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Encyclopedia > Quantum field theory

Quantum field theory (QFT) is the quantum theory of fields. It provides a theoretical framework, widely used in particle physics and condensed matter physics, in which to formulate consistent quantum theories of many-particle systems, especially in situations where particles may be created and destroyed. Non-relativistic quantum field theories are needed in condensed matter physics— for example in the BCS theory of superconductivity. Relativistic quantum field theories are indispensable in particle physics (see the standard model), although they are known to arise as effective field theories in condensed matter physics. Image File history File links Information_icon. ... Shortcut: WP:WIN Wikipedia is an online encyclopedia and, as a means to that end, also an online community. ... Shortcut: WP:CU Marking articles for cleanup This page is undergoing a transition to an easier-to-maintain format. ... This Manual of Style has the simple purpose of making things easy to read by following a consistent format — it is a style guide. ... Quantum theory is a theory of physics that uses Plancks constant. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Fig. ... In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... BCS theory (named for its creators, Bardeen, Cooper, and Schrieffer) successfully explains conventional superconductivity, the ability of certain metals at low temperatures to conduct electricity without resistance. ... A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... In physics, an effective field theory is an approximate theory (usually a quantum field theory) that contains the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter distances (or, equivalently, higher energies). ...

Contents

Origin

Quantum field theory originated in the problem of computing the energy radiated by an atom when it dropped from one quantum state to another of lower energy. This problem was first examined by Max Born and Pascual Jordan in 1925. In 1926, Max Born, Werner Heisenberg and Pascual Jordan wrote down the quantum theory of the electromagnetic field neglecting polarization and sources to obtain what would today be called a free field theory. In order to quantize this theory, they used the canonical quantization procedure. In 1927, Paul Dirac gave the first consistent treatment of this problem. Quantum field theory followed unavoidably from a quantum treatment of the only known classical field, viz. the electromagnetic field. The theory was required by the need to treat a situation where the number of particles changes. Here, one atom in the initial state becomes an atom and a photon in the final state. Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle still characterizing a chemical element. ... A quantum state is any possible state in which a quantum mechanical system can be. ... Max Born (December 11, 1882 in Breslau – January 5, 1970 in Göttingen) was a mathematician and physicist. ... Pascual Jordan (October 18, 1902 in Hanover - July 31, 1980 in Hamburg) was a German physicist. ... 1925 (MCMXXV) was a common year starting on Thursday (link will display the full calendar). ... Year 1926 (MCMXXVI) was a common year starting on Friday (link will display the full calendar). ... Max Born (December 11, 1882 in Breslau – January 5, 1970 in Göttingen) was a mathematician and physicist. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics, and acknowledged to be one of the most important physicists of the twentieth century. ... Pascual Jordan (October 18, 1902 in Hanover - July 31, 1980 in Hamburg) was a German physicist. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... Classically, a free field is a field described by linear partial differential equations which has a unique solution given initial data. ... In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... 1927 (MCMXXVII) was a common year starting on Saturday (link will display full calendar). ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Electromagnetism is the physics of the electromagnetic field; a field encompassing all of space which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ...


It was obvious from the beginning that the quantum treatment of the electromagnetic field required a proper treatment of relativity. Jordan and Wolfgang Pauli showed in 1928 that commutators of the field were actually Lorentz invariant. By 1933, Niels Bohr and Leon Rosenfeld had related these commutation relations to a limitation on the ability to measure fields at space-like separation. The development of the Dirac equation and the hole theory drove quantum field theory to explain these using the ideas of causality in relativity, work that was completed by Wendell Furry and Robert Oppenheimer using methods developed for this purpose by Vladimir Fock. This need to put together relativity and quantum mechanics was a second motivation which drove the development of quantum field theory. This thread was crucial to the eventual development of particle physics and the modern (partially) unified theory of forces called the standard model. This article is about Austrian-Swiss physicist Wolfgang Pauli. ... Year 1928 (MCMXXVIII) was a leap year starting on Sunday (link will display full calendar). ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ... Lorentz covariance is a term in physics for the property of space time, that in two different frames of reference, located at the same event in spacetime but moving relative to each other, all non-gravitational laws must make the same predictions for identical experiments. ... 1933 (MCMXXXIII) was a common year starting on Sunday. ... Niels (Henrik David) Bohr (October 7, 1885 – November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in 1922. ... Léon Rosenfeld (1904 – 1974) was a Belgian physicist. ... In the context of special relativity, space-like separated points (or events) in spacetime have a spacetime interval less than 0 (see sign convention). ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles possessing negative energy. ... Although causality, the relationship between causes and effects, is often examined in the fields of philosophy, computer science, and statistics, it has a place in the study of physics as well. ... J. Robert Oppenheimer, the father of the atomic bomb served as the first director of Los Alamos National Laboratory, beginning in 1943. ... Vladimir Aleksandrovich Fock (or Fok, Владимир Александрович Фок) (22 December 1898 - December 27, 1974) was a Soviet physicist, who did foundational work on quantum mechanics. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ...


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Description

Quantum mechanics in general deals with operators acting upon a (separable) Hilbert space. For a single nonrelativistic particle, the fundamental operators are its position and momentum, In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... Separable can refer to: Separable space in topology Separable sigma algebra in measure theory Separable differential equations This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

and .

These operators are time dependent in the Heisenberg picture, but we may also choose to work in the Schrödinger picture or (in the context of perturbation theory) the interaction picture. The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ... Heisenbergs form for the equations of motion We have seen that in Schrödingers scheme the dynamical variables of the system remain fixed during a period of undisturbed motion. ... Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the Schrödinger picture and the Heisenberg picture. ...


Quantum field theory is a special case of quantum mechanics in which the fundamental operators are an operator-valued field

.

A single scalar field describes a spinless particle. More fields are necessary for more types of particles, or for particles with spin. For example, particles with spin are usually described by higher order tensor or spinor-valued (or matrix-valued) tensor fields which in turn can be reinterpreted as a possibly large set of scalar fields with appropriate transformation rules as one changes the system of coordinates used. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


In quantum field theory, the energy is given by the Hamiltonian operator, which can be constructed from the quantum fields; it is the generator of infinitesimal time translations. (Being able to construct the generator of infinitesimal time translations out of quantum fields means many unphysical theories are ruled out, which is a good thing.)In order for the theory to be sensible, the Hamiltonian must be bounded from below. The lowest energy eigenstate (which may or may not be degenerate) is called the vacuum in particle physics and the ground state in condensed matter physics (QFT appears in the continuum limit of condensed matter systems). The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... The energy levels of two or more physical states are said to be degenerate when they have the same value. ...


Technical statement

Quantum field theory corrects several limitations of ordinary quantum mechanics. The time-independent Schrödinger equation, in its most commonly encountered form, is In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the space- and time-dependence of quantum mechanical systems. ...

where denotes the quantum state (notation) of a particle with mass m, in the presence of a potential V. A quantum state is any possible state in which a quantum mechanical system can be. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... Unsolved problems in physics: What causes anything to have mass? The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. Mass is the property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... It has been suggested that this article or section be merged with Scalar potential. ...


The first problem occurs when we seek to extend the equation to large numbers of particles. As described in the article on identical particles, quantum-mechanical particles of the same species are indistinguishable, in the sense that the state of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are extremely complicated to write. For example, the general quantum state of a system of N bosons is written as Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ... In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ... In particle physics, fermions are particles with half-integer spin. ...

where are the single-particle states, Nj is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases. Large numbers of particles are needed in condensed matter physics where typically the number of particles is on the order of Avogadro's number, approximately 1023. Template:Hellodablink Permutation is the rearrangement of objects or symbols into distinguishable sequences. ... For factorial rings in mathematics, see unique factorisation domain. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Avogadros number, also called Avogadros constant (NA), named after Amedeo Avogadro, is formally defined to be the number of carbon-12 atoms in 12 grams (0. ...


The second problem arises when trying to reconcile the Schrödinger equation with special relativity. It is possible to modify the Schrödinger equation to include the rest energy of a particle, resulting in the Klein-Gordon equation or the Dirac equation. 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. Such inconsistencies occur because these equations neglect 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. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory. This problem brings to the fore the notion that a consistent relativistic quantum theory, even of a single particle, must be a many particle theory. The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the Schrödinger equation. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... In physics, the ground state of a quantum mechanical system is its lowest-energy state. ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... Albert Einsteins equation E=mc2 is probably the most well-known equation of all time. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ...


Quantizing a classical field theory

Canonical quantization

Quantum field theory solves these problems by consistently quantizing a field. By interpreting the physical observables of the field appropriately, one can create a (rather successful) theory of many particles. Here is how it is: In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ...


1. Each normal mode oscillation of the field is interpreted as a particle with frequency f. Various normal modes in a 1D-lattice. ...


2. The quantum number n of each normal mode (which can be thought of as a harmonic oscillator) is interpreted as the number of particles. The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...


The energy associated with the mode of excitation is therefore which directly follows from the energy eigenvalues of a one dimensional harmonic oscillator in quantum mechanics. With some thought, one may similarly associate momenta and position of particles with observables of the field.


Having cleared up the correspondence between fields and particles (which is different from non-relativistic QM), we can proceed to define how a quantum field behaves.


Two caveats should be made before proceeding further:

  1. Each of these "particles" obeys the usual uncertainty principle of quantum mechanics. The "field" is an operator defined at each point of spacetime.
  2. Quantum field theory is not a wildly new theory. Classical field theory is the same as classical mechanics of an infinite number of dynamical quantities (say, tiny elements of rubber on a rubber sheet). Quantum field theory is the quantum mechanics of this infinite system.

The first method used to quantize field theory was the method now called canonical quantization (earlier known as second quantization). This method uses a Hamiltonian formulation of the classical problem. The later technique of Feynman path integrals uses a Lagrangian formulation. Many more methods are now in use; for an overview see the article on quantization. In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ... Fig. ... In physics, spacetime is a mathematical model that combines space and time into a single construct called the space-time continuum. ... This article is in need of attention from an expert on the subject. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Fig. ... In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... This article is about a formulation of quantum mechanics. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ...


Canonical quantization for bosons

Suppose we have a system of N bosons which can occupy mutually orthogonal single-particle states and so on. The usual method of writing a multi-particle state is to assign a state to each particle and then impose exchange symmetry. As we have seen, the resulting wavefunction is an unwieldy sum of N! terms. In contrast, in the second quantized approach we will simply list the number of particles in each of the single-particle states, with the understanding that the multi-particle wavefunction is symmetric. To be specific, suppose that N = 3, with one particle in state and two in state. The normal way of writing the wavefunction is

In second quantized form, we write this as

which means "one particle in state 1, two particles in state 2, and zero particles in all the other states."


Though the difference is entirely notational, the latter form makes it easy for us to define creation and annihilation operators, which add and subtract particles from multi-particle states. These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta. However, these operators literally create and annihilate particles with a given quantum state. The bosonic annihilation operator a2 and creation operator have the following effects: In physics, an annihilation operator is an operator that lowers the number of particles in a given state by one. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...

We may well ask whether these are operators in the usual quantum mechanical sense, i.e. linear operators acting on an abstract Hilbert space. In fact, the answer is yes: they are operators acting on a kind of expanded Hilbert space, known as a Fock space, composed of the space of a system with no particles (the so-called vacuum state), plus the space of a 1-particle system, plus the space of a 2-particle system, and so forth. Furthermore, the creation and annihilation operators are indeed Hermitian conjugates, which justifies the way we have written them. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... The Fock space is an algebraic system (Hilbert space) used in quantum mechanics to describe quantum states with a variable or unknown number of identical particles. ... In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ... In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator. ...


The bosonic creation and annihilation operators obey the commutation relation In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ...

where δ stands for the Kronecker delta. These are precisely the relations obeyed by the "ladder operators" for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...


The final step toward obtaining a quantum field theory is to re-write our original N-particle Hamiltonian in terms of creation and annihilation operators acting on a Fock space. For instance, the Hamiltonian of a field of free (non-interacting) bosons is

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

.

Canonical quantization for fermions

It turns out that the creation and annihilation operators for fermions must be defined differently, in order to satisfy the Pauli exclusion principle. For fermions, the occupation numbers Ni can only take on the value 0 or 1, since particles cannot share quantum states. We then define the fermionic annihilation operators c and creation operators by The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ...

The fermionic creation and annihilation operators obey an anticommutation relation, For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if...

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.


Significance of creation and annihilation operators

When we re-write a Hamiltonian using a Fock space and creation and annihilation operators, as in the previous example, the symbol N, which stands for the total number of particles, drops out. This means that the Hamiltonian is applicable to systems with any number of particles. Of course, in many common situations N is a physically important and perfectly well-defined quantity. For instance, if we are describing a gas of atoms sealed in a box, the number of atoms had better remain a constant at all times. This is certainly true for the above Hamiltonian. Viewing the Hamiltonian as the generator of time evolution, we see that whenever an annihilation operator ak destroys a particle during an infinitesimal time step, the creation operator to the left of it instantly puts it back. Therefore, if we start with a state of N non-interacting particles then we will always have N particles at a later time.


On the other hand, it is often useful to consider quantum states where the particle number is ill-defined, i.e. linear superpositions of vectors from the Fock space that possess different values of N. For instance, it may happen that our bosonic particles can be created or destroyed by interactions with a field of fermions. Denoting the fermionic creation and annihilation operators by and ck, we could add a "potential energy" term to our Hamiltonian such as:

This 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 is the expression for the interaction between phonons and conduction electrons in a solid. The interaction between photons and electrons is treated in a similar way; it is a little more complicated, because the role of spin must be taken into account. One thing to notice here is that even if we start out with a fixed number of bosons, we will generally end up with a superposition of states with different numbers of bosons at later times. On the other hand, the number of fermions is conserved in this case. Normals modes of vibration progression through a crystal. ... Electrical conduction is the current (movement of charged particles) through a material in response to an electric field. ... For other uses, see Solid (disambiguation). ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ...


In condensed matter physics, states with ill-defined particle numbers are also very important for describing the various superfluids. 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. Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Helium II will creep along surfaces in order to find its own level - after a short while, the levels in the two containers will equalize. ...


Field operators

We can now define field operators that create or destroy a particle at a particular point in space. In particle physics, these are often more convenient to work with than the creation and annihilation operators, because they make it easier to formulate theories that satisfy the demands of relativity.


Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator is In classical mechanics, momentum (pl. ... In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

The bosonic field operators obey the commutation relation

where δ(x) stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators. The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ...


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

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

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.


Quantization of classical fields

So far, we have shown how one goes from an ordinary quantum theory to a quantum field theory. There are certain systems for which no ordinary quantum theory exists. These are the "classical" fields, such as the electromagnetic field. There is no such thing as a wavefunction for a single photon in classical electromagnetism, so a quantum field theory must be formulated right from the start. This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ...


The essential difference between an ordinary system of particles and the electromagnetic field is the number of dynamical degrees of freedom. For a system of N particles, there are 3N coordinate variables corresponding to the position of each particle, and 3N conjugate momentum variables. One formulates a classical Hamiltonian using these variables, and obtains a quantum theory by turning the coordinate and position variables into quantum operators, and postulating commutation relations between them such as Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant . ...

For an electromagnetic field, the analogue of the coordinate variables are the values of the electrical potential and the vector potential at every point . This is an uncountable set of variables, because is continuous. This prevents us from postulating the same commutation relation as before. The way out is to replace the Kronecker delta with a Dirac delta function. This ends up giving us a commutation relation exactly like the one for field operators! We therefore end up treating "fields" and "particles" in the same way, using the apparatus of quantum field theory. Only by accident electrons were not regarded as de Broglie waves and photons governed by geometrical optics were not the dominant theory when QFT was developed. Electrical potential is the potential energy per unit charge associated with a static (time-invariant) electric field, also called the electrostatic potential or the electric potential, typically measured in volts. ... In vector calculus, a vector potential is a vector field whose curl is a given vector field. ... In mathematics, an uncountable or nondenumerable set is a set which is not countable. ... Louis de Broglie Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892–March 19, 1987), was a French physicist and Nobel Prize laureate. ... See also list of optical topics. ...


Path integral methods

The axiomatic approach

There have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. These attempts fall into two broad classes. This article does not adequately cite its references or sources. ...


The first class of axioms (most notably the Wightman, Osterwalder-Schrader, and Haag-Kastler systems) tried to formalize the physicists' notion of an "operator-valued field" within the context of functional analysis. These axioms enjoyed limited success. It was possible to prove that any QFT satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the PCT theorems. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory (e.g. quantum chromodynamics) satisfied these axioms. Most of the theories which could be treated with these analytic axioms were physically trivial: restricted to low-dimensions and lacking in interesting dynamics. Constructive quantum field theory is the construction of theories which satisfy one of these sets of axioms. Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others. In physics the Wightman axioms are an attempt of mathematically stringent, axiomatic formulation of quantum field theory. ... In physics the Wightman axioms are an attempt of mathematically stringent, axiomatic 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 to the ordered set of points in Euclidean space with no coinciding points. ... The Haag-Kastler axiomatic framework for quantum field theory is an application to local quantum physics of C-star algebra theory. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... In mathematical physics, constructive quantum field theory is the field devoted to attempts to put quantum field theory on a basis of completely defined concepts from functional analysis. ...


In the 1980s, a second wave of axioms were proposed. These axioms (associated most closely with Atiyah and Segal, and notably expanded upon by Witten, Borcherds, and Kontsevich) are more geometric in nature, and more closely resemble the path integrals of physics. They have not been exceptionally useful to physicists, as it is still extraordinarily difficult to show that any realistic QFTs satisfy these axioms, but have found many applications in mathematics, particularly in representation theory, algebraic topology, and geometry. A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. ...


Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. In fact, one of the Clay Millennium Prizes offers $1,000,000 to anyone who proves the existence of a mass gap in Yang-Mills theory. It seems likely that we have not yet understood the underlying structures which permit the Feynman path integrals to exist. The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. ... It has been suggested that this article or section be merged with Yang-Mills existence and mass Gap. ...


Renormalization

Some of the problems and phenomena eventually addressed by renormalization actually appeared earlier in the classical electrodynamics of point particles in the 19th and early 20th century. The basic problem is that the observable properties of an interacting particle cannot be entirely separated from the field that mediates the interaction. The standard classical example is the energy of a charged particle. To cram a finite amount of charge into a single point requires an infinite amount of energy; this manifests itself as the infinite energy of the particle's electric field. The energy density grows to infinity as one gets close to the charge. Figure 1. ... Classical electromagnetism is a theory of electromagnetism that was developed over the course of the 19th century, most prominently by James Clerk Maxwell. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999...


A single particle state in quantum field theory incorporates within it multiparticle states. This is most simply demonstrated by examining the evolution of a single particle state in the interaction picture In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the Schrödinger picture and the Heisenberg picture. ...

Taking the overlap with the initial state, one retains the even powers of HI. These terms are responsible for changing the number of particles during propagation, and are therefore quintessentially a product of quantum field theory. Corrections such as these are incorporated into wave function renormalization and mass renormalization. Similar corrections to the interaction Hamiltonian, HI, include vertex renormalization, or, in modern language, effective field theory. Wave function renormalization is the rescaling of quantum fields so that the Lehmann weight of its quanta is 1. ... In quantum field theory, mass renormalization refers to the quantum corrections to the mass of a particle through its self interactions, or through interactions with other particles. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... In physics, an effective field theory is an approximate theory (usually a quantum field theory) that contains the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter distances (or, equivalently, higher energies). ...


Gauge theories

A gauge theory is a theory which admits a symmetry with a local parameter. For example, in every quantum theory the global phase of the wave function is arbitrary and does not represent something physical, so the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is a global symmetry. In quantum electrodynamics, the theory is also invariant under a local change of phase, that is - one may shift the phase of all wave functions so that in every point in space-time the shift is different. This is a local symmetry. 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 quantum electrodynamics this gauge field is the electromagnetic field. The change of local change of variables is termed gauge transformation. In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... This article or section does not cite its references or sources. ... Fig. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... A wave function is the mathematical tool that quantum mechanics uses to describe any physical system. ... In quantum field theory, a global symmetry is any symmetry of a model which is not a gauge symmetry. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... A wave function is the mathematical tool that quantum mechanics uses to describe any physical system. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... This article or section does not cite its references or sources. ... In some places this article assumes an acquaintance with algebra, analytic geometry, or the limit. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...


In quantum field theory the excitations of fields represent particles. The particle associated with excitations of the gauge field is the gauge boson, which is the photon in the case of quantum electrodynamics. In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ...


The degrees of freedom in quantum field theory are local fluctuations of the fields. The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or gauge artifacts; usually some of them have a negative norm, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum anomaly. If a gauge symmetry is anomalous (i.e. not kept in the quantum theory) then the theory is non-consistent: for example, in quantum electrodynamics, had there been a gauge anomaly, this would require the appearance of 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). Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In physics, an anomaly is a classical symmetry — a symmetry of the Lagrangian — that is broken in quantum field theories. ... 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 invalidates the gauge symmetry of a quantum field theory i. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... 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 invalidates the gauge symmetry of a quantum field theory i. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ... Longitudinal waves are waves that have vibrations along or parallel to their direction of travel. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics and physics, unitarity is the property of an operator (or a matrix) that is unitary. ... In physics, the optical theorem is a very general law of wave scattering theory, which relates the forward scattering amplitude to the total cross section of the scatterer. ...


In general, the gauge transformations of a theory consist several different transformations, which may not be commutative. These transformations are together described by a mathematical object known as a gauge group. Infinitesimal gauge transformations are the gauge group generators. Therefore the number of gauge bosons is the group rank (i.e. number of generators forming an orthogonal basis). Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In Abstract Algebra, a generator is defined as follows: Let G be a group and , then a is called a generator and G is a cyclic group. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... Rank means a wide variety of things in mathematics, including: Rank (linear algebra) Rank of a tensor Rank of an array Rank of an abelian group Rank (set theory) Rank-into-rank Rank of a greedoid This is a disambiguation page — a navigational aid which lists other pages that... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...


All the fundamental interactions in nature are described by gauge theories. These are: A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation that of matrix multiplication. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ... Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... In particle physics, gluons are subatomic particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei. ... The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four fundamental interactions of nature. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation that of matrix multiplication. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... Gravity is a force of attraction that acts between bodies that have mass. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

Supersymmetry

Supersymmetry assumes that every fundamental fermion has a superpartner which is a boson and vice versa. 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...). 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. This article or section is in need of attention from an expert on the subject. ... In particle physics, fermions are particles with half-integer spin. ... In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ... In theoretical physics, supergravity (supergravity theory) refers to a field theory which combines the two theories of supersymmetry and general relativity. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point...


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. The Large Hadron Collider (LHC) is a particle accelerator and collider located at CERN, near Geneva, Switzerland ( ). Currently under construction, the LHC is scheduled to begin operation (at reduced energies) in November 2007. ...


History

More details can be found in the article on the history of quantum field theory. 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 field theory was created by Dirac when he attempted to quantize the electromagnetic field in the late 1920s. The early development of the field involved Fock, Jordan, Pauli, Heisenberg, Bethe, Tomonaga, Schwinger, Feynman, and Dyson. This phase of development culminated with the construction of the theory of quantum electrodynamics in the 1950s. Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... The 1920s is a decade sometimes referred to as the Jazz Age or the Roaring Twenties, usually applied to America. ... Vladimir Aleksandrovich Fock (or Fok, Владимир Александрович Фок) (December 22, 1898 - December 27, 1974) was a Soviet physicist, who did foundational work on quantum mechanics. ... This article is about Austrian-Swiss physicist Wolfgang Pauli. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics, and acknowledged to be one of the most important physicists of the twentieth century. ... Hans Bethe Hans Albrecht Bethe (pronounced Bay-tuh; July 2, 1906 – March 6, 2005), was a German-American physicist who won the Nobel Prize in Physics for 1967 for his discovery of stellar nucleosynthesis. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Julian Seymour Schwinger (February 12, 1918 -- July 16, 1994) was an American theoretical physicist. ... Richard Phillips Feynman (May 11, 1918 – February 15, 1988; surname pronounced ) was an American physicist known for expanding the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and particle theory. ... Freeman John Dyson (born December 15, 1923) is a British-born American physicist and mathematician, famous for his work in quantum mechanics, solid-state physics, nuclear weapons design and policy, and for his serious theorizing in futurism and science fiction concepts, including the search for extraterrestrial intelligence. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... // Recovering from World War I and its aftermath, the economic miracle emerged in West Germany and Italy. ...


Gauge theory was formulated and quantized, leading to the unification of forces embodied in the standard model of particle physics. This effort started in the 1950s with the work of Yang and Mills, was carried on by Martinus Veltman and a host of others during the 1960s and completed during the 1970s by the work of Gerard 't Hooft, Frank Wilczek, David Gross and David Politzer. In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... // Recovering from World War I and its aftermath, the economic miracle emerged in West Germany and Italy. ... Zhen-Ning Franklin Yang (Traditional Chinese: ; pinyin: ) (born 22 September[1], 1922) is a Chinese American physicist who worked on statistical mechanics and symmetry principles. ... Robert L. Mills (April 15, 1927 - October 27, 1999) was a physicist, specializing in quantum field theory, the theory of alloys, and many-body theory. ... Martinus J.G. Veltman (Tini for short) (born June 27, 1931) is a 1999 Nobel Prize in Physics laureate for elucidating the quantum structure of electroweak interactions in physics, work done at Utrecht University, The Netherlands. ... The 1960s decade refers to the years from January 1, 1960 to December 31, 1969, inclusive. ... The 1970s decade refers to the years from 1970 to 1979, inclusive. ... Gerard t Hooft at Harvard University Gerardus (Gerard) t Hooft (born July 5, 1946) is a professor in theoretical physics at Utrecht University, The Netherlands. ... Frank Wilczek (born May 15, 1951) is a Nobel prize winning American physicist. ... David Jonathan Gross (born February 19, 1941 in Washington, D.C.) is an American particle physicist and string theorist (although hes stated to the Brazilian newspaper Folha de São Paulo, on 09/27/2006, that the second area is included in the first one). ... Hugh David Politzer (born 31 August 1949) is an American theoretical physicist. ...


Parallel developments in the understanding of phase transitions in condensed matter physics led to the study of the renormalization group. This in turn led to the grand synthesis of theoretical physics which unified theories of particle and condensed matter physics through quantum field theory. This involved the work of Michael Fisher and Leo Kadanoff in the 1970s which led to the seminal reformulation of quantum field theory by Kenneth Wilson. In physics, a phase transition 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. ... In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... Leo Kadanoff is a professor of physics (emeritus as of 2004) at the University of Chicago. ... The 1970s decade refers to the years from 1970 to 1979, inclusive. ... Kenneth Geddes Wilson (born June 8, 1936) is an American physicist. ...


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. Physicists like Wilczek, Politzer, and Carl M. Bender are some of the foremost experts in the field. Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe. ... Many famous physicists of the 20th and 21st century are found on the list of recipients of the Nobel Prize in physics. ... Carl M. Bender is Professor of Physics at Washington University in St. ...


See also

List of quantum field theories: Phi to the fourth Quantum electrodynamics Schwinger model Yukawa model Wess-Zumino model Yang-Mills Quantum Yang-Mills theory Quantum chromodynamics Yang-Mills-Higgs model Nonlinear sigma model Chiral model Thirring model Sine-Gordon Chern-Simons model Topological quantum field theory Gross-Neveu Nambu-Jona... This article is about a formulation of quantum mechanics. ... Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... 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 of quantum field theory, is the notion that particles (such as electrons) can emit and absorb other particles (such as photons). ... The Abraham-Lorentz force is the average force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. ... 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 nonrelativistic particles. ... Invariance mechanics, in its simplest form, is the rewriting of the laws of quantum field theory in terms of invariant quantities only. ...

Suggested reading

  • Wilczek, Frank ; Quantum Field Theory, Review of Modern Physics 71 (1999) S85-S95. Review article written by a master of Q.C.D., Nobel laureate 2003. Full text available at : hep-th/9803075
  • Ryder, Lewis H. ; Quantum Field Theory (Cambridge University Press, 1985), [ISBN 0-521-33859-X]. Introduction to relativistic Q.F.T. for particle physics.
  • Zee, Anthony ; Quantum Field Theory in a Nutshell, Princeton University Press (2003) [ISBN 0-691-01019-6].
  • Peskin, M and Schroeder, D. ;An Introduction to Quantum Field Theory (Westview Press, 1995) [ISBN 0-201-50397-2]
  • Weinberg, Steven ; The Quantum Theory of Fields (3 volumes) Cambridge University Press (1995). A monumental treatise on Q.F.T. written by a leading expert, Nobel laureate 1979.
  • Loudon, Rodney ; The Quantum Theory of Light (Oxford University Press, 1983), [ISBN 0-19-851155-8]
  • D.A. Bromley (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3-540-67672-4. 
  • Paul H. Frampton , Gauge Field Theories, Frontiers in Physics, Addison-Wesley (1986), Second Edition, Wiley (2000).
  • Gordon L. Kane (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0-201-11749-5. 

Paul Frampton, Rubin Distinguished Professor. ...

External links



  Results from FactBites:
 
Quantum field theory - Wikipedia, the free encyclopedia (3793 words)
Quantum field theory originated in the problem of computing the power radiated by an atom when it dropped from one quantum state to another of lower energy.
QFT, on the other hand, deals with fields on a fundamental level and particles only emerge as localized excitations (aka quanta aka quasiparticles) of the ground state (aka the vacuum) and it's precisely these quantum fields which correspond to the operator valued functions.
QFT most definitely isn't the same thing as classical field theory or classical field theory with some "minor" quantum corrections, which is a mistake many high energy physicists are prone to making at times, especially when working in the semiclassical approximation.
  More results at FactBites »

 
 

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