between a set of quasiparticles. The Feynman diagram method of QFT was naturally well suited to the analysis of various phenomena in condensed matter systems. Gauge theory is used to describe the quantization of magnetic flux in superconductors, the resistivity in the quantum Hall effect, as well as the relation between frequency and voltage in the AC Josephson effect. Principles For simplicity, natural units are used in the following sections, in which the reduced Planck constant and the speed of light are both set to one. Classical fields A classical field is a function of spatial and time coordinates. Examples include the gravitational field in Newtonian gravity and the electric field and magnetic field in classical electromagnetism. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many degrees of freedom. Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the photoelectric effect are best explained by discrete particles (photons), rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields. Canonical quantization and path integrals are two common formulations of QFT. To motivate the fundamentals of QFT, an overview of classical field theory follows. The simplest classical field is a real scalar field — a real number at every point in space that changes in time. It is denoted as , where is the position vector, and is the time. Suppose the Lagrangian of the field, , is where is the Lagrangian density, is the time-derivative of the field, is the gradient operator, and is a real parameter (the "mass" of the field). Applying the Euler–Lagrange equation on the Lagrangian: we obtain the equations of motion for the field, which describe the way it varies in time and space: This is known as the Klein–Gordon equation. The Klein–Gordon equation is a wave equation, so its solutions can be expressed as a sum of normal modes (obtained via Fourier transform) as follows: where is a complex number (normalized by convention), denotes complex conjugation, and is the frequency of the normal mode: Thus each normal mode corresponding to a single can be seen as a classical harmonic oscillator with frequency . Canonical quantization The quantization procedure for the above classical field to a quantum operator