History Of Conformal Field Theory
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In
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, the history of quantum field theory starts with its creation by
Paul Dirac Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for bot ...
, when he attempted to quantize the
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
in the late 1920s. Major advances in the theory were made in the 1940s and 1950s, leading to the introduction of renormalized
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
(QED). The field theory behind QED was so accurate and successful in predictions that efforts were made to apply the same basic concepts for the other forces of nature. Beginning in 1954, the parallel was found by way of
gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
, leading by the late 1970s, to quantum field models of
strong nuclear force In nuclear physics and particle physics, the strong interaction, also called the strong force or strong nuclear force, is one of the four known fundamental interactions. It confines quarks into protons, neutrons, and other hadron particles, an ...
and
weak nuclear force In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is th ...
, united in the modern
Standard Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
of
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
. Efforts to describe
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
using the same techniques have, to date, failed. The study of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
is still flourishing, as are applications of its methods to many physical problems. It remains one of the most vital areas of
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
today, providing a common language to several different branches of
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
.


Early developments

Quantum field theory originated in the 1920s from the problem of creating a quantum mechanical theory of the
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
. In particular, de Broglie in 1924 introduced the idea of a wave description of elementary systems in the following way: "we proceed in this work from the assumption of the existence of a certain periodic phenomenon of a yet to be determined character, which is to be attributed to each and every isolated energy parcel". In 1925,
Werner Heisenberg Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist, one of the main pioneers of the theory of quantum mechanics and a principal scientist in the German nuclear program during World War II. He pub ...
,
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German-British theoretical physicist who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics, and supervised the work of a ...
, and
Pascual Jordan Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matri ...
constructed just such a theory by expressing the field's internal
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
as an infinite set of
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive const ...
s, and by then utilizing the
canonical quantization In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory to the greatest extent possible. Historically, this was not quit ...
procedure to these oscillators; their paper was published in 1926. This theory assumed that no electric charges or currents were present and today would be called a
free field theory In physics a free field is a field without interactions, which is described by the terms of motion and mass. Description In classical physics, a free field is a field whose equations of motion are given by linear partial differential equat ...
. The first reasonably complete theory of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
, which included both the electromagnetic field and electrically charged matter as quantum mechanical objects, was created by
Paul Dirac Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for bot ...
in 1927. This quantum field theory could be used to model important processes such as the emission of a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
by an electron dropping into a
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
of lower energy, a process in which the ''number of particles changes''—one atom in the initial state becomes an atom plus a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
in the final state. It is now understood that the ability to describe such processes is one of the most important features of quantum field theory. The final crucial step was
Enrico Fermi Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian and naturalized American physicist, renowned for being the creator of the world's first artificial nuclear reactor, the Chicago Pile-1, and a member of the Manhattan Project ...
's theory of β-decay (1934). In it, fermion species nonconservation was shown to follow from second quantization: creation and annihilation of fermions came to the fore and quantum field theory was seen to describe particle decays. (Fermi's breakthrough was somewhat foreshadowed in the abstract studies of Soviet physicists,
Viktor Ambartsumian Viktor Amazaspovich Ambartsumian (; , ''Viktor Hamazaspi Hambardzumyan''; 12 August 1996) was a Soviet and Armenian astrophysicist and science administrator. One of the 20th century's leading astronomers, he is widely regarded as the founder of ...
and
Dmitri Ivanenko Dmitri Dmitrievich Ivanenko (, ; July 29, 1904 – December 30, 1994) was a Soviet theoretical physicist of Ukrainian origin who made great contributions to the physical science of the twentieth century, especially to nuclear physics, field theo ...
, in particular the Ambarzumian–Ivanenko hypothesis of creation of massive particles (1930). The idea was that not only the quanta of the electromagnetic field, photons, but also other particles might emerge and disappear as a result of their interaction with other particles.)


Incorporating special relativity

It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate Einstein's
relativity theory The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phe ...
, which had grown out of the study of
classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of physics focused on the study of interactions between electric charges and electrical current, currents using an extension of the classical Newtonian model. It is, therefore, a ...
. This need to put together relativity and quantum mechanics was the second major motivation in the development of quantum field theory.
Pascual Jordan Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matri ...
and
Wolfgang Pauli Wolfgang Ernst Pauli ( ; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and a pioneer of quantum mechanics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics "for the ...
showed in 1928 that quantum fields could be made to behave in the way predicted by
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
during
coordinate transformations In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
(specifically, they showed that the field
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
s were
Lorentz invariant In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. While ...
). A further boost for quantum field theory came with the discovery of the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
, which was originally formulated and interpreted as a single-particle equation analogous to the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
, but unlike the Schrödinger equation, the Dirac equation satisfies both the Lorentz invariance, that is, the requirements of special relativity, and the rules of quantum mechanics. The Dirac equation accommodated the spin-1/2 value of the electron and accounted for its magnetic moment as well as giving accurate predictions for the spectra of hydrogen. The attempted interpretation of the Dirac equation as a single-particle equation could not be maintained long, however, and finally it was shown that several of its undesirable properties (such as negative-energy states) could be made sense of by reformulating and reinterpreting the Dirac equation as a true field equation, in this case for the quantized "Dirac field" or the "electron field", with the "negative-energy solutions" pointing to the existence of anti-particles. This work was performed first by Dirac himself with the invention of hole theory in 1930 and by Wendell Furry,
Robert Oppenheimer J. Robert Oppenheimer (born Julius Robert Oppenheimer ; April 22, 1904 – February 18, 1967) was an American theoretical physicist who served as the director of the Manhattan Project's Los Alamos Laboratory during World War II. He is often ...
,
Vladimir Fock Vladimir Aleksandrovich Fock (or Fok; ) (December 22, 1898 – December 27, 1974) was a Soviet physicist, who did foundational work on quantum mechanics and quantum electrodynamics. Biography He was born in St. Petersburg, Russia. In  ...
, and others.
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger ( ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was an Austrian-Irish theoretical physicist who developed fundamental results in quantum field theory, quantum theory. In particul ...
, during the same period that he discovered his equation in 1926, also independently found the relativistic generalization of it known as the
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is named after Oskar Klein and Walter Gordon. It is second-order i ...
but dismissed it since, without
spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...
, it predicted impossible properties for the hydrogen spectrum. (See
Oskar Klein Oskar Benjamin Klein (; 15 September 1894 – 5 February 1977) was a Swedish theoretical physics, theoretical physicist. Oskar Klein is known for his work on Kaluza–Klein theory, which is partially named after him. Biography Klein was born ...
and Walter Gordon.) All relativistic wave equations that describe spin-zero particles are said to be of the Klein–Gordon type.


Uncertainty, again

A subtle and careful analysis in 1933 by
Niels Bohr Niels Henrik David Bohr (, ; ; 7 October 1885 – 18 November 1962) was a Danish theoretical physicist who made foundational contributions to understanding atomic structure and old quantum theory, quantum theory, for which he received the No ...
and
Léon Rosenfeld Léon Rosenfeld (; 14 August 1904 in Charleroi – 23 March 1974) was a Belgian physicist and a communist activist. Rosenfeld was born into a secular Jewish family. He was a polyglot who knew eight or nine languages and was fluent in at lea ...
showed that there is a fundamental limitation on the ability to simultaneously measure the electric and magnetic field strengths that enter into the description of charges in interaction with radiation, imposed by the
uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
, which must apply to all canonically conjugate quantities. This limitation is crucial for the successful formulation and interpretation of a quantum field theory of photons and electrons (quantum electrodynamics), and indeed, any
perturbative In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which ...
quantum field theory. The analysis of Bohr and Rosenfeld explains fluctuations in the values of the electromagnetic field that differ from the classically "allowed" values distant from the sources of the field. Their analysis was crucial to showing that the limitations and physical implications of the uncertainty principle apply to all dynamical systems, whether fields or material particles. Their analysis also convinced most physicists that any notion of returning to a fundamental description of nature based on classical field theory, such as what Einstein aimed at with his numerous and failed attempts at a classical
unified field theory In physics, a Unified Field Theory (UFT) or “Theory of Everything” is a type of field theory that allows all fundamental forces of nature, including gravity, and all elementary particles to be written in terms of a single physical field. Ac ...
, was simply out of the question. ''Fields had to be quantized''.


Second quantization

The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease. In 1927,
Pascual Jordan Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matri ...
tried to extend the canonical quantization of fields to the many-body wave functions of
identical particles In quantum mechanics, indistinguishable particles (also called identical or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
using a formalism which is known as statistical transformation theory; this procedure is now sometimes called
second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
. Dirac is also credited with the invention, as he introduced the key ideas in a 1927 paper. In 1928, Jordan and
Eugene Wigner Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...
found that the quantum field describing electrons, or other
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s, had to be expanded using anti-commuting creation and annihilation operators due to the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle (German: Pauli-Ausschlussprinzip) states that two or more identical particles with half-integer spins (i.e. fermions) cannot simultaneously occupy the same quantum state within a system that o ...
(see
Jordan–Wigner transformation The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators. It was proposed by Pascual Jordan and Eugene Wigner for one-dimensional lattice models, but now two-dimensional a ...
). This thread of development was incorporated into
many-body theory The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. Terminology ''Microscopic'' here implies that quantum mechanics has to be ...
and strongly influenced
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
and
nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies th ...
.


The problem of infinities

Despite its early successes quantum field theory was plagued by several serious theoretical difficulties. Basic physical quantities, such as the self-energy of the electron, the energy shift of electron states due to the presence of the electromagnetic field, gave infinite, divergent contributions—a nonsensical result—when computed using the perturbative techniques available in the 1930s and most of the 1940s. The electron self-energy problem was already a serious issue in the classical electromagnetic field theory, where the attempt to attribute to the electron a finite size or extent (the classical electron-radius) led immediately to the question of what non-electromagnetic stresses would need to be invoked, which would presumably hold the electron together against the Coulomb repulsion of its finite-sized "parts". The situation was dire, and had certain features that reminded many of the "
Rayleigh–Jeans catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century and early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of ene ...
". What made the situation in the 1940s so desperate and gloomy, however, was the fact that the correct ingredients (the second-quantized Maxwell–Dirac field equations) for the theoretical description of interacting photons and electrons were well in place, and no major conceptual change was needed analogous to that which was necessitated by a finite and physically sensible account of the radiative behavior of hot objects, as provided by the Planck radiation law.


Renormalization procedures

Improvements in
microwave Microwave is a form of electromagnetic radiation with wavelengths shorter than other radio waves but longer than infrared waves. Its wavelength ranges from about one meter to one millimeter, corresponding to frequency, frequencies between 300&n ...
technology made it possible to take more precise measurements of the shift of the levels of a
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral hydrogen atom contains a single positively charged proton in the nucleus, and a single negatively charged electron bound to the nucleus by the Coulomb for ...
, now known as the
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is an anomalous difference in energy between two electron orbitals in a hydrogen atom. The difference was not predicted by theory and it cannot be derived from the Dirac equation, which pre ...
and
magnetic moment In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
of the electron. These experiments exposed discrepancies which the theory was unable to explain. A first indication of a possible way out was given by
Hans Bethe Hans Albrecht Eduard Bethe (; ; July 2, 1906 – March 6, 2005) was a German-American physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics and solid-state physics, and received the Nobel Prize in Physi ...
in 1947, after attending the
Shelter Island Conference The first Shelter Island Conference on the foundations of quantum mechanics was held from June 2–4, 1947 at the Ram's Head Inn in Shelter Island, New York. Shelter Island was the first major opportunity since Pearl Harbor and the Manhattan P ...
. While he was traveling by train from the conference to
Schenectady Schenectady ( ) is a City (New York), city in Schenectady County, New York, United States, of which it is the county seat. As of the United States Census 2020, 2020 census, the city's population of 67,047 made it the state's ninth-most populo ...
he made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by Lamb and Retherford. Despite the limitations of the computation, agreement was excellent. The idea was simply to attach infinities to corrections of
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
and
charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...
that were actually fixed to a finite value by experiments. In this way, the infinities get absorbed in those constants and yield a finite result in good agreement with experiments. This procedure was named
renormalization Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
. This "divergence problem" was solved in the case of quantum electrodynamics through the procedure known as
renormalization Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
in 1947–49 by
Hans Kramers Hendrik Anthony "Hans" Kramers (17 December 1894 – 24 April 1952) was a Dutch physicist who worked with Niels Bohr to understand how electromagnetic waves interact with matter and made important contributions to quantum mechanics and statistica ...
,
Hans Bethe Hans Albrecht Eduard Bethe (; ; July 2, 1906 – March 6, 2005) was a German-American physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics and solid-state physics, and received the Nobel Prize in Physi ...
,
Julian Schwinger Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize-winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
,
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...
, and Shin'ichiro Tomonaga; the procedure was systematized by
Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrix, random matrices, math ...
in 1949. Great progress was made after realizing that all infinities in quantum electrodynamics are related to two effects: the self-energy of the electron/positron, and vacuum polarization. Renormalization requires paying very careful attention to just what is meant by, for example, the very concepts "charge" and "mass" as they occur in the pure, non-interacting field-equations. The "vacuum" is itself polarizable and, hence, populated by
virtual particle A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle, which allows the virtual particles to spontaneously emer ...
(
on shell and off shell In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called on the mass shell (on shell); while those that do not are called off the mass shell (off shell). In quantu ...
) pairs, and, hence, is a seething and busy dynamical system in its own right. This was a critical step in identifying the source of "infinities" and "divergences". The "bare mass" and the "bare charge" of a particle, the values that appear in the free-field equations (non-interacting case), are abstractions that are simply not realized in experiment (in interaction). What we measure, and hence, what we must take account of with our equations, and what the solutions must account for, are the "renormalized mass" and the "renormalized charge" of a particle. That is to say, the "shifted" or "dressed" values these quantities must have when due systematic care is taken to include all deviations from their "bare values" is dictated by the very nature of quantum fields themselves.


Quantum electrodynamics

The first approach that bore fruit is known as the "interaction representation" (see the article
Interaction picture In quantum mechanics, the interaction picture (also known as the interaction representation or Dirac picture after Paul Dirac, who introduced it) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whe ...
), a Lorentz-covariant and gauge-invariant generalization of time-dependent perturbation theory used in ordinary quantum mechanics, and developed by Tomonaga and Schwinger, generalizing earlier efforts of Dirac, Fock and
Boris Podolsky Boris Yakovlevich Podolsky (; June 29, 1896 – November 28, 1966) was a Russian-American physicist of Jewish descent, noted for his work with Albert Einstein and Nathan Rosen on entangled wave functions and the EPR paradox. Education In 1 ...
. Tomonaga and Schwinger invented a relativistically covariant scheme for representing field commutators and field operators intermediate between the two main representations of a quantum system, the Schrödinger and the Heisenberg representations. Within this scheme, field commutators at separated points can be evaluated in terms of "bare" field creation and annihilation operators. This allows for keeping track of the time-evolution of both the "bare" and "renormalized", or perturbed, values of the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
and expresses everything in terms of the coupled, gauge invariant "bare" field-equations. Schwinger gave the most elegant formulation of this approach. The next development was due to
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...
, with his rules for assigning a graph to the terms in the scattering matrix (see
S-matrix In physics, the ''S''-matrix or scattering matrix is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...
and
Feynman diagrams In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
). These directly corresponded (through the Schwinger–Dyson equation) to the measurable physical processes (cross sections, probability amplitudes, decay widths and lifetimes of excited states) one needs to be able to calculate. This revolutionized how quantum field theory calculations are carried out in practice. Two classic text-books from the 1960s, James D. Bjorken, Sidney David Drell, ''Relativistic Quantum Mechanics'' (1964) and
J. J. Sakurai was a Japanese–American particle physicist and theorist. While a graduate student at Cornell University, Sakurai independently discovered the V-A theory of weak interactions. He authored the popular graduate text '' Modern Quantum Mechanics ...
, ''Advanced Quantum Mechanics'' (1967), thoroughly developed the Feynman graph expansion techniques using physically intuitive and practical methods following from the
correspondence principle In physics, a correspondence principle is any one of several premises or assertions about the relationship between classical and quantum mechanics. The physicist Niels Bohr coined the term in 1920 during the early development of quantum theory; ...
, without worrying about the technicalities involved in deriving the Feynman rules from the superstructure of quantum field theory itself. Although both Feynman's heuristic and pictorial style of dealing with the infinities, as well as the formal methods of Tomonaga and Schwinger, worked extremely well, and gave spectacularly accurate answers, the true analytical nature of the question of "renormalizability", that is, whether ANY theory formulated as a "quantum field theory" would give finite answers, was not worked-out until much later, when the urgency of trying to formulate finite theories for the strong and electro-weak (and gravitational) interactions demanded its solution. Renormalization in the case of QED was largely fortuitous due to the smallness of the coupling constant, the fact that the coupling has no dimensions involving mass, the so-called
fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...
, and also the zero-mass of the gauge boson involved, the photon, rendered the small-distance/high-energy behavior of QED manageable. Also, electromagnetic processes are very "clean" in the sense that they are not badly suppressed/damped and/or hidden by the other gauge interactions. By 1965 James D. Bjorken and Sidney David Drell observed: "Quantum electrodynamics (QED) has achieved a status of peaceful coexistence with its divergences ...". The unification of the electromagnetic force with the weak force encountered initial difficulties due to the lack of accelerator energies high enough to reveal processes beyond the Fermi interaction range. Additionally, a satisfactory theoretical understanding of hadron substructure had to be developed, culminating in the
quark model In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks that give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Eig ...
. Thanks to the somewhat brute-force, ''ad hoc'' and heuristic early methods of Feynman, and the abstract methods of Tomonaga and Schwinger, elegantly synthesized by
Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrix, random matrices, math ...
, from the period of early renormalization, the modern theory of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
(QED) has established itself. It is still the most accurate physical theory known, the prototype of a successful quantum field theory. Quantum electrodynamics is an example of what is known as an abelian gauge theory. It relies on the symmetry group ''U''(1) and has one massless gauge field, the ''U''(1) gauge symmetry, dictating the form of the interactions involving the electromagnetic field, with the photon being the gauge boson.


Yang-Mills theory

In the 1950s
Yang Yang may refer to: * Yang, in yin and yang, one half of the two symbolic polarities in Chinese philosophy * Korean yang, former unit of currency of Korea from 1892 to 1902 * YANG, a data modeling language for the NETCONF network configuration p ...
and
Mills Mills is the plural form of mill, but may also refer to: As a name * Mills (surname), a common family name of English or Gaelic origin * Mills (given name) *Mills, a fictional British secret agent in a trilogy by writer Manning O'Brine Places U ...
, following the previous lead of
Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
, explored the impact of symmetries and invariances on field theory. All field theories, including QED, were generalized to a class of quantum field theories known as
gauge theories In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
. That symmetries dictate, limit and necessitate the form of interaction between particles is the essence of the "gauge theory revolution". Yang and Mills formulated the first explicit example of a non-abelian gauge theory,
Yang–Mills theory Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special un ...
, with an attempted explanation of the
strong interactions In nuclear physics and particle physics, the strong interaction, also called the strong force or strong nuclear force, is one of the four known fundamental interactions. It confines quarks into protons, neutrons, and other hadron particles, a ...
in mind. The strong interactions were then (incorrectly) understood in the mid-1950s, to be mediated by the pi-mesons, the particles predicted by
Hideki Yukawa Hideki Yukawa (; ; 23 January 1907 – 8 September 1981) was a Japanese theoretical physicist who received the Nobel Prize in Physics in 1949 "for his prediction of the existence of mesons on the basis of theoretical work on nuclear forces". B ...
in 1935, based on his profound reflections concerning the reciprocal connection between the mass of any force-mediating particle and the range of the force it mediates. This was allowed by the
uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
. In the absence of dynamical information,
Murray Gell-Mann Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American theoretical physicist who played a preeminent role in the development of the theory of elementary particles. Gell-Mann introduced the concept of quarks as the funda ...
pioneered the extraction of physical predictions from sheer non-abelian symmetry considerations, and introduced non-abelian Lie groups to
current algebra Certain commutation relations among the current density operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra. Mathematically these are Lie algebras consisting of smooth maps from a manifold into a ...
and so the gauge theories that came to supersede it. The 1960s and 1970s saw the formulation of a gauge theory now known as the
Standard Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
of
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, which systematically describes the elementary particles and the interactions between them. The strong interactions are described by
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the study of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of ...
(QCD), based on "color" ''SU''(3). The weak interactions require the additional feature of
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion o ...
, elucidated by
Yoichiro Nambu was a Japanese-American physicist and professor at the University of Chicago. Known for his groundbreaking contributions to theoretical physics, Nambu was the originator of the theory of spontaneous symmetry breaking, a concept that revoluti ...
and the adjunct
Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the Mass generation, generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles ...
, considered next.


Electroweak unification

The
electroweak interaction In particle physics, the electroweak interaction or electroweak force is the unified description of two of the fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction. Although these two force ...
part of the Standard Model was formulated by
Sheldon Glashow Sheldon Lee Glashow (, ; born December 5, 1932) is a Nobel Prize-winning American theoretical physicist. He is the Metcalf Professor of Mathematics and Physics at Boston University, and a Eugene Higgins Professor of Physics, emeritus, at Harv ...
,
Abdus Salam Mohammad Abdus Salam Salam adopted the forename "Mohammad" in 1974 in response to the anti-Ahmadiyya decrees in Pakistan, similarly he grew his beard. (; ; 29 January 192621 November 1996) was a Pakistani theoretical physicist. He shared the 1 ...
, and
John Clive Ward John Clive Ward, (1 August 1924 – 6 May 2000) was an Anglo-Australian physicist who made significant contributions to quantum field theory, condensed-matter physics, and statistical mechanics. Andrei Sakharov called Ward one of the titans o ...
in 1959, with their discovery of the SU(2)xU(1) group structure of the theory. In 1967,
Steven Weinberg Steven Weinberg (; May 3, 1933 – July 23, 2021) was an American theoretical physicist and Nobel laureate in physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic inter ...
invoked the
Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the Mass generation, generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles ...
for the generation of the W and Z masses (the intermediate
vector boson In particle physics, a vector boson is a boson whose spin equals one. Vector bosons that are also elementary particles are gauge bosons, the force carriers of fundamental interactions. Some composite particles are vector bosons, for instance any ...
s responsible for the weak interactions and neutral-currents) and keeping the mass of the photon zero. The Goldstone and Higgs idea for generating mass in gauge theories was sparked in the late 1950s and early 1960s when a number of theoreticians (including
Yoichiro Nambu was a Japanese-American physicist and professor at the University of Chicago. Known for his groundbreaking contributions to theoretical physics, Nambu was the originator of the theory of spontaneous symmetry breaking, a concept that revoluti ...
,
Steven Weinberg Steven Weinberg (; May 3, 1933 – July 23, 2021) was an American theoretical physicist and Nobel laureate in physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic inter ...
,
Jeffrey Goldstone Jeffrey Goldstone (born 3 September 1933) is a Great Britain, British theoretical physicist and an ''emeritus'' physics faculty member at the MIT MIT Center for Theoretical Physics, Center for Theoretical Physics. He worked at the University of ...
,
François Englert François, Baron Englert (; born 6 November 1932) is a Belgian theoretical physicist and 2013 Nobel Prize laureate. Englert is professor emeritus at the Université libre de Bruxelles (ULB), where he is a member of the Service de Physique Thé ...
,
Robert Brout Robert Brout (; June 14, 1928 – May 3, 2011) was an Belgian theoretical physicist who made contributions in elementary particle physics. He was a professor of physics at the Université libre de Bruxelles where he created, together with Franço ...
, G. S. Guralnik, C. R. Hagen,
Tom Kibble Sir Thomas Walter Bannerman Kibble (; 23 December 1932 – 2 June 2016) was a British theoretical physicist, senior research investigator at the Blackett Laboratory and Emeritus Professor of Theoretical Physics at Imperial College London. His ...
and
Philip Warren Anderson Philip Warren Anderson (December 13, 1923 – March 29, 2020) was an American theoretical physicist and Nobel laureate. Anderson made contributions to the theories of Anderson localization, localization, antiferromagnetism, symmetry breaking ( ...
) noticed a possibly useful analogy to the (spontaneous) breaking of the U(1) symmetry of electromagnetism in the formation of the
BCS BCS may refer to: American football * Bowl Championship Series, a system that selected matchups for major college football bowl games between 1998 and 2013 * BCS conferences, the six FBS conferences with automatic major bowl bids under that sys ...
ground-state of a superconductor. The gauge boson involved in this situation, the photon, behaves as though it has acquired a finite mass. There is a further possibility that the physical vacuum (ground-state) does not respect the symmetries implied by the "unbroken" electroweak
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
from which one arrives at the field equations (see the article
Electroweak interaction In particle physics, the electroweak interaction or electroweak force is the unified description of two of the fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction. Although these two force ...
for more details). The electroweak theory of Weinberg and Salam was shown to be
renormalizable Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
(finite) and hence consistent by
Gerardus 't Hooft Gerardus "Gerard" 't Hooft (; born July 5, 1946) is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G. Veltman "for elucidating t ...
and
Martinus Veltman Martinus Justinus Godefriedus "Tini" Veltman (; 27 June 1931 – 4 January 2021) was a Dutch theoretical physicist. He shared the 1999 Nobel Prize in Physics with his former PhD student Gerardus 't Hooft for their work on particle theory. Bio ...
. The Glashow–Weinberg–Salam theory (GWS theory), in certain applications, gives an accuracy on a par with quantum electrodynamics.


Quantum chromodynamics

In the case of the strong interactions, progress concerning their short-distance/high-energy behavior was much slower and more frustrating. For strong interactions with the electro-weak fields, there were difficult issues regarding the strength of coupling, the mass generation of the force carriers as well as their non-linear, self interactions. Although there has been theoretical progress toward a grand unified quantum field theory incorporating the electro-magnetic force, the weak force and the strong force, empirical verification is still pending. Superunification, incorporating the gravitational force, is still very speculative, and is under intensive investigation by many of the best minds in contemporary theoretical physics. Gravitation is a
tensor field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...
description of a spin-2 gauge-boson, the "
graviton In theories of quantum gravity, the graviton is the hypothetical elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with re ...
", and is further discussed in the articles on
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
and
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
.


Quantum gravity

From the point of view of the techniques of (four-dimensional) quantum field theory, and as the numerous efforts to formulate a consistent quantum gravity theory attests, gravitational quantization has been the reigning champion for bad behavior. There are technical problems underlain by the fact that the
Newtonian constant of gravitation The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's theory of general relativity. It is also known as t ...
has dimensions involving inverse powers of mass, and, as a simple consequence, it is plagued by perturbatively badly behaved non-linear self-interactions. Gravity is itself a source of gravity, analogously to gauge theories (whose couplings, are, by contrast, dimensionless) leading to uncontrollable divergences at increasing orders of perturbation theory. Moreover, gravity couples to all energy equally strongly, as per the
equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...
, so this makes the notion of ever really "switching-off", "cutting-off" or separating, the gravitational interaction from other interactions ambiguous, since, with gravitation, we are dealing with the very structure of space-time itself. Moreover, it has not been established that a theory of quantum gravity is necessary (see
Quantum field theory in curved spacetime In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory uses a semi-classical approach; it treats spacetime as a fixed ...
).


Contemporary framework of renormalization

Parallel breakthroughs in the understanding of
phase transition In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic Sta ...
s in
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
led to novel insights based on the
renormalization group In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying p ...
. They involved the work of
Leo Kadanoff Leo Philip Kadanoff (January 14, 1937 – October 26, 2015) was an American physicist. He was a professor of physics (emeritus from 2004) at the University of Chicago and a former president of the American Physical Society (APS). He contributed t ...
(1966) and Kenneth Geddes Wilson
Michael Fisher Michael Ellis Fisher (3 September 1931 – 26 November 2021) was an English physicist, as well as chemist and mathematician, known for his many seminal contributions to statistical physics, including but not restricted to the theory of phase t ...
(1972)—extending the work of
Ernst Stueckelberg Ernst Carl Gerlach Stueckelberg (baptised as Johann Melchior Ernst Karl Gerlach Stückelberg, full name after 1911: Baron Ernst Carl Gerlach Stueckelberg von Breidenbach zu Breidenstein und Melsbach; 1 February 1905 – 4 September 1984) was a S ...
André Petermann Andreas Emil Petermann (27 September 1922– 21 August 2011), known as André Petermann, was a Swiss theoretical physicist known for introducing the renormalization group, suggesting a quark-like model, and work related to the anomalous magnetic ...
(1953) and
Murray Gell-Mann Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American theoretical physicist who played a preeminent role in the development of the theory of elementary particles. Gell-Mann introduced the concept of quarks as the funda ...
Francis Low (1954)—which led to the seminal reformulation of quantum field theory by Kenneth Geddes Wilson in 1975. This reformulation provided insights into the evolution of effective field theories with scale, which classified all field theories,
renormalizable Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
or not. The remarkable conclusion is that, in general, most observables are "irrelevant", i.e., the macroscopic physics is ''dominated by only a few observables'' in most systems. During the same period, Leo Kadanoff (1969) introduced an
operator algebra In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study o ...
formalism for the two-dimensional
Ising model The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...
, a widely studied mathematical model of
ferromagnetism Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromagne ...
in
statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
. This development suggested that quantum field theory describes its
scaling limit In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model (physics), lattice model characterizes its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to a ...
. Later, there developed the idea that a finite number of generating
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
could represent all the
correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables ...
s of the Ising model. The existence of a much stronger symmetry for the scaling limit of two-dimensional critical systems was suggested by Alexander Belavin,
Alexander Markovich Polyakov Alexander Markovich Polyakov (; born 27 September 1945) is a Russian theoretical physicist, formerly at the Landau Institute in Moscow and, since 1989, at Princeton University, where he is the Joseph Henry Professor of Physics Emeritus. Importa ...
and
Alexander Zamolodchikov Alexander Borisovich Zamolodchikov (; born September 18, 1952) is a Russian-American theoretical physicist, known for his contributions to conformal field theory, statistical mechanics, string theory and condensed matter physics. He is widel ...
in 1984, which eventually led to the development of
conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...
,Clément Hongler
''Conformal invariance of Ising model correlations''
Ph.D. thesis, Université of Geneva, 2010, p. 9.
a special case of quantum field theory, which is presently utilized in different areas of particle physics and condensed matter physics. The
renormalization group In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying p ...
spans a set of ideas and methods to monitor changes of the behavior of the theory with scale, providing a deep physical understanding which sparked what has been called the "grand synthesis" of theoretical physics, uniting the quantum field theoretical techniques used in particle physics and condensed matter physics into a single powerful theoretical framework. The gauge field theory of the
strong interaction In nuclear physics and particle physics, the strong interaction, also called the strong force or strong nuclear force, is one of the four known fundamental interaction, fundamental interactions. It confines Quark, quarks into proton, protons, n ...
s,
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the study of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of ...
, relies crucially on this renormalization group for its distinguishing characteristic features,
asymptotic freedom In quantum field theory, asymptotic freedom is a property of some gauge theory, gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases. (A ...
and
color confinement In quantum chromodynamics (QCD), color confinement, often simply called confinement, is the phenomenon that color-charged particles (such as quarks and gluons) cannot be isolated, and therefore cannot be directly observed in normal conditions b ...
.


See also

*
History of quantum mechanics The history of quantum mechanics is a fundamental part of the History of physics#20th century: birth of modern physics, history of modern physics. The major chapters of this history begin with the emergence of quantum ideas to explain individual ...
*
History of string theory The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantum ...
*
QED vacuum The QED vacuum or quantum electrodynamic vacuum is the field-theoretic vacuum of quantum electrodynamics. It is the lowest energy state (the ground state) of the electromagnetic field when the fields are quantized. When the Planck constant is ...


References


Further reading

* Pais, Abraham; ''Inward Bound – Of Matter & Forces in the Physical World'', Oxford University Press (1986) . Written by a former Einstein assistant at Princeton, this is a beautiful detailed history of modern fundamental physics, from 1895 (discovery of X-rays) to 1983 (discovery of vectors bosons at
CERN The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in Meyrin, western suburb of Gene ...
). * Weinberg, Steven; ''The Quantum Theory of Fields - Foundations (vol. I)'', Cambridge University Press (1995) The first chapter (pp. 1–40) of Weinberg's monumental treatise gives a brief history of Q.F.T., p. 608. *Weinberg, Steven; ''The Quantum Theory of Fields - Modern Applications'' (vol. II), Cambridge University Press:Cambridge, U.K. (1996) , pp. 489. *Weinberg, Steven; ''The Quantum Theory of Fields – Supersymmetry'' (vol. III), Cambridge University Press:Cambridge, U.K. (2000) , pp. 419. * Schweber, Silvan S.; ''QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga'', Princeton University Press (1994) *Ynduráin, Francisco José; ''Quantum Chromodynamics: An Introduction to the Theory of Quarks and Gluons'', Springer Verlag, New York, 1983. * Miller, Arthur I.; ''Early Quantum Electrodynamics : A Sourcebook'', Cambridge University Press (1995) * Schwinger, Julian; ''Selected Papers on Quantum Electrodynamics'', Dover Publications, Inc. (1958) * O'Raifeartaigh, Lochlainn; ''The Dawning of Gauge Theory'', Princeton University Press (May 5, 1997) * Cao, Tian Yu; ''Conceptual Developments of 20th Century Field Theories'', Cambridge University Press (1997) * Darrigol, Olivier; ''La genèse du concept de champ quantique'', Annales de Physique (France) 9 (1984) pp. 433–501. Text in French, adapted from the author's Ph.D. thesis. {{History of physics *
Quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...