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:''Not to be confused with [[Ferrimagnetism]]; for an overview see [[Magnetism]]''
[[Image:Feynmann Diagram Gluon Radiation.svg|thumb|287px|right|In this Feynman diagram, an [[electron]] and a [[positron]] [[annihilate]], producing a
[[photon]] (represented by the blue sine wave) that becomes a [[quark]]-[[antiquark]] pair. Then one radiates a [[gluon]] (represented by the green spiral).]]
'''Feynman diagrams''' are pictorial representations of the mathematical expressions governing the behavior of [[subatomic particle]]s. The scheme is named for its inventor, [[Nobel Prize]]-winning American physicist [[Richard Feynman]], and was first introduced in 1948. The interaction of sub-atomic particles can be complex and difficult to understand intuitively, and the Feynman diagrams allow for a simple visualization of what would otherwise be a rather arcane and abstract formula. As [[David Kaiser]] writes, "since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations," and as such "Feynman diagrams have revolutionized nearly every aspect of theoretical physics".<ref>[http://web.mit.edu/dikaiser/www/FdsAmSci.pdf "Physics and Feynman’s Diagrams" by David Kaiser, American Scientist, Volume 93, p. 156]</ref> While the diagrams are applied primarily to [[quantum field theory]], they can also be used in other fields, such as [[solid-state physics|solid-state theory]].
[[Image:MagnetEZ.jpg|thumb|A magnet made of [[alnico]], an iron alloy. Ferromagnetism is the physical theory which explains how materials become magnets.]]
'''Ferromagnetism''' is the basic mechanism by which certain materials (such as [[iron]]) form [[permanent magnet]]s, or are attracted to [[magnet]]s. In [[physics]], several different types of [[magnetism]] are distinguished. Ferromagnetism (including [[ferrimagnetism]])<ref>{{harvnb|Chikazumi|2009|p=118}}</ref> is the strongest type; it is the only type that creates forces strong enough to be felt, and is responsible for the common phenomena of magnetism [[Magnet#Common uses of magnets|encountered in everyday life]]. Other substances respond weakly to magnetic fields with two other types of magnetism, [[paramagnetism]] and [[diamagnetism]], but the forces are so weak that they can only be detected by sensitive instruments in a laboratory. An everyday example of ferromagnetism is a [[refrigerator magnet]] used to hold notes on a refrigerator door. The attraction between a magnet and ferromagnetic material is "the quality of magnetism first apparent to the ancient world, and to us today".<ref name="bozorth">Richard M. Bozorth, ''Ferromagnetism'', first published 1951, reprinted 1993 by [[IEEE]] Press, New York as a "Classic Reissue." ISBN 0-7803-1032-2.</ref>
The calculation of [[probability amplitude]]s in theoretical particle physics requires the use of rather large and complicated integrals over a large number of variables. These integrals do, however, have a regular structure, and may be represented graphically as Feynman diagrams. A Feynman diagram is a contribution of a particular class of particle paths, which join and split as described by the diagram. More precisely, and technically, a Feynman diagram is a graphical representation of a perturbative contribution to the [[transition amplitude]] or correlation function of a quantum mechanical or statistical field theory. Within the [[canonical quantization|canonical]] formulation of quantum field theory, a Feynman diagram represents a term in the [[Wick's theorem|Wick's expansion]] of the [[perturbative]] [[S-matrix]]. Alternatively, the [[path integral formulation]] of quantum field theory represents the transition amplitude as a weighted sum of all possible histories of the system from the initial to the final state, in terms of either particles or fields. The transition amplitude is then given as the matrix element of the S-matrix between the initial and the final states of the quantum system.
Permanent magnets (materials that can be [[Magnetization|magnetized]] by an external [[magnetic field]] and remain magnetized after the external field is removed) are either ferromagnetic or ferrimagnetic, as are other materials that are noticeably attracted to them. Only a few substances are ferromagnetic. The common ones are [[iron]], [[nickel]], [[cobalt]] and most of their alloys, some compounds of [[Rare earth magnet|rare earth metals]], and a few naturally-occurring minerals such as [[lodestone]].
{{Quantum field theory}}
Ferromagnetism is very important in industry and modern technology, and is the basis for many electrical and electromechanical devices such as [[electromagnet]]s, [[electric motor]]s, [[Electric generator|generators]], [[transformer]]s, and [[magnetic storage]] such as [[tape recorder]]s, and [[hard disk]]s.
== Motivation and history ==
==History and distinction from ferrimagnetism==
[[Image:Kaon-Decay.svg|301px|thumb|right|In this diagram, a [[kaon]], made of an up and anti-strange quark, decays both [[Weak interaction|weakly]] and strongly into three [[pion]]s, with intermediate steps involving a [[W and Z bosons|W boson]] and a [[gluon]] (represented by the green spiral).]]
Historically, the term ''ferromagnet'' was used for any material that could exhibit spontaneous magnetization: a net magnetic moment in the absence of an external magnetic field. This general definition is still in common use. More recently, however, different classes of spontaneous magnetization have been identified when there is more than one magnetic ion per [[primitive cell]] of the material, leading to a stricter definition of "ferromagnetism" that is often used to distinguish it from ferrimagnetism. In particular, a material is "ferromagnetic" in this narrower sense only if ''all'' of its magnetic ions add a positive contribution to the net magnetization. If some of the magnetic ions ''subtract'' from the net magnetization (if they are partially ''anti''-aligned), then the material is "ferrimagnetic".<ref>{{cite journal|last=Herrera|first=J. M.|coauthors=Bachschmidt, A, Villain, F, Bleuzen, A, Marvaud, V, Wernsdorfer, W, Verdaguer, M|title=Mixed valency and magnetism in cyanometallates and Prussian blue analogues|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|date=13 January 2008|volume=366|issue=1862|pages=127–138|doi=10.1098/rsta.2007.2145}}</ref> If the moments of the aligned and anti-aligned ions balance completely so as to have zero net magnetization, despite the magnetic [[Order (crystal lattice)|ordering]], then it is an [[antiferromagnet]]. These alignment effects only occur at [[temperature]]s below a certain critical temperature, called the [[Curie temperature]] (for ferromagnets and ferrimagnets) or the [[Néel temperature]] (for antiferromagnets).
When calculating [[scattering]] [[Cross section (physics)|cross section]]s in [[particle physics]], the interaction between particles can be described by starting from a [[free field]] which describes the incoming and outgoing particles, and including an interaction [[Hamiltonian (quantum mechanics)|Hamiltonian]] to describe how the particles deflect one another. The amplitude for scattering is the sum of each possible interaction history over all possible intermediate particle states. The number of times the interaction Hamiltonian acts is the order of the [[perturbation theory (quantum mechanics)|perturbation expansion]], and the time-dependent perturbation theory for fields is known as the [[Dyson series]]. When the intermediate states at intermediate times are energy [[Eigenvalues and eigenvectors|eigenstates]] (collections of particles with a definite momentum) the series is called [[Perturbation_theory_(quantum_mechanics)#Time-dependent perturbation theory|old-fashioned perturbation theory]].
Among the first investigations of ferromagnetism are the pioneering works of [[Aleksandr Stoletov]] on measurement of the [[magnetic permeability]] of ferromagnetics, known as the [[Stoletov curve]].
The Dyson series can be alternately rewritten as a sum over Feynman diagrams, where at each interaction vertex both the energy and momentum are conserved, but where the length of the energy momentum four vector is not equal to the mass. The Feynman diagrams are much easier to keep track of than old-fashioned terms, because the old-fashioned way treats the particle and antiparticle contributions as separate. Each Feynman diagram is the sum of exponentially many old-fashioned terms, because each internal line can separately represent either a particle or an antiparticle. In a non-relativistic theory, there are no antiparticles and there is no doubling, so each Feynman diagram includes only one term.
==Ferromagnetic materials==<!-- [[Ferromagnetic materials]] redirects here -->
{{See also|Category:Ferromagnetic materials}}
Feynman gave a prescription for calculating the amplitude for any given diagram from a field theory [[Lagrangian]] - the Feynman rules. Each internal line corresponds to a factor of the corresponding [[virtual particle]]'s [[propagator]]; each vertex where lines meet gives a factor derived from an interaction term in the Lagrangian, and incoming and outgoing lines carry an [[energy]], [[momentum]], and [[spin (physics)|spin]].
|+ style="font-size: 80%"|Curie temperatures for some crystalline ferromagnetic (* = ferrimagnetic) materials<ref>{{cite book|last=Kittel|first=Charles|author-link=Charles Kittel|title=Introduction to Solid State Physics|edition=sixth|publisher=[[John Wiley and Sons]]|year=1986|isbn=0-471-87474-4}}</ref>
| [[Yttrium iron garnet|Y<sub>3</sub>Fe<sub>5</sub>O<sub>12</sub>]]<sup>*</sup>
| 560
|-
| [[Chromium(IV) oxide|CrO<sub>2</sub>]]
| 386
|-
| Mn[[Arsenic|As]]
| 318
|-
| [[Gadolinium|Gd]]
| 292
|-
| [[Dysprosium|Dy]]
| 88
|-
| [[Europium|Eu]]O
| 69
|-
<!-- The numbers in this table currently come from Kittel, as referenced in the text. Please do not add new numbers without adding the corresponding reference. -->
|}
The table on the right lists a selection of ferromagnetic and ferrimagnetic compounds, along with the temperature above which they cease to exhibit spontaneous magnetization (see [[Ferromagnetism#Curie temperature|Curie temperature]]).
In addition to their value as a mathematical tool, Feynman diagrams provide deep physical insight into the nature of particle interactions. Particles interact in every way available; in fact, intermediate virtual particles are allowed to propagate faster than light. The probability of each final state is then obtained by summing over all such possibilities. This is closely tied to the [[functional integral]] formulation of [[quantum mechanics]], also invented by Feynman–see [[path integral formulation]].
Ferromagnetism is a property not just of the chemical make-up of a material, but of its crystalline structure and microscopic organization. There are ferromagnetic metal alloys whose constituents are not themselves ferromagnetic, called [[Heusler alloy]]s, named after [[Fritz Heusler]]. Conversely there are non-magnetic alloys, such as types of [[stainless steel]], composed almost exclusively of ferromagnetic metals.
The naïve application of such calculations often produces diagrams whose amplitudes are [[infinity|infinite]], because the short-distance particle interactions require a careful limiting procedure, to include particle [[self-interaction]]s. The technique of [[renormalization]], suggested by [[Ernst Stueckelberg]] and [[Hans Bethe]] and implemented by [[Freeman Dyson|Dyson]], Feynman, [[Julian Schwinger|Schwinger]], and [[Sin-Itiro Tomonaga|Tomonaga]] compensates for this effect and eliminates the troublesome infinities. After renormalization, calculations using Feynman diagrams match experimental results with very high accuracy.
One can also make amorphous (non-crystalline) ferromagnetic metallic alloys by very rapid [[quenching]] (cooling) of a liquid alloy. These have the advantage that their properties are nearly isotropic (not aligned along a crystal axis); this results in low coercivity, low [[hysteresis]] loss, high permeability, and high electrical resistivity. One such typical material is a transition metal-metalloid alloy, made from about 80% transition metal (usually Fe, Co, or Ni) and a metalloid component ([[Boron|B]], [[Carbon|C]], [[Silicon|Si]], [[Phosphorus|P]], or [[Aluminium|Al]]) that lowers the melting point.
<!--changes here are not correct; commenting out until sorted out on talk: One example of such an amorphous alloy is Fe<sub>80</sub>B<sub>20</sub> (Metglas 2605) which has a Curie temperature of 647 K and a room-temperature (300 K) saturation magnetization of 1.58 [[tesla (unit)|teslas]] (1,257 [[gauss]]), compared with 1,043 K and 2.15 T (1,707 gauss) for pure iron from above. The melting point, or more precisely the glass transition temperature, is only 714 K for the alloy versus a melting point of 1,811 K for pure iron.-->
Feynman diagram and path integral methods are also used in [[statistical mechanics]].
A relatively new class of exceptionally strong ferromagnetic materials are the [[rare-earth magnet]]s. They contain lanthanide elements that are known for their ability to carry large magnetic moments in well-localized f-orbitals.
=== Alternative names ===
===Actinide ferromagnets===
A number of [[actinide]] compounds are ferromagnets at room temperature or become ferromagnets below the Curie temperature (T<sub>C</sub>). [[Plutonium|Pu]][[Phosphorus|P]] is one actinide [[Nitrogen family|pnictide]] that is a paramagnet and has [[Cubic crystal system|cubic symmetry]] at room temperature, but upon cooling undergoes a lattice distortion to [[Tetragonal crystal system|tetragonal]] when cooled to below its T<sub>c</sub> = 125 K. PuP has an [[easy axis]] of <100>,<ref name=Lander>{{cite journal |author=Lander GH, Lam DJ |title=Neutron diffraction study of PuP: The electronic ground state |journal=Phys Rev B. |year=1976 |volume=14 |issue=9 |pages=4064–7 |doi=10.1103/PhysRevB.14.4064|bibcode = 1976PhRvB..14.4064L }}</ref> so that
[[Murray Gell-Mann]] always referred to Feynman diagrams as '''Stueckelberg diagrams''', after a Swiss physicist, [[Ernst Stueckelberg]], who devised a similar notation many years earlier. Stueckelberg was motivated by the need for a manifestly covariant formalism for quantum field theory, but did not provide as automated a way to handle symmetry factors and loops, although he was first to find the correct physical interpretation in terms of forward and backward in time particle paths, all without the path-integral.<ref>[http://www.theatlantic.com/issues/2000/07/johnson.htm The Jaguar and the Fox - 00.07<!-- Bot generated title -->]</ref> Historically they were sometimes called '''Feynman-Dyson diagrams''' or '''Dyson graphs''',<ref>Gribbin, John and Mary. ''Richard Feynman: A Life in Science'', Penguin-Putnam, 1997 Ch 5.</ref> because when they were introduced the path integral was unfamiliar, and [[Freeman Dyson]]'s derivation from old-fashioned perturbation theory was easier to follow for physicists trained in earlier methods. However, in 2006 Dyson himself confirmed that the diagrams should be called ''Feynman diagrams'' because "he taught us how to use them".{{Citation needed|date=April 2011}}
at 5 K.<ref name=Mueller>{{cite journal |author=Mueller MH, Lander GH, Hoff HA, Knott HW, Reddy JF |title=Lattice distortions measured in actinide ferromagnets PuP, NpFe<sub>2</sub>, and NpNi<sub>2</sub> |journal= J Phys Colloque C4, supplement |month=Apr |year=1979 |volume=40 |issue=4 |pages=C4–68–C4–69 |url=http://hal.archives-ouvertes.fr/docs/00/21/88/17/PDF/ajp-jphyscol197940C421.pdf}}</ref> The lattice distortion is presumably a consequence of strain induced by the magnetoelastic interactions as the [[magnetic moment]]s aligned parallel within [[magnetic domain]]s.
== Representation of physical reality ==
In [[Neptunium|Np]]Fe<sub>2</sub> the easy axis is <111>.<ref name=Aldred>{{cite journal |author=Aldred AT, Dunlap BD, Lam DJ, Lander GH, Mueller MH, Nowik I |title=Magnetic properties of neptunium Laves phases: NpMn<sub>2</sub>, NpFe<sub>2</sub>, NpCo<sub>2</sub>, and NpNi<sub>2</sub> |journal=Phys Rev B. |year=1975 |volume=11 |issue=1 |pages=530–44 |doi=10.1103/PhysRevB.11.530|bibcode = 1975PhRvB..11..530A }}</ref> Above T<sub>C</sub> ~500 K NpFe<sub>2</sub> is also paramagnetic and cubic. Cooling below the Curie temperature produces a rhombohedral distortion wherein the rhombohedral angle changes from 60° (cubic phase) to 60.53°. An alternate description of this distortion is to consider the length c along the unique trigonal axis (after the distortion has begun) and a as the distance in the plane perpendicular to c. In the cubic phase this reduces to <math>\scriptstyle\frac{c}{a}</math> = 1.00. Below the Curie temperature
In their presentations of [[fundamental interactions]],<ref>Gerardus 't Hooft, Martinus Veltman, ''Diagrammar'', CERN Yellow Report 1973, reprinted in G. ’t Hooft, Under the Spell of Gauge Principle (World Scientific, Singapore, 1994), Introduction [http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=1973-009 online]</ref><ref>Martinus Veltman, ''Diagrammatica: The Path to Feynman Diagrams'', Cambridge Lecture Notes in Physics, ISBN 0-521-45692-4</ref> written from the particle physics perspective, [[Gerard ’t Hooft]] and [[Martinus Veltman]] gave good arguments for taking the original, non-regularized Feynman diagrams as the most succinct representation of our present knowledge about the physics of quantum scattering of [[fundamental particles]]. Their motivations are consistent with the convictions of [[James Daniel Bjorken]] and [[Sidney Drell]]:<ref>{{cite journal |first=J. D. |last=Bjorken |first2=S. D. |last2=Drell |title=Relativistic Quantum Fields |publisher=McGraw-Hill |location=New York |year=1965 |page=viii }}</ref> ”The [[Feynman graphs]] and rules of calculation summarize [[quantum field theory]] in a form in close contact with the experimental numbers one wants to understand. Although the statement of the theory in terms of graphs may imply [[perturbation theory]], use of graphical methods in the [[many-body problem]] shows that this formalism is flexible enough to deal with phenomena of nonperturbative characters ... Some modification of the [[Feynman rules]] of calculation may well outlive the elaborate mathematical structure of local canonical quantum field theory ...” So far there are no opposing opinions.
In [[quantum field theories]] the [[Feynman diagrams]] are obtained from [[Lagrangian]] by [[#Feynman rules|Feynman Rules]].
A Feynman diagram is a representation of quantum field theory processes in terms of [[Elementary particle|particle]] paths. The particle trajectories are represented by the lines of the diagram, which can be squiggly or straight, with an arrow or without, depending on the type of particle. A point where lines connect to other lines is an interaction vertex, and this is where the particles meet and interact: by emitting or absorbing new particles, deflecting one another, or changing type.
which is the largest strain in any actinide compound.<ref name=Mueller/> NpNi<sub>2</sub> undergoes a similar lattice distortion below T<sub>C</sub> = 32 K, with a strain of (43 ± 5) × 10<sup>−4</sup>.<ref name=Mueller/> NpCo<sub>2</sub> is a ferrimagnet below 15 K.
There are three different types of lines: ''internal lines'' connect two vertices, ''incoming lines'' extend from "the past" to a vertex and represent an initial state, and ''outgoing lines'' extend from a vertex to "the future" and represent the final state. Sometimes, the bottom of the diagram is the past and the top the future; other times, the past is to the left and the future to the right. When calculating [[correlation functions]] instead of [[scattering amplitude]]s, there is no past and future and all the lines are internal. The particles then begin and end on little x's, which represent the positions of the operators whose correlation is being calculated.
===Lithium gas===
In 2009, a team of MIT physicists demonstrated that a lithium gas cooled to less than one Kelvin can exhibit ferromagnetism.<ref>{{cite journal |author=G-B Jo, Y-R Lee, J-H Choi, C. A. Christensen, T. H. Kim, J. H. Thywissen, D. E. Pritchard, and W. Ketterle |title=Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms |journal= Science |year=2009 |volume=325 |pages=1521–1524 |doi=10.1126/science.1177112 |pmid=19762638 |issue=5947 |bibcode = 2009Sci...325.1521J }}</ref> The team cooled [[fermion]]ic lithium-6 to less than 150 billionths of one Kelvin above absolute zero using infrared [[laser cooling]]. This demonstration is the first time that ferromagnetism has been demonstrated in a gas.
Feynman diagrams are a pictorial representation of a contribution to the total amplitude for a process which can happen in several different ways. When a group of incoming particles are to scatter off each other, the process can be thought of as one where the particles travel over all possible paths, including paths that go backward in time.
==Explanation==
Feynman diagrams are often confused with [[spacetime diagram]]s and [[bubble chamber]] images because they all describe particle scattering. Feynman diagrams are [[graph (mathematics)|graph]]s that represent the trajectories of particles in intermediate stages of a scattering process. Unlike a bubble chamber picture, only the sum of all the Feynman diagrams represent any given particle interaction; particles do not choose a particular diagram each time they interact. The law of summation is in accord with the [[principle of superposition]]--- every diagram contributes a factor to the total amplitude for the process.
The [[Bohr–van Leeuwen theorem]] shows that magnetism cannot occur in purely classical solids. Without [[quantum mechanics]], there would be no [[diamagnetism]], paramagnetism or ferromagnetism. The property of ferromagnetism is due to the direct influence of two effects from quantum mechanics: [[spin (physics)|spin]] and the [[Pauli exclusion principle]].<ref>{{cite book
|last = Feynman
|first = Richard P.
|coauthors = Robert Leighton, Matthew Sands
|title = The Feynman Lectures on Physics, Vol.2
|publisher = Addison-Wesley
|year = 1963
|location = USA
|pages = Ch. 37|isbn = 0-201-02011-4H}}</ref>
== Description ==
===Origin of magnetism===
One of the fundamental properties of an [[electron]] (besides that it carries charge) is that it has a [[Electron magnetic dipole moment|dipole moment]], i.e. it behaves itself as a tiny magnet. This dipole moment comes from the more fundamental property of the electron that it has quantum mechanical [[Spin (physics)|spin]]. The quantum mechanical nature of this spin causes the electron to only be able to be in two states, with the magnetic field either pointing "up" or "down" (for any choice of up and down). The spin of the electrons in atoms is the main source of ferromagnetism, although there is also a contribution from the [[planetary orbit|orbital]] [[angular momentum]] of the electron about the [[nucleus (atomic structure)|nucleus]]. When these tiny magnetic dipoles are aligned in the same direction, their individual magnetic fields add together to create a measurable macroscopic field.
A Feynman diagram represents a perturbative contribution to the amplitude of a quantum transition from some initial quantum state to some final quantum state.
However in materials with a filled [[electron shell]], the total dipole moment of the electrons is zero because the spins are in up/down pairs. Only atoms with partially filled shells (i.e., unpaired spins) can have a net magnetic moment, so ferromagnetism only occurs in materials with partially filled shells. Because of [[Hund's rules]], the first few electrons in a shell tend to have the same spin, thereby increasing the total dipole moment.
For example, in the process of electron-positron annihilation the initial state is one electron and one positron, the final state: two photons.
These unpaired dipoles (often called simply "spins" even though they also generally include angular momentum) tend to align in parallel to an external magnetic field, an effect called paramagnetism. Ferromagnetism involves an additional phenomenon, however: the dipoles tend to align spontaneously, giving rise to a [[spontaneous magnetization]], even when there is no applied field.
The initial state is often assumed to be at the left of the diagram and the final state at the right (although other conventions are also used quite often).
===Exchange interaction===
{{Main|Exchange interaction}}
According to classical [[electromagnetism]], two nearby magnetic dipoles will tend to align in ''opposite'' directions, so their magnetic fields will oppose one another and cancel out. However, this effect is very weak, because the magnetic fields generated by individual spins are small and the resulting alignment is easily destroyed by [[thermal fluctuations]]. In a few materials, a much stronger interaction between spins arises because the change in the direction of the spin leads to a change in [[electrostatic]] repulsion between neighboring electrons, due to a particular [[Quantum mechanics|quantum mechanical]] effect called the [[exchange interaction]]. At short distances, the exchange interaction is much stronger than the dipole-dipole magnetic interaction. As a result, in a few materials, the ferromagnetic ones, nearby spins tend to align in the same direction.
A Feynman diagram consists of points, called vertices, and lines attached to the vertices.
The exchange interaction is related to the [[Pauli exclusion principle]], which says that two electrons with the same spin cannot also have the same "position". Therefore, under certain conditions, when the [[atomic orbital|orbitals]] of the unpaired outer [[valence electron]]s from adjacent atoms overlap, the distributions of their electric charge in space are further apart when the electrons have parallel spins than when they have opposite spins. This reduces the [[electrostatic energy]] of the electrons when their spins are parallel compared to their energy when the spins are anti-parallel, so the parallel-spin state is more stable. In simple terms, the electrons, which repel one another, can move "further apart" by aligning their spins, so the spins of these electrons tend to line up. This difference in energy is called the [[exchange energy]].
The particles in the initial state are depicted by lines sticking out in the direction of the initial state (e.g., to the left), the particles in the final state are represented by lines sticking out in the direction of the final state (e.g., to the right).
The materials in which the exchange interaction is much stronger than the competing dipole-dipole interaction are frequently called ''magnetic materials''. For instance, in iron (Fe) the exchange force is about 1000 times stronger than the dipole interaction. Therefore below the Curie temperature virtually all of the dipoles in a ferromagnetic material will be aligned.
The [[exchange interaction]] is also responsible for the other types of spontaneous ordering of atomic magnetic moments occurring in magnetic solids, [[antiferromagnetism]] and ferrimagnetism.
There are different exchange interaction mechanisms which create the magnetism in different ferromagnetic, ferrimagnetic, and antiferromagnetic substances. These mechanisms include [[Exchange_interaction#Direct_exchange_interactions_in_solids|direct exchange]], [[RKKY interaction|RKKY exchange]], [[double exchange]], and [[superexchange]].
In [[Quantum electrodynamics|QED]] there are two types of particles: electrons/positrons (called [[fermions]]) and photons (called [[gauge bosons]]). They are represented in Feynman diagrams as follows:
===Magnetic anisotropy===
#Electron in the initial state is represented by a solid line with an arrow pointing toward the vertex (→•).
{{Main|Magnetic anisotropy}}
#Electron in the final state is represented by a line with an arrow pointing away from the vertex: (•→).
#Positron in the initial state is represented by a solid line with an arrow pointing away from the vertex: (←•).
#Positron in the final state is represented by a line with an arrow pointing toward the vertex: (•←).
#Photon in the initial and the final state is represented by a wavy line (<big>~•</big> and <big>•~</big>).
In QED a vertex always has three lines attached to it: one bosonic line, one fermionic line with arrow toward the vertex, and one fermionic line with arrow away from the vertex.
Although the exchange interaction keeps spins aligned, it does not align them in a particular direction. Without [[magnetic anisotropy]], the spins in a magnet randomly change direction in response to [[thermal fluctuations]] and the magnet is [[superparamagnetic]]. There are several kinds of magnetic anisotropy, the most common of which is [[magnetocrystalline anisotropy]]. This is a dependence of the energy on the direction of magnetization relative to the [[crystallographic lattice]]. Another common source of anisotropy, [[inverse magnetostriction]], is induced by internal [[deformation (mechanics)|strains]]. [[Single-domain (magnetic)|Single-domain magnets]] also can have a ''shape anisotropy'' due to the magnetostatic effects of the particle shape. As the temperature of a magnet increases, the anisotropy tends to decrease, and there is often a [[superparamagnetism#Blocking temperature|blocking temperature]] at which a transition to superparamagnetism occurs.<ref name=Aharoni>{{cite book|last = Aharoni|first = Amikam|author-link=Amikam Aharoni|title=Introduction to the Theory of Ferromagnetism|publisher=[[Clarendon Press]]|year = 1996|isbn=0-19-851791-2|url=http://www.oup.com/us/catalog/general/subject/Physics/ElectricityMagnetism/?view=usa&ci=9780198508090}}</ref>
The vertices might be connected by a bosonic or fermionic [[propagator]]. A bosonic propagator is represented by a wavy line connecting two vertexes (•~•). A fermionic propagator is represented by a solid line (with an arrow in one or another direction) connecting two vertexes, (•←•).
===Magnetic domains===
{{Main|Magnetic domain}}
The above would seem to suggest that every piece of ferromagnetic material should have a strong magnetic field, since all the spins are aligned, yet iron and other ferromagnets are often found in an "unmagnetized" state. [[Image:Weiss-Bezirke1.png|thumb|Weiss domains microstructure]] The reason for this is that a bulk piece of ferromagnetic material is divided into tiny ''[[magnetic domains]]''<ref name="Feynman">{{cite book
| isbn = 0-201-02117-XP}}</ref> (also known as ''Weiss domains''). Within each domain, the spins are aligned, but (if the bulk material is in its lowest energy configuration, i.e. ''unmagnetized''), the spins of separate domains point in different directions and their magnetic fields cancel out, so the object has no net large scale magnetic field.
The number of vertices gives the order of the term in the perturbation series expansion of the transition amplitude.
Ferromagnetic materials spontaneously divide into magnetic domains because the ''[[exchange interaction]]'' is a short-range force, so over long distances of many atoms the tendency of the magnetic dipoles to reduce their energy by orienting in opposite directions wins out. If all the dipoles in a piece of ferromagnetic material are aligned parallel, it creates a large magnetic field extending into the space around it. This contains a lot of [[magnetostatics|magnetostatic]] energy. The material can reduce this energy by splitting into many domains pointing in different directions, so the magnetic field is confined to small local fields in the material, reducing the volume of the field. The domains are separated by thin [[domain wall]]s a number of molecules thick, in which the direction of magnetization of the dipoles rotates smoothly from one domain's direction to the other.
===Electron-positron annihilation example===
Thus, a piece of iron in its lowest energy state ("unmagnetized") generally has little or no net magnetic field. However, if it is placed in a strong enough external magnetic field, the domain walls will move, reorienting the domains so more of the dipoles are aligned with the external field. The domains will remain aligned when the external field is removed, creating a magnetic field of their own extending into the space around the material, thus creating a "permanent" magnet. The domains do not go back to their original minimum energy configuration when the field is removed because the domain walls tend to become 'pinned' or 'snagged' on defects in the crystal lattice, preserving their parallel orientation. This is shown by the [[Barkhausen effect]]: as the magnetizing field is changed, the magnetization changes in thousands of tiny discontinuous jumps as the domain walls suddenly "snap" past defects.
[[Image:Feynman EP Annihilation.svg|thumb|Feynman Diagram of Electron-Positron Annihilation]]
The electron-positron annihilation interaction:
This magnetization as a function of the external field is described by a [[Hysteresis loop|hysteresis curve]]. Although this state of aligned domains found in a piece of magnetized ferromagnetic material is not a minimal-energy configuration, it is [[metastable]], and can persist for long periods, as shown by samples of [[magnetite]] from the sea floor which have maintained their magnetization for millions of years.
<math>e^+e^-\to2\gamma</math>
Alloys used for the strongest permanent magnets are "hard" alloys made with many defects in their crystal structure where the domain walls "catch" and stabilize. The net magnetization can be destroyed by heating and then cooling ([[Annealing (metallurgy)|annealing]]) the material without an external field, however. The thermal motion allows the domain boundaries to move, releasing them from any defects, to return to their low-energy unaligned state.
has a contribution from the second order Feynman diagram shown adjacent:
===Curie temperature===
{{Main|Curie temperature}}
As the temperature increases, thermal motion, or [[entropy]], competes with the ferromagnetic tendency for dipoles to align. When the temperature rises beyond a certain point, called the '''Curie temperature''', there is a second-order [[phase transition]] and the system can no longer maintain a spontaneous magnetization, although it still responds paramagnetically to an external field. Below that temperature, there is a [[spontaneous symmetry breaking]] and random domains form (in the absence of an external field). The Curie temperature itself is a [[critical point (thermodynamics)|critical point]], where the [[magnetic susceptibility]] is theoretically infinite and, although there is no net magnetization, domain-like spin correlations fluctuate at all length scales.
In the initial state (at the bottom; early time) there is one electron (e<sup>-</sup>) and one positron (e<sup>+</sup>) and in the final state (at the top; late time) there are two photons (γ).
The study of ferromagnetic phase transitions, especially via the simplified [[Ising model|Ising]] spin model, had an important impact on the development of statistical physics. There, it was first clearly shown that [[mean field theory]] approaches failed to predict the correct behavior at the critical point (which was found to fall under a ''universality class'' that includes many other systems, such as liquid-gas transitions), and had to be replaced by [[renormalization group]] theory.
{{clr}}
==See also==
*[[Ferromagnetic material properties]]
== Canonical quantization formulation ==
*[[Thermo-magnetic motor]]
Perturbative S-matrix
The [[probability amplitude]] for a transition of a quantum system from the initial state <math>|i\rangle</math> to the final state <math>|f\rangle</math> is given by the matrix element
:<math>S_{fi}=\langle f|S|i\rangle\;,</math>
where <math>S</math> is the [[S-matrix]].
In the canonical quantum field theory the S-matrix is represented within the [[interaction picture]] by the perturbation series in the powers of the interaction Lagrangian,
where <math>L_v</math> is the interaction Lagrangian and <math>T</math> signifies the time-ordered product of operators.
A Feynman diagram is a graphical representation of a term in the Wick's expansion of the time-ordered product in the <math>n</math>-th order term <math>S^{(n)}</math> of the S-matrix,
where <math>N</math> signifies the normal-product of the operators and <math>(\pm)</math> takes care of the possible sign change when commuting the fermionic operators to bring them together for a contraction (a [[propagator]]).
=== Feynman rules ===
The diagrams are drawn according to the Feynman rules which depend upon the interaction Lagrangian. For the [[Quantum electrodynamics|QED]] interaction Lagrangian, <math>L_v=-g\bar\psi\gamma^\mu\psi A_\mu</math>, describing the interaction of a fermionic field <math>\psi</math> with a bosonic gauge field <math>A_\mu</math>, the Feynman rules can be formulated in coordinate space as follows:
# Each integration coordinate <math>x_j</math> is represented by a point (sometimes called a vertex);
# A bosonic [[propagator]] is represented by a wiggly line connecting two points;
# A fermionic propagator is represented by a solid line connecting two points;
# A bosonic field <math>A_\mu(x_i)</math> is represented by a wiggly line attached to the point <math>x_i</math>;
# A fermionic field <math>\psi(x_i)</math> is represented by a solid line attached to the point <math>x_i</math> with an arrow toward the point;
# A fermionic field <math>\bar\psi(x_i)</math> is represented by a solid line attached to the point <math>x_i</math> with an arrow from the point;
=== Example: second order processes in QED ===
The second order perturbation term in the S-matrix is
|[[Image:Feynman-diagram-ee-scattering.png|right|thumb|360px|The Feynman diagram of the term <math>N\bar\psi(x)ie\gamma^\mu\psi(x)\bar\psi(x')ie\gamma^\nu\psi(x')\underline{A_\mu(x)A_\nu(x')}</math>]]
|}
The Wick's expansion of the integrand gives (among others) the following term
is the electromagnetic contraction (propagator) in the Feynman gauge. This term is represented by the Feynman diagram at the right. This diagram gives contributions to the following processes:
# <math>e^-e^-</math> scattering (initial state at the right, final state at the left of the diagram);
# <math>e^+e^+</math> scattering (initial state at the left, final state at the right of the diagram);
# <math>e^-e^+</math> scattering (initial state at the bottom/top, final state at the top/bottom of the diagram).
==== Compton scattering and annihilation/generation of <math>e^-e^+</math> pairs ====
In a path-integral, the field Lagrangian, integrated over all possible field histories, defines the probability amplitude to go from one field configuration to another. In order to make sense, the field theory should have a well-defined ground state, and the integral should be performed a little bit rotated into imaginary time.
=== Scalar Field Lagrangian ===
A simple example is the free relativistic scalar field in d-dimensions, whose action integral is:
where A and B are space-like hypersurfaces which define the boundary conditions. The collection of all the <math>\,\phi(A)</math> on the starting hypersurface give the initial value of the field, analogous to the starting position for a point particle, and the field values <math>\,\phi(B)</math> at each point of the final hypersurface defines the final field value, which is allowed to vary, giving a different amplitude to end up at different values. This is the field-to-field transition amplitude.
The path integral gives the expectation value of operators between the initial and final state:
and in the limit that A and B recede to the infinite past and the infinite future, the only contribution that matters is from the ground state (this is only rigorously true if the path-integral is defined slightly rotated into imaginary time). The path integral should be thought of as analogous to a probability distribution, and it is convenient to define it so that multiplying by a constant doesn't change anything:
The normalization factor on the bottom is called the ''partition function'' for the field, and it coincides with the statistical mechanical partition function at zero temperature when rotated into imaginary time.
The initial-to-final amplitudes are ill-defined if one thinks of the continuum limit right from the beginning, because the fluctuations in the field can become unbounded. So the path-integral should be thought of as on a discrete square lattice, with lattice spacing <math>a</math> and the limit <math>a\rightarrow 0</math> should be taken carefully. If the final results do not depend on the shape of the lattice or the value of a, then the continuum limit exists.
===On a lattice ...===
On a lattice, (i), the field can be expanded in Fourier modes:
Here the integration domain is over k restricted to a cube of side length <math>2\pi/a</math>, so that large values of k are not allowed. It is important to note that the k-measure contains the factors of <math>2\pi</math> from Fourier transforms, this is the best standard convention for k-integrals in QFT. The lattice means that fluctuations at large k are not allowed to contribute right away, they only start to contribute in the limit <math>a\rightarrow 0</math>. Sometimes, instead of a lattice, the field modes are just cut off at high values of k instead.
It is also convenient from time to time to consider the space-time volume to be finite, so that the k modes are also a lattice. This is not strictly as necessary as the space-lattice limit, because interactions in k are not localized, but it is convenient for keeping track of the factors in front of the k-integrals and the momentum-conserving delta functions which will arise.
On a lattice, (ii), the action needs to be discretized:
where <math><x,y></math> is a pair of nearest lattice neighbors <math>x</math> and <math>y</math>. The discretization should be thought of as defining what the derivative <math>\partial_\mu \phi</math> means.
In terms of the lattice Fourier modes, the action can be written:
Now we have the continuum Fourier transform of the original action. In finite volume, the quantity <math>d^dk</math> is not infinitesimal, but becomes the volume of a box made by neighboring Fourier modes, or <math>(2\pi/V)^d</math>.
The field <math>\,\phi</math> is real-valued, so the Fourier transform obeys:
::<math> \phi(k)^* = \phi(-k)\,.</math>
In terms of real and imaginary parts, the real part of <math>\,\phi(k)</math> is an [[even function]] of k, while the imaginary part is odd. The Fourier transform avoids double-counting, so that it can be written:
::<math> S = \int_k {1\over 2} k^2 \phi(k) \phi(-k)</math>
over an integration domain which integrates over each pair (k,-k) exactly once.
For a complex scalar field with action
::<math> S = \int {1\over 2} \partial_\mu\phi^* \partial^\mu\phi d^dx</math>
the Fourier transform is unconstrained:
::<math> S = \int_k {1\over 2} k^2 |\phi(k)|^2</math>
and the integral is over all k.
Integrating over all different values of <math>\,\phi(x)</math> is equivalent to integrating over all Fourier modes, because taking a Fourier transform is a unitary linear transformation of field coordinates. When you change coordinates in a multidimensional integral by a linear transformation, the value of the new integral is given by the determinant of the transformation matrix. If
so that <math>\det A = \pm 1</math>, and the sign depends on whether the rotation includes a reflection or not.
The matrix which changes coordinates from <math>\,\phi(x)</math> to <math>\,\phi(k)</math> can be read off from the definition of a Fourier transform.
::<math> A_{kx} = e^{ikx} \,</math>
and the Fourier inversion theorem tells you the inverse:
::<math> A^{-1}_{kx} = e^{-ikx} \,</math>
which is the complex conjugate-transpose, up to factors of <math>2\pi</math>. On a finite volume lattice, the determinant is nonzero and independent of the field values.
::<math> \det A = 1 \,</math>
and the path integral is a separate factor at each value of k.
The factor <math>\scriptstyle d^dk</math> is the infinitesimal volume of a discrete cell in k-space, in a square lattice box <math>\scriptstyle d^dk = {1/L}^d</math>, where L is the side-length of the box. Each separate factor is an oscillatory Gaussian, and the width of the Gaussian diverges as the volume goes to infinity.
In imaginary time, the ''Euclidean action'' becomes positive definite, and can be interpreted as a probability distribution. The probability of a field having values <math>\phi_k </math> is
Since the probability of <math>\,\phi_k</math> is a product, the value of <math>\,\phi(k)</math> at each separate value of k is independently Gaussian distributed. The variance of the Gaussian is 1/ (k^2 d^dk), which is formally infinite, but that just means that the fluctuations are unbounded in infinite volume. In any finite volume, the integral is replaced by a discrete sum, and the variance of the integral is <math>V/k^2</math>.
=== Monte-Carlo ===
The path integral defines a probabilistic algorithm to generate a Euclidean scalar field configuration. Randomly pick the real and imaginary parts of each Fourier mode at wavenumber k to be a gaussian random variable with variance <math>1/k^2</math>. This generates a configuration <math>\,\phi_C(k)</math> at random, and the Fourier transform gives <math>\,\phi_C(x)</math>. For real scalar fields, the algorithm must generate only one of each pair <math>\,\phi(k),\phi(-k)</math>, and make the second the complex conjugate of the first.
To find any correlation function, generate a field again and again by this procedure, and find the statistical average:
where <math>|C|</math> is the number of configurations, and the sum is of the product of the field values on each configuration. The Euclidean correlation function is just the same as the correlation function in statistics or statistical mechanics. The quantum mechanical correlation functions are an analytic continuation of the Euclidean correlation functions.
For free fields with a quadratic action, the probability distribution is a high-dimensional Gaussian, and the statistical average is given by an explicit formula. But the [[Monte Carlo method]] also works well for bosonic interacting field theories where there is no closed form for the correlation functions.
=== Scalar Propagator ===
Each mode is independently Gaussian distributed. The expectation of field modes is easy to calculate:
But near k=0, for field fluctuations long compared to the lattice spacing, the two forms coincide.
It is important to emphasize that the delta functions contain factors of <math>2\pi</math>, so that they cancel out the <math>2\pi</math> factors in the measure for k integrals.
where <math>\delta_D(k)</math> is the ordinary one-dimensional Dirac delta function. This convention for delta-functions is not universal--- some authors keep the factors of <math>2\pi</math> in the delta functions (and in the k-integration) explicit.
=== Equation of Motion ===
The form of the propagator can be more easily found by using the equation of motion for the field. From the Lagrangian, the equation of motion is:
Where the derivatives act on x, and the identity is true everywhere except when x and y coincide, and the operator order matters. The form of the singularity can be understood from the canonical commutation relations to be a delta-function. Defining the (euclidean) ''Feynman propagator'' <math>\Delta</math> as the Fourier transform of the time-ordered two-point function (the one that comes from the path-integral):
If the equations of motion are linear, the propagator will always be the reciprocal of the quadratic-form matrix which defines the free Lagrangian, since this gives the equations of motion. This is also easy to see directly from the Path integral. The factor of i disappears in the Euclidean theory.
==== Wick Theorem ====
Because each field mode is an independent Gaussian, the expectation values for the product of many field modes obeys ''Wick's theorem'':
is zero unless the field modes coincide in pairs. This means that it is zero for an odd number of <math>\phi</math>'s, and for an even number of phi's, it is equal to a contribution from each pair separately, with a delta function.
An interpretation of Wick's theorem is that each field insertion can be thought of as a dangling line, and the expectation value is calculated by linking up the lines in pairs, putting a delta function factor that ensures that the momentum of each partner in the pair is equal, and dividing by the propagator.
There is a subtle point left before Wick's theorem is proved--- what if more than two of the phi's have the same momentum? If its an odd number, the integral is zero, negative values cancel with the positive values, But if the number is even, the integral is positive. The previous demonstration assumed that the phi's would only match up in pairs.
But the theorem is correct even when arbitrarily many of the phis are equal, and this is a notable property of Gaussian integration:
::<math> I = \int e^{-ax^2/2} = \sqrt{2\pi\over a} </math>
where the x-s are all the same variable, the index is just to keep track of the number of ways to pair them. The first x can be paired with 2n-1 others, leaving 2n-2. The next unpaired x can be paired with 2n-3 different x's leaving 2n-4, and so on. This means that Wick's theorem, uncorrected, says that the expectation value of <math>x^{2n}</math> should be:
and this is in fact the correct answer. So Wick's theorem holds no matter how many of the momenta of the internal variables coincide.
==== Interaction ====
Interactions are represented by higher order contributions, since quadratic contributions are always Gaussian. The simplest interaction is the quartic self-interaction, with an action:
Where <math>S_F</math> is the free action, whose correlation functions are given by Wick's theorem. The exponential of S in the path integral can be expanded in powers of <math>\lambda</math>, giving a series of corrections to the free action.
::<math> e^{-S} = e^{-S_F} ( 1 + X + {1\over 2!} X X + {1\over 3!} X X X + ... ) </math>
The path integral for the interacting action is then a power series of corrections to the free action. The term represented by X should be thought of as four half-lines, one for each factor of <math>\phi(k)</math>. The half-lines meet at a vertex, which contributes a delta-function which ensures that the sum of the momenta are all equal.
To compute a correlation function in the interacting theory, there is a contribution from the X terms now. For example, the path-integral for the four-field correlator:
which in the free field was only nonzero when the momenta k were equal in pairs, is now nonzero for all values of the k. The momenta of the insertions <math>\phi(k_i)</math> can now match up with the momenta of the X's in the expansion. The insertions should also be thought of as half-lines, four in this case, which carry a momentum k, but one which is not integrated.
The lowest order contribution comes from the first nontrivial term <math> e^{-S_F} X </math> in the Taylor expansion of the action. Wick's theorem requires that the momenta in the X half-lines, the <math>\phi(k)</math> factors in X, should match up with the momenta of the external half-lines in pairs. The new contribution is equal to:
The 4! inside X is canceled because there are exactly 4! ways to match the half-lines in X to the external half-lines. Each of these different ways of matching the half-lines together in pairs contributes exactly once, regardless of the values of the k's, by Wick's theorem.
==== Feynman Diagrams ====
The expansion of the action in powers of X gives a series of terms with progressively higher number of X's. The contribution from the term with exactly n X's are called n-th order.
The n-th order terms has:
# 4n internal half-lines, which are the factors of <math>\phi(k)</math> from the X's. These all end on a vertex, and are integrated over all possible k.
# external half-lines, which are the come from the <math>\phi(k)</math> insertions in the integral.
By Wick's theorem, each pair of half-lines must be paired together to make a ''line'', and this line gives a factor of
::<math> \delta(k_1 + k_2) \over k_1^2 </math>
which multiplies the contribution. This means that the two half-lines that make a line are forced to have equal and opposite momentum. The line itself should be labelled by an arrow, drawn parallel to the line, and labeled by the momentum in the line k. The half-line at the tail end of the arrow carries momentum k, while the half-line at the head-end carries momentum -k. If one of the two half-lines is external, this kills the integral over the internal k, since it forces the internal k to be equal to the external k. If both are internal, the integral over k remains.
The diagrams which are formed by linking the half-lines in the X's with the external half-lines, representing insertions, are the Feynman diagrams of this theory. Each line carries a factor of <math>1\over k^2</math>, the propagator, and either goes from vertex to vertex, or ends at an insertion. If it is internal, it is integrated over. At each vertex, the total incoming k is equal to the total outgoing k.
The number of ways of making a diagram by joining half-lines into lines almost completely cancels the factorial factors coming from the Taylor series of the exponential and the 4! at each vertex.
==== Loop Order ====
A forest diagram is one where all the internal lines have momentum which is completely determined by the external lines and the condition that the incoming and outgoing momentum are equal at each vertex. The contribution of these diagrams is a product of propagators, without any integration. A tree diagram is a connected forest diagram.
An example of a tree diagram is the one where each of four external lines end on an X. Another is when three external lines end on an X, and the remaining half-line joins up with another X, and the remaining half-lines of this
X run off to external lines. These are all also forest diagrams (as every tree is a forest); an example of a forest which is not a tree is when eight external lines end on two X's.
It is easy to verify that in all these cases, the momenta on all the internal lines is determined by the external momenta and the condition of momentum conservation in each vertex.
A diagram which is not a forest diagram is called a ''loop'' diagram, and an example is one where two lines of an X are joined to external lines, while the remaining two lines are joined to each other. The two lines joined to each other can have any momentum at all, since they both enter and leave the same vertex. A more complicated example is one where two X's are joined to each other by matching the legs one to the other. This diagram has no external lines at all.
The reason loop diagrams are called loop diagrams is because the number of k-integrals which are left undetermined by momentum conservation is equal to the number of independent closed loops in the diagram, where independent loops are counted as in [[homology theory]]. The homology is real-valued (actually R^d valued), the value associated with each line is the momentum. The boundary operator takes each line to the sum of the end-vertices with a positive sign at the head and a negative sign at the tail. The condition that the momentum is conserved is exactly the condition that the boundary of the k-valued weighted graph is zero.
A set of k-values can be relabeled whenever there is a closed loop going from vertex to vertex, never revisiting the same vertex. Such a cycle can be thought of as the boundary of a 2-cell. The k-labelings of a graph which conserve momentum (which have zero boundary) up to redefinitions of k (up to boundaries of 2-cells) define the first homology of a graph. The number of independent momenta which are not determined is then equal to the number of independent homology loops. For many graphs, this is equal to the number of loops as counted in the most intuitive way.
==== Symmetry factors ====
The number of ways to form a given Feynman diagram by joining together half-lines is large, and by Wick's theorem, each way of pairing up the half-lines contributes equally. Often, this completely cancels the factorials in the denominator of each term, but the cancellation is sometimes incomplete.
The uncancelled denominator is called the ''symmetry factor'' of the diagram. The contribution of each diagram to the correlation function must be divided by its symmetry factor.
For example, consider the Feynman diagram formed from two external lines joined to one X, and the remaining two half-lines in the X joined to each other. There are 4*3 ways to join the external half-lines to the X, and then there is only one way to join the two remaining lines to each other. The X comes divided by 4!=4*3*2, but the number of ways to link up the X half lines to make the diagram is only 4*3, so the contribution of this diagram is divided by two.
For another example, consider the diagram formed by joining all the half-lines of one X to all the half-lines of another X. This diagram is called a ''vacuum bubble'', because it does not link up to any external lines. There are 4! ways to form this diagram, but the denominator includes a 2! (from the expansion of the exponential, there are two X's) and two factors of 4!. The contribution is multiplied by 4!/(2*4!*4!) = 1/48.
Another example is the Feynman diagram formed from two X's where each X links up to two external lines, and the remaining two half-lines of each X are joined to each other. The number of ways to link an X to two external lines is 4*3, and either X could link up to either pair, giving an additional factor of 2. The remaining two half-lines in the two X's can be linked to each other in two ways, so that the total number of ways to form the diagram is 4*3*4*3*2*2, while the denominator is 4!4!2!. The total symmetry factor is 2, and the contribution of this diagram is divided by two.
The symmetry factor theorem gives the symmetry factor for a general diagram: the contribution of each Feynman diagram must be divided by the order of its group of automorphisms, the number of symmetries that it has.
An [[automorphism]] of a Feynman graph is a permutation M of the lines and a permutation N of the vertices with the following properties:
# If a line l goes from vertex v to vertex v', then M(l) goes from N(v) to N(v'). If the line is undirected, as it is for a real scalar field, then M(l) can go from N(v') to N(v) too.
# If a line l ends on an external line, M(l) ends on the same external line.
# If there are different types of lines, M(l) should preserve the type.
This theorem has an interpretation in terms of particle-paths: when identical particles are present, the integral over all intermediate particles must not double-count states which only differ by interchanging identical particles.
Proof: To prove this theorem, label all the internal and external lines of a diagram with a unique name. Then form the diagram by linking the a half-line to a name and then to the other half line.
Now count the number of ways to form the named diagram. Each permutation of the X's gives a different pattern of linking names to half-lines, and this is a factor of n!. Each permutation of the half-lines in a single X gives a factor of 4!. So a named diagram can be formed in exactly as many ways as the denominator of the Feynman expansion.
But the number of unnamed diagrams is smaller than the number of named diagram by the order of the automorphism group of the graph.
Roughly speaking, a Feynman diagram is called ''connected'' if all vertices and propagator lines are linked by a sequence of vertices and propagators of the diagram itself. If one views it as a [[Graph (mathematics)|(undirected) graph]] it is connected. The remarkable relevance of such diagrams in QFTs is due to the fact that they are sufficient to determine the [[Partition function (quantum field theory)|quantum partition function]] <math>Z[J]</math>. More precisely, connected Feynman diagrams determine
::<math>i W[J]\equiv \ln Z[J].</math>
To see this, one should recall that
::<math> Z[J]\propto\sum_k{D_k}</math>
with <math> D_k </math> constructed from some (arbitrary) Feynman diagram which can be thought to consist of several connected components <math> C_i </math>. If one encounters <math> n_i </math> (identical) copies of a component <math> C_i </math> within the Feynman diagram <math> D_k </math> one has to include a ''symmetry factor'' <math> n_i! </math>. However, in the end each contribution of a Feynman diagram <math> D_k </math> to the partition function has the generic form
::<math>\prod_i {C_{i}^{n_i} \over n_i!} </math>
where <math>i</math> labels the (infinite) many connected Feynman diagrams possible.
A scheme to successively create such contributions from the <math> D_k </math> to <math> Z[J] </math> is obtained by
To establish the ''normalization'' <math>Z_0=\exp{W[0]}=1</math> one simply calculates all connected ''vacuum diagrams'', i.e., the diagrams without any ''sources'' <math>J</math> (sometimes referred to as ''external legs'' of a Feynman diagram).
==== Vacuum Bubbles ====
An immediate consequence of the linked-cluster theorem is that all vacuum bubbles, diagrams without external lines cancel when calculating correlation functions. A correlation function is given by a ratio of path-integrals:
The top is the sum over all Feynman diagrams, including disconnected diagrams which do not link up to external lines at all. In terms of the connected diagrams, the numerator includes the same contributions of vacuum bubbles as the denominator:
Where the sum over E diagrams includes only those diagrams each of whose connected components end on at least one external line. The vacuum bubbles are the same whatever the external lines, and give an overall multiplicative factor. The denominator is the sum over all vacuum bubbles, and dividing gets rid of the second factor.
The vacuum bubbles then are only useful for determining Z itself, which from the definition of the path integral is equal to:
where <math>\rho</math> is the energy density in the vacuum. Each vacuum bubble contains a factor of <math>\delta(k)</math> zeroing the total k at each vertex, and when there are no external lines, this contains a factor of <math>\delta(0)</math>, because the momentum conservation is over-enforced. In finite volume, this factor can be identified as the total volume of space time. Dividing by the volume, the remaining integral for the vacuum bubble has an interpretation: it is a contribution to the energy density of the vacuum.
==== Sources ====
Correlation functions are the sum of the connected Feynman diagrams, but the formalism treats the connected and disconnected diagrams differently. Internal lines end on vertices, while external lines go off to insertions. Introducing ''sources'' unifies the formalism, by making new vertices where one line can end.
Sources are external fields, fields which contribute to the action, but are not dynamical variables. A scalar field source is another scalar field h which contributes a term to the (Lorentz) Lagrangian:
In the Feynman expansion, this contributes H terms with one half-line ending on a vertex. Lines in a Feynman diagram can now end either on an X vertex, or on an H-vertex, and only one line enters an H vertex. The Feynman rule for an H-vertex is that a line from an H with momentum k gets a factor of h(k).
The sum of the connected diagrams in the presence of sources includes a term for each connected diagram in the absence of sources, except now the diagrams can end on the source. Traditionally, a source is represented by a little "x" with one line extending out, exactly as an insertion.
where <math> C(k_1,....,k_n) </math> is the connected diagram with n external lines carrying momentum as indicated. The sum is over all connected diagrams, as before.
The field h is not dynamical, which means that there is no path integral over h: h is just a parameter in the Lagrangian which varies from point to point. The path integral for the field is:
and it is a function of the values of h at every point. One way to interpret this expression is that it is taking the Fourier transform in field space. If there is a probability density on R^n, the Fourier transform of the probability density is:
::<math> \int \rho(y) e^{i k y} d^n y = \langle e^{i k y} \rangle = \langle \prod_{i=1}^{n} e^{ih_i y_i}\rangle \,</math>
The fourier transform is the expectation of an oscillatory exponential. The path integral in the presence of a source h(x) is:
Which integrates over the Fourier transform coordinate, over h. This expression is useful for formally changing field coordinates in the path integral, much as a delta function is used to change coordinates in an ordinary multi-dimensional integral.
The partition function is now a function of the field h, and the physical partition function is the value when h is the zero function:
The correlation functions are derivatives of the path integral with respect to the source:
In Euclidean space, source contributions to the action can still appear with a factor of "i", so that they still do a Fourier transform.
===Spin 1/2; "Photons" and "Ghosts"===
==== Spin 1/2: Grassmann integrals ====
The field path-integral can be extended to the Fermi case, but only if the notion of integration is expanded. A [[Berezin integral|Grassman integral]] of a free Fermi field is a high-dimensional [[determinant]] or [[Pfaffian]] which defines the new type of Gaussian integration appropriate for Fermi fields.
The two fundamental formulas of Grassmann integration are:
Where A is an antisymmetric matrix, <math>\scriptstyle \psi</math> is a collection of Grassmann variables, and the 1/2 is to prevent double-counting (since <math>\scriptstyle \psi^i\psi^j = -\psi^j\psi^i</math>). In matrix notation, where <math>\scriptstyle \bar\psi</math> and <math>\scriptstyle \bar\eta</math> are Grassman valued row vectors, <math>\scriptstyle \eta</math> and <math>\scriptstyle \psi</math> are Grassman valued column vectors, and M is a real valued matrix:
Where the last equality is a consequence of the translation invariance of the Grassman integral. The Grassman variables <math>\eta</math> are external sources for <math>\psi</math>, and differentiating with respect to <math>\eta</math> pulls down factors of <math>\scriptstyle \bar\psi</math>.
again, in a schematic matrix notation. The meaning of the formula above is that the derivative with respect to the appropriate component of <math>\eta</math> and <math>\scriptstyle\bar\eta</math> gives the matrix element of <math>\scriptstyle M^{-1}</math>. This is exactly analogous to the Bosonic path integration formula for a Gaussian integral of a complex Bosonic field:
::<math> \int e^{\phi^* M \phi + h^* \phi + \phi^* h } D\phi^* D\phi = {e^{h^* M^{-1} h} \over \mathrm{Det}(M)}</math>
So that the propagator is the inverse of the matrix in the quadratic part of the action in both the Bose and Fermi case.
For real Grassmann fields, for [[Majorana fermion]]s, the path integral is a Pfaffian times a source quadratic form, and the formulas give the square root of the determinant, just as they do for real Bosonic fields. The propagator is still the inverse of the quadratic part.
The free Dirac Lagrangian:
::<math> \int \bar\psi(\gamma^\mu \partial_{\mu} - m ) \psi </math>
formally gives the equations of motion and the anticommutation relations of the Dirac field, just as the Klein Gordon Lagrangian in an ordinary path integral gives the equations of motion and commutation relations of the scalar field. By using the spatial fourier-transform of the Dirac field as a new basis for the Grassmann algebra, the quadratic part of the Dirac action becomes simple to invert:
The propagator is the inverse of the matrix M linking <math>\psi(k)</math> and <math>\scriptstyle \bar\psi(k)</math>, since different values of k do not mix together.
where S is the sign of the permutation which reorders the sequence of psi-bars and psis to put the ones which are paired up to make the delta-functions next to each other, with the psi-bar coming right before the psi. Since a psi-psi-bar pair is a commuting element of the Grassman algebra, it doesn't matter what order the pairs are in. If more than one psi/psi-bar pair have the same k, the integral is zero, and it is easy to check that the sum over pairings gives zero in this case (there are always an even number of them). This is the Grassman analog of the higher Gaussian moments which completed the Bosonic Wick's theorem earlier.
The rules for spin-1/2 Dirac particles are as follows: The propagator is the inverse of the Dirac operator, the lines have arrows just as for a complex scalar field, and the diagram acquires an overall factor of -1 for each closed Fermi loop. If there are an odd number of Fermi loops, the diagram changes sign. Historically, the -1 rule was very difficult for Feynman to discover. He discovered it after a long process of trial and error, since he lacked a proper theory of Grassman integration.
The rule follows from the observation that the number of Fermi lines at a vertex is always even. Each term in the Lagrangian must always be Bosonic. A Fermi loops is counted by following Fermionic lines until one comes back to the starting point, then removing those lines from the diagram. Repeating this process eventually erases all the Fermionic lines: this is the Euler algorithm to 2-color a graph, which works whenever each vertex has even degree. Note that the number of steps in the Euler algorithm is only equal to the number of independent Fermionic homology cycles in the common special case that all terms in the Lagrangian are exactly quadratic in the Fermi fields, so that each vertex has exactly two Fermionic lines. When there are four-Fermi interactions (like in the Fermi effective theory of the Weak interactions) there are more k-integrals than Fermi loops. In this case, the counting rule should apply the Euler algorithm by pairing up the Fermi lines at each vertex into pairs which together form a bosonic factor of the term in the Lagrangian, and when entering a vertex by one line, the algorithm should always leave with the partner line.
To clarify and prove the rule, consider a Feynman diagram formed from vertices, terms in the Lagrangian, with Fermion fields. The full term is Bosonic, it is a commuting element of the Grassman algebra, so the order in which the vertices appear is not important. The Fermi lines are linked into loops, and when traversing the loop, one can reorder the vertex terms one after the other as one goes around without any sign cost. The exception is when you return to the starting point, and the final half-line must be joined with the unlinked first half-line. This requires one permutation to move the last psi-bar to go in front of the first psi, and this gives the sign.
This rule is the only visible effect of the exclusion principle in internal lines. When there are external lines, the amplitudes are antisymmetric when two Fermi insertions for identical particles are interchanged. This is automatic in the source formalism, because the sources for Fermi fields are themselves Grassman valued.
==== Spin 1: Photons ====
The naive propagator for photons is infinite, since the Lagrangian for the A-field is:
The quadratic form defining the propagator is non-invertible. The reason is the [[gauge invariance]] of the field, adding a gradient to A does not change the physics.
To fix this problem, one needs to fix a gauge. The most convenient way is to demand that the divergence of A is some function f, whose value is random from point to point. It does no harm to integrate over the values of f, since it only determines the choice of gauge. This procedure inserts the following factor into the path integral for A:
The first factor, the delta function, fixes the gauge. The second factor sums over different values of f which are inequivalent gauge fixings. This is simply
The one difference is that the sign of one propagator is wrong in the Lorentz case: the timelike component has an opposite sign propagator. This means that these particle states have negative norm--- they are not physical states. In the case of photons, it is easy to show by diagram methods that these states are not physical--- their contribution cancels with longitudinal photons to only leave two physical photon polarization contributions for any value of k.
If the averaging over f is done with a coefficient different from 1/2, the two terms don't cancel completely. This gives a covariant Lagrangian with a coefficient <math>\lambda</math> which does not affect anything:
To find the Feynman rules for nonabelian Gauge fields, the procedure which performs the Gauge fixing must be carefully corrected to account for a change of variables in the path-integral.
The gauge fixing factor has an extra determinant from popping the delta function:
To find the form of the determinant, consider first a simple two-dimensional integral, of a function f which depends only on r, not on the angle <math>\scriptstyle \theta</math>. Inserting an integral over theta:
::<math> \int f(r) dx dy = \int f(r) \int d\theta \delta(y) |{dy \over d\theta}| dx dy </math>
The derivative-factor ensures that popping the delta function in <math>\scriptstyle \theta</math> removes the integral. Exchanging the order of integration,
::<math> \int f(r) dx dy = \int d\theta \int f(r) \delta(y) |{dy\over d\theta}| dx dy </math>
The integral over <math>\scriptstyle\theta</math> just gives an overall factor of <math>\scriptstyle 2\pi</math>, while the rate of change of <math>\scriptstyle y</math> with a change in <math>\theta</math> is just x, so this exercise reproduces the standard formula for polar integration of a radial function:
::<math> \int f(r) dx dy = 2\pi \int f(x) x dx </math>
In the path-integral for a nonabelian gauge field, the analogous manipulation is:
The factor in front is the volume of the gauge group, and it contributes a constant which can be discarded. The remaining integral is over the gauge fixed action.
::<math> \int \mathrm{Det}({\partial F\over \partial G})e^{iS_{GF}} DA \,</math>
To get a covariant gauge, the gauge fixing condition is the same as in the Abelian case:
::<math> \partial_\mu A^\mu = f \,</math>
Whose variation under an infinitesimal gauge transformation is given by:
::<math> \partial_\mu D_\mu \alpha \,</math>
where <math>\scriptstyle \alpha</math> is the adjoint valued element of the Lie algebra at every point which performs the infinitesimal gauge transformation. This adds the Faddeev Popov determinant to the action:
::<math> Det(\partial_\mu D_\mu) \,</math>
which can be rewritten as a Grassman integral by introducing ghost fields:
The determinant is independent of f, so the path-integral over f can give the Feynman propagator (or a covariant propagator) by choosing the measure for f as in the abelian case. The full gauge fixed action is then the Yang Mills action in Feynman gauge with an additional ghost action:
The diagrams are derived from this action. The propagator for the spin-1 fields has the usual Feynman form. There are vertices of degree 3 with momentum factors whose couplings are the structure constants, and vertices of degree 4 whose couplings are products of structure constants. There are additional ghost loops, which cancel out timelike and logitudinal states in A loops.
In the Abelian case, the determinant for covariant gauges does not depend on A, so the ghosts do not contribute to the connected diagrams.
== Particle-Path representation ==
Feynman diagrams were originally discovered by Feynman, by trial and error, as a way to represent the contribution to the S-matrix from different classes of particle trajectories.
=== Schwinger representation ===
The Euclidean scalar propagator has a suggestive representation:
The contribution at any one value of <math>\tau</math> to the propagator is a Gaussian of width <math>\scriptstyle \sqrt{\tau}</math>. The total propagation function from 0 to x is a weighted sum over all proper times <math>\tau</math> of a normalized Gaussian, the probability of ending up at x after a random walk of time <math>\tau</math>.
The path-integral representation for the propagator is then:
which is a path-integral rewrite of the Schwinger representation.
The Schwinger representation is both useful for making manifest the particle aspect of the propagator, and for symmetrizing denominators of loop diagrams.
=== Combining Denominators ===
The Schwinger representation has an immediate practical application to loop diagrams. For example, For the diagram in the phi-4 theory formed by joining two X-s together in two half-lines, and making the remaining lines external, the integral over the internal propagators in the loop is:
::<math> \int_{t,t'} e^{-(t+t')(k^2+m^2) - t' 2p\cdot k -t' p^2}\,, </math>
except for the asymmetrical little bit. Defining the variable u=(t+t') and <math>\scriptstyle v</math>= t'/u, the variable u goes from 0 to infinity, while <math>\scriptstyle v</math> goes from 0 to 1. The variable u is the total proper time for the loop, while <math>\scriptstyle v </math> parametrizes the fraction of the proper time on the top of the loop vs. the bottom.
The Jacobian for this transformation of variables is easy to work out from the identities:
::<math> d(uv)= dt'\;\;\;du = dt+dt'\,,</math>
and "wedging" gives
::<math> u du \wedge dv = dt \wedge dt'\,</math>.
This allows the u integral to be evaluated explicitly:
::<math> \int_{u,v} u e^{-u ( k^2+m^2 + v 2p\cdot k + v p^2)} = \int {1\over (k^2 + m^2 + v 2p\cdot k - v p^2)^2} dv </math>
leaving only the <math>\,v</math>-integral. This method, invented by Schwinger but usually attributed to Feynman, is called ''combining denominator''. Abstractly, it is the elementary identity:
::<math> {1\over AB}= \int_0^1 {1\over( vA+ (1-v)B)^2} dv </math>
But this form does not provide the physical motivation for introducing <math>\,v</math> --- <math>\,v</math> is the proportion of proper time on one of the legs of the loop.
Once the denominators are combined, a shift in k to <math>k'=k+vp</math> symmetrizes everything:
::<math> \int_0^1 \int{1\over (k^2 + m^2 + v 2p \cdot k + v p^2)^2} dk dv = \int_0^1 \int {1\over (k'^2 + m^2 + v(1-v)p^2)^2} dk' dv</math>
This form shows that the moment that p<sup>2</sup> is more negative than 4 times the mass of the particle in the loop, which happens in a physical region of Lorentz space, the integral has a cut. This is exactly when the external momentum can create physical particles.
When the loop has more vertices, there are more denominators to combine:
The integral over the Schwinger parameters <math>\scriptstyle u_i</math> can be split up as before into an integral over the total proper time <math>\scriptstyle u = u_0 + u_1 ... + u_n </math> and an integral over the fraction of the proper time in all but the first segment of the loop <math>\scriptstyle v_i = u_i/u </math> for <math>\scriptstyle i\in \{ 1,2,...,n\}</math>. The v's are positive and add up to less than 1, so that the v integral is over an n dimensional simplex.
The Jacobian for the coordinate transformation can be worked out as before:
::<math> du = du_0 + du_1 ... + du_n \,</math>
::<math> d(uv_i) = d u_i \,.</math>
"Wedging" all these equation together, one obtains
where the simplex is the region defined by the conditions <math>\scriptstyle v_i>0 </math> and <math> \scriptstyle \sum_{i=1}^n v_i < 1 </math> as well as <math>\scriptstyle v_0 = 1-\sum_{i=1}^n v_i</math>. Performing the u integral gives the general prescription for combining denominators:
Since the numerator of the integrand is not involved, the same prescription works for any loop, no matter what the spins are carried by the legs. The interpretation of the parameters <math>\scriptstyle v_i </math> is that they are the fraction of the total proper time spent on each leg.
=== Scattering ===
The correlation functions of a quantum field theory describe the scattering of particles. The definition of "particle" in relativistic field theory is not self-evident, because if you try to determine the position so that the uncertainty is less than the [[compton wavelength]], the uncertainty in energy is large enough to produce more particles and antiparticles of the same type from the vacuum. This means that the notion of a single-particle state is to some extent incompatible with the notion of an object localized in space.
In the 1930s, [[Eugene Wigner|Wigner]] gave a mathematical definition for single-particle states: they are a collection of states which form an irreducible representation of the Poincaré group. Single particle states describe an object with a finite mass, a well defined momentum, and a spin. This definition is fine for protons and neutrons, electrons and photons, but it excludes quarks, which are permanently confined, so the modern point of view is more accommodating: a particle is anything whose interaction can be described in terms of Feynman diagrams, which have an interpretation as a sum over particle trajectories.
A field operator can act to produce a one-particle state from the vacuum, which means that the field operator <math>\phi(x)</math> produces a superposition of Wigner particle states. In the free field theory, the field produces one particle states only. But when there are interactions, the field operator can also produce 3-particle,5-particle (if there is no +/- symmetry also 2,4,6 particle) states too. To compute the scattering amplitude for single particle states only requires a careful limit, sending the fields to infinity and integrating over space to get rid of the higher-order corrections.
The relation between scattering and correlation functions is the LSZ-theorem: The scattering amplitude for n particles to go to m-particles in a scattering event is the given by the sum of the Feynman diagrams that go into the correlation function for n+m field insertions, leaving out the propagators for the external legs.
For example, for the <math>\lambda \phi^4</math> interaction of the previous section, the order <math>\lambda</math> contribution to the (Lorentz) correlation function is:
Stripping off the external propagators, that is, removing the factors of <math>i/k^2</math>, gives the invariant scattering amplitude M:
::<math> M = i\lambda \,</math>
which is a constant, independent of the incoming and outgoing momentum. The interpretation of the scattering amplitude is that the sum of <math>|M|^2</math> over all possible final states is the probability for the scattering event. The normalization of the single-particle states must be chosen carefully, however, to ensure that M is a relativistic invariant.
Non-relativistic single particle states are labeled by the momentum k, and they are chosen to have the same norm at every value of k. This is because the nonrelativistic unit operator on single particle states is:
::<math> \int dk |k\rangle\langle k|\, </math>
In relativity, the integral over the k-states for a particle of mass m integrates over a hyperbola in E,k space defined by the energy-momentum relation:
::<math> E^2 - k^2 = m^2 \,</math>
If the integral weighs each k point equally, the measure is not Lorentz invariant. The invariant measure integrates over all values of k and E, restricting to the hyperbola with a Lorentz invariant delta function:
So the normalized k-states are different from the relativistically normalized k-states by a factor of <math>\sqrt{E} = (k^2-m^2)^{1\over 4}</math>
The invariant amplitude M is then the probability amplitude for relativistically normalized incoming states to become relativistically normalized outgoing states.
For nonrelativistic values of k, the relativistic normalization is the same as the nonrelativistic normalization (up to a constant factor <math>\sqrt{m}</math> ). In this limit, the <math>\phi^4</math> invariant scattering amplitude is still constant. The particles created by the field phi scatter in all directions with equal amplitude.
The nonrelativistic potential which scatters in all directions with an equal amplitude (in the [[Born approximation]]) is one whose Fourier transform is constant--- a delta-function potential. The lowest order scattering of the theory reveals the non-relativistic interpretation of this theory--- it describes a collection of particles with a delta-function repulsion. Two such particles have an aversion to occupying the same point at the same time.
== Nonperturbative effects ==
Thinking of Feynman diagrams as a perturbation series, nonperturbative effects like tunneling do not show up, because any effect which goes to zero faster than any polynomial does not affect the Taylor series. Even bound states are absent, since at any finite order particles are only exchanged a finite number of times, and to make a bound state, the binding force must last forever.
But this point of view is misleading, because the diagrams not only describe scattering, but they also are a representation of the short-distance field theory correlations. They encode not only asymptotic processes like particle scattering, they also describe the multiplication rules for fields, the [[operator product expansion]]. Nonperturbative tunneling processes involve field configurations which on average get big when the coupling constant gets small, but each configuration is a coherent superposition of particles whose local interactions are described by Feynman diagrams. When the coupling is small, these become collective processes which involve large numbers of particles, but where the interactions between each of the particles is simple.
This means that nonperturbative effects show up asymptotically in resummations of infinite classes of diagrams, and these diagrams can be locally simple. The graphs determine the local equations of motion, while the allowed large-scale configurations describe non-perturbative physics. But because Feynman propagators are nonlocal in time, translating a field process to a coherent particle language is not completely intuitive, and has only been explicitly worked out in certain special cases. In the case of nonrelativistic bound states, the [[Bethe-Salpeter equation]] describes the class of diagrams to include to describe a relativistic atom. For quantum chromodynamics, the Shifman Vainshtein Zakharov sum rules describe non-perturbatively excited long-wavelength field modes in particle language, but only in a phenomenological way.
The number of Feynman diagrams at high orders of perturbation theory is very large, because there are as many diagrams as there are graphs with a given number of nodes. Nonperturbative effects leave a signature on the way in which the number of diagrams and resummations diverge at high order. It is only because non-perturbative effects appear in hidden form in diagrams that it was possible to analyze nonperturbative effects in string theory, where in many cases a Feynman description is the only one available.
== In popular culture ==
*The use of the above diagram of the Virtual Particle producing a [[quark]]-[[antiquark]] pair, was featured in the Television sit-com, ''[[The Big Bang Theory]]'' episode ''"The Bat Jar Conjecture"''
*[[Phd comics]] of January 11, 2012 shows Feynman diagrams that ''visualize and describe quantum academic interactions'', i.e. the paths followed by Ph.D. students when interacting with their advisors<ref>[[Jorge Cham]], [http://www.phdcomics.com/comics.php?f=1461 Academic Interaction - Feynman Diagrams], January 11, 2012.</ref>
==See also==
*[[Schwinger#Schwinger and Feynman]]
*[[Stueckelberg-Feynman interpretation]]
*[[Invariance mechanics]]
*[[Penguin diagram]]
*[[Path integral formulation]]
*[[Propagator]]s
==Notes==
==Bibliography==
{{reflist}}
{{Refbegin}}
*{{cite book|last=Ashcroft|first=Neil W.|first2=N. David |last2=Mermin |title=Solid state physics|year=1977|publisher=Holt, Rinehart and Winston|location=New York|isbn=978-0-03-083993-1|edition=27. repr.}}
*{{cite book|last=Chikazumi|first=Sōshin|title=Physics of ferromagnetism|year=2009|publisher=Oxford University Press|location=Oxford|isbn=9780199564811|edition=2nd |others= English edition prepared with the assistance of C.D. Graham, Jr |ref=harv}}
*[http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html Electromagnetism] - a chapter from an online textbook
* David Kaiser, ''Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics'', Chicago: University of Chicago Press, 2005. ISBN 0-226-42266-6
*{{cite web
* Martinus Veltman, ''Diagrammatica: The Path to Feynman Diagrams'', Cambridge Lecture Notes in Physics, ISBN 0-521-45692-4 (expanded, updated version of above)
|last = Sandeman
* Mark Srednicki, ''Quantum Field Theory'', [http://www.physics.ucsb.edu/~mark/qft.html online] Script (2006)
|first = Karl
|title = Ferromagnetic Materials
|work = DoITPoMS
|publisher = Dept. of Materials Sci. and Metallurgy, Univ. of Cambridge
A magnet made of alnico, an iron alloy. Ferromagnetism is the physical theory which explains how materials become magnets.
Ferromagnetism is the basic mechanism by which certain materials (such as iron) form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished. Ferromagnetism (including ferrimagnetism)[1] is the strongest type; it is the only type that creates forces strong enough to be felt, and is responsible for the common phenomena of magnetism encountered in everyday life. Other substances respond weakly to magnetic fields with two other types of magnetism, paramagnetism and diamagnetism, but the forces are so weak that they can only be detected by sensitive instruments in a laboratory. An everyday example of ferromagnetism is a refrigerator magnet used to hold notes on a refrigerator door. The attraction between a magnet and ferromagnetic material is "the quality of magnetism first apparent to the ancient world, and to us today".[2]
Permanent magnets (materials that can be magnetized by an external magnetic field and remain magnetized after the external field is removed) are either ferromagnetic or ferrimagnetic, as are other materials that are noticeably attracted to them. Only a few substances are ferromagnetic. The common ones are iron, nickel, cobalt and most of their alloys, some compounds of rare earth metals, and a few naturally-occurring minerals such as lodestone.
Historically, the term ferromagnet was used for any material that could exhibit spontaneous magnetization: a net magnetic moment in the absence of an external magnetic field. This general definition is still in common use. More recently, however, different classes of spontaneous magnetization have been identified when there is more than one magnetic ion per primitive cell of the material, leading to a stricter definition of "ferromagnetism" that is often used to distinguish it from ferrimagnetism. In particular, a material is "ferromagnetic" in this narrower sense only if all of its magnetic ions add a positive contribution to the net magnetization. If some of the magnetic ions subtract from the net magnetization (if they are partially anti-aligned), then the material is "ferrimagnetic".[3] If the moments of the aligned and anti-aligned ions balance completely so as to have zero net magnetization, despite the magnetic ordering, then it is an antiferromagnet. These alignment effects only occur at temperatures below a certain critical temperature, called the Curie temperature (for ferromagnets and ferrimagnets) or the Néel temperature (for antiferromagnets).
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Curie temperatures for some crystalline ferromagnetic (* = ferrimagnetic) materials[4]
The table on the right lists a selection of ferromagnetic and ferrimagnetic compounds, along with the temperature above which they cease to exhibit spontaneous magnetization (see Curie temperature).
Ferromagnetism is a property not just of the chemical make-up of a material, but of its crystalline structure and microscopic organization. There are ferromagnetic metal alloys whose constituents are not themselves ferromagnetic, called Heusler alloys, named after Fritz Heusler. Conversely there are non-magnetic alloys, such as types of stainless steel, composed almost exclusively of ferromagnetic metals.
One can also make amorphous (non-crystalline) ferromagnetic metallic alloys by very rapid quenching (cooling) of a liquid alloy. These have the advantage that their properties are nearly isotropic (not aligned along a crystal axis); this results in low coercivity, low hysteresis loss, high permeability, and high electrical resistivity. One such typical material is a transition metal-metalloid alloy, made from about 80% transition metal (usually Fe, Co, or Ni) and a metalloid component (B, C, Si, P, or Al) that lowers the melting point.
A relatively new class of exceptionally strong ferromagnetic materials are the rare-earth magnets. They contain lanthanide elements that are known for their ability to carry large magnetic moments in well-localized f-orbitals.
Actinide ferromagnets
A number of actinide compounds are ferromagnets at room temperature or become ferromagnets below the Curie temperature (TC). PuP is one actinide pnictide that is a paramagnet and has cubic symmetry at room temperature, but upon cooling undergoes a lattice distortion to tetragonal when cooled to below its Tc = 125 K. PuP has an easy axis of <100>,[5] so that
at 5 K.[6] The lattice distortion is presumably a consequence of strain induced by the magnetoelastic interactions as the magnetic moments aligned parallel within magnetic domains.
In NpFe2 the easy axis is <111>.[7] Above TC ~500 K NpFe2 is also paramagnetic and cubic. Cooling below the Curie temperature produces a rhombohedral distortion wherein the rhombohedral angle changes from 60° (cubic phase) to 60.53°. An alternate description of this distortion is to consider the length c along the unique trigonal axis (after the distortion has begun) and a as the distance in the plane perpendicular to c. In the cubic phase this reduces to = 1.00. Below the Curie temperature
which is the largest strain in any actinide compound.[6] NpNi2 undergoes a similar lattice distortion below TC = 32 K, with a strain of (43 ± 5) × 10−4.[6] NpCo2 is a ferrimagnet below 15 K.
Lithium gas
In 2009, a team of MIT physicists demonstrated that a lithium gas cooled to less than one Kelvin can exhibit ferromagnetism.[8] The team cooled fermionic lithium-6 to less than 150 billionths of one Kelvin above absolute zero using infrared laser cooling. This demonstration is the first time that ferromagnetism has been demonstrated in a gas.
One of the fundamental properties of an electron (besides that it carries charge) is that it has a dipole moment, i.e. it behaves itself as a tiny magnet. This dipole moment comes from the more fundamental property of the electron that it has quantum mechanical spin. The quantum mechanical nature of this spin causes the electron to only be able to be in two states, with the magnetic field either pointing "up" or "down" (for any choice of up and down). The spin of the electrons in atoms is the main source of ferromagnetism, although there is also a contribution from the orbitalangular momentum of the electron about the nucleus. When these tiny magnetic dipoles are aligned in the same direction, their individual magnetic fields add together to create a measurable macroscopic field.
However in materials with a filled electron shell, the total dipole moment of the electrons is zero because the spins are in up/down pairs. Only atoms with partially filled shells (i.e., unpaired spins) can have a net magnetic moment, so ferromagnetism only occurs in materials with partially filled shells. Because of Hund's rules, the first few electrons in a shell tend to have the same spin, thereby increasing the total dipole moment.
These unpaired dipoles (often called simply "spins" even though they also generally include angular momentum) tend to align in parallel to an external magnetic field, an effect called paramagnetism. Ferromagnetism involves an additional phenomenon, however: the dipoles tend to align spontaneously, giving rise to a spontaneous magnetization, even when there is no applied field.
Exchange interaction
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According to classical electromagnetism, two nearby magnetic dipoles will tend to align in opposite directions, so their magnetic fields will oppose one another and cancel out. However, this effect is very weak, because the magnetic fields generated by individual spins are small and the resulting alignment is easily destroyed by thermal fluctuations. In a few materials, a much stronger interaction between spins arises because the change in the direction of the spin leads to a change in electrostatic repulsion between neighboring electrons, due to a particular quantum mechanical effect called the exchange interaction. At short distances, the exchange interaction is much stronger than the dipole-dipole magnetic interaction. As a result, in a few materials, the ferromagnetic ones, nearby spins tend to align in the same direction.
The exchange interaction is related to the Pauli exclusion principle, which says that two electrons with the same spin cannot also have the same "position". Therefore, under certain conditions, when the orbitals of the unpaired outer valence electrons from adjacent atoms overlap, the distributions of their electric charge in space are further apart when the electrons have parallel spins than when they have opposite spins. This reduces the electrostatic energy of the electrons when their spins are parallel compared to their energy when the spins are anti-parallel, so the parallel-spin state is more stable. In simple terms, the electrons, which repel one another, can move "further apart" by aligning their spins, so the spins of these electrons tend to line up. This difference in energy is called the exchange energy.
The materials in which the exchange interaction is much stronger than the competing dipole-dipole interaction are frequently called magnetic materials. For instance, in iron (Fe) the exchange force is about 1000 times stronger than the dipole interaction. Therefore below the Curie temperature virtually all of the dipoles in a ferromagnetic material will be aligned.
The exchange interaction is also responsible for the other types of spontaneous ordering of atomic magnetic moments occurring in magnetic solids, antiferromagnetism and ferrimagnetism.
There are different exchange interaction mechanisms which create the magnetism in different ferromagnetic, ferrimagnetic, and antiferromagnetic substances. These mechanisms include direct exchange, RKKY exchange, double exchange, and superexchange.
Magnetic anisotropy
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
Although the exchange interaction keeps spins aligned, it does not align them in a particular direction. Without magnetic anisotropy, the spins in a magnet randomly change direction in response to thermal fluctuations and the magnet is superparamagnetic. There are several kinds of magnetic anisotropy, the most common of which is magnetocrystalline anisotropy. This is a dependence of the energy on the direction of magnetization relative to the crystallographic lattice. Another common source of anisotropy, inverse magnetostriction, is induced by internal strains. Single-domain magnets also can have a shape anisotropy due to the magnetostatic effects of the particle shape. As the temperature of a magnet increases, the anisotropy tends to decrease, and there is often a blocking temperature at which a transition to superparamagnetism occurs.[10]
Magnetic domains
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
The above would seem to suggest that every piece of ferromagnetic material should have a strong magnetic field, since all the spins are aligned, yet iron and other ferromagnets are often found in an "unmagnetized" state.
Weiss domains microstructure
The reason for this is that a bulk piece of ferromagnetic material is divided into tiny magnetic domains[11] (also known as Weiss domains). Within each domain, the spins are aligned, but (if the bulk material is in its lowest energy configuration, i.e. unmagnetized), the spins of separate domains point in different directions and their magnetic fields cancel out, so the object has no net large scale magnetic field.
Ferromagnetic materials spontaneously divide into magnetic domains because the exchange interaction is a short-range force, so over long distances of many atoms the tendency of the magnetic dipoles to reduce their energy by orienting in opposite directions wins out. If all the dipoles in a piece of ferromagnetic material are aligned parallel, it creates a large magnetic field extending into the space around it. This contains a lot of magnetostatic energy. The material can reduce this energy by splitting into many domains pointing in different directions, so the magnetic field is confined to small local fields in the material, reducing the volume of the field. The domains are separated by thin domain walls a number of molecules thick, in which the direction of magnetization of the dipoles rotates smoothly from one domain's direction to the other.
Thus, a piece of iron in its lowest energy state ("unmagnetized") generally has little or no net magnetic field. However, if it is placed in a strong enough external magnetic field, the domain walls will move, reorienting the domains so more of the dipoles are aligned with the external field. The domains will remain aligned when the external field is removed, creating a magnetic field of their own extending into the space around the material, thus creating a "permanent" magnet. The domains do not go back to their original minimum energy configuration when the field is removed because the domain walls tend to become 'pinned' or 'snagged' on defects in the crystal lattice, preserving their parallel orientation. This is shown by the Barkhausen effect: as the magnetizing field is changed, the magnetization changes in thousands of tiny discontinuous jumps as the domain walls suddenly "snap" past defects.
This magnetization as a function of the external field is described by a hysteresis curve. Although this state of aligned domains found in a piece of magnetized ferromagnetic material is not a minimal-energy configuration, it is metastable, and can persist for long periods, as shown by samples of magnetite from the sea floor which have maintained their magnetization for millions of years.
Alloys used for the strongest permanent magnets are "hard" alloys made with many defects in their crystal structure where the domain walls "catch" and stabilize. The net magnetization can be destroyed by heating and then cooling (annealing) the material without an external field, however. The thermal motion allows the domain boundaries to move, releasing them from any defects, to return to their low-energy unaligned state.
Curie temperature
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
As the temperature increases, thermal motion, or entropy, competes with the ferromagnetic tendency for dipoles to align. When the temperature rises beyond a certain point, called the Curie temperature, there is a second-order phase transition and the system can no longer maintain a spontaneous magnetization, although it still responds paramagnetically to an external field. Below that temperature, there is a spontaneous symmetry breaking and random domains form (in the absence of an external field). The Curie temperature itself is a critical point, where the magnetic susceptibility is theoretically infinite and, although there is no net magnetization, domain-like spin correlations fluctuate at all length scales.
The study of ferromagnetic phase transitions, especially via the simplified Ising spin model, had an important impact on the development of statistical physics. There, it was first clearly shown that mean field theory approaches failed to predict the correct behavior at the critical point (which was found to fall under a universality class that includes many other systems, such as liquid-gas transitions), and had to be replaced by renormalization group theory.
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
↑Richard M. Bozorth, Ferromagnetism, first published 1951, reprinted 1993 by IEEE Press, New York as a "Classic Reissue." ISBN 0-7803-1032-2.
↑One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
↑One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ 6.06.16.2One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
↑20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
↑20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
