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Quantum Theory: Quantum Field Theories
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Quantum field theory started with a theoretical framework that was built in analogy to quantum mechanics. Although there was no unique and fully developed theory, quantum field theoretical tools could be applied to concrete processes. Examples are the scattering of radiation by free electrons (“Compton scattering”), the collision between relativistic electrons or the production of electron-positron pairs by photons. Calculations to the first order of approximation were quite successful, but most people working in the field thought that QFT still had to undergo a major change. On the one side some calculations of effects for cosmic rays clearly differed from measurements. On the other side and, from a theoretical point of view more threatening, calculations of higher orders of the perturbation series led to infinite results.
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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 quantum field theory seminar will meet Mondays, 2:00-3:00, room 939, starting 2004 January 26. This will be an instructional seminar for graduate students and anyone else interested. The goal is to give a rigorous account of the standard model for mathematicians. This goal will not be achieved. Richard Borcherds.
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If you look at general relativity and quantum field theory (QFT), both represent fields using calculus: they both use differential equations describing continuous variables to represent fields which should strictly be sigma sums for the action in discrete interactions. This is why differential QFT leads to perturbative expansions with an infinite number of terms, each term corresponding to a Feynman diagram:
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This text offers a balanced treatment of quantum field theory, providing bothformal presentation and numerous examples. It begins with the standard quantization of electrodynamics, culminating in the perturbative renormalization, and proceeds tofunctional methods, relativistic bound states, broken symmetries, nonabelian gauge fields, and asymptotic behavior. 157 figures. 1980 edition.
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Quantum Field Theories