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Quantum field theory (QFT) is the application of quantum mechanics to fields. It provides a theoretical framework widely used in particle physics and condensed matter physics. In particular, the quantum theory of the electromagnetic field, known as quantum electrodynamics, is one of the most well-tested and successful theories in physics. The fundamentals of quantum field theory were developed between the late 1920s and the 1950s, notably by Dirac, Fock, Pauli, Tomonaga, Schwinger, Feynman, and Dyson.1 Shortcomings of ordinary quantum mechanics
Quantum field theory corrects several deficiencies of ordinary quantum mechanics, which we will briefly discuss. The Schrödinger equationIn physics, the Schrodinger equation proposed by the Austrian physicist Erwin Schrodinger in 1925, describes the time-dependence of quantum mechanical systems. It is of central importance to the theory of quantum mechanics, playing a role analogous to New, in its most commonly encountered form, is
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where |ψ> denotes the quantum stateQuite literally, quantum state describes the state of a quantum system. In quantum mechanics this is described using a mathematical representation such as a state vector (also called a wave function for some quantum mechanical systems) or a density operat of a particle with massMass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. It is a central concept of classical mechanics and related subjects. Strictly speaking, there are two different quantities called mass Inertial mass m, acted on by a potential energyPotential energy U or E , a kind of scalar potential, is energy by virtue of matter being able to move to a lower-energy state, releasing energy in some form. For example a mass released above the Earth has energy resulting from the gravitational attracti V.
There are two problems with this equation. Firstly, it is not relativisticSpecial relativity (SR or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. It replaced Newtonian notions of space and time, and incorporated electromagnetism as represented by Maxwell's equations. The theory is, reducing to classical mechanicsClassical mechanics is a model of the physics of forces acting upon bodies. It is often referred to as Newtonian mechanics after Newton and his laws of motion. Classical mechanics is subdivided into statics (which models objects at rest), kinematics (whic rather than relativistic mechanics in the correspondence limitIn physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. The rules of quantum mechanics ar. To see this, we note that the first term on the left is only the classical kinetic energy p²/2m, with the rest energy mc² omitted. It is possible to modify the Schrödinger equation to include the rest energy, resulting in the Klein-Gordon equationPartial differential equations The Klein-Gordon equation Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation is a relativistic version (describing scalar (or pseudoscalar) spinless particles) of the Schrodinger equation. The Schrodinger equ or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy spectra which extend to -∞, so that there is no ground state. Such inconsistencies occur because these equations neglect the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory.
The second problem occurs when we seek to extend the equation to large numbers of particles. As described in the article on identical particles, quantum mechanical particles of the same species are indistinguishable, in the sense that the state of the entire system must be symmetric ( bosons) or antisymmetric ( fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are extremely complicated to write. For example, the general quantum state of a system of N bosons is written as
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where |φi> are the single-particle states, Nj is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases.
