just like the wave function, the amplitude has the magnitude, with its probabilistic interpretation, and the phase, which cannot be measured directly. But it's the way the amplitude is calculated that will have direct impact on our understanding of the world.

Amplitude of a transition from an initial state to some final state is calculated in QFT by summing up the amplitudes of all possible paths between the two states. The paths not only correspond to different trajectories of particles, but they involve the creation and annihilation of multitudes of intermediate virtual particles. The physics of virtual particles is different than that of ordinary particles. A virtual photon or electron may have any mass, including imaginary (real photons are massless). They still have to obey the fundamental laws of physics, like the conservation of energy and momentum, conservation of charge, etc.

The various possible intermediate paths (or histories) are described using Feynman diagrams. For instance, to calculate the amplitude of an electron going from point A to point B, you have to sum up the amplitudes corresponding to (see Fig 6):

  • A single non-virtual electron moving from A to B
  • The electron emitting a virtual photon and re-absorbing it
  • The electron emitting two virtual photons and re-absorbing them
  • The electron emitting a virtual photon which turns into an electron-positron pair, which then annihilates, creating another virtual photon that is absorbed by the electron
  • And so on

There are infinitely many amplitudes to sum up, but the more interaction vertices (branching points) they have, the smaller their contribution. In all practical calculations it's enough to calculate just a few simple diagrams to get a pretty good agreement with experiment. And once again, the calculations of amplitudes are fully deterministic. It's the probabilistic interpretation of the amplitudes that introduces uncertainties (we'll come back to it later).


Fig 6. Some of the Feynman diagrams used in the calculation of the amplitude of an electron moving from point A to point B. Solid lines with arrows correspond to electrons (positron arrows point in the opposite direction). Wavy lines correspond to photons. Black dots correspond to interaction vertices. Virtual particles are drawn in red.

What should we make of it? Do virtual particles really exist? The name "virtual" suggests that they don't, but without them, we wouldn't be able to calculate any of the physical, measurable quantities. Once more we have to ask the question: "Is this only a mathematical device?" Some physicists would indeed argue that it is. They would argue that Feynman diagrams are an artifact of perturbation theory—a scheme in which a quantity to be calculated is expanded into an infinite series. But the truth is that we don't know of any other scheme to calculate these quantities. Are we therefore to believe that all these paths and virtual particles have an existence of their own? That would be an even stronger case for a many-universe interpretation.

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