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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, canonical quantization is a procedure for quantizing a
classical theory Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, while attempting to preserve the formal structure, such as
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, of the classical theory, to the greatest extent possible. Historically, this was not quite
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent serie ...
's route to obtaining
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, but
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
introduced it in his 1926 doctoral thesis, the "method of classical analogy" for quantization, and detailed it in his classic text. The word ''canonical'' arises from the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
approach to classical mechanics, in which a system's dynamics is generated via canonical Poisson brackets, a structure which is ''only partially preserved'' in canonical quantization. This method was further used in the context of quantum field theory by
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
, in his construction of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
. In the field theory context, it is also called the
second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
of fields, in contrast to the semi-classical
first quantization A first quantization of a physical system is a possibly semiclassical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well o ...
of single particles.


History

When it was first developed, quantum physics dealt only with the quantization of the
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
of particles, leaving the electromagnetic field classical, hence the name
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
. Later the electromagnetic field was also quantized, and even the particles themselves became represented through quantized fields, resulting in the development of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED) and quantum field theory in general. Thus, by convention, the original form of particle quantum mechanics is denoted
first quantization A first quantization of a physical system is a possibly semiclassical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well o ...
, while quantum field theory is formulated in the language of
second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
.


First quantization


Single particle systems

The following exposition is based on Dirac's treatise on quantum mechanics. In the
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
of a particle, there are dynamic variables which are called coordinates () and momenta (). These specify the ''state'' of a classical system. The canonical structure (also known as the symplectic structure) of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
consists of Poisson brackets enclosing these variables, such as = 1. All transformations of variables which preserve these brackets are allowed as
canonical transformation In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canon ...
s in classical mechanics. Motion itself is such a canonical transformation. By contrast, in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, all significant features of a particle are contained in a state , \psi\rangle, called a
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
. Observables are represented by operators acting on a Hilbert space of such
quantum states In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
. The eigenvalue of an operator acting on one of its eigenstates represents the value of a measurement on the particle thus represented. For example, the
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
is read off by the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
operator \hat acting on a state , \psi_n\rangle, yielding :\hat, \psi_n\rangle=E_n, \psi_n\rangle, where is the characteristic energy associated to this , \psi_n\rangle eigenstate. Any state could be represented as a linear combination of eigenstates of energy; for example, :, \psi\rangle=\sum_^ a_n, \psi_n\rangle, where are constant coefficients. As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones, \hat and \hat, respectively. The connection between this representation and the more usual
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
representation is given by the eigenstate of the position operator \hat representing a particle at position x, which is denoted by an element , x\rangle in the Hilbert space, and which satisfies \hat, x\rangle = x, x\rangle. Then, \psi(x)= \langle x, \psi\rangle. Likewise, the eigenstates , p\rangle of the momentum operator \hat specify the momentum representation: \psi(p)= \langle p, \psi\rangle. The central relation between these operators is a quantum analog of the above Poisson bracket of classical mechanics, the
canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p_ ...
, : hat,\hat= \hat\hat-\hat\hat = i\hbar. This relation encodes (and formally leads to) the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
, in the form . This algebraic structure may be thus considered as the quantum analog of the ''canonical structure'' of classical mechanics.


Many-particle systems

When turning to N-particle systems, i.e., systems containing N
identical particles In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
(particles characterized by the same
quantum numbers In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be k ...
such as
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
,
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
and spin), it is necessary to extend the single-particle state function \psi(\mathbf) to the N-particle state function \psi(\mathbf_1,\mathbf_2,...,\mathbf_N). A fundamental difference between classical and quantum mechanics concerns the concept of indistinguishability of identical particles. Only two species of particles are thus possible in quantum physics, the so-called
bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
and
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
which obey the rules: \psi(\mathbf_1,...,\mathbf_j,...,\mathbf_k,...,\mathbf)=+\psi(\mathbf_1,...,\mathbf_k,...,\mathbf_j,...,\mathbf_N) (bosons), \psi(\mathbf_1,...,\mathbf_j,...,\mathbf_k,...,\mathbf)=-\psi(\mathbf_1,...,\mathbf_k,...,\mathbf_j,...,\mathbf_N) (fermions). Where we have interchanged two coordinates (\mathbf_j, \mathbf_k) of the state function. The usual wave function is obtained using the
Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two elect ...
and the
identical particles In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
theory. Using this basis, it is possible to solve various many-particle problems.


Issues and limitations


Classical and quantum brackets

Dirac's book details his popular rule of supplanting Poisson brackets by commutators: One might interpret this proposal as saying that we should seek a "quantization map" Q mapping a function f on the classical phase space to an operator Q_f on the quantum Hilbert space such that :Q_=\frac _f,Q_g/math> It is now known that there is no reasonable such quantization map satisfying the above identity exactly for all functions f and g.


Groenewold's theorem

One concrete version of the above impossibility claim is Groenewold's theorem (after Dutch theoretical physicist Hilbrand J. Groenewold), which we describe for a system with one degree of freedom for simplicity. Let us accept the following "ground rules" for the map Q. First, Q should send the constant function 1 to the identity operator. Second, Q should take x and p to the usual position and momentum operators X and P. Third, Q should take a polynomial in x and p to a "polynomial" in X and P, that is, a finite linear combinations of products of X and P, which may be taken in any desired order. In its simplest form, Groenewold's theorem says that there is no map satisfying the above ground rules and also the bracket condition :Q_=\frac _f,Q_g/math> for all polynomials f and g. Actually, the nonexistence of such a map occurs already by the time we reach polynomials of degree four. Note that the Poisson bracket of two polynomials of degree four has degree six, so it does not exactly make sense to require a map on polynomials of degree four to respect the bracket condition. We ''can'', however, require that the bracket condition holds when f and g have degree three. Groenewold's theorem can be stated as follows: :Theorem: There is no quantization map Q (following the above ground rules) on polynomials of degree less than or equal to four that satisfies :\quad Q_=\frac _f,Q_g/math> :whenever f and g have degree less than or equal to three. (Note that in this case, \ has degree less than or equal to four.) The proof can be outlined as follows. Suppose we first try to find a quantization map on polynomials of degree less than or equal to three satisfying the bracket condition whenever f has degree less than or equal to two and g has degree less than or equal to two. Then there is precisely one such map, and it is the Weyl quantization. The impossibility result now is obtained by writing the same polynomial of degree four as a Poisson bracket of polynomials of degree three ''in two different ways''. Specifically, we have :x^2p^2=\frac\=\frac\ On the other hand, we have already seen that if there is going to be a quantization map on polynomials of degree three, it must be the Weyl quantization; that is, we have already determined the only possible quantization of all the cubic polynomials above. The argument is finished by computing by brute force that :\frac (x^3),Q(p^3)/math> does not coincide with :\frac (x^2p),Q(xp^2)/math>. Thus, we have two incompatible requirements for the value of Q(x^2p^2).


Axioms for quantization

If represents the quantization map that acts on functions in classical phase space, then the following properties are usually considered desirable: #Q_x \psi = x \psi and Q_p \psi = -i\hbar \partial_x \psi ~~   (elementary position/momentum operators) #f \longmapsto Q_f ~~   is a linear map # _f,Q_gi\hbar Q_~~   (Poisson bracket) #Q_=g(Q_f)~~   (von Neumann rule). However, not only are these four properties mutually inconsistent, ''any three'' of them are also inconsistent! As it turns out, the only pairs of these properties that lead to self-consistent, nontrivial solutions are 2 & 3, and possibly 1 & 3 or 1 & 4. Accepting properties 1 & 2, along with a weaker condition that 3 be true only asymptotically in the limit (see Moyal bracket), leads to deformation quantization, and some extraneous information must be provided, as in the standard theories utilized in most of physics. Accepting properties 1 & 2 & 3 but restricting the space of quantizable observables to exclude terms such as the cubic ones in the above example amounts to
geometric quantization In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a wa ...
.


Second quantization: field theory

Quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
was successful at describing non-relativistic systems with fixed numbers of particles, but a new framework was needed to describe systems in which particles can be created or destroyed, for example, the electromagnetic field, considered as a collection of photons. It was eventually realized that
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
was inconsistent with single-particle quantum mechanics, so that all particles are now described relativistically by
quantum field In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
s. When the canonical quantization procedure is applied to a field, such as the electromagnetic field, the classical
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
variables become ''
quantum operator In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Bec ...
s''. Thus, the normal modes comprising the amplitude of the field are simple oscillators, each of which is quantized in standard first quantization, above, without ambiguity. The resulting quanta are identified with individual particles or excitations. For example, the quanta of the electromagnetic field are identified with photons. Unlike first quantization, conventional second quantization is completely unambiguous, in effect a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
, since the constituent set of its oscillators are quantized unambiguously. Historically, quantizing the classical theory of a single particle gave rise to a wavefunction. The classical equations of motion of a field are typically identical in form to the (quantum) equations for the wave-function of ''one of its quanta''. For example, the
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant ...
is the classical equation of motion for a free scalar field, but also the quantum equation for a scalar particle wave-function. This meant that quantizing a field ''appeared'' to be similar to quantizing a theory that was already quantized, leading to the fanciful term second quantization in the early literature, which is still used to describe field quantization, even though the modern interpretation detailed is different. One drawback to canonical quantization for a relativistic field is that by relying on the Hamiltonian to determine time dependence, relativistic invariance is no longer manifest. Thus it is necessary to check that relativistic invariance is not lost. Alternatively, the Feynman integral approach is available for quantizing relativistic fields, and is manifestly invariant. For non-relativistic field theories, such as those used in condensed matter physics, Lorentz invariance is not an issue.


Field operators

Quantum mechanically, the variables of a field (such as the field's amplitude at a given point) are represented by operators on a Hilbert space. In general, all observables are constructed as operators on the Hilbert space, and the time-evolution of the operators is governed by the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, which must be a positive operator. A state , 0\rangle annihilated by the Hamiltonian must be identified as the
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
, which is the basis for building all other states. In a non-interacting (free) field theory, the vacuum is normally identified as a state containing zero particles. In a theory with interacting particles, identifying the vacuum is more subtle, due to
vacuum polarization In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs that change the distribution of charges and curr ...
, which implies that the physical vacuum in quantum field theory is never really empty. For further elaboration, see the articles on the quantum mechanical vacuum and the vacuum of quantum chromodynamics. The details of the canonical quantization depend on the field being quantized, and whether it is free or interacting.


Real scalar field

A scalar field theory provides a good example of the canonical quantization procedure.This treatment is based primarily on Ch. 1 in Classically, a scalar field is a collection of an infinity of oscillator normal modes. It suffices to consider a 1+1-dimensional space-time \mathbb \times S_1, in which the spatial direction is compactified to a circle of circumference 2, rendering the momenta discrete. The classical
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
density describes an infinity of coupled harmonic oscillators, labelled by which is now a ''label'' (and not the displacement dynamical variable to be quantized), denoted by the classical field , :\mathcal(\phi) = \frac(\partial_t \phi)^2 - \frac(\partial_x \phi)^2 - \frac m^2\phi^2 - V(\phi), where is a potential term, often taken to be a polynomial or monomial of degree 3 or higher. The action functional is :S(\phi) = \int \mathcal(\phi) dx dt = \int L(\phi, \partial_t\phi) dt. The canonical momentum obtained via the
Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
using the action is \pi = \partial_t\phi, and the classical
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
is found to be :H(\phi,\pi) = \int dx \left frac \pi^2 + \frac (\partial_x \phi)^2 + \frac m^2 \phi^2 + V(\phi)\right Canonical quantization treats the variables and as operators with
canonical commutation relations In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat ...
at time = 0, given by : phi(x),\phi(y)= 0, \ \ pi(x), \pi(y)= 0, \ \ phi(x),\pi(y)= i\hbar \delta(x-y). Operators constructed from and can then formally be defined at other times via the time-evolution generated by the Hamiltonian, : \mathcal(t) = e^ \mathcal e^. However, since and no longer commute, this expression is ambiguous at the quantum level. The problem is to construct a representation of the relevant operators \mathcal on a Hilbert space \mathcal and to construct a positive operator as a
quantum operator In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Bec ...
on this Hilbert space in such a way that it gives this evolution for the operators \mathcal as given by the preceding equation, and to show that \mathcal contains a vacuum state , 0\rangle on which has zero eigenvalue. In practice, this construction is a difficult problem for interacting field theories, and has been solved completely only in a few simple cases via the methods of
constructive quantum field theory In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum field theory can be defined in terms of precise mathematical structures. This demonstration requires new mathematics, in a sense analogous to ...
. Many of these issues can be sidestepped using the Feynman integral as described for a particular in the article on scalar field theory. In the case of a free field, with = 0, the quantization procedure is relatively straightforward. It is convenient to Fourier transform the fields, so that : \phi_k = \int \phi(x) e^ dx, \ \ \pi_k = \int \pi(x) e^ dx. The reality of the fields implies that :\phi_ = \phi_k^\dagger, ~~~ \pi_ = \pi_k^\dagger. The classical Hamiltonian may be expanded in Fourier modes as : H=\frac\sum_^\left pi_k \pi_k^\dagger + \omega_k^2\phi_k\phi_k^\dagger\right where \omega_k = \sqrt. This Hamiltonian is thus recognizable as an infinite sum of classical normal mode oscillator excitations , each one of which is quantized in the
standard Standard may refer to: Symbols * Colours, standards and guidons, kinds of military signs * Standard (emblem), a type of a large symbol or emblem used for identification Norms, conventions or requirements * Standard (metrology), an object th ...
manner, so the free quantum Hamiltonian looks identical. It is the s that have become operators obeying the standard commutation relations, = sup>†, = ''iħ'', with all others vanishing. The collective Hilbert space of all these oscillators is thus constructed using creation and annihilation operators constructed from these modes, : a_k = \frac\left(\omega_k\phi_k + i\pi_k\right), \ \ a_k^\dagger = \frac\left(\omega_k\phi_k^\dagger - i\pi_k^\dagger\right), for which = 1 for all , with all other commutators vanishing. The vacuum , 0\rangle is taken to be annihilated by all of the , and \mathcal is the Hilbert space constructed by applying any combination of the infinite collection of creation operators to , 0\rangle. This Hilbert space is called
Fock space The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intr ...
. For each , this construction is identical to a quantum harmonic oscillator. The quantum field is an infinite array of quantum oscillators. The quantum Hamiltonian then amounts to : H = \sum_^ \hbar\omega_k a_k^\dagger a_k = \sum_^ \hbar\omega_k N_k, where may be interpreted as the ''
number operator In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles. The number operator acts on Fock space. Let :, \Psi\rangle_\nu=, \phi_1,\phi_2 ...
'' giving the
number of particles The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter ''N'', is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is ...
in a state with momentum . This Hamiltonian differs from the previous expression by the subtraction of the zero-point energy of each harmonic oscillator. This satisfies the condition that must annihilate the vacuum, without affecting the time-evolution of operators via the above exponentiation operation. This subtraction of the zero-point energy may be considered to be a resolution of the quantum operator ordering ambiguity, since it is equivalent to requiring that ''all creation operators appear to the left of annihilation operators'' in the expansion of the Hamiltonian. This procedure is known as Wick ordering or normal ordering.


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 In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be ''continuo ...
, 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 to avoid over-counting equivalent configurations, and gauge-fixing may be applied if needed. It turns out that commutation relations are useful only for quantizing ''bosons'', for which the occupancy number of any state is unlimited. To quantize ''fermions'', which satisfy the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
, anti-commutators are needed. These are defined by . When quantizing fermions, the fields are expanded in creation and annihilation operators, , , which satisfy :\ = \delta_, \ \ \ = 0, \ \ \ = 0. The states are constructed on a vacuum , 0> annihilated by the , and the
Fock space The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intr ...
is built by applying all products of creation operators to , 0>. Pauli's exclusion principle is satisfied, because (\theta_k^\dagger)^2, 0\rangle = 0, by virtue of the anti-commutation relations.


Condensates

The construction of the scalar field states above assumed that the potential was minimized at = 0, so that the vacuum minimizing the Hamiltonian satisfies 〈 〉= 0, indicating that the
vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
(VEV) of the field is zero. In cases involving
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
, it is possible to have a non-zero VEV, because the potential is minimized for a value = . This occurs for example, if with > 0 and 2 > 0, for which the minimum energy is found at . The value of in one of these vacua may be considered as ''condensate'' of the field . Canonical quantization then can be carried out for the ''shifted field'' , and particle states with respect to the shifted vacuum are defined by quantizing the shifted field. This construction is utilized in the
Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other be ...
in the standard model of
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
.


Mathematical quantization


Deformation quantization

The classical theory is described using a
spacelike In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
with the state at each slice being described by an element of a symplectic manifold with the time evolution given by the
symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
generated by a
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified 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 expressed in the
phase space formulation The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...
is . (Here, the curly braces denote the Poisson bracket. The subleading terms are all encoded in the Moyal bracket, the suitable quantum deformation of the Poisson bracket.) In general, for the quantities (observables) involved, and providing the arguments of such brackets, ''ħ''-deformations are highly nonunique—quantization is an "art", and is specified by the physical context. (Two ''different'' quantum systems may represent two different, inequivalent, deformations of the same
classical limit The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
,.) Now, one looks for
unitary representation 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 ...
s of this quantum algebra. With respect to such a unitary representation, a symplectomorphism in the classical theory would now deform to a (metaplectic)
unitary transformation In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, ...
. In particular, the time evolution symplectomorphism generated by the classical Hamiltonian deforms to a unitary transformation generated by the corresponding quantum Hamiltonian. A further generalization is to consider a
Poisson manifold In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalent ...
instead of a symplectic space for the classical theory and perform an ''ħ''-deformation of the corresponding
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
or even Poisson supermanifolds.


Geometric quantization

In contrast to the theory of deformation quantization described above, geometric quantization seeks to construct an actual Hilbert space and operators on it. Starting with a symplectic manifold M, one first constructs a prequantum Hilbert space consisting of the space of square-integrable sections of an appropriate line bundle over M. On this space, one can map ''all'' classical observables to operators on the prequantum Hilbert space, with the commutator corresponding exactly to the Poisson bracket. The prequantum Hilbert space, however, is clearly too big to describe the quantization of M. One then proceeds by choosing a polarization, that is (roughly), a choice of n variables on the 2n-dimensional phase space. The ''quantum'' Hilbert space is then the space of sections that depend only on the n chosen variables, in the sense that they are covariantly constant in the other n directions. If the chosen variables are real, we get something like the traditional Schrödinger Hilbert space. If the chosen variables are complex, we get something like the
Segal–Bargmann space In mathematics, the Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions ''F'' in ''n'' complex variables satisfying the square-integr ...
.


See also

* Correspondence principle *
Creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
* Dirac bracket * Moyal bracket *
Phase space formulation The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...
(of quantum mechanics) *
Geometric quantization In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a wa ...


References


Historical References

* Silvan S. Schweber: ''QED and the men who made it'', Princeton Univ. Press, 1994,


General Technical References

*Alexander Altland, Ben Simons: ''Condensed matter field theory'', Cambridge Univ. Press, 2009, *James D. Bjorken, Sidney D. Drell: ''Relativistic quantum mechanics'', New York, McGraw-Hill, 1964 * . *''An introduction to quantum field theory'', by M.E. Peskin and H.D. Schroeder, *Franz Schwabl: ''Advanced Quantum Mechanics'', Berlin and elsewhere, Springer, 2009


External links


What is "Relativistic Canonical Quantization"?Pedagogic Aides to Quantum Field Theory
Click on the links for Chaps. 1 and 2 at this site to find an extensive, simplified introduction to second quantization. See Sect. 1.5.2 in Chap. 1. See Sect. 2.7 and the chapter summary in Chap. 2. {{DEFAULTSORT:Canonical Quantization Quantum field theory Mathematical quantization