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	PlanetMath: canonical quantization
		Canonical quantization is a method of turning a classical system of the form
		The first formulations of the theory of quantum mechanics used this method, and it continues to be the most practical and accessible method of quantization.
		This is version 1 of canonical quantization, born on 2006-05-02.
	planetmath.org /encyclopedia/CanonicalQuantization.html (242 words)

	PlanetMath: Quantization
		Quantization is the process of obtaining a quantum system from a classical system.
		It should be emphasized the result of quantization is not unique; in fact, methods of quantization usually possess inherent ambiguities, in the sense that, while performing quantization, one usually must make choices at certain points of the process.
		This is version 1 of Quantization, born on 2005-12-25.
	planetmath.org /encyclopedia/ClassicalState.html (366 words)

	Canonical quantization and covariance
		In conventional canonical quantization, the momenta are defined by p_i = dL/d(\dot q^i).
		Upon quantization, we replace the Poisson brackets by commutators.
		Quantization dutifully eliminates half of the classical oscillators (those with negative energy), as expected.
	www.lns.cornell.edu /spr/2005-06/msg0069783.html (1826 words)

	Proper Dirac Quantization of Free Particle on D-Dimensional Sphere
		We show that an unambiguous and correct quantization of the second-class constrained system of a free particle on a sphere in D dimensions is possible only by converting the constraints to abelian gauge constraints, which are of first class in Dirac's classification scheme.
		Quantization of a free point particle in curved space is a long-standing and controversial problem in quantum mechanics.
		Although the canonical variables are Cartesian, the canonical quantization is not applicable, since the constraints (3), (4) cannot be enforced for the associated operators -- the conditions
	www.physik.fu-berlin.de /~kleinert/kleiner_re253/partsph.html (2308 words)

	Quantization (physics) - Wikipedia, the free encyclopedia
		Quantization converts classical fields into operators acting on quantum states of the field theory.
		However, the use of canonical quantization has left its mark on the language and interpretation of quantum field theory.
		The classical field is treated as a dynamical variable called the canonical coordinate, and its time-derivative is the canonical momentum.
	en.wikipedia.org /wiki/Quantization_(physics) (668 words)

	Louisiana State University Physics & Astronomy
		String theorists believe that one cannot quantize general relativity because it is not a fundamental theory and one has to replace it with string theory in order to quantize it.
		Those of us who work in canonical quantization believe that it has not been properly shown that one cannot quantize plain old general relativity in four space-time dimensions.
		Canonical quantum gravity picked momentum when Ashtekar introduced a new set of variables that make the theory resemble Yang-Mills theories in 1986.
	www.phys.lsu.edu /dept/text/people/pullin.html (797 words)

	Canonical quantization - Wikipedia, the free encyclopedia
		The word canonical refers actually to a certain structure of the classical theory (called the symplectic structure) which is preserved in the quantum theory.
		When the canonical quantization procedure is applied to quantum field theory, the classical field variable becomes a quantum operator which acts on a quantum state of the field theory to increase or decrease the number of particles by one.
		Thus, the canonical quantization procedure does not allow for a field condensate in the vacuum state, irrespective of the Lagrangian.
	en.wikipedia.org /wiki/Canonical_quantization (1470 words)

	Dorothea Bahns thesis
		In this spirit, the graded Poisson algebra of invariant charges of the Nambu-Goto string is quantized by application of the correspondence principle, replacing the Poisson brackets by commutators and allowing for particular (observable) quantum corrections which are restricted by demanding structural similarity of the classical and the quantum algebra.
		In this paper, which is an exposition of results gained some years ago [1], it is shown that canonical quantization does not yield a representation of the algebra of invariant charges.
		I do believe that there is a canonical quantization of the Pohlmeyer invariants which does yield a rep of an algebra of invariant charges and which furthermore reproduces the results obtained by the standard way of quantization.
	www.physicsforums.com /showthread.php?p=162922 (2015 words)

	1+1D LQG is really about boundary states \| The String Coffee Table
		This is the postulate that ‘to quantize’ a theory does not require to have an operator representation of canonical coordinates and canonical momenta.
		In the discussion Thomas Thiemann conceded that hence the use of ‘relaxed canonical quantization’ is a step based on the hope that experiment will (or would) find that at the Planck scale we have to modify standard quantum theory.
		Since Thomas Thiemann has originally suggested that Pohlmeyer invariants serve as observables for the LQG-like quantization of the string it is maybe interesting to observe that Pohlmeyer invariants, do play a role in the standard quantization of (super-)string as boundary state deformation operators, which turn on non-abelian gauge fields on D-branes.
	golem.ph.utexas.edu /string/archives/000420.html (960 words)

	Poisson Brackets
		Canonical quantization is a procedure which converts a classical field theory or a classical mechanical theory into the corresponding Quantum theory.
		But the truth about canonical quantization carried out like that is that it has to be interfered with frequently in order to deliver a meaningful Quantum theory, and the reason for that is that Quantum theories cannot be derived formally from classical theories.
		But canonical quantization was useful in its day in providing a bridge between XIXth century classical physics and XXth century quantum physics.
	beige.ucs.indiana.edu /B673/node47.html (242 words)

	Geometry of Constrained Dynamical Systems - Cambridge University Press
		Canonical structure of the non-linear sigma model in a polynomial formulation; 8.
		Quantization of 2+1 gravity for g=1 and g=2; 29.
		Generalized canonical quantization of gauge theories with polarized second-class constraints; 34.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521482712 (456 words)

	A CONCEPT OF QUANTUM GRAVITY
		The expression of QT is currently restricted to "methods of quantization", of which there many besides the ever popular canonical quantization that begins with the Hamiltonian formulation of classical mechanics.
		Such a quantization method starts with a mathematical formulation of the kinematics and dynamics of a classical system and then applies quantization rules of transformation to express the quantized version of the kinematics and dynamics of the same system.
		While loop quantum gravity is still an approach that is a quantization procedure with a goal of constructing a projective Hilbert space of states, its two main features are that it does avoid the assumption of a "background metric", and concommitantly that field on which attention is focused is an affine connection.
	graham.main.nc.us /~bhammel/FCCR/qg.html (9795 words)

	Canonical Science Project
		In canonical science, the status is still poor, since one need know several highly sophisticated fields of physics, math, chemistry, etc. Moreover, there is no still true experts in the world in canonical science.
		Our canonical cosmology also is. However, it is illustrative to consider the questions that could be answered from canonical science and other methods: 1) Perhaps there is not Big Bang, since that cosmological redshift could be another effect.
		Canonical science research and education are done outside of official funding and support.
	www.canonical.chemicalforums.com (3868 words)

	No Title
		We discussed canonical quantization of Fermi fields (that is, fields obeying a Dirac equation).
		Thus we obtain, upon quantization, creation and annihilation operators for particles and antiparticles in the spin states (r).
		The Dirac equation, along with canonical anticommutation relations, has automatically led to a theory with multiple particles and antiparticles, which are all fermions.
	www.emory.edu /PHYSICS/Faculty/Benson/380-96/notes/34/34.html (664 words)

	Thiemann's quantization of the Nambu-Goto action \| The String Coffee Table
		Since these are the same methods used in LQG for quantizing the gravitational field in 3+1 dimensions it is somewhat interesting to see how vastly different the results obtained this way are from the standard lore.
		It must therefore be emphasized that LQG is not canonical quantization in the sense that the classical first-class constraints are not promoted to hold as expectation value equations in the quantum theory.
		It means that, in the strinct sense of the word ‘canonical’, LQG is not canonical at all but rather similar in spirit to other proposed modifications of quantum theory, like for instance those proposed to explain away the fl hole information loss problem by modifying Schroedinger’s equation.
	golem.ph.utexas.edu /string/archives/000299.html (8917 words)

	No Title
		We discussed the Fock space of the quantized Klein-Gordon field, then canonical quantization of other boson fields.
		, the particles we have obtained by canonical quantization are necessarily bosons.
		With that done, canonical quantization goes through as usual, giving a Fock space of multiphoton states of both polarizations, with photons being bosons.
	www.emory.edu /PHYSICS/Faculty/Benson/380-96/notes/33/33.html (426 words)

	Canonical quantization of constrained theories on discrete spacetime lattices
		Canonical quantization of constrained theories on discrete spacetime lattices
		We propose a new canonical formulation and quantization for such systems in terms of discrete canonical transformations.
		The whole discussion is framed in such a way so as to provide a formalism that would allow a consistent, well-defined, canonical formulation and quantization of discrete general relativity, which we will discuss in a forthcoming paper.
	stacks.iop.org /0264-9381/19/5275 (383 words)

	DEPARAMETRIZATION AND PATH INTEGRAL QUANTIZATION OF COSMOLOGICAL MODELS
		The problem of time is a central feature of quantum cosmology: differing from ordinary quantum mechanics, in cosmology there is nothing "outside" the system which plays the role of clock, and this makes difficult the obtention of a consistent quantization.
		A possible solution is to assume that a subset of the variables describing the state of the universe can be a clock for the remaining of the system.
		Both path integral and canonical formulations are studied for relativistic and string cosmologies; in particular, a complete chapter about low energy string cosmology is included.
	www.worldscibooks.com /physics/4804.html (234 words)

	quant.htm
		Apparently, F is an essential usage of the quantum axiomatics, in particular, application of the Heisenberg representation and the second quantization.
		This became clear quite recently, although the incompatibility of the energy-Hamiltonian identification with the dynamic equations of the QFT is known for as long as twenty five years (see.
		Now, when there are more arguments suggesting that the QFT should be constructed on the dynamic base without any reference to the quantum axiomatics and second quantization, the present paper can be considered as an additional argument in favour of a revision of the QFT.
	pavel.physics.sunysb.edu /~rylov/quant.htm (1295 words)

	q-Quantum Gravity Abstract
		This dissertation explores one approach to this theory known as nonperturbative canonical quantization.
		Included is a discussion of the recent ``Immirzi ambiguity.'' The review of nonperturbative quantum gravity continues in Chapter 3 with a discussion on the origins of one of the key structures in nonfigurative quantum gravity, spin networks.
		This allows an alternative nonperturbative quantization which is suitable for describing a phase of quantum gravity characterized by states which are normalizable in the measure of Chern-Simons theory.
	academics.hamilton.edu /physics/smajor/Papers/d_abs.html (556 words)

	Canonical quantization and covariance
		path integrals, and it is not suited for canonical quantization.
		In conventional canonical quantization, the momenta are defined by
		quantization are equivalent), it is by no means easy to do so.
	physicsforums.com /showthread.php?t=136138 (2184 words)

	[No title]
		This is one rigorous interpretation of the old issue of quantum mechanics known as "canonical quantization", namely how to replace a classical Hamiltonian with a quantum-mechanical one.
		A coherent picture of quantizations of all Hamiltonians requires a deformation quantization, i.e., a deformation of the commutative algebra of coordinate functions on the Poisson manifold to a non-commutative algebra, where the commutators are to leading order given by the Poisson structure.
		The first big question is whether a deformation quantization always exists; Kontsevich proved that it does.
	www.math.niu.edu /~rusin/known-math/00_incoming/featured (793 words)

	"canonical" - Shopping.com
		Sangharakshita - The Eternal Legacy: An Introduction to the Canonical Literature of Buddhism
		Archibald Alexander - Evidences of the Authenticity, Inspiration and Canonical Authority of the Holy Scriptures 1836
		C. MacKenzie Brown - The Triumph of the Goddess: The Canonical Models and Theological Visions of the Devi- Bhagavata Purana
	www.shopping.com /xCC-canonical-price_range_20_30 (506 words)

	week11
		Both the technical problems of "canonical" quantum gravity and one of the main conceptual problems - the problem of time - stem from the fact that general relativity is a system in which the initial data have constraints.
		In canonical quantization, we treat this a lot like the space R^2 of pairs (q,p) describing the initial position and momentum of a particle.
		The problem in quantizing general relativity in the "canonical" approach is largely figuring out what to do with the constraints.
	math.ucr.edu /home/baez/week11.html (987 words)

	Alejandro Corichi's research
		The particular approach to quantum gravity I am most familiar with is the non-perturbative canonical quantization program, also known as ``loop quantum gravity'' (or ``quantum geometry'').
		This canonical quantization program has become in recent years a serious candidate for a quantum theory of geometry (without any other matter field), thanks to the introduction of connection and loop variables by Ashtekar, Rovelli and Smolin.
		I have also been involved in the study of the quantum collapse of a spherically symmetric dust shell, and in the midi-superspace quantization of Gowdy models.
	www.phy.olemiss.edu /~corichi/research.html (809 words)

	LECTURES ON NON-PERTURBATIVE CANONICAL GRAVITY
		It is now generally recognized that perturbative field theoretical methods that have been highly successful in the quantum description of non-gravitational interactions cannot be used as a means of constructing a quantum theory of gravity.
		The primary aim of the book is to present an up-to-date account of a non-perturbative, canonical quantization program for gravity.
		provides an excellent, up-to-date account of his reformulation of general relativity and of the new approaches to canonical quantum gravity that it has suggested.
	www.worldscibooks.com /physics/1321.html (260 words)

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