Quantum field theory

Quantum field theory

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Quantum field theory (QFT) is a theoretical framework for constructing quantum mechanical models of system classically described by fields (having an infinite number of degrees of freedom) or of many-body systems (having a finite number of degrees of freedom). Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described The history of quantum field theory starts with its creation by Dirac when he attempted to quantize the Electromagnetic field in the late 1920s Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, a field is a Physical quantity associated to each point of Spacetime. The many-body problem may be defined as the study of the effects of interaction between bodies on the behaviour of a many-body system i It is widely used in particle physics and condensed matter physics. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. Most theories in modern particle physics, including the Standard Model of elementary particles and their interactions, are formulated as relativistic quantum field theories. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In condensed matter physics, quantum field theories are used in many circumstances, especially those where the number of particles is allowed to fluctuate—for example, in the BCS theory of superconductivity. Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance

History

Main article: History of quantum field theory

Quantum field theory originated in the 1920s from the problem of creating a quantum mechanical theory of the electromagnetic field. The history of quantum field theory starts with its creation by Dirac when he attempted to quantize the Electromagnetic field in the late 1920s The 1920s is sometimes referred to as the " Jazz Age " or the " Roaring Twenties " when speaking about the United States and Canada Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The electromagnetic field is a physical field produced by electrically charged objects. In 1926, Max Born, Pascual Jordan, and Werner Heisenberg constructed such a theory by expressing the field's internal degrees of freedom as an infinite set of harmonic oscillators and by employing the usual procedure for quantizing those oscillators (canonical quantization). Year 1926 ( MCMXXVI) was a Common year starting on Friday (link will display the full calendar of the Gregorian calendar. Max Born (11 December 1882 &ndash 5 January 1970 was a German Physicist and Mathematician who was instrumental in the development of Quantum Pascual Jordan (b October 18, 1902 in Hanover, Germany; d July 31, 1980 in Hamburg, Federal Republic Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the For information on degrees of freedom in other sciences see Degrees of freedom. This article is about the harmonic oscillator in classical mechanics In Physics, canonical quantization is one of many procedures for quantizing a Classical theory. This theory assumed that no electric charges or currents were present and today would be called a free field theory. Classically a free field has Equations of motion given by Linear Partial differential equations Such linear PDE's have a unique solution for a given The first reasonably complete theory of quantum electrodynamics, which included both the electromagnetic field and electrically charged matter (specifically, electrons) as quantum mechanical objects, was created by Paul Dirac in 1927. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Year 1927 ( MCMXXVII) was a Common year starting on Saturday (link will display full calendar of the Gregorian calendar. This quantum field theory could be used to model important processes such as the emission of a photon by an electron dropping into a quantum state of lower energy, a process in which the number of particles changes — one atom in the initial state becomes an atom plus a photon in the final state. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena It is now understood that the ability to describe such processes is one of the most important features of quantum field theory.

It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate Einstein's relativity theory, which had after all grown out of the study of classical electromagnetism. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently This need to put together relativity and quantum mechanics was the second major motivation in the development of quantum field theory. Pascual Jordan and Wolfgang Pauli showed in 1928 that quantum fields could be made to behave in the way predicted by special relativity during coordinate transformations (specifically, they showed that the field commutators were Lorentz invariant), and in 1933 Niels Bohr and Leon Rosenfeld showed that this result could be interpreted as a limitation on the ability to measure fields at space-like separations, exactly as required by relativity. Pascual Jordan (b October 18, 1902 in Hanover, Germany; d July 31, 1980 in Hamburg, Federal Republic Year 1928 ( MCMXXVIII) was a Leap year starting on Sunday (link will display full calendar of the Gregorian calendar. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial For other uses of "covariant" or "contravariant" see Covariance and contravariance. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally Year 1933 ( MCMXXXIII) was a Common year starting on Sunday (link will display full calendar of the Gregorian calendar. Niels Henrik David Bohr (nels ˈb̥oɐ̯ˀ in Danish 7 October 1885 – 18 November 1962 was a Danish Physicist who made fundamental contributions to understanding Léon Rosenfeld (1904 &ndash 1974 was a Belgian Physicist. He obtained a PhD at the University of Liege in 1926 and he was a collaborator of the physicist SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS A further boost for quantum field theory came with the discovery of the Dirac equation, a single-particle equation obeying both relativity and quantum mechanics, when it was shown that several of its undesirable properties (such as negative-energy states) could be eliminated by reformulating the Dirac equation as a quantum field theory. In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides This work was performed by Wendell Furry, Robert Oppenheimer, Vladimir Fock, and others. Vladimir Aleksandrovich Fock (or Fok, Владимир Александрович Фoк ( December 22 1898 &ndash December 27 1974

The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease. In 1927, Jordan tried to extend the canonical quantization of fields to the many-body wavefunctions of identical particles, a procedure that is sometimes called second quantization. Year 1927 ( MCMXXVII) was a Common year starting on Saturday (link will display full calendar of the Gregorian calendar. Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another even in principle In Physics, canonical quantization is one of many procedures for quantizing a Classical theory. In 1928, Jordan and Eugene Wigner found that the quantum field describing electrons, or other fermions, had to be expanded using anti-commuting creation and annihilation operators due to the Pauli exclusion principle. Year 1928 ( MCMXXVIII) was a Leap year starting on Sunday (link will display full calendar of the Gregorian calendar. Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 This thread of development was incorporated into many-body theory, and strongly influenced condensed matter physics and nuclear physics. Many-Body Theory (or Many-body physics) is an area of Physics which provides the framework for understanding the collective behavior of vast assemblies of Interacting Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. Nuclear physics is the field of Physics that studies the building blocks and interactions of Atomic nuclei.

Despite its early successes, quantum field theory was plagued by several serious theoretical difficulties. Many seemingly-innocuous physical quantities, such as the energy shift of electron states due to the presence of the electromagnetic field, gave infinity — a nonsensical result — when computed using quantum field theory. This "divergence problem" was solved during the 1940s by Bethe, Tomonaga, Schwinger, Feynman, and Dyson, through the procedure known as renormalization. The 1940s decade ran from 1940 to 1949 Events and trends The 1940s was a period between the radical 1930s and the conservative 1950s which also leads the period to be Hans Albrecht Bethe (/hans ˈalbʀɛçt ˈbeːtə/ ( July 2 1906 &ndash March 6, 2005) was a German - American Physicist Sin-Itiro Tomonaga or Shinichirō Tomonaga (朝永振一郎 Tomonaga Shin'ichirō, March 31, 1906 Julian Seymour Schwinger ( February 12, 1918 &ndash July 16, 1994) was an American Theoretical physicist. Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum Freeman John Dyson FRS (born December 15, 1923) is an English-born American theoretical Physicist and Mathematician, famous for his In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection This phase of development culminated with the construction of the modern theory of quantum electrodynamics (QED). Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Beginning in the 1950s with the work of Yang and Mills, QED was generalized to a class of quantum field theories known as gauge theories. The 1950s Decade refers to the years of 1950 to 1959 inclusive Chen-Ning Franklin Yang ( (born October 1, 1922) is a Chinese -born American Physicist who worked on Statistical mechanics Robert L Mills ( April 15, 1927 - October 27, 1999) was a Physicist, specializing in Quantum field theory, the theory Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations The 1960s and 1970s saw the formulation of a gauge theory now known as the Standard Model of particle physics, which describes all known elementary particles and the interactions between them. The 1960s decade refers to the years from the beginning of 1960 to the end of 1969 This article is about the Decade 1970-1979 For the Year 1970 see 1970. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them The weak interaction part of the standard model was formulated by Sheldon Glashow, with the Higgs mechanism added by Steven Weinberg and Abdus Salam. Sheldon Lee Glashow (born December 5, 1932) is an American physicist. The Higgs mechanism is Spontaneous symmetry breaking in a Gauge theory. Steven Weinberg (born May 3, 1933) is an American Physicist, and Nobel laureate in Physics for his contributions with Abdus Salam Abdus Salam ( Urdu: محمد عبد السلام) ( January 29, 1926; Jhang Punjab &ndash November 21, The theory was shown to be consistent by Gerardus 't Hooft and Martinus Veltman. Gerardus 't Hooft (xeːrɑrt ət hoːft (born July 5, 1946, Den Helder) is a professor in Theoretical physics at Utrecht University Martinus Justinus Godefriedus Veltman (born June 27, 1931 in Waalwijk) is a Dutch theoretical physicist

Also during the 1970s, parallel developments in the study of phase transitions in condensed matter physics led Leo Kadanoff, Michael Fisher and Kenneth Wilson (extending work of Ernst Stueckelberg, Andre Peterman, Murray Gell-Mann and Francis Low) to a set of ideas and methods known as the renormalization group. This article is about the Decade 1970-1979 For the Year 1970 see 1970. In Thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. Leo Kadanoff is a professor of physics (emeritus as of 2004 at the University of Chicago and the current President of the American Physical Society (APS Michael Ellis Fisher (born 3 September 1931 in Trinidad) is a physicist as well as chemist and mathematician known for his many seminal contributionsto Kenneth Geddes Wilson (born June 8, 1936) is an American Theoretical physicist. This article is about the physicist for his grandfather the Swiss artist see Ernst Alfred Stueckelberg Ernst Carl Gerlach Stueckelberg ( Murray Gell-Mann (born September 15, 1929) is an American Physicist who received the 1969 Nobel Prize in physics for his work In Theoretical physics, renormalization group (RG refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views By providing a better physical understanding of the renormalization procedure invented in the 1940s, the renormalization group sparked what has been called the "grand synthesis" of theoretical physics, uniting the quantum field theoretical techniques used in particle physics and condensed matter physics into a single theoretical framework. The 1940s decade ran from 1940 to 1949 Events and trends The 1940s was a period between the radical 1930s and the conservative 1950s which also leads the period to be

The study of quantum field theory is alive and flourishing, as are applications of this method to many physical problems. It remains one of the most vital areas of theoretical physics today, providing a common language to many branches of physics. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

Principles of quantum field theory

Classical fields and quantum fields

Quantum mechanics, in its most general formulation, is a theory of abstract operators (observables) acting on an abstract state space (Hilbert space), where the observables represent physically-observable quantities and the state space represents the possible states of the system under study. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Mathematics, an operator is a function which operates on (or modifies another function This article assumes some familiarity with Analytic geometry and the concept of a limit. Furthermore, each observable corresponds, in a technical sense, to the classical idea of a degree of freedom. This article discusses quantum theory For other uses see Correspondence principle (disambiguation. For information on degrees of freedom in other sciences see Degrees of freedom. For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators $\hat{x}$ and $\hat{p}$ . Ordinary quantum mechanics deals with systems such as this, which possess a small set of degrees of freedom.

(It is important to note, at this point, that this article does not use the word "particle" in the context of wave–particle duality. In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and In quantum field theory, "particle" is a generic term for any discrete quantum mechanical entity, such as an electron, which can behave like classical particles or classical waves under different experimental conditions. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. )

A quantum field is a quantum mechanical system containing a large, and possibly infinite, number of degrees of freedom. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness This is not as exotic a situation as one might think. A classical field contains a set of degrees of freedom at each point of space; for instance, the classical electromagnetic field defines two vectors — the electric field and the magnetic field — that can in principle take on distinct values for each position $r$ . In Physics, a field is a Physical quantity associated to each point of Spacetime. The electromagnetic field is a physical field produced by electrically charged objects. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges When the field as a whole is considered as a quantum mechanical system, its observables form an infinite (in fact uncountable) set, because $r$ is continuous.

Furthermore, the degrees of freedom in a quantum field are arranged in "repeated" sets. For example, the degrees of freedom in an electromagnetic field can be grouped according to the position $r$ , with exactly two vectors for each $r$ . Note that $r$ is an ordinary number that "indexes" the observables; it is not to be confused with the position operator $\hat{x}$ encountered in ordinary quantum mechanics, which is an observable. (Thus, ordinary quantum mechanics is sometimes referred to as "zero-dimensional quantum field theory", because it contains only a single set of observables. ) It is also important to note that there is nothing special about $r$ because, as it turns out, there is generally more than one way of indexing the degrees of freedom in the field.

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

Single-particle and many-particle quantum mechanics

In ordinary quantum mechanics, the time-dependent Schrödinger equation describing the motion of a single non-relativistic particle is

$\left[ \frac{|\mathbf{p}|^2}{2m} + V(\mathbf{r}) \right] |\psi(t)\rang = i \hbar \frac{\partial}{\partial t} |\psi(t)\rang,$

where $m$ is the particle's mass, $V$ is the applied potential, and $|\psi\rang$ denotes the quantum state (we are using bra-ket notation). In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object The Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical

We wish to consider how this problem generalizes to $N$ particles. There are two motivations for studying the many-particle problem. The first is a straightforward need in condensed matter physics, where typically the number of particles is on the order of Avogadro's number (6. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. The Avogadro constant (symbols L, N A also called Avogadro's number, is the number of "elementary entities" (usually Atoms 0221415 x 10²³). The second motivation for the many-particle problem arises from particle physics and the desire to incorporate the effects of special relativity. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial If one attempts to include the relativistic rest energy into the above equation, the result is either the Klein-Gordon equation or the Dirac equation. The rest energy E or rest mass-energy of a particle is its energy when it is at rest relative to a given Inertial reference frame. The Klein–Gordon equation ( Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides However, these equations have many unsatisfactory qualities; for instance, they possess energy eigenvalues which extend to –∞, so that there seems to be no easy definition of a ground state. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Quantum mechanics, a stationary state is an Eigenstate of a Hamiltonian, or in other words a state of definite energy It turns out that such inconsistencies arise from neglecting the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical In Physics, mass–energy equivalence is the concept that for particles slower than light any Mass has an associated Energy and vice versa. For example, an electron and a positron can annihilate each other to create photons. The positrons or antielectron is the Antiparticle or the Antimatter counterpart of the Electron. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena Thus, a consistent relativistic quantum theory must be formulated as a many-particle theory.

Furthermore, we will assume that the $N$ particles are indistinguishable. Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another even in principle As described in the article on identical particles, this implies that the state of the entire system must be either symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another even in principle In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. These multi-particle states are rather complicated to write. For example, the general quantum state of a system of $N$ bosons is written as

$|\phi_1 \cdots \phi_N \rang = \sqrt{\frac{\prod_j N_j!}{N!}} \sum_{p\in S_N} |\phi_{p(1)}\rang \cdots |\phi_{p(N)} \rang,$

where $|\phi_i\rang$ are the single-particle states, $N j$ is the number of particles occupying state $j$ , and the sum is taken over all possible permutations $p$ acting on $N$ elements. In several fields of Mathematics the term permutation is used with different but closely related meanings In general, this is a sum of $N!$ ( $N$ factorial) distinct terms, which quickly becomes unmanageable as $N$ increases. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k The way to simplify this problem is to turn it into a quantum field theory.

Second quantization

Main article: Second quantization

In this section, we will describe a method for constructing a quantum field theory called second quantization. In Physics, canonical quantization is one of many procedures for quantizing a Classical theory. In Physics, canonical quantization is one of many procedures for quantizing a Classical theory. This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identical-particle states. It is based on the Hamiltonian formulation of quantum mechanics; several other approaches exist, such as the Feynman path integral^[1], which uses a Lagrangian formulation. In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system For an overview, see the article on quantization. In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory.

Second quantization of bosons

For simplicity, we will first discuss second quantization for bosons, which form perfectly symmetric quantum states. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein Let us denote the mutually orthogonal single-particle states by $|\phi_1\rang, |\phi_2\rang, |\phi_3\rang,$ and so on. For example, the 3-particle state with one particle in state $|\phi_1\rang$ and two in state $|\phi_2\rang$ is

$\frac{1}{\sqrt{3}} \left[ |\phi_1\rang |\phi_2\rang |\phi_2\rang + |\phi_2\rang |\phi_1\rang |\phi_2\rang + |\phi_2\rang |\phi_2\rang |\phi_1\rang \right].$

The first step in second quantization is to express such quantum states in terms of occupation numbers, by listing the number of particles occupying each of the single-particle states $|\phi_1\rang, |\phi_2\rang,$ etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as

$|1, 2, 0, 0, 0, \cdots \rangle.$

The next step is to expand the $N$ -particle state space to include the state spaces for all possible values of $N$ . This extended state space, known as a Fock space, is composed of the state space of a system with no particles (the so-called vacuum state), plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. The Fock space is an Algebraic system ( Hilbert space) used in Quantum mechanics to describe Quantum states with a variable or unknown number of In Quantum field theory, the vacuum state (also called the vacuum) is the Quantum state with the lowest possible Energy. It is easy to see that there is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space.

At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number $j\cdots$ , indicating which of the single-particle states $|\phi_1\rang, |\phi_2\rang, \cdots|\phi_j\rang\cdots$ it refers to.

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

$a_2 | N_1, N_2, N_3, \cdots \rangle = \sqrt{N_2} \mid N_1, (N_2 - 1), N_3, \cdots \rangle,$

$a_2^\dagger | N_1, N_2, N_3, \cdots \rangle = \sqrt{N_2 + 1} \mid N_1, (N_2 + 1), N_3, \cdots \rangle.$

It can be shown that these are operators in the usual quantum mechanical sense, i. e. linear operators acting on the Fock space. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Furthermore, they are indeed Hermitian conjugates, which justifies the way we have written them. In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator. They can be shown to obey the commutation relation

$\left[a_i , a_j \right] = 0 \quad,\quad \left[a_i^\dagger , a_j^\dagger \right] = 0 \quad,\quad \left[a_i , a_j^\dagger \right] = \delta_{ij},$

where $δ$ stands for the Kronecker delta. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two These are precisely the relations obeyed by the ladder operators for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator.

The Hamiltonian of the quantum field (which, through the Schrödinger equation, determines its dynamics) can be written in terms of creation and annihilation operators. In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system For instance, the Hamiltonian of a field of free (non-interacting) bosons is

$H = \sum_k E_k \, a^\dagger_k \,a_k,$

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

$a_k^\dagger\,a_k|\cdots, N_k, \cdots \rangle=N_k| \cdots, N_k, \cdots \rangle$ .

Second quantization of fermions

It turns out that a different definition of creation and annihilation must be used for describing fermions. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. According to the Pauli exclusion principle, fermions cannot share quantum states, so their occupation numbers $N i$ can only take on the value 0 or 1. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 The fermionic annihilation operators $c$ and creation operators $c^\dagger$ are defined by

$c_j | N_1, N_2, \cdots, N_j = 0, \cdots \rangle = 0$

$c_j | N_1, N_2, \cdots, N_j = 1, \cdots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \cdots, N_j = 0, \cdots \rangle$

$c_j^\dagger | N_1, N_2, \cdots, N_j = 0, \cdots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \cdots, N_j = 1, \cdots \rangle$

$c_j^\dagger | N_1, N_2, \cdots, N_j = 1, \cdots \rangle = 0$

These obey an anticommutation relation:

$\left\{c_i , c_j \right\} = 0 \quad,\quad \left\{c_i^\dagger , c_j^\dagger \right\} = 0 \quad,\quad \left\{c_i , c_j^\dagger \right\} = \delta_{ij}$

One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative.

Field operators

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

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

Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a problem consisting of a single particle inside ) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and For example, the bosonic field annihilation operator $\phi(\mathbf{r})$ is

$\phi(\mathbf{r}) \ \stackrel{\mathrm{def}}{=}\ \sum_{j} e^{i\mathbf{k}_j\cdot \mathbf{r}} a_{j}$

The bosonic field operators obey the commutation relation

$\left[\phi(\mathbf{r}) , \phi(\mathbf{r'}) \right] = 0 \quad,\quad \left[\phi^\dagger(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = 0 \quad,\quad \left[\phi(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = \delta^3(\mathbf{r} - \mathbf{r'})$

where $δ(x)$ stands for the Dirac delta function. The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.

It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say

$H = - \frac{\hbar^2}{2m} \sum_i \nabla_i^2 + \sum_{i < j} U(|\mathbf{r}_i - \mathbf{r}_j|)$

where the indices $i$ and $j$ run over all particles, then the field theory Hamiltonian is

$H = - \frac{\hbar^2}{2m} \int d^3\!r \; \phi^\dagger(\mathbf{r}) \nabla^2 \phi(\mathbf{r}) + \int\!d^3\!r \int\!d^3\!r' \; \phi(\mathbf{r})^\dagger \phi(\mathbf{r}')^\dagger U(|\mathbf{r} - \mathbf{r}'|) \phi(\mathbf{r'}) \phi(\mathbf{r})$

This looks remarkably like an expression for the expectation value of the energy, with $φ$ playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.

Implications of quantum field theory

Unification of fields and particles

The "second quantization" procedure that we have outlined in the previous section takes a set of single-particle quantum states as a starting point. Sometimes, it is impossible to define such single-particle states, and one must proceed directly to quantum field theory. For example, a quantum theory of the electromagnetic field must be a quantum field theory, because it is impossible (for various reasons) to define a wavefunction for a single photon. The electromagnetic field is a physical field produced by electrically charged objects. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena In such situations, the quantum field theory can be constructed by examining the mechanical properties of the classical field and guessing the corresponding quantum theory. This article discusses quantum theory For other uses see Correspondence principle (disambiguation. The quantum field theories obtained in this way have the same properties as those obtained using second quantization, such as well-defined creation and annihilation operators obeying commutation or anticommutation relations.

Quantum field theory thus provides a unified framework for describing "field-like" objects (such as the electromagnetic field, whose excitations are photons) and "particle-like" objects (such as electrons, which are treated as excitations of an underlying electron field).

Physical meaning of particle indistinguishability

The second quantization procedure relies crucially on the particles being identical. Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another even in principle We would not have been able to construct a quantum field theory from a distinguishable many-particle system, because there would have been no way of separating and indexing the degrees of freedom.

Many physicists prefer to take the converse interpretation, which is that quantum field theory explains what identical particles are. In ordinary quantum mechanics, there is not much theoretical motivation for using symmetric (bosonic) or antisymmetric (fermionic) states, and the need for such states is simply regarded as an empirical fact. From the point of view of quantum field theory, particles are identical if and only if they are excitations of the same underlying quantum field. ↔ Thus, the question "why are all electrons identical?" arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental.

Particle conservation and non-conservation

During second quantization, we started with a Hamiltonian and state space describing a fixed number of particles ( $N$ ), and ended with a Hamiltonian and state space for an arbitrary number of particles. Of course, in many common situations $N$ is an important and perfectly well-defined quantity, e. g. if we are describing a gas of atoms sealed in a box. In Physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a problem consisting of a single particle inside From the point of view of quantum field theory, such situations are described by quantum states that are eigenstates of the number operator $\hat{N}$ , which measures the total number of particles present. In Quantum mechanics, for systems where the total number of particles may not be preserved the number operator is the observable that counts the number of particles As with any quantum mechanical observable, $\hat{N}$ is conserved if it commutes with the Hamiltonian. In that case, the quantum state is trapped in the $N$ -particle subspace of the total Fock space, and the situation could equally well be described by ordinary $N$ -particle quantum mechanics. In Linear algebra, an Euclidean subspace (or subspace of R n) is a set of vectors that is closed under addition

For example, we can see that the free-boson Hamiltonian described above conserves particle number. Whenever the Hamiltonian operates on a state, each particle destroyed by an annihilation operator $a k$ is immediately put back by the creation operator $a_k^\dagger$ .

On the other hand, it is possible, and indeed common, to encounter quantum states that are not eigenstates of $\hat{N}$ , which do not have well-defined particle numbers. Such states are difficult or impossible to handle using ordinary quantum mechanics, but they can be easily described in quantum field theory as quantum superpositions of states having different values of $N$ . Quantum superposition is the fundamental law of Quantum mechanics. For example, suppose we have a bosonic field whose particles can be created or destroyed by interactions with a fermionic field. The Hamiltonian of the combined system would be given by the Hamiltonians of the free boson and free fermion fields, plus a "potential energy" term such as

$H_I = \sum_{k,q} V_q (a_q + a_{-q}^\dagger) c_{k+q}^\dagger c_k$ ,

where $a_k^\dagger$ and $a k$ denotes the bosonic creation and annihilation operators, $c_k^\dagger$ and $c k$ denotes the fermionic creation and annihilation operators, and $V q$ is a parameter that describes the strength of the interaction. This "interaction term" describes processes in which a fermion in state $k$ either absorbs or emits a boson, thereby being kicked into a different eigenstate $k + q$ . (In fact, this type of Hamiltonian is used to describe interaction between conduction electrons and phonons in metals. Electrical conduction is the movement of electrically charged particles through a Transmission medium ( Electrical conductor) In Physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the Atomic lattice of a Solid The M acro E xpansion T emplate A ttribute L anguage complements TAL, providing macros which allow the reuse of code across The interaction between electrons and photons is treated in a similar way, but is a little more complicated because the role of spin must be taken into account. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin ) One thing to notice here is that even if we start out with a fixed number of bosons, we will typically end up with a superposition of states with different numbers of bosons at later times. The number of fermions, however, is conserved in this case.

In condensed matter physics, states with ill-defined particle numbers are particularly important for describing the various superfluids. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. Superfluidity is a phase of matter or description of Heat capacity in which unusual effects are observed when Liquids, typically of Helium-4 Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers.

Axiomatic approaches

The preceding description of quantum field theory follows the spirit in which most physicists approach the subject. A physicist is a Scientist who studies or practices Physics. Physicists study a wide range of physical phenomena in many branches of physics spanning However, it is not mathematically rigorous. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse Over the past several decades, there have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject These attempts fall into two broad classes.

The first class of axioms, first proposed during the 1950s, include the Wightman, Osterwalder-Schrader, and Haag-Kastler systems. The 1950s Decade refers to the years of 1950 to 1959 inclusive In Physics the Wightman axioms are an attempt at a mathematically rigorous formulation of Quantum field theory. In Quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted The Haag-Kastler Axiomatic framework for Quantum field theory, named after Rudolf Haag and Daniel Kastler, is an application to local They attempted to formalize the physicists' notion of an "operator-valued field" within the context of functional analysis, and enjoyed limited success. For functional analysis as used in psychology see the Functional analysis (psychology article It was possible to prove that any quantum field theory satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the CPT theorem. The spin-statistics theorem in Quantum mechanics relates the spin of a particle to the statistics obeyed by that particle CPT symmetry is a fundamental symmetry of Physical laws under transformations that involve the inversions of charge, parity and Unfortunately, it proved extraordinarily difficult to show that any realistic field theory, including the Standard Model, satisfied these axioms. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Most of the theories that could be treated with these analytic axioms were physically trivial, being restricted to low-dimensions and lacking interesting dynamics. The construction of theories satisfying one of these sets of axioms falls in the field of constructive quantum field theory. In Mathematical physics, constructive quantum field theory is the field devoted to showing that quantum theory is mathematically compatible with Special Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others. This article is about the Decade 1970-1979 For the Year 1970 see 1970.

During the 1980s, a second set of axioms based on geometric ideas was proposed. The 1980s was the decade spanning from January 1 1980 to December 31 1989. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position This line of investigation, which restricts its attention to a particular class of quantum field theories known as topological quantum field theories, is associated most closely with Michael Atiyah and Graeme Segal, and was notably expanded upon by Edward Witten, Richard Borcherds, and Maxim Kontsevich. A topological quantum field theory (or topological field theory or TQFT) is a Quantum field theory which computes Topological invariants Sir Michael Francis Atiyah, OM, FRS, FRSE (b April 22, 1929) is a British Mathematician, and one of the Graeme B Segal (born 1942 is a British Mathematician, and Professor at the University of Oxford. Edward Witten (born August 26, 1951) is an American Theoretical physicist and Professor at the Institute for Advanced Study Richard Ewen Borcherds (born November 29, 1959) is a British Mathematician specializing in lattices, Number theory, Maxim Lvovich Kontsevich (Максим Львович Концевич (born August 25, 1964) is a Russian Mathematician. However, most physically-relevant quantum field theories, such as the Standard Model, are not topological quantum field theories; the quantum field theory of the fractional quantum Hall effect is a notable exception. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles The quantum Hall effect (or integer quantum Hall effect) is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems The main impact of axiomatic topological quantum field theory has been on mathematics, with important applications in representation theory, algebraic topology, and differential geometry. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry

Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the Millennium Prize Problems—proving the existence of a mass gap in Yang-Mills theory—is linked to this issue. The Millennium Prize Problems are seven problems in Mathematics that were stated by the Clay Mathematics Institute in 2000 The Clay Mathematics Institute has offered the prize of USD 1000000 for each of 7 great problems in mathematics

Phenomena associated with quantum field theory

In the previous part of the article, we described the most general properties of quantum field theories. Some of the quantum field theories studied in various fields of theoretical physics possess additional special properties, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections.

Renormalization

Main article: Renormalization

Early in the history of quantum field theory, it was found that many seemingly innocuous calculations, such as the perturbative shift in the energy of an electron due to the presence of the electromagnetic field, give infinite results. In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection In Quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system Many of these problems are related to failures in classical electrodynamics that were identified (but unsolved) as far back as the 19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field with which it interacts. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar To illustrate this, recall from the preceding discussion that the interaction Hamiltonian between two quantum fields, such as the electron field and the electromagnetic field, need not conserve particle number. Thus, even if we start out with a single electron and no photons, the quantum state will rapidly evolve into a superposition of states that can include one or more photons. Therefore, the energy carried by that "single" electron—its self energy—is not simply the "bare" value, but also includes the energy contained in an attendant cloud of photons. In Theoretical physics and Quantum field theory a particle's self-energy \Sigma represents the contribution to the particle's Energy, or When this self energy is computed, one finds that the contribution of photons possessing arbitrarily high energies (or, equivalently, arbitrarily short wavelengths) leads to a formally infinite value. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency.

The solution to this problem, first given by Julian Schwinger, is called renormalization. Julian Seymour Schwinger ( February 12, 1918 &ndash July 16, 1994) was an American Theoretical physicist. In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection The idea is to impose a "cutoff" for the photonic contribution, e. In Theoretical physics, cutoff is the maximal or minimal value of Energy, Momentum, or Length, so that the objects with even larger or smaller g. by postulating that photons cannot possess energies above some extremely high value. Any quantity we wish to compute, such as the rest energy, is now finite but dependent on the cutoff. We then recast the result in terms of physically-observable quantities such as the observed electron mass, instead of unobservable quantities such as the cutoff energy and the bare electron mass. The final result is independent of all details of the cutoff procedure, including the value of the cutoff energy, provided the relevant processes occur at energies far below the cutoff.

The renormalization procedure only works for a certain class of quantum field theories, called renormalizable quantum field theories. The Standard Model of particle physics is renormalizable, and so are its component theories (quantum electrodynamics/electroweak theory and quantum chromodynamics). The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. In Particle physics, the electroweak interaction is the unified description of two of the four Fundamental interactions of nature Electromagnetism and the Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the According to the theory of the renormalization group, each renormalizable theory is a unique low-energy limit (i. In Theoretical physics, renormalization group (RG refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views e. a so-called "effective field theory") for a broad range of high-energy theories. In Physics, an effective field theory is an approximate theory (usually a Quantum field theory) that includes appropriate degrees of freedom to describe Renormalizable theories are therefore independent of the precise nature of the underlying high-energy phenomena.

Gauge freedom

A gauge theory is a theory that admits a symmetry with a local parameter. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these For example, in every quantum theory the global phase of the wave function is arbitrary and does not represent something physical. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system Consequently, the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is a global symmetry. A global symmetry is a symmetry that holds for all points in the Spacetime under consideration as opposed to a Local symmetry that only holds for an In quantum electrodynamics, the theory is also invariant under a local change of phase, that is - one may shift the phase of all wave functions so that the shift may be different at every point in space-time. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS This is a local symmetry. In quantum field theory (QFT the forces between particles are mediated by other particles However, in order for a well-defined derivative operator to exist, one must introduce a new field, the gauge field, which also transforms in order for the local change of variables (the phase in our example) not to affect the derivative. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Physics, a field is a Physical quantity associated to each point of Spacetime. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In quantum electrodynamics this gauge field is the electromagnetic field. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations The electromagnetic field is a physical field produced by electrically charged objects. The change of local gauge of variables is termed gauge transformation. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations

In quantum field theory the excitations of fields represent particles. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made The particle associated with excitations of the gauge field is the gauge boson, which is the photon in the case of quantum electrodynamics. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Particle physics, gauge bosons are Bosonic particles that act as carriers of the fundamental forces of nature In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics.

The degrees of freedom in quantum field theory are local fluctuations of the fields. For information on degrees of freedom in other sciences see Degrees of freedom. The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Such fluctuations are usually called "non-physical degrees of freedom" or gauge artifacts; usually some of them have a negative norm, making them inadequate for a consistent theory. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i. e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum anomaly. In Quantum physics an anomaly or quantum anomaly is the failure of a Symmetry of a theory's classical action to be a symmetry of any regularization If a gauge symmetry is anomalous (i. In Theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of Quantum mechanics —usually a One-loop diagram —that e. not kept in the quantum theory) then the theory is non-consistent: for example, in quantum electrodynamics, had there been a gauge anomaly, this would require the appearance of photons with longitudinal polarization and polarization in the time direction, the latter having a negative norm, rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any interaction, making the theory non unitary and again inconsistent (see optical theorem). Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. In Theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of Quantum mechanics —usually a One-loop diagram —that In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena Longitudinal waves are waves that have vibrations along or parallel to their direction of travel that is waves in which the motion of the medium is in the same direction as the motion Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Functional analysis, a branch of Mathematics, a unitary operator is a Bounded linear operator U H → In Physics, the optical theorem is a very general law of Wave Scattering theory, which relates the forward Scattering amplitude to the total

In general, the gauge transformations of a theory consist several different transformations, which may not be commutative. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Mathematics, commutativity is the ability to change the order of something without changing the end result These transformations are together described by a mathematical object known as a gauge group. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Infinitesimal gauge transformations are the gauge group generators. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings Therefore the number of gauge bosons is the group dimension (i. In Particle physics, gauge bosons are Bosonic particles that act as carriers of the fundamental forces of nature In Mathematics, the dimension of a Vector space V is the cardinality (i e. number of generators forming a basis). Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

All the fundamental interactions in nature are described by gauge theories. In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations These are:

Quantum electrodynamics, whose gauge transformation is a local change of phase, so that the gauge group is U(1). Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex The gauge boson is the photon. In Particle physics, gauge bosons are Bosonic particles that act as carriers of the fundamental forces of nature In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena
Quantum chromodynamics, whose gauge group is SU(3). Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n The gauge bosons are eight gluons. In Particle physics, gauge bosons are Bosonic particles that act as carriers of the fundamental forces of nature Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the
The electroweak Theory, whose gauge group is $U(1)\times SU(2)$ (a direct product of U(1) and SU(2)). The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n
Gravity, whose classical theory is general relativity, admits the equivalence principle which is a form of gauge symmetry. Gravitation is a natural Phenomenon by which objects with Mass attract one another General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

Supersymmetry

Supersymmetry assumes that every fundamental fermion has a superpartner that is a boson and vice versa. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein It was introduced in order to solve the so-called Hierarchy Problem, that is, to explain why particles not protected by any symmetry (like the Higgs boson) do not receive radiative corrections to its mass driving it to the larger scales (GUT, Planck. The Higgs Boson is a hypothetical massive scalar Elementary particle predicted to exist by the Standard Model of Particle physics . . ). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity (Supergravity), and it is a key ingredient for the consistency of string theory. In Theoretical physics, supergravity ( supergravity theory) is a field theory that combines the principles of Supersymmetry and General relativity String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite.

Since no superpartners have yet been observed, if supersymmetry exists it must be broken (through a so-called soft term, which breaks supersymmetry without ruining its helpful features). The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the Large Hadron Collider.

Notes

^ Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World ISBN 0198519974. List of quantum field theories: Chern-Simons model Chiral model Complex Quantum Mechanicshttp//tech This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. In Quantum mechanics, quantum flavordynamics (or "flavourdynamics" is a mathematical model used to describe the interaction of flavored particles ( Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. The Schwinger-Dyson equation, named after Julian Schwinger and Freeman Dyson, is an equation of Quantum field theory (QFT Many first principles in Quantum field theory are explained or get further insight in String theory: Emission and absorption one of the most basic building blocks In the Physics of Electromagnetism, the Abraham-Lorentz force is the Recoil Force on an accelerating Charged particle caused Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of In physics invariance mechanics, in its simplest form is the rewriting of the laws of Quantum field theory in terms of invariant quantities only Green–Kubo relations give exact mathematical expression for transport coefficients in terms of integrals of time correlation functions In Many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with Correlation function, but refers specifically Pais recounts how his astonishment at the rapidity with which Feynman could calculate using his method. Feynman's method is now part of the standard methods for physicists.

External links

Siegel, Warren ; Fields (also available from arXiv:hep-th/9912205)
't Hooft, Gerard ; The Conceptual Basis of Quantum Field Theory, Handbook of the Philosophy of Science, Elsevier (to be published). Hagen Kleinert (born 1941 is Professor of Theoretical Physics at the Free University of Berlin, Germany (since 1968 Honorary Professor at the Kyrgyz-Russian Review article written by a master of gauge theories, Nobel laureate 1999. Full text available in pdf.
Srednicki, Mark ; Quantum Field Theory
Kuhlmann, Meinard ; Quantum Field Theory, Stanford Encyclopedia of Philosophy
Quantum field theory textbooks: a list with links to amazon.com

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