Quantum field theory

From Wikipedia, the free encyclopedia

Jump to: navigation, search

Quantum field theory (QFT) provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized (represented) by an infinite number of degrees of freedom, that is, fields and (in a condensed matter context) many-body systems. It is the natural and quantitative language of particle physics and condensed matter physics. Most theories in modern particle physics, including the Standard Model of elementary particles and their interactions, are formulated as relativistic quantum field theories. Quantum field theories are used in many contexts, and are especially vital in elementary particle physics, where the particle count/number may change over the course of a reaction. They are also used in the description of critical phenomena and quantum phase transitions, such as in the BCS theory of superconductivity. Quantum field theory is thought by many^[who?] to be the unique and correct outcome of combining the rules of quantum mechanics with special relativity.

In perturbative quantum field theory, the forces between particles are mediated by other particles. The electromagnetic force between two electrons is caused by an exchange of photons. Intermediate vector bosons mediate the weak force and gluons mediate the strong force. There is currently no complete quantum theory of the remaining fundamental force, gravity, but many of the proposed theories postulate the existence of a graviton particle that mediates it. These force-carrying particles are virtual particles and, by definition, cannot be detected while carrying the force, because such detection will imply that the force is not being carried. In addition, the notion of force mediating particle comes from perturbation theory, and thus does not make sense in a context of bound states.

In QFT, photons are not thought of as little billiard balls but are rather viewed as field quanta – necessarily chunked ripples in a field, or excitations, that look like particles. Fermions, like the electron, can also be described as ripples/excitations in a field, where each kind of fermion has its own field. In summary, the classical visualisation of everything is particles and field, in quantum field theory, resolves into everything is particles, which then resolves into everything is fields. In the end, particles are regarded as excited states of a field (field quanta). The gravitational field and the electromagnetic field are the only two fundamental fields in Nature that have infinite range and a corresponding classical low-energy limit, which greatly diminishes and hides their particle-like excitations. Albert Einstein, in 1905, attributed particle-like and discrete exchanges of momenta and energy, characteristic of field quanta, to the electromagnetic field. Originally, his principal motivation was to explain the thermodynamics of radiation. Although it is often claimed that the photoelectric and Compton effects require a quantum description of the EM field, this is now understood to be untrue, and proper proof of the quantum nature of radiation is now taken up into modern quantum optics as in the antibunching effect.^[1] The word photon was coined in 1926 by physical chemist Gilbert Newton Lewis (see also the articles photon antibunching and laser).

In the low-energy limit, the quantum field-theoretic description of the electromagnetic field, quantum electrodynamics, does not exactly reduce to James Clerk Maxwell's 1864 theory of classical electrodynamics. Small quantum corrections due to virtual electron positron pairs give rise to small non-linear corrections to the Maxwell equations, although the classical limit of quantum electrodynamics has not been as widely explored as that of quantum mechanics. This (super)classical limit is the Maxwell-Dirac System, and is under intense investigation. Going further to the Einstein-Maxwell-Dirac System gives a Unified Field Theory having all the properties Einstein sought.

Presumably, the as yet unknown correct quantum field-theoretic treatment of the gravitational field will become and look exactly like Einstein's general theory of relativity in the low-energy limit, or, more generally, like the Einstein-Yang-Mills-Dirac System. Indeed, quantum field theory itself is possibly the low-energy-effective-field-theory limit of a more fundamental theory such as superstring theory. Compare in this context the article effective field theory.

[edit] History

Main article: History of quantum field theory

Quantum field theory originated in the 1920s from the problem of creating a quantum mechanical theory of the electromagnetic field. In particlular de Broglie in 1924 introduced the idea of a wave description of elementary systems in the following way: we proceed in this work from the assumption of existence of a certain periodic phenomenon of a yet to be determined character, which is to be attributed to each and every isolated energy parcel.^[2] In 1925, Werner Heisenberg, Max Born, and Pascual Jordan constructed such a theory by expressing the field's internal degrees of freedom as an infinite set of harmonic oscillators and by employing the canonical quantization procedure to those oscillators. This theory assumed that no electric charges or currents were present and today would be called a free field theory. The first reasonably complete theory of quantum electrodynamics, which included both the electromagnetic field and electrically charged matter (specifically, electrons) as quantum mechanical objects, was created by Paul Dirac in 1927.^[3] This quantum field theory could be used to model important processes such as the emission of a photon by an electron dropping into a quantum state of lower energy, a process in which the number of particles changes—one atom in the initial state becomes an atom plus a photon in the final state. It is now understood that the ability to describe such processes is one of the most important features of quantum field theory.

It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate Einstein's relativity theory, which had grown out of the study of classical electromagnetism. This need to put together relativity and quantum mechanics was the second major motivation in the development of quantum field theory. Pascual Jordan and Wolfgang Pauli showed in 1928 that quantum fields could be made to behave in the way predicted by special relativity during coordinate transformations (specifically, they showed that the field commutators were Lorentz invariant). A further boost for quantum field theory came with the discovery of the Dirac equation, which was originally formulated and interpreted as a single-particle equation analogous to the Schrödinger equation, but unlike the Schrödinger equation, the Dirac equation satisfies both the Lorentz invariance, that is, the requirements of special relativity, and the rules of quantum mechanics. The Dirac equation accommodated the spin-1/2 value of the electron and accounted for its magnetic moment as well as giving accurate predictions for the spectra of hydrogen. The attempted interpretation of the Dirac equation as a single-particle equation could not be maintained long, however, and finally it was shown that several of its undesirable properties (such as negative-energy states) could be made sense of by reformulating and reinterpreting the Dirac equation as a true field equation, in this case for the quantized Dirac field or the electron field, with the negative-energy solutions pointing to the existence of anti-particles. This work was performed first by Dirac himself with the invention of hole theory in 1930 and by Wendell Furry, Robert Oppenheimer, Vladimir Fock, and others. Schrödinger, during the same period that he discovered his famous equation in 1926, also independently found the relativistic generalization of it known as the Klein-Gordon equation but dismissed it since, without spin, it predicted impossible properties for the hydrogen spectrum. (See Oskar Klein and Walter Gordon.) All relativistic wave equations that describe spin-zero particles are said to be of the Klein-Gordon type.

Of great importance are the studies of Soviet physicists, Viktor Ambartsumian and Dmitri Ivanenko, in particular the Ambarzumian-Ivanenko hypothesis of creation of massive particles (published in 1930) which is the cornerstone of the contemporary quantum field theory.^[4] The idea is that not only the quanta of the electromagnetic field, photons, but also other particles (including particles having nonzero rest mass) may be born and disappear as a result of their interaction with other particles. This idea of Ambartsumian and Ivanenko formed the basis of modern quantum field theory and theory of elementary particles.^[5]^[6]

A subtle and careful analysis in 1933 and later in 1950 by Niels Bohr and Leon Rosenfeld showed that there is a fundamental limitation on the ability to simultaneously measure the electric and magnetic field strengths that enter into the description of charges in interaction with radiation, imposed by the uncertainty principle, which must apply to all canonically conjugate quantities. This limitation is crucial for the successful formulation and interpretation of a quantum field theory of photons and electrons (quantum electrodynamics), and indeed, any perturbative quantum field theory. The analysis of Bohr and Rosenfeld explains fluctuations in the values of the electromagnetic field that differ from the classically allowed values distant from the sources of the field. Their analysis was crucial to showing that the limitations and physical implications of the uncertainty principle apply to all dynamical systems, whether fields or material particles. Their analysis also convinced most people that any notion of returning to a fundamental description of nature based on classical field theory, such as what Einstein aimed at with his numerous and failed attempts at a classical unified field theory, was simply out of the question.

The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease. In 1927, Jordan tried to extend the canonical quantization of fields to the many-body wave functions of identical particles, a procedure that is sometimes called second quantization. In 1928, Jordan and Eugene Wigner found that the quantum field describing electrons, or other fermions, had to be expanded using anti-commuting creation and annihilation operators due to the Pauli exclusion principle. This thread of development was incorporated into many-body theory and strongly influenced condensed matter physics and nuclear physics.

Despite its early successes quantum field theory was plagued by several serious theoretical difficulties. Basic physical quantities, such as the self-energy of the electron, the energy shift of electron states due to the presence of the electromagnetic field, gave infinite, divergent contributions—a nonsensical result—when computed using the perturbative techniques available in the 1930s and most of the 1940s. The electron self-energy problem was already a serious issue in the classical electromagnetic field theory, where the attempt to attribute to the electron a finite size or extent (the classical electron-radius) led immediately to the question of what non-electromagnetic stresses would need to be invoked, which would presumably hold the electron together against the Coulomb repulsion of its finite-sized parts. The situation was dire, and had certain features that reminded many of the Rayleigh-Jeans difficulty. What made the situation in the 1940s so desperate and gloomy, however, was the fact that the correct ingredients (the second-quantized Maxwell-Dirac field equations) for the theoretical description of interacting photons and electrons were well in place, and no major conceptual change was needed analogous to that which was necessitated by a finite and physically sensible account of the radiative behavior of hot objects, as provided by the Planck radiation law.

This divergence problem was solved in the case of quantum electrodynamics during the late 1940s and early 1950s by Hans Bethe, Tomonaga, Schwinger, Feynman, and Dyson, through the procedure known as renormalization. Great progress was made after realizing that ALL infinities in quantum electrodynamics are related to two effects: the self-energy of the electron/positron and vacuum polarization. Renormalization concerns the business of paying very careful attention to just what is meant by, for example, the very concepts charge and mass as they occur in the pure, non-interacting field-equations. The vacuum is itself polarizable and, hence, populated by virtual particle (on shell and off shell) pairs, and, hence, is a seething and busy dynamical system in its own right. This was a critical step in identifying the source of infinities and divergences. The bare mass and the bare charge of a particle, the values that appear in the free-field equations (non-interacting case), are abstractions that are simply not realized in experiment (in interaction). What we measure, and hence, what we must take account of with our equations, and what the solutions must account for, are the renormalized mass and the renormalized charge of a particle. That is to say, the shifted or dressed values these quantities must have when due care is taken to include all deviations from their bare values is dictated by the very nature of quantum fields themselves.

The first approach that bore fruit is known as the interaction representation, (see the article Interaction picture) a Lorentz covariant and gauge-invariant generalization of time-dependent perturbation theory used in ordinary quantum mechanics, and developed by Tomonaga and Schwinger, generalizing earlier efforts of Dirac, Fock and Podolsky. Tomonaga and Schwinger invented a relativistically covariant scheme for representing field commutators and field operators intermediate between the two main representations of a quantum system, the Schrödinger and the Heisenberg representations (see the article on quantum mechanics). Within this scheme, field commutators at separated points can be evaluated in terms of bare field creation and annihilation operators. This allows for keeping track of the time-evolution of both the bare and renormalized, or perturbed, values of the Hamiltonian and expresses everything in terms of the coupled, gauge invariant bare field-equations. Schwinger gave the most elegant formulation of this approach. The next and most famous development is due to Feynman, who, with his brilliant rules for assigning a graph/diagram to the terms in the scattering matrix (See S-Matrix Feynman diagrams). These directly corresponded (through the Schwinger-Dyson equation) to the measurable physical processes (cross sections, probability amplitudes, decay widths and lifetimes of excited states) one needs to be able to calculate. This revolutionized how quantum field theory calculations are carried-out in practice.

Two classic text-books from the 1960s, J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (1964) and J.J. Sakurai, Advanced Quantum Mechanics (1967), thoroughly developed the Feynman graph expansion techniques using physically intuitive and practical methods following from the correspondence principle, without worrying about the technicalities involved in deriving the Feynman rules from the superstructure of quantum field theory itself. Although both Feynman's heuristic and pictorial style of dealing with the infinities, as well as the formal methods of Tomonaga and Schwinger, worked extremely well, and gave spectacularly accurate answers, the true analytical nature of the question of renormalizability, that is, whether ANY theory formulated as a quantum field theory would give finite answers, was not worked-out till much later, when the urgency of trying to formulate finite theories for the strong and electro-weak (and gravitational interactions) demanded its solution.

Renormalization in the case of QED was largely fortuitous due to the smallness of the coupling constant, the fact that the coupling has no dimensions involving mass, the so-called fine structure constant, and also the zero-mass of the gauge boson involved, the photon, rendered the small-distance/high-energy behavior of QED manageable. Also, electromagnetic processess are very clean in the sense that they are not badly suppressed/damped and/or hidden by the other gauge interactions. By 1958 Sidney Drell observed: Quantum electrodynamics (QED) has achieved a status of peaceful coexistence with its divergences....

The unification of the electromagnetic force with the weak force encountered initial difficulties due to the lack of accelerator energies high enough to reveal processes beyond the Fermi interaction range. Additionally, a satisfactory theoretical understanding of hadron substructure had to be developed, culminating in the quark model.

In the case of the strong interactions, progress concerning their short-distance/high-energy behavior was much slower and more frustrating. For strong interactions with the electro-weak fields, there were difficult issues regarding the strength of coupling, the mass generation of the force carriers as well as their non-linear, self interactions. Although there has been theoretical progress toward a grand unified quantum field theory incorporating the electro-magnetic force, the weak force and the strong force, empirical verification is still pending. Superunification, incorporating the gravitational force, is still very speculative, and is under intensive investigation by many of the best minds in contemporary theoretical physics. Gravitation is a tensor field description of a spin-2 gauge-boson, the graviton, and is further discussed in the articles on general relativity and quantum gravity.

From the point of view of the techniques of (four-dimensional) quantum field theory, and as the numerous and heroic efforts to formulate a consistent quantum gravity theory by some very able minds attests, gravitational quantization was, and is still, the reigning champion for bad behavior. There are problems and frustrations stemming from the fact that the gravitational coupling constant has dimensions involving inverse powers of mass, and as a simple consequence, it is plagued by badly behaved (in the sense of perturbation theory) non-linear and violent self-interactions. Gravity, basically, gravitates, which in turn...gravitates...and so on, (i.e., gravity is itself a source of gravity,...,) thus creating a nightmare at all orders of perturbation theory. Also, gravity couples to all energy equally strongly, as per the equivalence principle, so this makes the notion of ever really switching-off, cutting-off or separating, the gravitational interaction from other interactions ambiguous and impossible since, with gravitation, we are dealing with the very structure of space-time itself. (See general covariance and, for a modest, yet highly non-trivial and significant interplay between (QFT) and gravitation (spacetime), see the article Hawking radiation and references cited therein. Also quantum field theory in curved spacetime).

Thanks to the somewhat brute-force, clanky and heuristic methods of Feynman, and the elegant and abstract methods of Tomonaga/Schwinger, from the period of early renormalization, we do have the modern theory of quantum electrodynamics (QED). It is still the most accurate physical theory known, the prototype of a successful quantum field theory. Beginning in the 1950s with the work of Yang and Mills, as well as Ryoyu Utiyama, following the previous lead of Weyl and Pauli, deep explorations illuminated the types of symmetries and invariances any field theory must satisfy. QED, and indeed, all field theories, were generalized to a class of quantum field theories known as gauge theories. Quantum electrodynamics is the most famous example of what is known as an Abelian gauge theory. It relies on the symmetry group U(1) and has one massless gauge field, the U(1) gauge symmetry, dictating the form of the interactions involving the electromagnetic field, with the photon being the gauge boson. That symmetries dictate, limit and necessitate the form of interaction between particles is the essence of the gauge theory revolution. Yang and Mills formulated the first explicit example of a non-Abelian gauge theory, Yang-Mills theory, with an attempted explanation of the strong interactions in mind. The strong interactions were then (incorrectly) understood in the mid-1950s, to be mediated by the pi-mesons, the particles predicted by Hideki Yukawa in 1935, based on his profound reflections concerning the reciprocal connection between the mass of any force-mediating particle and the range of the force it mediates. This was allowed by the uncertainty principle. The 1960s and 1970s saw the formulation of a gauge theory now known as the Standard Model of particle physics, which systematically describes the elementary particles and the interactions between them.

The electroweak interaction part of the standard model was formulated by Sheldon Glashow in the years 1958-60 with his discovery of the SU(2)xU(1) group structure of the theory. Steven Weinberg and Abdus Salam brilliantly invoked the Anderson-Higgs mechanism for the generation of the W's and Z masses (the intermediate vector boson(s) responsible for the weak interactions and neutral-currents) and keeping the mass of the photon zero. The Goldstone/Higgs idea for generating mass in gauge theories was sparked in the late 1950s and early 1960s when a number of theoreticians (including Yoichiro Nambu, Steven Weinberg, Jeffrey Goldstone, François Englert, Robert Brout, G. S. Guralnik, C. R. Hagen, Tom Kibble and Philip Warren Anderson) noticed a possibly useful analogy to the (spontaneous) breaking of the U(1) symmetry of electromagnetism in the formation of the BCS ground-state of a superconductor. The gauge boson involved in this situation, the photon, behaves as though it has acquired a finite mass. There is a further possibility that the physical vacuum (ground-state) does not respect the symmetries implied by the unbroken electroweak Lagrangian (see the article Electroweak interaction for more details) from which one arrives at the field equations. The electroweak theory of Weinberg and Salam was shown to be renormalizable (finite) and hence consistent by Gerardus 't Hooft and Martinus Veltman. The Glashow-Weinberg-Salam theory (GWS-Theory) is a triumph and, in certain applications, gives an accuracy on a par with quantum electrodynamics.

Also during the 1970s, parallel developments in the study of phase transitions in condensed matter physics led Leo Kadanoff, Michael Fisher and Kenneth Wilson (extending work of Ernst Stueckelberg, Andre Peterman, Murray Gell-Mann, and Francis Low) to a set of ideas and methods known as the renormalization group. By providing a better physical understanding of the renormalization procedure invented in the 1940s, the renormalization group sparked what has been called the grand synthesis of theoretical physics, uniting the quantum field theoretical techniques used in particle physics and condensed matter physics into a single theoretical framework.

[edit] Principles of quantum field theory

[edit] Classical fields and quantum fields

Quantum mechanics, in its most general formulation, is a theory of abstract operators (observables) acting on an abstract state space (Hilbert space), where the observables represent physically observable quantities and the state space represents the possible states of the system under study. Furthermore, each observable corresponds, in a technical sense, to the classical idea of a degree of freedom. For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators $\hat{x}$ and $\hat{p}$ . Ordinary quantum mechanics deals with systems such as this, which possess a small set of degrees of freedom.

(It is important to note, at this point, that this article does not use the word particle in the context of wave–particle duality. In quantum field theory, particle is a generic term for any discrete quantum mechanical entity, such as an electron or photon, which can behave like classical particles or classical waves under different experimental conditions, such that one could say 'this particle can behave like a wave or a particle'.)

A quantum field is a quantum mechanical system containing a large, and possibly infinite, number of degrees of freedom. A classical field contains a set of degrees of freedom at each point of space; for instance, the classical electromagnetic field defines two vectors — the electric field and the magnetic field — that can in principle take on distinct values for each position r. When the field as a whole is considered as a quantum mechanical system, its observables form an infinite (in fact uncountable) set, because r is continuous.

Furthermore, the degrees of freedom in a quantum field are arranged in repeated sets. For example, the degrees of freedom in an electromagnetic field can be grouped according to the position r, with exactly two vectors for each r. Note that r is an ordinary number that indexes the observables; it is not to be confused with the position operator $\hat{x}$ encountered in ordinary quantum mechanics, which is an observable. (Thus, ordinary quantum mechanics is sometimes referred to as zero-dimensional quantum field theory, because it contains only a single set of observables.)

It is also important to note that there is nothing special about r because, as it turns out, there is generally more than one way of indexing the degrees of freedom in the field.

In the following sections, we will show how these ideas can be used to construct a quantum mechanical theory with the desired properties. We will begin by discussing single-particle quantum mechanics and the associated theory of many-particle quantum mechanics. Then, by finding a way to index the degrees of freedom in the many-particle problem, we will construct a quantum field and study its implications.

[edit] Single-particle and many-particle quantum mechanics

In quantum mechanics, the time-dependent Schrödinger equation for a single particle in one dimension is

$-\frac{{\hbar}^2}{2m}\frac{{\partial}^2}{\partial$

where m is the particle's mass, V is the applied potential, and $ψ(x, t)$ denotes the wavefunction.

We wish to consider how this problem generalizes to N particles. There are two motivations for studying the many-particle problem. The first is a straightforward need in condensed matter physics, where typically the number of particles is on the order of Avogadro's number (6.0221415 x 10²³). The second motivation for the many-particle problem arises from particle physics and the desire to incorporate the effects of special relativity. If one attempts to include the relativistic rest energy into the above equation (in quantum mechanics where position is an observable), the result is either the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy eigenvalues that extend to –∞, so that there seems to be no easy definition of a ground state. It turns out that such inconsistencies arise from relativistic wavefunctions having a probabilistic interpretation in position space, as probability conservation is not a relativistically covariant concept. In quantum field theory, unlike in quantum mechanics, position is not an observable, and thus, one does not need the concept of a position-space probability density. For quantum fields whose interaction can be treated perturbatively, this is equivalent to neglecting the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativistic quantum theory. Einstein's famous mass-energy relation allows for the possibility that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. This suggests that a consistent relativistic quantum theory should be able to describe many-particle dynamics.

Furthermore, we will assume that the N particles are indistinguishable. As described in the article on identical particles, this implies that the state of the entire system must be either symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are rather complicated to write. For example, the general quantum state of a system of N bosons is written as

$|\phi_1$

where $|\phi_i\rang$ are the single-particle states, N_j is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases. The way to simplify this problem is to turn it into a quantum field theory.

[edit] Second quantization

Main article: Second quantization

In this section, we will describe a method for constructing a quantum field theory called second quantization. This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identical-particle states. It is based on the Hamiltonian formulation of quantum mechanics; several other approaches exist, such as the Feynman path integral,^[7] which uses a Lagrangian formulation. For an overview, see the article on quantization.

[edit] Second quantization of bosons

For simplicity, we will first discuss second quantization for bosons, which form perfectly symmetric quantum states. Let us denote the mutually orthogonal single-particle states by $|\phi_1\rang,$ and so on. For example, the 3-particle state with one particle in state $|\phi_1\rang$ and two in state $|\phi_2\rang$ is

$\frac{1}{\sqrt{3}}$

The first step in second quantization is to express such quantum states in terms of occupation numbers, by listing the number of particles occupying each of the single-particle states $|\phi_1\rang,$ etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as

$|1,$

The next step is to expand the N-particle state space to include the state spaces for all possible values of N. This extended state space, known as a Fock space, is composed of the state space of a system with no particles (the so-called vacuum state), plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. It is easy to see that there is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space.

At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number $j\cdots$ , indicating which of the single-particle states $|\phi_1\rang,$ it refers to.

The properties of this quantum field can be explored by defining creation and annihilation operators, which add and subtract particles. They are analogous to ladder operators in the quantum harmonic oscillator problem, which added and subtracted energy quanta. However, these operators literally create and annihilate particles of a given quantum state. The bosonic annihilation operator $a 2$ and creation operator $a_2^\dagger$ have the following effects:

$a_2$

$a_2^\dagger$

It can be shown that these are operators in the usual quantum mechanical sense, i.e. linear operators acting on the Fock space. Furthermore, they are indeed Hermitian conjugates, which justifies the way we have written them. They can be shown to obey the commutation relation

$\left[a_i$

where $δ$ stands for the Kronecker delta. These are precisely the relations obeyed by the ladder operators for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator.

The Hamiltonian of the quantum field (which, through the Schrödinger equation, determines its dynamics) can be written in terms of creation and annihilation operators. For instance, the Hamiltonian of a field of free (non-interacting) bosons is

$H$

where $E k$ is the energy of the k-th single-particle energy eigenstate. Note that

$a_k^\dagger\,a_k|\dots,$

Hence, $a_k^\dagger\,a_k$ is known as the number operator for the k-th eigenstate.

[edit] Second quantization of fermions

It turns out that a different definition of creation and annihilation must be used for describing fermions. According to the Pauli exclusion principle, fermions cannot share quantum states, so their occupation numbers N_i can only take on the value 0 or 1. The fermionic annihilation operators c and creation operators $c^\dagger$ are defined by their actions on a Fock state thus

$c_j$

$c_j^\dagger$

These obey an anticommutation relation:

$\left\{c_i$

One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.

[edit] Field operators

We have previously mentioned that there can be more than one way of indexing the degrees of freedom in a quantum field. Second quantization indexes the field by enumerating the single-particle quantum states. However, as we have discussed, it is more natural to think about a field, such as the electromagnetic field, as a set of degrees of freedom indexed by position.

To this end, we can define field operators that create or destroy a particle at a particular point in space. In particle physics, these operators turn out to be more convenient to work with, because they make it easier to formulate theories that satisfy the demands of relativity.

Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator $\phi(\mathbf{r})$ is

$\phi(\mathbf{r})$

The bosonic field operators obey the commutation relation

$\left[\phi(\mathbf{r})$

where $δ(x)$ stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.

The field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is a quantum-mechanical amplitude for finding a particle in some position. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say

$H$

where the indices i and j run over all particles, then the field theory Hamiltonian (in the non-relativistic limit and for negligible self-interactions) is

$H$

<

[1]

[2]

[3]