| Canonical Quantization |
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HISTORY Commutators were introduced by Werner Heisenberg ; Wavefunctions , by Erwin Schrödinger . The connection between the two was discovered by Paul Dirac , who was also the first person to apply this technique to the quantization of the Electromagnetic Field . Eugene Wigner and Pascual Jordan were the first to quantize the electron field, whose quantum mechanics was first investigated by Dirac . The name ''canonical quantization'' may have been first coined by Pascual Jordan . The exposition here leans heavily on Dirac's influential book on quantum mechanics. This route to Quantum Mechanics is through the Uncertainty Principle . A later development was the Feynman Path Integral , a formulation of Quantum Theory which emphasizes the role of superposition of quantum amplitudes. The two methods give the same results. QUANTUM MECHANICS In the Classical Mechanics of a particle one has dynamical variables which are called coordinates (q) and momenta (p). These specify the ''state'' of a classical system. The canonical structure (also known as the Symplectic structure) of classical mechanics consists of Poisson Bracket s between these variables. All transformations which keep these brackets unchanged are allowed as Canonical Transformation s in classical mechanics. In quantum mechanics, these dynamical variables become operators acting on a Hilbert Space of Quantum States . The Poisson Bracket s are replaced by Commutator s, {Link without Title} = qp-pq = 1. This readily yields the Uncertainty Principle in the form ΔpΔq ≥ 1. This algebraic structure corresponds to a generalization of the ''canonical structure'' of classical mechanics. | ||
| One Basic Notion In This Technique Is Of A | "http://wwwinformationdelightinfo/encyclopedia/entry/vacuum_state" class="copylinks">Vacuum State of a Quantum Field Theory This is a quantum state containing zero particles For further elaboration and niceties, see the articles on The Quantum Mechanical Vacuum and The Vacuum Of Quantum Chromodynamics We shall represent this quantum state as <b>0></b> |
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| <b>a<sub>k</sub>0>  |  0</b>, since the vacuum state has no particles, and therefore a state with smaller number of particles cannot exist |
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| <b>a<sup>+</sup><sub>k</sub>0>  |  1(k)></b>, where we have introduced the notation <b>n(k)></b> to denote the state with <b>n</b> particles of momentum <b>k</b> |
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| Note That The | "http://wwwinformationdelightinfo/encyclopedia/entry/vacuum_expectation_value" class="copylinks">Vacuum Expectation Value (VEV) <b><0φ0> = 0</b> Thus, the canonical quantization procedure does not allow for a field condensate in the Vacuum State , irrespective of the Lagrangian The only exception to this is to shift the field by a constant before embarking on the process above, ie, quantize the field <b>φ(x, t)-v</b>, where <b>v</b> is a number and not an operator The quantity <b>v</b> then denotes the condensate of the field φ, and the particle states become the excitations over the new vacuum defined with this condensate The VEV of any power (or other function) of <b>φ</b> can then be expressed in terms of <b>v</b> Thus, this procedure allows only a single condensate This construction is used in the Higgs Mechanism which is needed to construct the Standard Model of Particle Physics |
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