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Quantum Field Theory: Quantization
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Quantum field theory originated in the 1920s from the problem of creating a quantum mechanical theory of the electromagnetic field. In 1926, Max Born, Pascual Jordan, and Werner Heisenberg constructed such a theory by expressing the field's internal degrees of freedom as an infinite set of harmonic oscillators and employing the usual procedure for quantizing those oscillators (canonical quantization). This theory assumed that no
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The crucial step towards quantum field theory is in some respects analogous to the corresponding quantization in quantum mechanics by imposing the commutation relations. Its starting point is the classical Lagrangian formulation of mechanics, which is a so-called analytical formulation as opposed to the standard version of Newtonian mechanics. A generalized notion of momentum (the conjugate or canonical momentum) is defined by setting p = ∂L/∂q̇, where L is the Lagrange function L = T − V (T is the kinetic energy and V the potential) and q̇ ≡ dq/dt. This definition can be motivated by looking at the special case of a Lagrange function with a potential V which depends only on the position so that (using Cartesian coordinates) ∂L/∂ẋ = (∂/∂ẋ)·(mẋ2/2) = mẋ = px. Under these conditions the generalized momentum coincides with the usual mechanical momentum. In classical Lagrangian field theory one associates with the given field φ a second field, namely the conjugate field
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The first method used to quantize field theory was the method now called canonical quantization (earlier known as second quantization). This method uses a Hamiltonian formulation of the classical problem. The later technique of Feynman path integrals uses a Lagrangian formulation. Many more methods are now in use; for an overview see the article on quantization.
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