necessity of field theory

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Contents

Historical viewpoint

Here we shall discuss the breakdown of single-particle quantum mechanics when special relativity is demanded. The root of the problem being the lack of finite-dimensional unitary representations of the Lorentz group which therefore necessitates a system with infinitely many degrees of freedom - a quantum field.

From first to second quantization

The development of single-particle quantum mechanics, in which the position of a particle is interpreted as an operator \hat\mathbf{x}\, acting on a wavefunction \psi(\mathbf{x})\,, is sometimes referred to as first quantization for historical reasons. In order to get a sensible relativistic quantum theory in which particles can be created and destroyed, we are led to introduce a field, which, when quantized, itself becomes an operator \hat\psi(\mathbf{x})\, acting on a much larger, multi-particle Hilbert space, or Fock space. This procedure is then referred to as second quantization; however, \mathbf{x}\, here is merely a label and \hat\psi(\mathbf{x})\, should first be considered as a classical field, just like the electromagnetic field. The system is therefore only quantized once!

The prototype classical field \psi(\mathbf{x})\, for the electron obeys a classical equation of motion known as the Dirac equation. As we will see, this equation can be written as a Schrödinger equation for a single particle with spin. Much of the mathematical structure developed in single-particle quantum mechanics is in fact well-suited to the study of any linear partial differential equation, without the need of a probabilistic interpretation, and the Dirac equation is no exception. It is a fortuitous accident, therefore, that the Schrödinger equation be so successful in describing the quantum behaviour of single particles. The reason for this is that correlation functions such as \left\langle \psi(\mathbf{x}) \psi(\mathbf{y})\right\rangle\, may be written as an expansion in powers of \hbar\,, with the leading term in the expansion being the classical Green's function of the Dirac equation[1]. Higher quantum mechanical corrections result from interactions between particles, and when the field contains only a single, well isolated particle, these corrections may be neglected[2]. We may therefore say that single-particle quantum mechanics inherits its probabilistic interpretation from the quantum field that it is an approximation to. There are other fields, however, for which quantum corrections are never small; that is, fields that can't be described by single, well-isolated particles. The most striking example of this is found in the strong interaction, and is known as confinement.

Condensed matter viewpoint

Quantum field theory can also be a useful tool for describing many particle quantum mechanics.

Modern viewpoint

Here we shall discuss the modern viewpoint that the low-energy limit of a relativistic quantum theory theory is a quantum field theory. A whirlwind mention of renormalization group flow and string theory will be made.

References

  1. ↑ The reader is invited to look at the Schwinger proper time formalism
  2. ↑ Actually, they must be absorbed into redefinitions of the mass and charge of the electron, a process known as renormalization
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