- Canonical quantization
In
physics , canonical quantization is one of many procedures for quantizing aclassical theory . Historically, this was the earliest method to be used to buildquantum mechanics . When applied to a classical field theory it is also called second quantization. The word "canonical" refers actually to a certain structure of the classical theory (called thesymplectic structure) which is preserved in the quantum theory. This was first emphasized byPaul Dirac , in his attempt to buildquantum field theory .History
Commutators were introduced byWerner Heisenberg ;wavefunctions , byErwin Schrödinger . The connection between the two was discovered byPaul Dirac , who was also the first to apply this technique to the quantization of theelectromagnetic field .Eugene Wigner andPascual 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 theuncertainty principle . A later development was theFeynman 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 () and momenta (). These specify the "state" of a classical system. The canonical structure (also known as thesymplectic structure) of classical mechanics consists ofPoisson bracket s between these variables. All transformations which keep these brackets unchanged are allowed ascanonical transformation s in classical mechanics.In quantum mechanics, these dynamical variables become operators acting on a
Hilbert space ofquantum states . ThePoisson bracket s (more generally theDirac bracket s) are replaced bycommutator s, . This readily yields theuncertainty principle in the form. 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
eigenvalue s of any operator. For example, one may write for a state which is aneigenvector of witheigenvalue . Notationally, one would write this as . The wavefunction of a state is .In
quantum mechanics one deals with thequantum state s 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 ofquantum field theory .econd 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 aquantum 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.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 theFeynman path integral .Field operator
One basic notion in this technique is of a
vacuum state of aquantum 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>.Then one introduces single particle
creation and annihilation operators , a†k and ak respectively, which act on quantum states to increase or decrease the number of particles of the given momentum k. For example—
*ak|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 aFock space and these kinds of states are calledFock state s. They are a useful basis with which to discuss quantum field theory, although strictly, their use is limited tofree field theory only.Real scalar field
A classical scalar field can now be written as a quantum field operator by the following simple recipe—
#Make aFourier transformation of the classical field to find theFourier coefficients φ(k) and φ*(k). The first corresponds to positive frequencies, and the second, to negative.
#Convert each Fourier coefficient into an operator φ(k)→φ(k) ak and φ*(k)→φ*(k) a†k.
#Reconstruct the field operator by putting together thisoperator valued Fourier expansion .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 agauge symmetry , then the number of independent components of the field must be carefully analyzed. This usually involvesgauge fixing .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 [ak,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 {ak,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 thevacuum state , irrespective of theLagrangian . 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 theHiggs mechanism which is needed to construct thestandard model ofparticle physics .A bosonic condensate is a
coherent state of zerowavenumber bosons.Why "canonical"?
Why is this process called canonical quantization? This is because of the strong connection that
classical field theory has withclassical 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 thecanonical coordinate . Then thecanonical momentum is the partial derivative of the Lagrangian density with respect to the time derivative of φ. Inclassical dynamics , thePoisson bracket between these quantities should be unity. Inquantum mechanics , the canonical coordinate and momentum become operators, and a Poisson bracket becomes a commutator. This is exactly what happens here.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 thatrelativistic invariance is hidden, but not lost. Field theories used incondensed matter physics are not required to havePoincare invariance , and for them canonical quantization does not suffer from this drawback.Mathematical quantization
The classical theory is described using a
spacelike foliation ofspacetime with the state at each slice being described by an element of asymplectic manifold with the time evolution given by thesymplectomorphism 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 thecommutator is . (Here, the curly braces denote thePoisson bracket .) In general, this -deformation is highly nonunique, which explains the claim that quantization is an art. Now, we look forunitary representation s of this quantum algebra. With respect to such a unitary rep, a symplectomorphism in the classical theory would now correspond to aunitary transformation . 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 deformation of the correspondingPoisson algebra or evenPoisson supermanifold s.ee also
*
Correspondence principle
*Creation and annihilation operators
*Dirac bracket References
Historical
*"QED and the men who made it", by S.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 0-201-50397-2External links
* [http://daarb.narod.ru/wircq-eng.html What is "Relativistic Canonical Quantization"?]
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