Canonical quantization

In physics, canonical quantization is one of many procedures for quantizing a classical theory. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory. Classical theory has at least two distinct meanings in Physics: In the context of Quantum mechanics, "classical theory" refers to Historically, this was the earliest method to be used to build quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons When applied to a classical field theory it is also called second quantization. In Physics, a field is a Physical quantity associated to each point of Spacetime. The word canonical refers actually to a certain structure of the classical theory (called the symplectic structure) which is preserved in the quantum theory. Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a This was first emphasized by Paul Dirac, in his attempt to build quantum field theory. In quantum field theory (QFT the forces between particles are mediated by other particles

1 History
2 Quantum mechanics
3 Second quantization: field theory
4 Mathematical quantization
5 See also
6 References
- 6.1 Historical
- 6.2 Technical

History

Commutators were introduced by Werner Heisenberg; wavefunctions, by Erwin Schrödinger. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system The connection between the two was discovered by Paul Dirac, who was also the first to apply this technique to the quantization of the electromagnetic field. The electromagnetic field is a physical field produced by electrically charged objects. Eugene Wigner and Pascual Jordan were the first to quantize the electron field, whose quantum mechanics was first investigated by Dirac. Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a Pascual Jordan (b October 18, 1902 in Hanover, Germany; d July 31, 1980 in Hamburg, Federal Republic 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. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain A later development was the Feynman path integral, a formulation of quantum theory which emphasizes the role of superposition of quantum amplitudes. This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The two methods give the same results.

Quantum mechanics

In the classical mechanics of a particle, one has dynamical variables which are called coordinates ( $x$ ) and momenta ( $p$ ). Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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. Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition All transformations which keep these brackets unchanged are allowed as canonical transformations in classical mechanics. In Hamiltonian mechanics, a canonical transformation is a change of Canonical coordinates (\mathbf{q} \mathbf{p} t \rightarrow (\mathbf{Q} \mathbf{P} t

In quantum mechanics, these dynamical variables become operators acting on a Hilbert space of quantum states. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. The Poisson brackets (more generally the Dirac brackets) are replaced by commutators, $[X,P] = XP-PX = i\hbar$ . In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac to correctly treat systems with second class constraints in Hamiltonian In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. This readily yields the uncertainty principle in the form $\Delta x \Delta p \geq \frac{\hbar}{2}$ . In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain This algebraic structure corresponds to a generalization of the canonical structure of classical mechanics.

The states of a quantum system can be labelled by the eigenvalues of any operator. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes For example, one may write $|x\rangle$ for a state which is an eigenvector of $A$ with eigenvalue $x$ . In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes Notationally, one would write this as $A|x\rangle = x|x\rangle$ . The wavefunction of a state $|\varphi\rangle$ is $\varphi (x)=\langle x|\varphi\rangle$ .

In quantum mechanics one deals with the quantum states of a system of a fixed number of particles. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. 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. In quantum field theory (QFT the forces between particles are mediated by other particles

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 quantum field theory (QFT the forces between particles are mediated by other particles In Physics, a field is a Physical quantity associated to each point of Spacetime. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. 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.

The rest of this article deals with canonical quantization of field theory. It would also be useful to consult the companion articles on quantum field theory, quantization and the Feynman path integral. In quantum field theory (QFT the forces between particles are mediated by other particles 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 For integrals along a path also known as line or contour integrals see Line integral.

Field operator

One basic notion in this technique is of a vacuum state of a quantum field theory. In Quantum field theory, the vacuum state (also called the vacuum) is the Quantum state with the lowest possible Energy. In quantum field theory (QFT the forces between particles are mediated by other particles 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. This vacuum means "absence of matter" or "an empty area or space" for the cleaning appliance see Vacuum cleaner. The QCD vacuum is the Vacuum state of Quantum chromodynamics (QCD We shall represent this quantum state as |0>.

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. In Physics, an annihilation operator is an Operator that lowers the number of particles in a given state by one For example—

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. This article assumes some familiarity with Analytic geometry and the concept of a limit. The Fock space is an Algebraic system ( Hilbert space) used in Quantum mechanics to describe Quantum states with a variable or unknown number of A Fock state, in Quantum mechanics, is any state of the Fock space with a well-defined number of particles in each state They are a useful basis with which to discuss quantum field theory, although strictly, their use is limited to free field theory only. Classically a free field has Equations of motion given by Linear Partial differential equations Such linear PDE's have a unique solution for a given

Real scalar field

A classical scalar field can now be written as a quantum field operator by the following simple recipe—

Make a Fourier transformation of the classical field to find the Fourier coefficients φ(k) and φ^*(k). In Physics, a field is a Physical quantity associated to each point of Spacetime. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions The first corresponds to positive frequencies, and the second, to negative.
Convert each Fourier coefficient into an operator φ(k)→φ(k) a_k and φ^*(k)→φ^*(k) a^†_k.
Reconstruct the field operator by putting together this operator valued Fourier expansion. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions

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. In Physics, a field is a Physical quantity associated to each point of Spacetime. If there is a gauge symmetry, then the number of independent components of the field must be carefully analyzed. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations This usually involves gauge fixing. In the Physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees

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 until 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. In Quantum field theory the vacuum expectation value (also called condensate) of an operator is its average Expected value in the vacuum Thus, the canonical quantization procedure does not allow for a field condensate in the vacuum state, irrespective of the Lagrangian. In Quantum field theory, the vacuum state (also called the vacuum) is the Quantum state with the lowest possible Energy. The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system 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 Higgs mechanism is Spontaneous symmetry breaking in a Gauge theory. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them

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 Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters

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. A classical field theory is a Physical theory that describes the study of how one or more physical fields interact with matter Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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. In Physics, the canonical commutation relation is the relation between Canonical conjugate quantities (quantities which are related by definition such that one is Then the canonical momentum is the time derivative of φ. In Mathematics and Classical mechanics, canonical coordinates are particular sets of coordinates on the Phase space, or equivalently on the Cotangent In classical dynamics, the Poisson bracket between these quantities should be unity. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition In quantum mechanics, the canonical coordinate and momentum become operators, and a Poisson bracket becomes a commutator. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons This is exactly what happens here.

The one major drawback of this procedure is that Poincare invariance is no longer manifest. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime 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. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime 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. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime

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. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Mathematics, a foliation is a geometric device used to study manifolds Informally speaking a foliation is a kind of "clothing" worn on a manifold SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the In Mathematics, a symplectomorphism is an Isomorphism in the category of Symplectic manifolds Formal definition Specifically Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. The quantum algebra of "operators" is an $\hbar$ -deformation of the algebra of smooth functions over the symplectic space such that the leading term in the Taylor expansion over $\hbar$ of the commutator $[A, B]$ is $i\hbar\{A,B\}$ . In Mathematics and Physics, in the area of Quantum mechanics, Weyl quantization is a method for associating a "quantum mechanical" Hermitian In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. (Here, the curly braces denote the Poisson bracket. In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition ) In general, this $\hbar$ -deformation is highly nonunique, which explains the claim that quantization is an art. Now, we look for unitary representations of this quantum algebra. In Mathematics, a unitary representation of a group G is a Linear representation π of G on a complex Hilbert space With respect to such a unitary rep, a symplectomorphism in the classical theory would now correspond to a unitary transformation. Informally a unitary transformation is a transformation that respects the Dot product: the dot product of two vectors before the transformation is equal to their In particular, the time evolution symplectomorphism generated by the classical Hamiltonian is now a unitary transformation generated by the corresponding quantum Hamiltonian.

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 $\hbar$ deformation of the corresponding Poisson algebra or even Poisson supermanifolds. In Mathematics, a Poisson manifold is a Differential manifold M such that the algebra C &infin( M) of Smooth functions In Mathematics, a Poisson algebra is an Associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is the bracket In Differential geometry a Poisson supermanifold is a differential Supermanifold M such that the Supercommutative algebra of Smooth functions

References

Historical

QED and the men who made it, by S. In Physics, an annihilation operator is an Operator that lowers the number of particles in a given state by one The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac to correctly treat systems with second class constraints in Hamiltonian S. Schweber, ISBN 0-691-03327-7

Technical

Principles of quantum mechanics, by P. A. M. Dirac, ISBN 0-19-852011-5
An introduction to quantum field theory, by M. E. Peskin and H. D. Schroeder, ISBN 0-201-50397-2

Quantum field theory

Feynman diagram
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In quantum field theory (QFT the forces between particles are mediated by other particles Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described The history of quantum field theory starts with its creation by Dirac when he attempted to quantize the Electromagnetic field in the late 1920s

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