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| {{Redirect|Relativistic quantum field theory|other uses|Relativity (disambiguation){{!}}Relativity}}
| | Greetings! I am Marvella and I feel comfy when individuals use the full title. For a whilst I've been in South Dakota and my mothers and fathers reside close by. Doing ceramics is what adore performing. Bookkeeping is what I do.<br><br>My web-site: [http://ustanford.com/index.php?do=/profile-36410/info/ std testing at home] |
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| {{Quantum field theory}}
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| In [[theoretical physics]], '''quantum field theory''' ('''QFT''') is a theoretical framework for constructing [[quantum mechanics|quantum mechanical]] models of [[subatomic particle]]s in [[particle physics]] and [[quasiparticle]]s in [[condensed matter physics]]. A QFT treats particles as [[excited state]]s of an underlying [[field (physics)|physical field]], so these are called [[field quanta]].
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| For example, [[quantum electrodynamics]] (QED) has one [[electron]] field and one [[photon]] field; [[quantum chromodynamics]] (QCD) has one field for each type of [[quark]]; and, in condensed matter, there is an atomic displacement field that gives rise to [[phonon]] particles. [[Edward Witten]] describes QFT as "by far" the most difficult theory in modern physics.<ref>{{Cite web|title=Beautiful Minds, Vol. 20: Ed Witten|url=http://temi.repubblica.it/iniziative-beautifulminds/|publisher=[[la Repubblica]]|year=2010|accessdate=22 June 2012}} [http://www.youtube.com/watch?v=zPganhQDnzM&t=2m22s Here].</ref>
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| In QFT, quantum mechanical interactions between particles are described by interaction terms between the corresponding underlying fields. QFT interaction terms are similar in spirit to those between charges with electric and magnetic fields in [[Maxwell's equations]]. However, unlike the classical fields of Maxwell's theory, fields in QFT generally exist in [[quantum superposition]]s of states and are subject to the laws of quantum mechanics.
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| Quantum mechanical systems have a fixed number of particles, with each particle having a finite number of [[degrees of freedom (physics and chemistry)|degrees of freedom]]. In contrast, the excited states of a QFT can represent any number of particles. This makes quantum field theories especially useful for describing systems where the particle count/number may change over time, a crucial feature of relativistic dynamics.
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| Because the fields are continuous quantities over space, there exist excited states with arbitrarily large numbers of particles in them, providing QFT systems with an effectively infinite number of degrees of freedom. Infinite degrees of freedom can easily lead to divergences of calculated quantities (i.e., the quantities become infinite). Techniques such as [[renormalization]] of QFT parameters or discretization of [[spacetime]], as in [[lattice QCD]], are often used to avoid such infinities so as to yield physically meaningful results.
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| Most theories in standard particle physics are formulated as '''relativistic quantum field theories''', such as QED, QCD, and the [[Standard Model]]. QED, the quantum field-theoretic description of the electromagnetic field, approximately reproduces [[James Clerk Maxwell|Maxwell]]'s theory of electrodynamics in the low-energy limit, with [[Euler–Heisenberg Lagrangian|small non-linear corrections to the Maxwell equations]] required due to [[virtual particle|virtual]] electron–positron pairs.
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| In the [[Perturbation theory (quantum mechanics)|perturbative]] approach to quantum field theory, the full field interaction terms are approximated as a perturbative expansion in the number of particles involved. Each term in the expansion can be thought of as forces between particles being mediated by other particles. In QED, the [[electromagnetic force]] between two [[electrons]] is caused by an exchange of [[photons]]. Similarly, [[intermediate vector boson]]s mediate the [[weak force]] and [[gluons]] mediate the [[strong force]] in QCD. The notion of a force-mediating particle comes from perturbation theory, and does not make sense in the context of [[non-perturbative]] approaches to QFT, such as with [[bound states]].
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| 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 the [[photoelectric effect]] and [[Compton scattering]] strongly suggest the existence of the photon, it is now understood that they can be explained without invoking a quantum electromagnetic field; therefore, a more definitive proof of the quantum nature of radiation is now taken up into modern [[quantum optics]] as in the [[Photon antibunching|antibunching]] effect.<ref>J. J. Thorn et al. (2004). [http://people.whitman.edu/~beckmk/QM/grangier/Thorn_ajp.pdf Observing the quantum behavior of light in an undergraduate laboratory.] . J. J. Thorn, M. S. Neel, V. W. Donato, G. S. Bergreen, R. E. Davies, and M. Beck. American Association
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| of Physics Teachers, 2004.DOI: 10.1119/1.1737397.</ref>
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| There is currently no complete quantum theory of the remaining [[fundamental interaction|fundamental force]], [[gravity]]. Many of the [[quantum gravity|proposed theories]] to describe gravity as a QFT postulate the existence of a [[graviton]] particle that mediates the gravitational force. Presumably, the as yet unknown correct quantum field-theoretic treatment of the gravitational field will behave like Einstein's [[general theory of relativity]] in the low-energy limit. Quantum field theory of the fundamental forces itself has been postulated to be the low-energy [[effective field theory]] limit of a more fundamental theory such as [[superstring theory]].
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| ==History==
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| {{Main|History of quantum field theory}}
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| === Foundations ===
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| The early development of the field involved [[Paul Dirac|Dirac]], [[Vladimir Aleksandrovich Fock|Fock]], [[Wolfgang Ernst Pauli|Pauli]], [[Werner Heisenberg|Heisenberg]] and [[Nikolay Bogolyubov|Bogolyubov]]. This phase of development culminated with the construction of the theory of [[quantum electrodynamics]] in the 1950s.
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| === Gauge theory ===
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| [[Gauge theory]] was formulated and [[quantization (physics)|quantized]], leading to the '''unification of forces''' embodied in the [[standard model]] of [[particle physics]]. This effort started in the 1950s with the work of [[Chen Ning Yang|Yang]] and [[Robert Mills (physicist)|Mills]], was carried on by [[Martinus Veltman]] and a host of others during the 1960s and completed by the 1970s through the work of [[Gerard 't Hooft]], [[Frank Wilczek]], [[David Gross]] and [[David Politzer]].
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| === Grand synthesis ===
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| Parallel developments in the understanding of [[phase transitions]] in [[condensed matter physics]] led to the study of the [[renormalization group]]. This in turn led to the '''grand synthesis''' of theoretical physics which unified theories of particle and condensed matter physics through quantum field theory. This involved the work of [[Michael Fisher]] and [[Leo Kadanoff]] in the 1970s which led to the seminal reformulation of quantum field theory by [[Kenneth G. Wilson]].
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| == Principles ==
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| === Classical and quantum fields ===
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| {{main|Classical field theory}}
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| A classical field is a function defined over some region of space and time.<ref name="tong1">David Tong, ''[http://www.damtp.cam.ac.uk/user/tong/qft.html Lectures on Quantum Field Theory]'', chapter 1.</ref> Two physical phenomena which are described by classical fields are [[Newtonian gravitation]], described by [[Newtonian gravitational field]] '''g'''('''x''', ''t''), and [[classical electromagnetism]], described by the [[electric field|electric]] and [[magnetic fields]] '''E'''('''x''', ''t'') and '''B'''('''x''', ''t''). Because such fields can in principle take on distinct values at each point in space, they are said to have infinite [[Degrees of freedom (physics and chemistry)|degrees of freedom]].<ref name="tong1" />
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| Classical field theory does not, however, account for the quantum-mechanical aspects of such physical phenomena. For instance, it is known from quantum mechanics that certain aspects of electromagnetism involve discrete particles—[[photons]]—rather than continuous fields. The business of ''quantum'' field theory is to write down a field that is, like a classical field, a function defined over space and time, but which also accommodates the observations of quantum mechanics. This is a ''quantum field''.
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| It is not immediately clear ''how'' to write down such a quantum field, since [[quantum mechanics]] has a structure very unlike a field theory. In its most general formulation, quantum mechanics is a theory of abstract [[Operator (physics)|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.<ref name="srednicki19">{{cite book|last=Srednicki|first=Mark|title=Quantum Field Theory|edition=1st|page=19}}</ref> For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the [[position operator|position]] and [[momentum operator]]s <math>\hat{x}</math> and <math>\hat{p}</math>. Field theory, in contrast, treats ''x'' as a way to index the field rather than as an operator.<ref name="srednicki25">{{cite book|last=Srednicki|first=Mark|title=Quantum Field Theory|edition=1st|pages=25–6}}</ref>
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| There are two common ways of developing a quantum field: the [[path integral formalism]] and [[canonical quantization]].<ref name="zee61">{{cite book|last=Zee|first=Anthony|title=Quantum Field Theory in a Nutshell|edition=2nd|page=61}}</ref> The latter of these is pursued in this article.
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| ==== Lagrangian formalism ====
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| Quantum field theory frequently makes use of the [[Lagrangian]] formalism from [[classical field theory]]. This formalism is analogous to the Lagrangian formalism used in [[classical mechanics]] to solve for the motion of a particle under the influence of a field. In classical field theory, one writes down a [[Lagrangian density]], <math>\mathcal{L}</math>, involving a field, φ('''x''',''t''), and possibly its first derivatives (∂φ/∂''t'' and ∇φ), and then applies a field-theoretic form of the [[Euler–Lagrange equation]]. Writing coordinates (''t'', '''x''') = (''x''<sup>0</sup>, ''x''<sup>1</sup>, ''x''<sup>2</sup>, ''x''<sup>3</sup>) = ''x''<sup>μ</sup>, this form of the Euler–Lagrange equation is<ref name="tong1" />
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| :<math>\frac{\partial}{\partial x^\mu} \left[\frac{\partial\mathcal{L}}{\partial(\partial\phi/\partial x^\mu)}\right] - \frac{\partial\mathcal{L}}{\partial\phi} = 0,</math> | |
| where a sum over μ is performed according to the rules of [[Einstein notation]].
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| By solving this equation, one arrives at the "equations of motion" of the field.<ref name="tong1" /> For example, if one begins with the Lagrangian density
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| :<math> \mathcal{L}(\phi,\nabla\phi) = -\rho(t,\mathbf{x})\,\phi(t,\mathbf{x}) - \frac{1}{8\pi G}|\nabla\phi|^2,</math>
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| and then applies the Euler–Lagrange equation, one obtains the equation of motion
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| :<math> 4\pi G \rho(t,\mathbf{x}) = \nabla^2 \phi.</math>
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| This equation is [[Newton's law of universal gravitation]], expressed in differential form in terms of the [[gravitational potential]] φ(''t'', '''x''') and the [[mass density]] ρ(''t'', '''x'''). Despite the nomenclature, the "field" under study is the gravitational potential, φ, rather than the gravitational field, '''g'''. Similarly, when classical field theory is used to study electromagnetism, the "field" of interest is the electromagnetic [[four-potential]] (''V''/''c'', '''A'''), rather than the electric and magnetic fields '''E''' and '''B'''.
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| Quantum field theory uses this same Lagrangian procedure to determine the equations of motion for quantum fields. These equations of motion are then supplemented by [[commutation relation]]s derived from the canonical quantization procedure described below, thereby incorporating quantum mechanical effects into the behavior of the field.
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| === Single- and many-particle quantum mechanics ===
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| {{main|Quantum mechanics|First quantization}}
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| In quantum mechanics, a particle (such as an electron or proton) is described by a [[complex number|complex]] [[wavefunction]], ψ(''x'', ''t''), whose time-evolution is governed by the [[Schrödinger equation]]:
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| :<math>-\frac{{\hbar}^2}{2m}\frac{{\partial}^2}{\partial x^2}\psi(x,t) + V(x)\psi(x,t) = i \hbar \frac{\partial}{\partial t} \psi(x,t).</math>
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| Here ''m'' is the particle's [[mass]] and ''V''(''x'') is the applied [[potential]]. Physical information about the behavior of the particle is extracted from the wavefunction by constructing [[probability density function]]s for various quantities; for example, the p.d.f. for the particle's position is ψ<sup>*</sup>(''x'') ''x'' ψ(''x''), and the p.d.f. for the particle's [[momentum]] is −''iħ''ψ<sup>*</sup>(''x'')dψ/d''x''; integrating over ''x'' yields the expectation for the position, and the momentum respectively. This treatment of quantum mechanics, where a particle's wavefunction evolves against a classical background potential ''V''(''x''), is sometimes called ''first quantization''.
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| This description of quantum mechanics can be extended to describe the behavior of multiple particles, so long as the number and the type of particles remain fixed. The particles are described by a wavefunction ψ(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>N</sub>'', ''t'') which is governed by an extended version of the Schrödinger equation.
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| Often one is interested in the case where ''N'' particles are all of the same type (for example, the 18 electrons orbiting a neutral [[argon]] nucleus). As described in the article on [[identical particles]], this implies that the state of the entire system must be either symmetric ([[boson]]s) or antisymmetric ([[fermion]]s) when the coordinates of its constituent particles are exchanged. This is achieved by using a [[Slater determinant]] as the wavefunction of a fermionic system (and a Slater [[permanent]] for a bosonic system), which is equivalent to an element of the symmetric or antisymmetric subspace of a tensor product.
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| For example, the general quantum state of a system of ''N'' bosons is written as
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| :<math> |\phi_1 \cdots \phi_N \rang = \sqrt{\frac{\prod_j N_j!}{N!}} \sum_{p\in S_N} |\phi_{p(1)}\rang \otimes \cdots \otimes |\phi_{p(N)} \rang,</math>
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| where <math>|\phi_i\rang</math> are the single-particle states, ''N<sub>j</sub>'' is the number of particles occupying state ''j'', and the sum is taken over all possible [[permutation]]s ''p'' acting on ''N'' elements. In general, this is a sum of ''N''! (''N'' [[factorial]]) distinct terms. <math>\sqrt{\frac{\prod_j N_j!}{N!}}</math> is a normalizing factor.
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| There are several shortcomings to the above description of quantum mechanics which are addressed by quantum field theory. First, it is unclear how to extend quantum mechanics to include the effects of [[special relativity]].<ref name="tong0">David Tong, ''[http://www.damtp.cam.ac.uk/user/tong/qft.html Lectures on Quantum Field Theory]'', Introduction.</ref> Attempted replacements for the Schrödinger equation, such as the [[Klein–Gordon equation]] or the [[Dirac equation]], 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 [[Lorentz covariance|relativistically covariant]] concept. The second shortcoming, related to the first, is that in quantum mechanics there is no mechanism to describe particle creation and annihilation;<ref name="zee3">{{cite book|last=Zee|first=Anthony|title=Quantum Field Theory in a Nutshell|edition=2nd|page=3}}</ref> this is crucial for describing phenomena such as [[pair production]] which result from the conversion between mass and energy according to the relativistic relation [[Mass–energy equivalence|''E'' = ''mc''<sup>2</sup>]].
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| === Second quantization ===
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| {{Main|Second quantization}}
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| 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 (quantum mechanics)|Hamiltonian]] formulation of quantum mechanics
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| Several other approaches exist, such as the [[Feynman path integral]],<ref>
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| Abraham Pais, ''Inward Bound: Of Matter and Forces in the Physical World'' ISBN 0-19-851997-4. Pais recounts how his astonishment at the rapidity with which [[Feynman]] could calculate using his method. Feynman's method is now part of the standard methods for physicists.
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| </ref> which uses a [[Lagrangian]] formulation. For an overview of some of these approaches, see the article on [[quantization (physics)|quantization]].
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| ==== Bosons ====
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| For simplicity, we will first discuss second quantization for [[boson]]s, which form perfectly symmetric quantum states. Let us denote the mutually orthogonal single-particle states which are possible in the system by <math>|\phi_1\rang, |\phi_2\rang, |\phi_3\rang,</math> and so on. For example, the 3-particle state with one particle in state <math>|\phi_1\rang</math> and two in state <math>|\phi_2\rang</math> is
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| :<math> \frac{1}{\sqrt{3}} \left[ |\phi_1\rang |\phi_2\rang
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| |\phi_2\rang + |\phi_2\rang |\phi_1\rang |\phi_2\rang + |\phi_2\rang
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| |\phi_2\rang |\phi_1\rang \right]. </math>
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| 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 <math>|\phi_1\rang, |\phi_2\rang, </math> etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as
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| :<math> |1, 2, 0, 0, 0, \dots \rangle.</math>
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| An ''N''-particle state belongs to a space of states describing systems of ''N'' particles. The next step is to combine the individual ''N''-particle state spaces into an extended state space, known as [[Fock space]], which can describe systems of any number of particles. This is composed of the state space of a system with no particles (the so-called [[vacuum state]], written as <math>|0\rang</math>), plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. States describing a definite number of particles are known as [[Fock state]]s: a general element of Fock space will be a linear combination of Fock states. There is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space.
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| 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 <math>j</math> indicating which of the single-particle states <math>|\phi_1\rang, |\phi_2\rang,\dots,|\phi_j\rang,\dots</math> it refers to:
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| :<math> | N_1, N_2, N_3, \dots, N_j, \dots \rang .</math>
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| 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 <math>a_2</math> and creation operator <math>a_2^\dagger</math> are easily defined in the occupation number representation as having the following effects:
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| :<math> a_2 | N_1, N_2, N_3, \dots \rang = \sqrt{N_2} \mid N_1, (N_2 - 1), N_3, \dots \rang,</math>
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| :<math> a_2^\dagger | N_1, N_2, N_3, \dots \rang = \sqrt{N_2 + 1} \mid N_1, (N_2 + 1), N_3, \dots \rang.</math>
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| It can be shown that these are operators in the usual quantum mechanical sense, i.e. [[linear operator]]s acting on the Fock space. Furthermore, they are indeed [[Hermitian adjoint|Hermitian conjugates]], which justifies the way we have written them. They can be shown to obey the [[commutator|commutation relation]]
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| :<math>
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| \left[a_i , a_j \right] = 0 \quad,\quad
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| \left[a_i^\dagger , a_j^\dagger \right] = 0 \quad,\quad
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| \left[a_i , a_j^\dagger \right] = \delta_{ij},
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| </math>
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| where <math>\delta</math> stands for the [[Kronecker delta]]. These are precisely the relations obeyed by the ladder operators for an infinite set of independent [[quantum harmonic oscillator]]s, 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.
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| Applying an annihilation operator <math>a_k</math> followed by its corresponding creation operator <math>a_k^\dagger</math> returns the number <math>N_k</math> of particles in the ''k''<sup>th</sup> single-particle eigenstate:
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| :<math>a_k^\dagger\,a_k|\dots, N_k, \dots \rangle=N_k| \dots, N_k, \dots \rangle.</math>
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| The combination of operators <math>a_k^\dagger a_k</math> is known as the [[number operator]] for the ''k''<sup>th</sup> eigenstate.
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| The [[Hamiltonian (quantum mechanics)|Hamiltonian operator]] of the quantum field (which, through the [[Schrödinger equation]], determines its dynamics) can be written in terms of creation and annihilation operators. For instance, for a field of free (non-interacting) bosons, the total energy of the field is found by summing the energies of the bosons in each energy eigenstate. If the ''k''<sup>th</sup> single-particle energy eigenstate has energy <math>E_k</math> and there are <math>N_k</math> bosons in this state, then the total energy of these bosons is <math>E_k N_k</math>. The energy in the ''entire'' field is then a sum over <math>k</math>:
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| :<math> E_\mathrm{tot} = \sum_k E_k N_k</math>
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| This can be turned into the Hamiltonian operator of the field by replacing <math>N_k</math> with the corresponding number operator, <math>a_k^\dagger a_k</math>. This yields
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| :<math>H = \sum_k E_k \, a^\dagger_k \,a_k.</math>
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| ==== Fermions ====
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| It turns out that a different definition of creation and annihilation must be used for describing [[fermion]]s. According to the [[Pauli exclusion principle]], fermions cannot share quantum states, so their occupation numbers ''N<sub>i</sub>'' can only take on the value 0 or 1. The fermionic annihilation operators ''c'' and creation operators <math>c^\dagger</math> are defined by their actions on a Fock state thus
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| :<math> c_j | N_1, N_2, \dots, N_j = 0, \dots \rangle = 0 </math>
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| :<math> c_j | N_1, N_2, \dots, N_j = 1, \dots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \dots, N_j = 0, \dots \rangle </math>
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| :<math> c_j^\dagger | N_1, N_2, \dots, N_j = 0, \dots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \dots, N_j = 1, \dots \rangle </math>
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| :<math> c_j^\dagger | N_1, N_2, \dots, N_j = 1, \dots \rangle = 0. </math>
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| These obey an [[anticommutator|anticommutation relation]]:
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| :<math>
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| \left\{c_i , c_j \right\} = 0 \quad,\quad
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| \left\{c_i^\dagger , c_j^\dagger \right\} = 0 \quad,\quad
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| \left\{c_i , c_j^\dagger \right\} = \delta_{ij}.
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| </math>
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| 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.
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| ==== Field operators ====
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| 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.
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| 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.
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| Single-particle states are usually enumerated in terms of their [[momentum|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 <math>\phi(\mathbf{r})</math> is
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| :<math>\phi(\mathbf{r}) \ \stackrel{\mathrm{def}}{=}\ \sum_{j} e^{i\mathbf{k}_j\cdot \mathbf{r}} a_{j}. </math>
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| The bosonic field operators obey the commutation relation
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| :<math>
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| \left[\phi(\mathbf{r}) , \phi(\mathbf{r'}) \right] = 0 \quad,\quad
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| \left[\phi^\dagger(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = 0 \quad,\quad
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| \left[\phi(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = \delta^3(\mathbf{r} - \mathbf{r'})
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| </math>
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| where <math>\delta(x)</math> stands for the [[Dirac delta function]]. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.
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| 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
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| :<math>H = - \frac{\hbar^2}{2m} \sum_i \nabla_i^2 + \sum_{i < j} U(|\mathbf{r}_i - \mathbf{r}_j|) </math>
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| 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
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| :<math>H = - \frac{\hbar^2}{2m} \int d^3\!r \ \phi^\dagger(\mathbf{r}) \nabla^2 \phi(\mathbf{r}) + \frac{1}{2}\int\!d^3\!r \int\!d^3\!r' \; \phi^\dagger(\mathbf{r}) \phi^\dagger(\mathbf{r}') U(|\mathbf{r} - \mathbf{r}'|) \phi(\mathbf{r'}) \phi(\mathbf{r}). </math>
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| This looks remarkably like an expression for the expectation value of the energy, with <math>\phi</math> playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.
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| === Dynamics ===
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| Once the Hamiltonian operator is obtained as part of the [[canonical quantization]] process, the time dependence of the state is described with the [[Schrödinger equation]], just as with other quantum theories. Alternatively, the [[Heisenberg picture]] can be used where the time dependency is in the operators rather than in the states.
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| === Implications ===
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| ==== Unification of fields and particles ====
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| {{Expand section|date=July 2009}}
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| The "second quantization" procedure that we have outlined in the previous section takes a set of single-particle quantum states as a starting point. Sometimes, it is impossible to define such single-particle states, and one must proceed directly to quantum field theory. For example, a quantum theory of the [[electromagnetic field]] ''must'' be a quantum field theory, because it is impossible (for various reasons) to define a [[wavefunction]] for a single [[photon]].<ref>
| |
| {{cite journal
| |
| |last = Newton|first = T.D.
| |
| |coauthors = [[Eugene Wigner|Wigner, E.P.]]
| |
| |year = 1949
| |
| |title = Localized states for elementary particles
| |
| |journal = [[Reviews of Modern Physics]]
| |
| |volume = 21|pages = 400–406
| |
| |doi = 10.1103/RevModPhys.21.400
| |
| |bibcode=1949RvMP...21..400N
| |
| |issue = 3
| |
| }}</ref> In such situations, the quantum field theory can be constructed by examining the mechanical properties of the classical field and guessing the [[correspondence principle|corresponding]] quantum theory. For free (non-interacting) quantum fields, the quantum field theories obtained in this way have the same properties as those obtained using second quantization, such as well-defined creation and annihilation operators obeying commutation or anticommutation relations.
| |
| | |
| Quantum field theory thus provides a unified framework for describing "field-like" objects (such as the electromagnetic field, whose excitations are photons) and "particle-like" objects (such as electrons, which are treated as excitations of an underlying electron field), so long as one can treat interactions as "perturbations" of free fields. There are still unsolved problems relating to the more general case of interacting fields that may or may not be adequately described by perturbation theory. For more on this topic, see [[Haag's theorem]].
| |
| | |
| ==== Physical meaning of particle indistinguishability ====
| |
| The second quantization procedure relies crucially on the particles being [[identical particles|identical]]. We would not have been able to construct a quantum field theory from a distinguishable many-particle system, because there would have been no way of separating and indexing the degrees of freedom.
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| | |
| Many physicists prefer to take the converse interpretation, which is that ''quantum field theory explains what identical particles are''. In ordinary quantum mechanics, there is not much theoretical motivation for using symmetric (bosonic) or antisymmetric (fermionic) states, and the need for such states is simply regarded as an empirical fact. From the point of view of quantum field theory, particles are identical [[if and only if]] they are excitations of the same underlying quantum field. Thus, the question "why are all electrons identical?" arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental.
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| | |
| ==== Particle conservation and non-conservation ====
| |
| During second quantization, we started with a Hamiltonian and state space describing a fixed [[Particle number|number of particles]] (''N''), and ended with a Hamiltonian and state space for an arbitrary number of particles. Of course, in many common situations ''N'' is an important and perfectly well-defined quantity, e.g. if we are describing a gas of [[particle in a box|atoms sealed in a box]]. From the point of view of quantum field theory, such situations are described by quantum states that are eigenstates of the [[particle number operator|number operator]] <math>\hat{N}</math>, which measures the total number of particles present. As with any quantum mechanical observable, <math>\hat{N}</math> is conserved if it commutes with the Hamiltonian. In that case, the quantum state is trapped in the ''N''-particle [[Euclidean subspace|subspace]] of the total Fock space, and the situation could equally well be described by ordinary ''N''-particle quantum mechanics. (Strictly speaking, this is only true in the noninteracting case or in the low energy density limit of renormalized quantum field theories)
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| | |
| For example, we can see that the free-boson Hamiltonian described above conserves particle number. Whenever the Hamiltonian operates on a state, each particle destroyed by an annihilation operator ''a<sub>k</sub>'' is immediately put back by the creation operator <math>a_k^\dagger</math>.
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| | |
| On the other hand, it is possible, and indeed common, to encounter quantum states that are ''not'' eigenstates of <math>\hat{N}</math>, which do not have well-defined particle numbers. Such states are difficult or impossible to handle using ordinary quantum mechanics, but they can be easily described in quantum field theory as [[quantum superposition]]s of states having different values of ''N''. For example, suppose we have a bosonic field whose particles can be created or destroyed by interactions with a fermionic field. The Hamiltonian of the combined system would be given by the Hamiltonians of the free boson and free fermion fields, plus a "potential energy" term such as
| |
| | |
| :<math>H_I = \sum_{k,q} V_q (a_q + a_{-q}^\dagger) c_{k+q}^\dagger c_k,</math>
| |
| | |
| where <math>a_k^\dagger</math> and ''a<sub>k</sub>'' denotes the bosonic creation and annihilation operators, <math>c_k^\dagger</math> and ''c<sub>k</sub>'' denotes the fermionic creation and annihilation operators, and ''V<sub>q</sub>'' is a parameter that describes the strength of the interaction. This "interaction term" describes processes in which a fermion in state ''k'' either absorbs or emits a boson, thereby being kicked into a different eigenstate ''k+q''. (In fact, this type of Hamiltonian is used to describe interaction between [[conduction electron]]s and [[phonon]]s in [[metal]]s. The interaction between electrons and [[photon]]s is treated in a similar way, but is a little more complicated because the role of [[spin (physics)|spin]] must be taken into account.) One thing to notice here is that even if we start out with a fixed number of bosons, we will typically end up with a superposition of states with different numbers of bosons at later times. The number of fermions, however, is conserved in this case.
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| | |
| In [[condensed matter physics]], states with ill-defined particle numbers are particularly important for describing the various [[superfluid]]s. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers. In addition, the concept of a coherent state (used to model the laser and the BCS ground state) refers to a state with an ill-defined particle number but a well-defined phase.
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| | |
| === Axiomatic approaches ===
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| The preceding description of quantum field theory follows the spirit in which most [[physicist]]s approach the subject. However, it is not [[rigour|mathematically rigorous]]. Over the past several decades, there have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of [[axiom]]s for it. These attempts fall into two broad classes.
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| | |
| The first class of axioms, first proposed during the 1950s, include the [[Wightman axioms|Wightman]], [[Osterwalder–Schrader theorem|Osterwalder–Schrader]], and [[Haag–Kastler]] systems. They attempted to formalize the physicists' notion of an "operator-valued field" within the context of [[functional analysis]], and enjoyed limited success. It was possible to prove that any quantum field theory satisfying these axioms satisfied certain general theorems, such as the [[spin-statistics theorem]] and the [[CPT symmetry|CPT theorem]]. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory, including the [[Standard Model]], satisfied these axioms. Most of the theories that could be treated with these analytic axioms were physically trivial, being restricted to low-dimensions and lacking interesting dynamics. The construction of theories satisfying one of these sets of axioms falls in the field of [[constructive quantum field theory]]. Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others.
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| | |
| During the 1980s, a second set of axioms based on [[geometry|geometric]] ideas was proposed. This line of investigation, which restricts its attention to a particular class of quantum field theories known as [[topological quantum field theory|topological quantum field theories]], is associated most closely with [[Michael Atiyah]] and [[Graeme Segal]], and was notably expanded upon by [[Edward Witten]], [[Richard Borcherds]], and [[Maxim Kontsevich]]. However, most of the physically relevant quantum field theories, such as the [[Standard Model]], are not topological quantum field theories; the quantum field theory of the [[Quantum Hall effect|fractional quantum Hall effect]] is a notable exception. The main impact of axiomatic topological quantum field theory has been on mathematics, with important applications in [[representation theory]], [[algebraic topology]], and [[differential geometry]].
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| | |
| Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the [[Millennium Prize Problems]]—proving the existence of a [[Yang–Mills existence and mass gap|mass gap in Yang–Mills theory]]—is linked to this issue.
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| | |
| == Associated phenomena ==
| |
| In the previous part of the article, we described the most general properties of quantum field theories. Some of the quantum field theories studied in various fields of theoretical physics possess additional special properties, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections.
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| | |
| === Renormalization ===
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| {{Main|Renormalization}}
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| Early in the history of quantum field theory, it was found that many seemingly innocuous calculations, such as the [[perturbation theory (quantum mechanics)|perturbative]] shift in the energy of an electron due to the presence of the electromagnetic field, give infinite results. The reason is that the perturbation theory for the shift in an energy involves a sum over all other energy levels, and there are infinitely many levels at short distances that each give a finite contribution.
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| Many of these problems are related to failures in [[classical electrodynamics]] that were identified but unsolved in the 19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field that it carries around with it. The energy carried by a single electron—its [[self energy]]—is not simply the bare value, but also includes the energy contained in its electromagnetic field, its attendant cloud of photons. The energy in a field of a spherical source diverges in both classical and quantum mechanics, but as discovered by [[Victor Weisskopf|Weisskopf]] with help from [[Wendell Furry|Furry]], in quantum mechanics the divergence is much milder, going only as the logarithm of the radius of the sphere.
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| | |
| The solution to the problem, presciently suggested by [[Ernst Stueckelberg|Stueckelberg]], independently by [[Hans Bethe|Bethe]] after the crucial experiment by [[Willis Lamb|Lamb]], implemented at one loop by [[Julian Schwinger|Schwinger]], and systematically extended to all loops by [[Richard Feynman|Feynman]] and [[Freeman Dyson|Dyson]], with converging work by [[Sin-Itiro Tomonaga|Tomonaga]] in isolated postwar Japan, comes from recognizing that all the infinities in the interactions of photons and electrons can be isolated into redefining a finite number of quantities in the equations by replacing them with the observed values: specifically the electron's mass and charge: this is called [[renormalization]]. The technique of renormalization recognizes that the problem is essentially purely mathematical, that extremely short distances are at fault. In order to define a theory on a continuum, first place a [[cutoff (physics)|cutoff]] on the fields, by postulating that quanta cannot have energies above some extremely high value. This has the effect of replacing continuous space by a structure where very short wavelengths do not exist, as on a lattice. Lattices break rotational symmetry, and one of the crucial contributions made by Feynman, Pauli and Villars, and modernized by [[Gerardus 't Hooft|'t Hooft]] and [[Martinus Veltman|Veltman]], is a symmetry-preserving cutoff for perturbation theory (this process is called [[regularization (physics)|regularization]]). There is no known symmetrical cutoff outside of perturbation theory, so for rigorous or numerical work people often use an actual lattice.
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| | |
| On a lattice, every quantity is finite but depends on the spacing. When taking the limit of zero spacing, we make sure that the physically observable quantities like the observed electron mass stay fixed, which means that the constants in the Lagrangian defining the theory depend on the spacing. Hopefully, by allowing the constants to vary with the lattice spacing, all the results at long distances become insensitive to the lattice, defining a [[continuum limit]].
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| | |
| The renormalization procedure only works for a certain class of quantum field theories, called '''renormalizable quantum field theories'''. A theory is '''perturbatively renormalizable''' when the constants in the Lagrangian only diverge at worst as logarithms of the lattice spacing for very short spacings. The continuum limit is then well defined in perturbation theory, and even if it is not fully well defined non-perturbatively, the problems only show up at distance scales that are exponentially small in the inverse coupling for weak couplings. The [[Standard Model]] of [[particle physics]] is perturbatively renormalizable, and so are its component theories ([[quantum electrodynamics]]/[[electroweak interaction|electroweak theory]] and [[quantum chromodynamics]]). Of the three components, quantum electrodynamics is believed to not have a continuum limit, while the [[asymptotic freedom|asymptotically free]] SU(2) and SU(3) weak hypercharge and strong color interactions are nonperturbatively well defined.
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| | |
| The [[renormalization group]] describes how renormalizable theories emerge as the long distance low-energy [[effective field theory]] for any given high-energy theory. Because of this, renormalizable theories are insensitive to the precise nature of the underlying high-energy short-distance phenomena. This is a blessing because it allows physicists to formulate low energy theories without knowing the details of high energy phenomenon. It is also a curse, because once a renormalizable theory like the standard model is found to work, it gives very few clues to higher energy processes. The only way high energy processes can be seen in the standard model is when they allow otherwise forbidden events, or if they predict quantitative relations between the coupling constants.
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| | |
| === Haag's theorem ===
| |
| {{See also|Haag's theorem}}
| |
| From a mathematically rigorous perspective, there exists no [[interaction picture]] in a Lorentz-covariant quantum field theory. This implies that the perturbative approach of [[Feynman diagram]]s in QFT is not strictly justified, despite producing vastly precise predictions validated by experiment. This is called [[Haag's theorem]], but most particle physicists relying on QFT largely shrug it off.
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| | |
| === Gauge freedom ===
| |
| A [[gauge theory]] is a theory that admits a [[Symmetry in physics|symmetry]] with a local parameter. For example, in every [[quantum mechanics|quantum]] theory the global [[Phase (waves)|phase]] of the [[wave function]] is arbitrary and does not represent something physical. Consequently, the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is a [[global symmetry]]. In [[quantum electrodynamics]], the theory is also invariant under a ''local'' change of phase, that is – one may shift the phase of all [[wave function]]s so that the shift may be different at every point in space-time. This is a ''local'' symmetry. However, in order for a well-defined [[derivative]] operator to exist, one must introduce a new [[field (physics)|field]], the [[gauge field]], which also transforms in order for the local change of variables (the phase in our example) not to affect the derivative. In quantum electrodynamics this gauge field is the [[electromagnetic field]]. The change of local gauge of variables is termed [[gauge transformation]].
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| | |
| In quantum field theory the excitations of fields represent [[Elementary particle|particles]]. The particle associated with excitations of the [[gauge field]] is the [[gauge boson]], which is the [[photon]] in the case of quantum electrodynamics.
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| | |
| The [[Degrees of freedom (physics and chemistry)|degrees of freedom]] in quantum field theory are local fluctuations of the fields. The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or ''gauge artifacts''; usually some of them have a negative [[norm (mathematics)|norm]], making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum [[Anomaly (physics)|anomaly]]. If a gauge symmetry is [[Gauge anomaly|anomalous]] (i.e. not kept in the quantum theory) then the theory is non-consistent: for example, in quantum electrodynamics, had there been a [[gauge anomaly]], this would require the appearance of [[photon]]s with [[Longitudinal wave|longitudinal]] [[photon polarization|polarization]] and polarization in the time direction, the latter having a negative [[norm (mathematics)|norm]], rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any interaction, making the theory non-[[unitarity|unitary]] and again inconsistent (see [[optical theorem]]).
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| | |
| In general, the gauge transformations of a theory consist of several different transformations, which may not be [[commutative]]. These transformations are together described by a mathematical object known as a [[gauge group]]. [[Infinitesimal]] [[gauge transformation]]s are the [[gauge group]] [[generator (mathematics)|generator]]s. Therefore the number of gauge bosons is the group [[Dimension (vector space)|dimension]] (i.e. number of generators forming a [[Basis (linear algebra)|basis]]).
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| | |
| All the [[fundamental interaction]]s in nature are described by gauge theories. These are:
| |
| * [[Quantum chromodynamics]], whose [[gauge group]] is [[SU(3)|'''SU'''(3)]]. The [[gauge boson]]s are eight [[gluon]]s.
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| * The [[Electroweak interaction|electroweak theory]], whose [[gauge group]] is '''U'''(1) × '''SU'''(2), (a [[direct product of groups|direct product]] of '''U'''(1) and [[SU(2)|'''SU'''(2)]]).
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| * [[Gravity]], whose classical theory is [[general relativity]], admits the [[equivalence principle]], which is a form of gauge symmetry. However, it is explicitly non-renormalizable.
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| | |
| === Multivalued gauge transformations ===
| |
| The gauge transformations which leave the theory invariant involve by definition only single-valued gauge functions <math>\Lambda(x_i)</math> which satisfy the [[Symmetry of second derivatives|Schwarz integrability criterion]]
| |
| | |
| :<math>\partial_{x_i x_j} \Lambda = \partial_{x_jx_i} \Lambda.</math>
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| | |
| An interesting extension of gauge transformations arises if the gauge functions <math>\Lambda(x_i)</math> are allowed to be multivalued functions which violate the integrability criterion. These are capable of changing the physical field strengths
| |
| and are therefore no proper symmetry transformations. Nevertheless, the transformed field equations describe correctly the physical laws in the presence of the newly generated field strengths. See the textbook by [[Hagen Kleinert|H. Kleinert]] cited below
| |
| for the applications to phenomena in physics.
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| | |
| === Supersymmetry ===
| |
| {{Main|Supersymmetry}}
| |
| | |
| [[Supersymmetry]] assumes that every fundamental [[fermion]] has a superpartner that is a [[boson]] and vice versa. It was introduced in order to solve the so-called [[Hierarchy problem|Hierarchy Problem]], that is, to explain why particles not protected by any symmetry (like the [[Higgs boson]]) do not receive radiative corrections to its mass driving it to the larger scales (GUT, Planck...). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity ([[Supergravity]]), and it is a key ingredient for the consistency of [[string theory]].
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| | |
| The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite.
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| | |
| Since no superpartners have yet been observed, if supersymmetry exists it must be broken (through a so-called soft term, which breaks supersymmetry without ruining its helpful features). The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the [[Large Hadron Collider]]. The Higgs particle has been detected at the LHC, and no such superparticles have been discovered.
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| | |
| == See also ==
| |
| {{Portal|Mathematics|Physics}}
| |
| {{columns-list|2|
| |
| * [[Abraham–Lorentz force]]
| |
| * [[Basic concepts of quantum mechanics]]
| |
| * [[Common integrals in quantum field theory]]
| |
| * [[Constructive quantum field theory]]
| |
| * [[Einstein–Maxwell–Dirac equations]]
| |
| * [[Feynman path integral]]
| |
| * [[Form factor (quantum field theory)]]
| |
| * [[Green–Kubo relations]]
| |
| * [[Green's function (many-body theory)]]
| |
| * [[Invariance mechanics]]
| |
| * [[List of quantum field theories]]
| |
| * [[Pauli exclusion principle]]
| |
| * [[Photon polarization]]
| |
| * [[Quantum field theory in curved spacetime]]
| |
| * [[Quantum flavordynamics]]
| |
| * Quantum [[geometrodynamics]]
| |
| * [[Quantum hydrodynamics]]
| |
| * [[Quantum magnetodynamics]]
| |
| * [[Quantum triviality]]
| |
| * [[Relation between Schrödinger's equation and the path integral formulation of quantum mechanics]]
| |
| * [[Relationship between string theory and quantum field theory]]
| |
| * [[Schwinger–Dyson equation]]
| |
| * [[Static forces and virtual-particle exchange]]
| |
| * [[Symmetry in quantum mechanics]]
| |
| * [[Theoretical and experimental justification for the Schrödinger equation]]
| |
| * [[Ward–Takahashi identity]]
| |
| * [[Wheeler–Feynman absorber theory]]
| |
| * [[Wigner's classification]]
| |
| * [[Wigner's theorem]]
| |
| }}
| |
| | |
| ==Notes==
| |
| {{reflist|group="note"}}
| |
| | |
| == References ==
| |
| {{Reflist}}
| |
| | |
| == Further reading ==
| |
| ;General readers
| |
| | |
| *{{cite book
| |
| |last=Feynman |first=R.P. |authorlink=Richard Feynman
| |
| |year=2001 |origyear=1964
| |
| |title=[[The Character of Physical Law]]
| |
| |publisher=[[MIT Press]]
| |
| |isbn=0-262-56003-8
| |
| }}
| |
| *{{cite book
| |
| |last=Feynman |first=R.P.
| |
| |year=2006 |origyear=1985
| |
| |title=[[QED: The Strange Theory of Light and Matter]]
| |
| |publisher=[[Princeton University Press]]
| |
| |isbn=0-691-12575-9
| |
| }}
| |
| *{{cite book
| |
| |last=Gribbin |first=J.
| |
| |year=1998
| |
| |title=Q is for Quantum: Particle Physics from A to Z
| |
| |publisher=[[Weidenfeld & Nicolson]]
| |
| |isbn=0-297-81752-3
| |
| }}
| |
| *Schumm, Bruce A. (2004) ''Deep Down Things''. Johns Hopkins Univ. Press. Chpt. 4.
| |
| | |
| ;Introductory texts
| |
| | |
| *{{cite book
| |
| |last=McMahon |first=D.
| |
| |year=2008
| |
| |title=Quantum Field Theory
| |
| |publisher=[[McGraw-Hill]]
| |
| |isbn=978-0-07-154382-8
| |
| }}
| |
| *{{cite book
| |
| |last=Bogoliubov |first=N. |authorlink=Nikolay Bogoliubov
| |
| |last2=Shirkov |first2=D. |authorlink2=Dmitry Shirkov
| |
| |year=1982
| |
| |title=Quantum Fields
| |
| |publisher=[[Benjamin-Cummings]]
| |
| |isbn=0-8053-0983-7
| |
| }}
| |
| *{{cite book
| |
| |last=Frampton |first=P.H. |authorlink=Paul Frampton
| |
| |year=2000
| |
| |title=Gauge Field Theories''
| |
| |edition=2nd
| |
| |series=Frontiers in Physics
| |
| |publisher=[[John Wiley & Sons|Wiley]]
| |
| }}
| |
| *{{cite book
| |
| |last=Greiner |first=W |last2=Müller |first2=B.
| |
| |year=2000
| |
| |title=Gauge Theory of Weak Interactions
| |
| |publisher=[[Springer Science+Business Media|Springer]]
| |
| |isbn=3-540-67672-4
| |
| }}
| |
| *{{cite book
| |
| |last=Itzykson |first=C. |last2=Zuber |first2=J.-B.
| |
| |year=1980
| |
| |title=Quantum Field Theory
| |
| |publisher=[[McGraw-Hill]]
| |
| |isbn=0-07-032071-3
| |
| }}
| |
| *{{cite book
| |
| |last=Kane |first=G.L.
| |
| |year=1987
| |
| |title=Modern Elementary Particle Physics
| |
| |publisher=[[Perseus Books]]
| |
| |isbn=0-201-11749-5
| |
| }}
| |
| *{{cite book
| |
| |last=Kleinert |first=H. |authorlink=Hagen Kleinert
| |
| |last2=Schulte-Frohlinde |first2=Verena
| |
| |year=2001
| |
| |title=Critical Properties of φ<sup>4</sup>-Theories
| |
| |url=http://users.physik.fu-berlin.de/~kleinert/re.html#B6
| |
| |publisher=[[World Scientific]]
| |
| |isbn=981-02-4658-7
| |
| }}
| |
| *{{cite book
| |
| |last=Kleinert |first=H.
| |
| |year=2008
| |
| |title=Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation
| |
| |url=http://users.physik.fu-berlin.de/~kleinert/public_html/kleiner_reb11/psfiles/mvf.pdf
| |
| |publisher=[[World Scientific]]
| |
| |isbn=978-981-279-170-2
| |
| }}
| |
| *{{cite book
| |
| |last=Loudon |first=R
| |
| |year=1983
| |
| |title=The Quantum Theory of Light
| |
| |publisher=Oxford University Press
| |
| |isbn=0-19-851155-8
| |
| }}
| |
| *{{cite book
| |
| |last=Mandl |first=F. |last2=Shaw |first2=G.
| |
| |year=1993
| |
| |title=Quantum Field Theory
| |
| |publisher=[[John Wiley & Sons]]
| |
| |isbn=978-0-471-94186-6
| |
| }}
| |
| *{{cite book
| |
| |last=Peskin |first=M. |last2=Schroeder |first2=D.
| |
| |year=1995
| |
| |title=An Introduction to Quantum Field Theory
| |
| |publisher=[[Westview Press]]
| |
| |isbn=0-201-50397-2
| |
| }}
| |
| *{{cite book
| |
| |last=Ryder |first=L.H.
| |
| |year=1985
| |
| |title=Quantum Field Theory
| |
| |publisher=[[Cambridge University Press]]
| |
| |isbn=0-521-33859-X
| |
| }}
| |
| * Srednicki, Mark (2007) ''[http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=0521864496 Quantum Field Theory.]'' Cambridge Univ. Press.
| |
| *{{cite book
| |
| |last=Ynduráin |first=F.J.
| |
| |year=1996
| |
| |title=Relativistic Quantum Mechanics and Introduction to Field Theory
| |
| |edition=1st
| |
| |publisher=[[Springer Science+Business Media|Springer]]
| |
| |isbn=978-3-540-60453-2
| |
| }}
| |
| *{{cite book
| |
| |last=Zee |first=A. |authorlink=Anthony Zee
| |
| |year=2003
| |
| |title=Quantum Field Theory in a Nutshell
| |
| |publisher=[[Princeton University Press]]
| |
| |isbn=0-691-01019-6
| |
| }}
| |
| | |
| ;Advanced texts
| |
| | |
| *{{cite book
| |
| |author=Bogoliubov, N.; [[Anatoly Logunov|Logunov, A.A.]]; Oksak, A.I.; Todorov, I.T.
| |
| |year=1990
| |
| |title=General Principles of Quantum Field Theory
| |
| |publisher=[[Kluwer Academic Publishers]]
| |
| |isbn=978-0-7923-0540-8
| |
| }}
| |
| *{{cite book
| |
| |last=Weinberg |first=S. |authorlink=Steven Weinberg
| |
| |title=The Quantum Theory of Fields
| |
| |volume=1–3
| |
| |publisher=[[Cambridge University Press]]
| |
| |year=1995
| |
| |isbn=
| |
| }}
| |
| '''Articles''':
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| * [[Gerard 't Hooft]] (2007) "[http://www.phys.uu.nl/~thooft/lectures/basisqft.pdf The Conceptual Basis of Quantum Field Theory]" in Butterfield, J., and [[John Earman]], eds., ''Philosophy of Physics, Part A''. Elsevier: 661–730.
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| * [[Frank Wilczek]] (1999) "[http://arxiv.org/abs/hep-th/9803075 Quantum field theory]", ''Reviews of Modern Physics'' 71: S83–S95. Also doi=10.1103/Rev. Mod. Phys. 71.
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| ==External links==
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| * {{springer|title=Quantum field theory|id=p/q076300}}
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| * [[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/quantum-field-theory/ Quantum Field Theory]", by Meinard Kuhlmann.
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| * Siegel, Warren, 2005. ''[http://insti.physics.sunysb.edu/%7Esiegel/errata.html Fields.]'' A free text, also available from [[arXiv:hep-th/9912205]].
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| * [http://www.nat.vu.nl/~mulders/QFT-0.pdf Quantum Field Theory] by P. J. Mulders
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| [[Category:Quantum field theory| ]]
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