In physics, canonical quantization is one of many procedures for quantizing a classical theory. Historically, this was the earliest method to be used to build quantum mechanics. When applied to a classical field theory it was initially called second quantization. This name has now fallen out of fashion. The word canonical refers actually to a certain structure of the classical theory (called the symplectic structure) which is preserved in the quantum theory. This was first emphasized by Paul Dirac, in his attempt to build quantum field theory. Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... In physics, a classical theory usually refers to a theory that does not obey the principles of quantum mechanics (classical theory vs. ... Fig. ... Field (physics) - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... Symplectic topology (also called symplectic geometry) is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with closed, nondegenerate, 2-forms. ... Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, OM (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

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. For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if and only... Werner Heisenberg Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics. ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space and normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude... Erwin Schrödinger, as depicted on the former Austrian 1000 Schilling bank note. ... Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, OM (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... In the physics of electromagnetism, an electromagnetic field is a field composed of two related vector fields: the electric field and the magnetic field. ... Eugene Wigner (left) and Alvin Weinberg Eugene Paul Wigner (Hungarian Wigner Pál JenÅ‘) (November 17, 1902 – January 1, 1995) was a Hungarian physicist and mathematician. ... Pascual Jordan (October 18, 1902 in Hanover - July 31, 1980 in Hamburg) was a German physicist. ... Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, OM (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Pascual Jordan (October 18, 1902 in Hanover - July 31, 1980 in Hamburg) was a German physicist. ...

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. Needless to say, the two methods give the same results. Fig. ... In quantum physics, the Heisenberg uncertainty principle states that one cannot assign with full precision values for certain pairs of observable variables, including the position and momentum, of a single particle at the same time. ... This article is about a formulation of quantum mechanics. ... Fig. ...

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 brackets between these variables. All transformations which keep these brackets unchanged are allowed as canonical transformations in classical mechanics. In physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies, and the forces that cause them. ... Symplectic topology (also called symplectic geometry) is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with closed, nondegenerate, 2-forms. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...

In quantum mechanics, these dynamical variables become operators acting on a Hilbert space of quantum states. The Poisson brackets are replaced by commutators, [q,p] = qp-pq = 1. This readily yields up the uncertainty principle in the form ΔpΔq ≥ 1. This algebraic structure corresponds to a generalization of the canonical structure of classical mechanics. In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... Quite literally, quantum state describes the state of a quantum system. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how badly a certain binary operation fails to be commutative. ... In quantum physics, the Heisenberg uncertainty principle states that one cannot assign with full precision values for certain pairs of observable variables, including the position and momentum, of a single particle at the same time. ...

The states of a quantum system can be labelled by the eigenvalues of any operator. For example, one may write |x> for a state which is an eigenvector of q with eigenvalue x. Notationally, one would write this as q|x> = x|x>. The wavefunction of a state |φ> is φ(x)=<x|φ>. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

In quantum mechanics one deals with the quantum states of a system of a fixed number of particles. This is inadequate for the study of systems in which particles are created and destroyed. Historically, this problem was solved through the introduction of quantum field theory. Fig. ... A quantum state is any possible state in which a quantum mechanical system can be. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

Second quantization: field 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. In one way of viewing things, quantizing the classical theory of a fixed number of particles gave rise to a wavefunction. This wavefunction is a field variable which could then be quantized to deal with the theory of many particles. So the process of canonical quantization of a field theory was called second quantization in the early literature. Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Field (physics) - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... A quantum state is any possible state in which a quantum mechanical system can be. ...

The rest of this article deals with canonical quantization of field theory. It would be useful to also consult the companion articles on quantum field theory, quantization and the Feynman path integral. Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... This article is about a formulation of quantum mechanics. ...

Field operator

One basic notion in this technique is of a 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 |0>. In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... For other uses, see vacuum cleaner and Vacuum (musical group). ... The QCD vacuum is the vacuum state of Quantum chromodynamics (QCD). ...

Then one introduces single particle creation and annihilation operators, a⁺_k and a_k respectively, which act on quantum states to increase or decrease the number of particles of the given momentum k. For example— This article needs to be cleaned up to conform to a higher standard of quality. ...

• a_k|0>=0, since the vacuum state has no particles, and therefore a state with smaller number of particles cannot exist
• a⁺_k|0>=|1(k)>, where we have introduced the notation |n(k)> to denote the state with n particles of momentum k.

The Hilbert space of states of this kind is called a Fock space and these kinds of states are called Fock states. They are an useful basis with which to discuss quantum field theory, although strictly, their use is limited only to free field theory. In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... The Fock space is an algebraic system (Hilbert space) used in quantum mechanics to describe quantum states with a variable or unknown number of particles. ... A Fock state, in quantum mechanics, is any state of the Fock space with a well-defined number of particles in each state. ... Classically, a free field is a field described by linear partial differential equations which has a unique solution given initial data. ...

Real scalar field

A classical scalar field can be written as a quantum field operator now by the following simple recipe— Field (physics) - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...

1. Make a Fourier transformation of the classical field to find the Fourier coefficients φ(k) and φ^*(k). The first corresponds to positive frequencies, and the second, to negative.
2. Convert each Fourier coefficient into an operator φ(k)→φ(k) a_k and φ^*(k)→φ^*(k) a⁺_k.
3. Reconstruct the field operator by putting together this operator valued Fourier expansion.

The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... In mathematics, a Fourier series, named in honor of Joseph Fourier (1768-1830), is a representation of a periodic function with period 2π as a sum of periodic functions of the form which are the harmonics of ei x. ... In mathematics, a Fourier series, named in honor of Joseph Fourier (1768-1830), is a representation of a periodic function (often taken to have period 2π — in a sense, the simplest case) as a sum of periodic functions of the form which are harmonics of ei x. ...

Other fields

All other fields can be quantized by a generalization of this procedure. Vector or tensor fields simply have more components, and independent creation and destruction operators must be introduced for each independent component. If a field has any internal symmetry, then creation and destruction operators must be introduced for each component of the field related to this symmetry as well. If there is a gauge symmetry, then the number of independent components of the field must be carefully analyzed. This usually involves gauge fixing. In physics, a field is an assignment of a quantity to every point in space. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes the act of removing redundant field variables. ...

We have introduced the commutator of two operators, [A,B]. Before proceeding further we need the anti-commutator, which is {A,B} = AB+BA. Note that [A,B]=-[B,A], but {A,B}={B,A}.

For all the fields we have named till now, one uses boson creation and annihilation operators. This means that the operators satisfy the commutation relations [a_k,a⁺_k]=1. All other commutators vanish. To quantize spinor fields, corresponding to fermions, we need to use operators which satisfy the anti-commutation relations {a_k,a⁺_k}=1, and that all other anti-commutators vanish.

Condensates

Note that the vacuum expectation value (VEV) <0|φ|0>=0. 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 φ(x,t)-v, where v is a number and not an operator. The quantity v 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 φ can then be expressed in terms of v. 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. The vacuum expectation value (also called vacuum condensate) of an operator is its average, expected value in the vacuum. ... In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a functional of the dynamical variables which concisely describes the equations of motion of the system. ... The Higgs mechanism, originally discovered by the British physicist Peter Higgs (building on a previous suggestion by Philip Anderson in condensed matter physics), is the mechanism that gives masses to all elementary particles in particle physics. ... The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ... Particles erupt from the collision point of two relativistic (100 GeV) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...

A bosonic condensate is a coherent state of zero wavenumber bosons. In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system. ... Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length. ...

Why "canonical"?

Why is this process called canonical quantization? This is because of the strong connection that classical field theory has with classical mechanics, and which is sought to be preserved here. In classical field theory, the field φ(x,t) is the analogue of a dynamical variable, one at each point of spacetime, x,t. Consider this to be the canonical coordinate. Then the canonical momentum is the time derivative of φ. In classical dynamics, the Poisson bracket between these quantities should be unity. In quantum mechanics, the canonical coordinate and momentum become operators, and a Poisson bracket becomes a commutator. This is exactly what happens here. In physics, a field is an assignment of a quantity to every point in space. ... In physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies, and the forces that cause them. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... In physics, Classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies, and the forces that cause them. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... Fig. ...

The one major drawback of this procedure is that Poincare invariance is no longer manifest. That is because to define the time coordinate, one must choose an inertial frame to work with. At the end of the computation one is required to check that relativistic invariance is hidden, but not lost. Field theories used in condensed matter physics are not required to have Poincare invariance, and for them canonical quantization does not suffer from this drawback. Poincare symmetry is the full symmetry of special relativity and includes translations (ie, displacements) in time and space (these form the Abelian Lie group of translations on space-time) rotations in space (this forms the non-Abelian Lie group of 3-dimensional rotations) boosts, ie, transformations connecting two uniformly moving... Poincare symmetry is the full symmetry of special relativity and includes translations (ie, displacements) in time and space (these form the Abelian Lie group of translations on space-time) rotations in space (this forms the non-Abelian Lie group of 3-dimensional rotations) boosts, ie, transformations connecting two uniformly moving... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Poincare symmetry is the full symmetry of special relativity and includes translations (ie, displacements) in time and space (these form the Abelian Lie group of translations on space-time) rotations in space (this forms the non-Abelian Lie group of 3-dimensional rotations) boosts, ie, transformations connecting two uniformly moving...

Mathematical quantization

The classical theory is described using a spacelike foliation of spacetime with the state at each slice being described by an element of a symplectic manifold with the time evolution given by the symplectomorphism generated by a Hamiltonian function over the symplectic manifold. The quantum algebra of "operators" is an -deformation of the algebra of smooth functions over the symplectic space such that the leading term in the Taylor expansion over of the commutator [A,B] is . (Here, the curly braces denote the Poisson bracket.) In general, this -deformation is highly nonunique, which explains the claim that quantization is an art. Now, we look for unitary representations of this quantum algebra. With respect to such a unitary rep, a symplectomorphism in the classical theory would now correspond to a unitary transformation. In particular, the time evolution symplectomorphism generated by the classical Hamiltonian is now a unitary transformation generated by the corresponding quantum Hamiltonian. In the context of special relativity, space-like separated points (or events) in spacetime have a spacetime interval less than 0 (see sign convention). ... In mathematics, informally speaking, a foliation is a kind of clothing worn on a manifold, cut from a stripy fabric. ... This article may be too technical for most readers to understand. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a symplectomorphism (or Hamiltonian flow) is an isomorphism in the category of symplectic manifolds. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how badly a certain binary operation fails to be commutative. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group... A unitary transformation is an isomorphism (but not an antiisomorphism; that corresponds to an antiunitary transformation) between two Hilbert spaces or an automorphism of a single Hilbert space. ...

We could be more general than this. We can work with a Poisson manifold instead of a symplectic space for the classical theory and perform a deformation of the corresponding Poisson algebra or even Poisson supermanifolds. (The literal classical interpretation of this, of course, does not exist. This is a purely formal procedure.) A Poisson manifold is a differential manifold M such that the algebra of smooth functions over it, is equipped with a bilinear map called the Poisson bracket turning it into a Poisson algebra. ... A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz law. ... A Poisson supermanifold is a differential supermanifold M such that the supercommutative algebra of smooth functions over it (let me clarify a bit. ...

See also

• Creation and annihilation operators

This article needs to be cleaned up to conform to a higher standard of quality. ...

References and external links

Historical

• QED and the men who made it, by S.S.Schweber [ISBN 0691033277]

Technical

• Principles of quantum mechanics, by P.A.M.Dirac [ISBN 0198520115]
• An introduction to quantum field theory, by M.E.Peskin and H.D.Schroeder [ISBN 0201503972]