Quantum field theory
Quantum field theory |
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History |
In
QFT treats particles as
History
Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between
Theoretical background
Quantum field theory results from the combination of classical field theory, quantum mechanics, and special relativity.[1]: xi A brief overview of these theoretical precursors follows.
The earliest successful classical field theory is one that emerged from Newton's law of universal gravitation, despite the complete absence of the concept of fields from his 1687 treatise Philosophiæ Naturalis Principia Mathematica. The force of gravity as described by Isaac Newton is an "action at a distance"—its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with Richard Bentley, however, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact".[2]: 4 It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields—a numerical quantity (a vector in the case of gravitational field) assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered merely a mathematical trick.[3]: 18
Fields began to take on an existence of their own with the development of electromagnetism in the 19th century. Michael Faraday coined the English term "field" in 1845. He introduced fields as properties of space (even when it is devoid of matter) having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines of force". This description of fields remains to this day.[2][4]: 301 [5]: 2
The theory of
Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in
In 1913,
In the same year as his paper on the photoelectric effect, Einstein published his theory of
Two difficulties remained. Observationally, the
Quantum electrodynamics
Quantum field theory naturally began with the study of electromagnetic interactions, as the electromagnetic field was the only known classical field as of the 1920s.[8]: 1
Through the works of Born, Heisenberg, and Pascual Jordan in 1925–1926, a quantum theory of the free electromagnetic field (one with no interactions with matter) was developed via canonical quantization by treating the electromagnetic field as a set of quantum harmonic oscillators.[8]: 1 With the exclusion of interactions, however, such a theory was yet incapable of making quantitative predictions about the real world.[3]: 22
In his seminal 1927 paper The quantum theory of the emission and absorption of radiation, Dirac coined the term quantum electrodynamics (QED), a theory that adds upon the terms describing the free electromagnetic field an additional interaction term between electric current density and the electromagnetic vector potential. Using first-order perturbation theory, he successfully explained the phenomenon of spontaneous emission. According to the uncertainty principle in quantum mechanics, quantum harmonic oscillators cannot remain stationary, but they have a non-zero minimum energy and must always be oscillating, even in the lowest energy state (the ground state). Therefore, even in a perfect vacuum, there remains an oscillating electromagnetic field having zero-point energy. It is this quantum fluctuation of electromagnetic fields in the vacuum that "stimulates" the spontaneous emission of radiation by electrons in atoms. Dirac's theory was hugely successful in explaining both the emission and absorption of radiation by atoms; by applying second-order perturbation theory, it was able to account for the scattering of photons, resonance fluorescence and non-relativistic Compton scattering. Nonetheless, the application of higher-order perturbation theory was plagued with problematic infinities in calculations.[6]: 71
In 1928, Dirac wrote down a wave equation that described relativistic electrons: the Dirac equation. It had the following important consequences: the spin of an electron is 1/2; the electron g-factor is 2; it led to the correct Sommerfeld formula for the fine structure of the hydrogen atom; and it could be used to derive the Klein–Nishina formula for relativistic Compton scattering. Although the results were fruitful, the theory also apparently implied the existence of negative energy states, which would cause atoms to be unstable, since they could always decay to lower energy states by the emission of radiation.[6]: 71–72
The prevailing view at the time was that the world was composed of two very different ingredients: material particles (such as electrons) and quantum fields (such as photons). Material particles were considered to be eternal, with their physical state described by the probabilities of finding each particle in any given region of space or range of velocities. On the other hand, photons were considered merely the excited states of the underlying quantized electromagnetic field, and could be freely created or destroyed. It was between 1928 and 1930 that Jordan, Eugene Wigner, Heisenberg, Pauli, and Enrico Fermi discovered that material particles could also be seen as excited states of quantum fields. Just as photons are excited states of the quantized electromagnetic field, so each type of particle had its corresponding quantum field: an electron field, a proton field, etc. Given enough energy, it would now be possible to create material particles. Building on this idea, Fermi proposed in 1932 an explanation for beta decay known as Fermi's interaction. Atomic nuclei do not contain electrons per se, but in the process of decay, an electron is created out of the surrounding electron field, analogous to the photon created from the surrounding electromagnetic field in the radiative decay of an excited atom.[3]: 22–23
It was realized in 1929 by Dirac and others that negative energy states implied by the Dirac equation could be removed by assuming the existence of particles with the same mass as electrons but opposite electric charge. This not only ensured the stability of atoms, but it was also the first proposal of the existence of antimatter. Indeed, the evidence for positrons was discovered in 1932 by Carl David Anderson in cosmic rays. With enough energy, such as by absorbing a photon, an electron-positron pair could be created, a process called pair production; the reverse process, annihilation, could also occur with the emission of a photon. This showed that particle numbers need not be fixed during an interaction. Historically, however, positrons were at first thought of as "holes" in an infinite electron sea, rather than a new kind of particle, and this theory was referred to as the Dirac hole theory.[6]: 72 [3]: 23 QFT naturally incorporated antiparticles in its formalism.[3]: 24
Infinities and renormalization
A series of papers was published between 1934 and 1938 by Ernst Stueckelberg that established a relativistically invariant formulation of QFT. In 1947, Stueckelberg also independently developed a complete renormalization procedure. Such achievements were not understood and recognized by the theoretical community.[6]
Faced with these infinities,
In 1947, Willis Lamb and Robert Retherford measured the minute difference in the 2S1/2 and 2P1/2 energy levels of the hydrogen atom, also called the Lamb shift. By ignoring the contribution of photons whose energy exceeds the electron mass, Hans Bethe successfully estimated the numerical value of the Lamb shift.[6][3]: 28 Subsequently, Norman Myles Kroll, Lamb, James Bruce French, and Victor Weisskopf again confirmed this value using an approach in which infinities cancelled other infinities to result in finite quantities. However, this method was clumsy and unreliable and could not be generalized to other calculations.[6]
The breakthrough eventually came around 1950 when a more robust method for eliminating infinities was developed by
Since those parts of the modified mass and charge due to field reactions [become infinite], it is impossible to calculate them by the theory. However, the mass and charge observed in experiments are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite. On the other hand, the mass and charge appearing in the theory are… the values modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental values for them phenomenologically... This procedure is called the renormalization of mass and charge… After long, laborious calculations, less skillful than Schwinger's, we obtained a result... which was in agreement with [the] Americans'.[9]
By applying the renormalization procedure, calculations were finally made to explain the electron's
At the same time, Feynman introduced the
It was with the invention of the renormalization procedure and Feynman diagrams that QFT finally arose as a complete theoretical framework.[8]: 2
Non-renormalizability
Given the tremendous success of QED, many theorists believed, in the few years after 1949, that QFT could soon provide an understanding of all microscopic phenomena, not only the interactions between photons, electrons, and positrons. Contrary to this optimism, QFT entered yet another period of depression that lasted for almost two decades.[3]: 30
The first obstacle was the limited applicability of the renormalization procedure. In perturbative calculations in QED, all infinite quantities could be eliminated by redefining a small (finite) number of physical quantities (namely the mass and charge of the electron). Dyson proved in 1949 that this is only possible for a small class of theories called "renormalizable theories", of which QED is an example. However, most theories, including the Fermi theory of the weak interaction, are "non-renormalizable". Any perturbative calculation in these theories beyond the first order would result in infinities that could not be removed by redefining a finite number of physical quantities.[3]: 30
The second major problem stemmed from the limited validity of the Feynman diagram method, which is based on a series expansion in perturbation theory. In order for the series to converge and low-order calculations to be a good approximation, the coupling constant, in which the series is expanded, must be a sufficiently small number. The coupling constant in QED is the fine-structure constant α ≈ 1/137, which is small enough that only the simplest, lowest order, Feynman diagrams need to be considered in realistic calculations. In contrast, the coupling constant in the strong interaction is roughly of the order of one, making complicated, higher order, Feynman diagrams just as important as simple ones. There was thus no way of deriving reliable quantitative predictions for the strong interaction using perturbative QFT methods.[3]: 31
With these difficulties looming, many theorists began to turn away from QFT. Some focused on symmetry principles and conservation laws, while others picked up the old S-matrix theory of Wheeler and Heisenberg. QFT was used heuristically as guiding principles, but not as a basis for quantitative calculations.[3]: 31
Source theory
Schwinger, however, took a different route. For more than a decade he and his students had been nearly the only exponents of field theory,[10] but in 1951[11][12] he found a way around the problem of the infinities with a new method using external sources as currents coupled to gauge fields.[13] Motivated by the former findings, Schwinger kept pursuing this approach in order to "quantumly" generalize the classical process of coupling external forces to the configuration space parameters known as Lagrange multipliers. He summarized his source theory in 1966[14] then expanded the theory's applications to quantum electrodynamics in his three volume-set titled: Particles, Sources, and Fields.[15][16][17] Developments in pion physics, in which the new viewpoint was most successfully applied, convinced him of the great advantages of mathematical simplicity and conceptual clarity that its use bestowed.[15]
In source theory there are no divergences, and no renormalization. It may be regarded as the calculational tool of field theory, but it is more general.[18] Using source theory, Schwinger was able to calculate the anomalous magnetic moment of the electron, which he had done in 1947, but this time with no ‘distracting remarks’ about infinite quantities.[19]
Schwinger also applied source theory to his QFT theory of gravity, and was able to reproduce all four of Einstein's classic results: gravitational red shift, deflection and slowing of light by gravity, and the perihelion precession of Mercury.[20] The neglect of source theory by the physics community was a major disappointment for Schwinger:
The lack of appreciation of these facts by others was depressing, but understandable. -J. Schwinger[15]
See "the shoes incident" between J. Schwinger and S. Weinberg.[21]
Standard-Model
In 1954, Yang Chen-Ning and Robert Mills generalized the local symmetry of QED, leading to non-Abelian gauge theories (also known as Yang–Mills theories), which are based on more complicated local symmetry groups.[22]: 5 In QED, (electrically) charged particles interact via the exchange of photons, while in non-Abelian gauge theory, particles carrying a new type of "charge" interact via the exchange of massless gauge bosons. Unlike photons, these gauge bosons themselves carry charge.[3]: 32 [23]
Sheldon Glashow developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in 1960. In 1964, Abdus Salam and John Clive Ward arrived at the same theory through a different path. This theory, nevertheless, was non-renormalizable.[24]
By combining the earlier theory of Glashow, Salam, and Ward with the idea of spontaneous symmetry breaking, Steven Weinberg wrote down in 1967 a theory describing electroweak interactions between all leptons and the effects of the Higgs boson. His theory was at first mostly ignored,[24][22]: 6 until it was brought back to light in 1971 by Gerard 't Hooft's proof that non-Abelian gauge theories are renormalizable. The electroweak theory of Weinberg and Salam was extended from leptons to quarks in 1970 by Glashow, John Iliopoulos, and Luciano Maiani, marking its completion.[24]
Harald Fritzsch, Murray Gell-Mann, and Heinrich Leutwyler discovered in 1971 that certain phenomena involving the strong interaction could also be explained by non-Abelian gauge theory. Quantum chromodynamics (QCD) was born. In 1973, David Gross, Frank Wilczek, and Hugh David Politzer showed that non-Abelian gauge theories are "asymptotically free", meaning that under renormalization, the coupling constant of the strong interaction decreases as the interaction energy increases. (Similar discoveries had been made numerous times previously, but they had been largely ignored.) [22]: 11 Therefore, at least in high-energy interactions, the coupling constant in QCD becomes sufficiently small to warrant a perturbative series expansion, making quantitative predictions for the strong interaction possible.[3]: 32
These theoretical breakthroughs brought about a renaissance in QFT. The full theory, which includes the electroweak theory and chromodynamics, is referred to today as the Standard Model of elementary particles.[25] The Standard Model successfully describes all fundamental interactions except gravity, and its many predictions have been met with remarkable experimental confirmation in subsequent decades.[8]: 3 The Higgs boson, central to the mechanism of spontaneous symmetry breaking, was finally detected in 2012 at CERN, marking the complete verification of the existence of all constituents of the Standard Model.[26]
Other developments
The 1970s saw the development of non-perturbative methods in non-Abelian gauge theories. The 't Hooft–Polyakov monopole was discovered theoretically by 't Hooft and Alexander Polyakov, flux tubes by Holger Bech Nielsen and Poul Olesen, and instantons by Polyakov and coauthors. These objects are inaccessible through perturbation theory.[8]: 4
Supersymmetry also appeared in the same period. The first supersymmetric QFT in four dimensions was built by Yuri Golfand and Evgeny Likhtman in 1970, but their result failed to garner widespread interest due to the Iron Curtain. Supersymmetry only took off in the theoretical community after the work of Julius Wess and Bruno Zumino in 1973.[8]: 7
Among the four fundamental interactions, gravity remains the only one that lacks a consistent QFT description. Various attempts at a theory of quantum gravity led to the development of string theory,[8]: 6 itself a type of two-dimensional QFT with conformal symmetry.[27] Joël Scherk and John Schwarz first proposed in 1974 that string theory could be the quantum theory of gravity.[28]
Condensed-matter-physics
Although quantum field theory arose from the study of interactions between elementary particles, it has been successfully applied to other physical systems, particularly to
Historically, the Higgs mechanism of spontaneous symmetry breaking was a result of
Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first quasiparticle—phonons. Lev Landau claimed that low-energy excitations in many condensed matter systems could be described in terms of interactions between a set of quasiparticles. The Feynman diagram method of QFT was naturally well suited to the analysis of various phenomena in condensed matter systems.[30]
Gauge theory is used to describe the quantization of
Principles
For simplicity,
Classical fields
A classical
Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the photoelectric effect are best explained by discrete particles (photons), rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields.
Canonical quantization and path integrals are two common formulations of QFT.[33]: 61 To motivate the fundamentals of QFT, an overview of classical field theory follows.
The simplest classical field is a real scalar field — a real number at every point in space that changes in time. It is denoted as ϕ(x, t), where x is the position vector, and t is the time. Suppose the Lagrangian of the field, , is
where is the Lagrangian density, is the time-derivative of the field, ∇ is the gradient operator, and m is a real parameter (the "mass" of the field). Applying the Euler–Lagrange equation on the Lagrangian:[1]: 16
we obtain the equations of motion for the field, which describe the way it varies in time and space:
This is known as the Klein–Gordon equation.[1]: 17
The Klein–Gordon equation is a wave equation, so its solutions can be expressed as a sum of normal modes (obtained via Fourier transform) as follows:
where a is a
Thus each normal mode corresponding to a single p can be seen as a classical harmonic oscillator with frequency ωp.[1]: 21,26
Canonical quantization
The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator.
The displacement of a classical harmonic oscillator is described by
where a is a complex number (normalized by convention), and ω is the oscillator's frequency. Note that x is the displacement of a particle in simple harmonic motion from the equilibrium position, not to be confused with the spatial label x of a quantum field.
For a quantum harmonic oscillator, x(t) is promoted to a
Complex numbers a and a* are replaced by the
The Hamiltonian of the simple harmonic oscillator can be written as
The
and has energy One can easily check that which implies that increases the energy of the simple harmonic oscillator by . For example, the state is an eigenstate of energy . Any energy eigenstate state of a single harmonic oscillator can be obtained from by successively applying the creation operator :[1]: 20 and any state of the system can be expressed as a linear combination of the states
A similar procedure can be applied to the real scalar field ϕ, by promoting it to a quantum field operator , while the annihilation operator , the creation operator and the angular frequency are now for a particular p:
Their commutation relations are:[1]: 21
where δ is the Dirac delta function. The vacuum state is defined by
Any quantum state of the field can be obtained from by successively applying creation operators (or by a linear combination of such states), e.g. [1]: 22
While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a Fock space, which can account for the fact that particle numbers are not fixed in relativistic quantum systems.[34] The process of quantizing an arbitrary number of particles instead of a single particle is often also called second quantization.[1]: 19
The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields,
The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a quartic interaction term could be introduced to the Lagrangian of the real scalar field:[1]: 77
where μ is a spacetime index, , etc. The summation over the index μ has been omitted following the Einstein notation. If the parameter λ is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.
Path integrals
The path integral formulation of QFT is concerned with the direct computation of the scattering amplitude of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the probability amplitude for a system to evolve from some initial state at time t = 0 to some final state at t = T, the total time T is divided into N small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let H be the
Taking the limit N → ∞, the above product of integrals becomes the Feynman path integral:[1]: 282 [33]: 12
where L is the Lagrangian involving ϕ and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian H via Legendre transformation. The initial and final conditions of the path integral are respectively
In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.
Two-point correlation function
In calculations, one often encounters expression like
The free two-point function, also known as the
In an interacting theory, where the Lagrangian or Hamiltonian contains terms or that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the free two-point function.
In canonical quantization, the two-point correlation function can be written as:[1]: 87
where ε is an infinitesimal number and ϕI is the field operator under the free theory. Here, the exponential should be understood as its power series expansion. For example, in -theory, the interacting term of the Hamiltonian is ,[1]: 84 and the expansion of the two-point correlator in terms of becomes
In the path integral formulation, the two-point correlation function can be written[1]: 284
where is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in λ, reducing the interacting two-point function to quantities in the free theory.
Wick's theorem further reduce any n-point correlation function in the free theory to a sum of products of two-point correlation functions. For example,
Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory.[1]: 90 This makes the Feynman propagator one of the most important quantities in quantum field theory.
Feynman diagram
Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a Feynman diagram. For example, the λ1 term in the two-point correlation function in the ϕ4 theory is
After applying Wick's theorem, one of the terms is
This term can instead be obtained from the Feynman diagram
The diagram consists of
- external vertices connected with one edge and represented by dots (here labeled and ).
- internal vertices connected with four edges and represented by dots (here labeled ).
- edges connecting the vertices and represented by lines.
Every vertex corresponds to a single field factor at the corresponding point in spacetime, while the edges correspond to the propagators between the spacetime points. The term in the perturbation series corresponding to the diagram is obtained by writing down the expression that follows from the so-called Feynman rules:
- For every internal vertex , write down a factor .
- For every edge that connects two vertices and , write down a factor .
- Divide by the symmetry factor of the diagram.
With the symmetry factor , following these rules yields exactly the expression above. By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space.[1]: 91–94
In order to compute the n-point correlation function to the k-th order, list all valid Feynman diagrams with n external points and k or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise,
is equal to the sum of (expressions corresponding to) all connected diagrams with n external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the ϕ4 interaction theory discussed above, every vertex must have four legs.[1]: 98
In realistic applications, the scattering amplitude of a certain interaction or the
Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing n loops are referred to as n-loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction.[33]: 44 Lines whose end points are vertices can be thought of as the propagation of virtual particles.[1]: 31
Renormalization
Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The
Parameters appearing in the Lagrangian, such as the mass m and the coupling constant λ, have no physical meaning — m, λ, and the field strength ϕ are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off Λ, obtain expressions for the physical quantities, and then take the limit Λ → ∞. This is an example of regularization, a class of methods to treat divergences in QFT, with Λ being the regulator.
The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of ϕ4 theory, the field strength is first redefined:
where ϕ is the bare field, ϕr is the renormalized field, and Z is a constant to be determined. The Lagrangian density becomes:
where mr and λr are the experimentally measurable, renormalized, mass and coupling constant, respectively, and
are constants to be determined. The first three terms are the ϕ4 Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularization scheme (such as the cut-off regularization introduced above or dimensional regularization); call the regulator Λ. Compute Feynman diagrams, in which divergent terms will depend on Λ. Then, define δZ, δm, and δλ such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit Λ → ∞ is taken. In this way, meaningful finite quantities are obtained.[1]: 323–326
It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The Standard Model of elementary particles is a renormalizable QFT,[1]: 719–727 while quantum gravity is non-renormalizable.[1]: 798 [33]: 421
Renormalization group
The renormalization group, developed by Kenneth Wilson, is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales.[1]: 393 The way in which each parameter changes with scale is described by its β function.[1]: 417 Correlation functions, which underlie quantitative physical predictions, change with scale according to the Callan–Symanzik equation.[1]: 410–411
As an example, the coupling constant in QED, namely the elementary charge e, has the following β function:
where Λ is the energy scale under which the measurement of e is performed. This differential equation implies that the observed elementary charge increases as the scale increases.[36] The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant.[1]: 420
The coupling constant g in quantum chromodynamics, a non-Abelian gauge theory based on the symmetry group SU(3), has the following β function:
where Nf is the number of quark flavours. In the case where Nf ≤ 16 (the Standard Model has Nf = 6), the coupling constant g decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as asymptotic freedom.[1]: 531
According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off Λ, i.e. that the theory is no longer valid at energies higher than Λ, and all degrees of freedom above the scale Λ are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable effective field theory.[1]: 402–403 The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them.[8]: 2 According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off Λ from calculations in such a theory merely indicates that new physical phenomena appear at scales above Λ, where a new theory is necessary.[33]: 156
Other theories
The quantization and renormalization procedures outlined in the preceding sections are performed for the free theory and
As an example, quantum electrodynamics contains a Dirac field ψ representing the electron field and a vector field Aμ representing the electromagnetic field (photon field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical electromagnetic four-potential, rather than the classical electric and magnetic fields.) The full QED Lagrangian density is:
where γμ are
Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an
Gauge symmetry
If the following transformation to the fields is performed at every spacetime point x (a local transformation), then the QED Lagrangian remains unchanged, or invariant:
where α(x) is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the
at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations and is yet another symmetry transformation . For any α(x), is an element of theU(1) is an
where Dμ is the gauge covariant derivative:
where g is the coupling constant, ta are the eight generators of SU(3) in the fundamental representation (3×3 matrices),
and fabc are the structure constants of SU(3). Repeated indices i,j,a are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation:
where U(x) is an element of SU(3) at every spacetime point x:
The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called anomaly. For instance, in the path integral formulation, despite the invariance of the Lagrangian density under a certain local transformation of the fields, the measure of the path integral may change.[33]: 243 For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group SU(3) × SU(2) × U(1), in which all anomalies exactly cancel.[1]: 705–707
The theoretical foundation of general relativity, the equivalence principle, can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the Lorentz group.[39]
Noether's theorem states that every continuous symmetry, i.e. the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding conservation law.[1]: 17–18 [33]: 73 For example, the U(1) symmetry of QED implies charge conservation.[40]
Gauge-transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field Aμ, being a four-vector, has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the polarization. The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing Aμ can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description.[33]: 168
To account for the gauge redundancy in the path integral formulation, one must perform the so-called Faddeev–Popov gauge fixing procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally.[1]: 512–515 A more rigorous generalization of the Faddeev–Popov procedure is given by BRST quantization.[1]: 517
Spontaneous symmetry-breaking
Spontaneous symmetry breaking is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it.[1]: 347
To illustrate the mechanism, consider a linear sigma model containing N real scalar fields, described by the Lagrangian density:
where μ and λ are real parameters. The theory admits an O(N) global symmetry:
The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field ϕ0 satisfying
Without loss of generality, let the ground state be in the N-th direction:
The original N fields can be rewritten as:
and the original Lagrangian density as:
where k = 1, ..., N − 1. The original O(N) global symmetry is no longer manifest, leaving only the subgroup O(N − 1). The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken.[1]: 349–350
On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson.[1]: 743–744
In the QFT of ferromagnetism, spontaneous symmetry breaking can explain the alignment of magnetic dipoles at low temperatures.[33]: 199 In the Standard Model of elementary particles, the W and Z bosons, which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the Higgs boson, a process called the Higgs mechanism.[1]: 690
Supersymmetry
All experimentally known symmetries in nature relate bosons to bosons and fermions to fermions. Theorists have hypothesized the existence of a type of symmetry, called supersymmetry, that relates bosons and fermions.[1]: 795 [33]: 443
The Standard Model obeys
The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group.[33]: 448 Examples of such theories include: Minimal Supersymmetric Standard Model (MSSM), N = 4 supersymmetric Yang–Mills theory,[33]: 450 and superstring theory. In a supersymmetric theory, every fermion has a bosonic superpartner and vice versa.[33]: 444
If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called supergravity.[44]
Supersymmetry is a potential solution to many current problems in physics. For example, the
Nevertheless, as of 2018[update], experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.[1]: 797 [33]: 443
Other spacetimes
The ϕ4 theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional Minkowski space (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT a priori imposes no restriction on the number of dimensions nor the geometry of spacetime.
In
In Minkowski space, the flat metric ημν is used to raise and lower spacetime indices in the Lagrangian, e.g.
where ημν is the inverse of ημν satisfying ημρηρν = δμν. For QFTs in curved spacetime on the other hand, a general metric (such as the Schwarzschild metric describing a black hole) is used:
where gμν is the inverse of gμν. For a real scalar field, the Lagrangian density in a general spacetime background is
where g = det(gμν), and ∇μ denotes the covariant derivative.[47] The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background.
Topological quantum field theory
The correlation functions and physical predictions of a QFT depend on the spacetime metric gμν. For a special class of QFTs called
Perturbative and non-perturbative methods
Using
Mathematical rigor
In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to Haag's theorem, there does not exist a well-defined interaction picture for QFT, which implies that perturbation theory of QFT, which underlies the entire Feynman diagram method, is fundamentally ill-defined.[55]
However, perturbative quantum field theory, which only requires that quantities be computable as a formal power series without any convergence requirements, can be given a rigorous mathematical treatment. In particular,
Since the 1950s,
Compared to ordinary QFT, topological quantum field theory and conformal field theory are better supported mathematically — both can be classified in the framework of representations of cobordisms.[63]
Yang–Mills existence and mass gap, one of the Millennium Prize Problems, concerns the well-defined existence of Yang–Mills theories as set out by the above axioms. The full problem statement is as follows.[64]
Prove that for any
gauge groupG, a non-trivial quantum Yang–Mills theory exists on and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).
See also
- Abraham–Lorentz force
- AdS/CFT correspondence
- Axiomatic quantum field theory
- Introduction to quantum mechanics
- Common integrals in quantum field theory
- Conformal field theory
- Constructive quantum field theory
- Dirac's equation
- Form factor (quantum field theory)
- Feynman diagram
- Green–Kubo relations
- Green's function (many-body theory)
- Group field theory
- Lattice field theory
- List of quantum field theories
- Local quantum field theory
- Noncommutative quantum field theory
- Quantization of a field
- Quantum electrodynamics
- Quantum field theory in curved spacetime
- Quantum chromodynamics
- Quantum flavordynamics
- Quantum hadrodynamics
- Quantum hydrodynamics
- 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
- Topological quantum field theory
- Ward–Takahashi identity
- Wheeler–Feynman absorber theory
- Wigner's classification
- Wigner's theorem
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Further reading
- General readers
- ISBN 978-0198519973.
- ISBN 9780691033273.
- ISBN 978-0-262-56003-0.
- Feynman, R.P. (2006) [1985]. QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 978-0-691-12575-6.
- ISBN 978-0-297-81752-9.
- Introductory texts
- McMahon, D. (2008). Quantum Field Theory. ISBN 978-0-07-154382-8.
- ISBN 978-0-8053-0983-6.
- ISBN 3527408355.
- ISBN 978-3-540-67672-0.
- ISBN 978-0-07-032071-0.
- ISBN 978-0-201-11749-3.
- ISBN 978-981-02-4658-7.
- Kleinert, H. (2008). Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation (PDF). World Scientific. ISBN 978-981-279-170-2.
- Lancaster, T., & Blundell, S. J. (2014). Quantum field theory for the gifted amateur. OUP Oxford. ISBN 9780199699339
- ISBN 978-0-19-851155-7.
- ISBN 978-0-471-94186-6.
- ISBN 978-0-521-33859-2.
- Schwartz, M.D. (2014). Quantum Field Theory and the Standard Model. Cambridge University Press. ISBN 978-1107034730. Archived from the originalon 2018-03-22. Retrieved 2020-05-13.
- ISBN 978-3-540-60453-2.
- ISBN 978-3-540-59179-5.
- ISBN 978-0-201-50397-5.
- Scharf, Günter (2014) [1989]. Finite Quantum Electrodynamics: The Causal Approach (third ed.). Dover Publications. ISBN 978-0486492735.
- Srednicki, M. (2007). Quantum Field Theory. ISBN 978-0521-8644-97.
- Tong, David (2015). "Lectures on Quantum Field Theory". Retrieved 2016-02-09.
- Williams, A.G. (2022). Introduction to Quantum Field Theory: Classical Mechanics to Gauge Field Theories. ISBN 978-1108470902.
- ISBN 978-0691140346.
- Advanced texts
- ISBN 978-0-521-46946-3.
- Bogoliubov, N.; ISBN 978-0-7923-0540-8.
- ISBN 978-0521550017.
External links
- Media related to Quantum field theory at Wikimedia Commons
- "Quantum field theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Stanford Encyclopedia of Philosophy: "Quantum Field Theory", by Meinard Kuhlmann.
- Siegel, Warren, 2005. Fields. arXiv:hep-th/9912205.
- Quantum Field Theory by P. J. Mulders