- Quantum field theory
In

**quantum field theory (QFT)**the forces between particles are mediated by other particles. For instance, the electromagnetic force between two electrons is caused by an exchange of photons. But quantum field theory applies to all fundamental forces. Intermediate vector bosons mediate theweak force , gluons mediate thestrong force , andgravitons mediate the gravitational force. These force carrying particles arevirtual particles and, by definition, cannot be detected while carrying the force, because such detection will imply that the force is not being carried.In QFT photons are not thought of as 'little billiard balls', they are considered to befield quanta - necessarily chunked ripples in a field that 'look like' particles.Fermions , like the electron, can also be described as ripples in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field (field quanta).Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in

particle physics andcondensed matter physics . Most theories in modern particle physics, including theStandard Model of elementary particles and their interactions, are formulated as relativistic quantum field theories. In condensed matter physics, quantum field theories are used in many circumstances, especially those where the number of particles is allowed to fluctuate—for example, in theBCS theory ofsuperconductivity .**History**Quantum field theory originated in the 1920s from the problem of creating a quantum mechanical theory of the

electromagnetic field . In 1926,Max Born ,Pascual Jordan , andWerner Heisenberg constructed such a theory by expressing the field's internal degrees of freedom as an infinite set ofharmonic oscillator s and by employing the usual procedure for quantizing those oscillators (canonical quantization ). This theory assumed that no electric charges or currents were present and today would be called afree field theory . The first reasonably complete theory ofquantum electrodynamics , which included both the electromagnetic field and electrically charged matter (specifically,electron s) as quantum mechanical objects, was created byPaul Dirac in 1927. This quantum field theory could be used to model important processes such as the emission of aphoton by an electron dropping into aquantum state of lower energy, a process in which the "number of particles changes" — one atom in the initial state becomes an atom plus aphoton in the final state. It is now understood that the ability to describe such processes is one of the most important features of quantum field theory.It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate Einstein's relativity theory, which had after all grown out of the study of

classical electromagnetism . This need to "put together relativity and quantum mechanics" was the second major motivation in the development of quantum field theory.Pascual Jordan andWolfgang Pauli showed in 1928 that quantum fields could be made to behave in the way predicted byspecial relativity during coordinate transformations (specifically, they showed that the fieldcommutator s wereLorentz invariant ), and in 1933Niels Bohr andLeon Rosenfeld showed that this result could be interpreted as a limitation on the ability to measure fields atspace-like separations, exactly as required by relativity. A further boost for quantum field theory came with the discovery of theDirac equation , a single-particle equation obeying both relativity and quantum mechanics, when it was shown that several of its undesirable properties (such as negative-energy states) could be eliminated by reformulating the Dirac equation as a quantum field theory. This work was performed byWendell Furry ,Robert Oppenheimer ,Vladimir Fock , and others.The third thread in the development of quantum field theory was the need to "handle the statistics of many-particle systems" consistently and with ease. In 1927, Jordan tried to extend the canonical quantization of fields to the many-body wavefunctions of

identical particles , a procedure that is sometimes calledsecond quantization . In 1928, Jordan andEugene Wigner found that the quantum field describing electrons, or otherfermion s, had to be expanded using anti-commuting creation and annihilation operators due to thePauli exclusion principle . This thread of development was incorporated intomany-body theory , and strongly influencedcondensed matter physics andnuclear physics .Despite its early successes, quantum field theory was plagued by several serious theoretical difficulties. Many seemingly-innocuous physical quantities, such as the energy shift of electron states due to the presence of the electromagnetic field, gave infinity — a nonsensical result — when computed using quantum field theory. This "divergence problem" was solved during the 1940s by

Bethe , Tomonaga, Schwinger, Feynman, and Dyson, through the procedure known asrenormalization . This phase of development culminated with the construction of the modern theory ofquantum electrodynamics (QED). Beginning in the 1950s with the work of Yang and Mills, QED was generalized to a class of quantum field theories known as gauge theories. The 1960s and 1970s saw the formulation of a gauge theory now known as theStandard Model ofparticle physics , which describes all known elementary particles and the interactions between them. The weak interaction part of the standard model was formulated bySheldon Glashow , with theHiggs mechanism added bySteven Weinberg andAbdus Salam . The theory was shown to be consistent byGerardus 't Hooft andMartinus Veltman .Also during the 1970s, parallel developments in the study of

phase transitions incondensed matter physics ledLeo Kadanoff ,Michael Fisher andKenneth Wilson (extending work ofErnst Stueckelberg ,Andre Peterman ,Murray Gell-Mann andFrancis Low ) to a set of ideas and methods known as therenormalization group . By providing a better physical understanding of the renormalization procedure invented in the 1940s, the renormalization group sparked what has been called the "grand synthesis" of theoretical physics, uniting the quantum field theoretical techniques used in particle physics and condensed matter physics into a single theoretical framework.The study of quantum field theory is alive and flourishing, as are applications of this method to many physical problems. It remains one of the most vital areas of

theoretical physics today, providing a common language to many branches ofphysics .**Principles of quantum field theory****Classical fields and quantum fields**Quantum mechanics , in its most general formulation, is a theory of abstractoperator s (observables) acting on an abstract state space (Hilbert space ), where the observables represent physically-observable quantities and the state space represents the possible states of the system under study. Furthermore, each observable corresponds, in a technical sense, to the classical idea of a degree of freedom. For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators $hat\{x\}$ and $hat\{p\}$. Ordinary quantum mechanics deals with systems such as this, which possess a small set of degrees of freedom.(It is important to note, at this point, that this article does not use the word "

particle " in the context ofwave–particle duality . In quantum field theory, "particle" is a generic term for any discrete quantum mechanical entity, such as an electron, which can behave like classical particles or classical waves under different experimental conditions.)A

**quantum field**is a quantum mechanical system containing a large, and possibly infinite, number of degrees of freedom. This is not as exotic a situation as one might think. A classical field contains a set of degrees of freedom at each point of space; for instance, the classicalelectromagnetic field defines two vectors — theelectric field and themagnetic field — that can in principle take on distinct values for each position $r$. When the field "as a whole" is considered as a quantum mechanical system, its observables form an infinite (in fact uncountable) set, because $r$ is continuous.Furthermore, the degrees of freedom in a quantum field are arranged in "repeated" sets. For example, the degrees of freedom in an electromagnetic field can be grouped according to the position $r$, with exactly two vectors for each $r$. Note that $r$ is an ordinary number that "indexes" the observables; it is not to be confused with the position operator $hat\{x\}$ encountered in ordinary quantum mechanics, which is an observable. (Thus, ordinary quantum mechanics is sometimes referred to as "zero-dimensional quantum field theory", because it contains only a single set of observables.) It is also important to note that there is nothing special about $r$ because, as it turns out, there is generally more than one way of indexing the degrees of freedom in the field.

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

**Single-particle and many-particle quantum mechanics**In ordinary quantum mechanics, the time-dependent

Schrödinger equation describing the time evolution of the quantum state of a single non-relativistic particle is:$left\; [\; frac\{|mathbf\{p\}|^2\}\{2m\}\; +\; V(mathbf\{r\})\; ight]$

psi(t) ang = i hbar frac{partial}{partial t} |psi(t) ang,where $m$ is the particle's

mass , $V$ is the appliedpotential , and $|psi\; ang$ denotes thequantum state (we are usingbra-ket notation ).We wish to consider how this problem generalizes to $N$ particles. There are two motivations for studying the many-particle problem. The first is a straightforward need in

condensed matter physics , where typically the number of particles is on the order ofAvogadro's number (6.0221415 x 10^{23}). The second motivation for the many-particle problem arises fromparticle physics and the desire to incorporate the effects ofspecial relativity . If one attempts to include the relativisticrest energy into the above equation, the result is either theKlein-Gordon equation or theDirac equation . However, these equations have many unsatisfactory qualities; for instance, they possess energyeigenvalues which extend to –∞, so that there seems to be no easy definition of aground state . It turns out that such inconsistencies arise from neglecting the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and apositron can annihilate each other to createphoton s. Thus, a consistent relativistic quantum theory must be formulated as a many-particle theory.Furthermore, we will assume that the $N$ particles are indistinguishable. As described in the article on

identical particles , this implies that the state of the entire system must be either symmetric (boson s) or antisymmetric (fermion s) when the coordinates of its constituent particles are exchanged. These multi-particle states are rather complicated to write. For example, the general quantum state of a system of $N$ bosons is written as:$|phi\_1\; cdots\; phi\_N\; ang\; =\; sqrt\{frac\{prod\_j\; N\_j!\}\{N!\; sum\_\{pin\; S\_N\}\; |phi\_\{p(1)\}\; ang\; cdots\; |phi\_\{p(N)\}\; ang,$

where $|phi\_i\; ang$ are the single-particle states, $N\_j$ is the number of particles occupying state $j$, and the sum is taken over all possible

permutation s $p$ acting on $N$ elements. In general, this is a sum of $N!$ ($N$factorial ) distinct terms, which quickly becomes unmanageable as $N$ increases. The way to simplify this problem is to turn it into a quantum field theory.**Second quantization**In this section, we will describe a method for constructing a quantum field theory called

. This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identical-particle states. It is based on the Hamiltonian formulation of quantum mechanics; several other approaches exist, such as thesecond quantization Feynman path integral [*Abraham Pais, "Inward Bound: Of Matter and Forces in the Physical World" ISBN 0198519974. Pais recounts how his astonishment at the rapidity with which*] , which uses aFeynman could calculate using his method. Feynman's method is now part of the standard methods for physicists.Lagrangian formulation. For an overview, see the article on quantization.**Second quantization of bosons**For simplicity, we will first discuss second quantization for

boson s, which form perfectly symmetric quantum states. Let us denote the mutually orthogonal single-particle states by $|phi\_1\; ang,\; |phi\_2\; ang,\; |phi\_3\; ang,$ and so on. For example, the 3-particle state with one particle in state $|phi\_1\; ang$ and two in state$|phi\_2\; ang$ is:$frac\{1\}\{sqrt\{3\; left\; [\; |phi\_1\; ang\; |phi\_2\; ang$

phi_2 ang + |phi_2 ang |phi_1 ang |phi_2 ang + |phi_2 ang

phi_2 ang |phi_1 ang ight] .The first step in second quantization is to express such quantum states in terms of

**occupation numbers**, by listing the number of particles occupying each of the single-particle states $|phi\_1\; ang,\; |phi\_2\; ang,$ etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as:$|1,\; 2,\; 0,\; 0,\; 0,\; cdots\; angle.$

The next step is to expand the $N$-particle state space to include the state spaces for all possible values of $N$. This extended state space, known as a

Fock space , is composed of the state space of a system with no particles (the so-calledvacuum state ), plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. It is easy to see that there is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space.At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number $jcdots$, indicating which of the single-particle states $|phi\_1\; ang,\; |phi\_2\; ang,\; cdots|phi\_j\; angcdots$ it refers to.

The properties of this quantum field can be explored by defining

creation and annihilation operators , which add and subtract particles. They are analogous to "ladder operators" in thequantum harmonic oscillator problem, which added and subtracted energy quanta. However, these operators literally create and annihilate particles of a given quantum state. The bosonic annihilation operator $a\_2$ and creation operator $a\_2^dagger$ have the following effects::$a\_2\; |\; N\_1,\; N\_2,\; N\_3,\; cdots\; angle\; =\; sqrt\{N\_2\}\; mid\; N\_1,\; (N\_2\; -\; 1),\; N\_3,\; cdots\; angle,$:$a\_2^dagger\; |\; N\_1,\; N\_2,\; N\_3,\; cdots\; angle\; =\; sqrt\{N\_2\; +\; 1\}\; mid\; N\_1,\; (N\_2\; +\; 1),\; N\_3,\; cdots\; angle.$

It can be shown that these are operators in the usual quantum mechanical sense, i.e.

linear operator s acting on the Fock space. Furthermore, they are indeed Hermitian conjugates, which justifies the way we have written them. They can be shown to obey the commutation relation:$left\; [a\_i\; ,\; a\_j\; ight]\; =\; 0\; quad,quadleft\; [a\_i^dagger\; ,\; a\_j^dagger\; ight]\; =\; 0\; quad,quadleft\; [a\_i\; ,\; a\_j^dagger\; ight]\; =\; delta\_\{ij\},$

where $delta$ stands for the

Kronecker delta . These are precisely the relations obeyed by the ladder operators for an infinite set of independentquantum harmonic oscillator s, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator.The Hamiltonian of the quantum field (which, through the

Schrödinger equation , determines its dynamics) can be written in terms of creation and annihilation operators. For instance, the Hamiltonian of a field of free (non-interacting) bosons is:$H\; =\; sum\_k\; E\_k\; ,\; a^dagger\_k\; ,a\_k,$

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

:$a\_k^dagger,a\_k|cdots,\; N\_k,\; cdots\; angle=N\_k|\; cdots,\; N\_k,\; cdots\; angle$.

**Second quantization of fermions**It turns out that a different definition of creation and annihilation must be used for describing

fermion s. According to thePauli exclusion principle , fermions cannot share quantum states, so their occupation numbers $N\_i$ can only take on the value 0 or 1. The fermionic annihilation operators $c$ and creation operators $c^dagger$ are defined by:$c\_j\; |\; N\_1,\; N\_2,\; cdots,\; N\_j\; =\; 0,\; cdots\; angle\; =\; 0$:$c\_j\; |\; N\_1,\; N\_2,\; cdots,\; N\_j\; =\; 1,\; cdots\; angle\; =\; (-1)^\{(N\_1\; +\; cdots\; +\; N\_\{j-1\})\}\; |\; N\_1,\; N\_2,\; cdots,\; N\_j\; =\; 0,\; cdots\; angle$:$c\_j^dagger\; |\; N\_1,\; N\_2,\; cdots,\; N\_j\; =\; 0,\; cdots\; angle\; =\; (-1)^\{(N\_1\; +\; cdots\; +\; N\_\{j-1\})\}\; |\; N\_1,\; N\_2,\; cdots,\; N\_j\; =\; 1,\; cdots\; angle$:$c\_j^dagger\; |\; N\_1,\; N\_2,\; cdots,\; N\_j\; =\; 1,\; cdots\; angle\; =\; 0$

These obey an anticommutation relation:

:$left\{c\_i\; ,\; c\_j\; ight\}\; =\; 0\; quad,quadleft\{c\_i^dagger\; ,\; c\_j^dagger\; ight\}\; =\; 0\; quad,quadleft\{c\_i\; ,\; c\_j^dagger\; ight\}\; =\; delta\_\{ij\}$

One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.

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

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

Single-particle states are usually enumerated in terms of their momenta (as in the

particle in a box problem.) We can construct field operators by applying theFourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator $phi(mathbf\{r\})$ is:$phi(mathbf\{r\})\; stackrel\{mathrm\{def\{=\}\; sum\_\{j\}\; e^\{imathbf\{k\}\_jcdot\; mathbf\{r\; a\_\{j\}$

The bosonic field operators obey the commutation relation

:$left\; [phi(mathbf\{r\})\; ,\; phi(mathbf\{r\text{'}\})\; ight]\; =\; 0\; quad,quadleft\; [phi^dagger(mathbf\{r\})\; ,\; phi^dagger(mathbf\{r\text{'}\})\; ight]\; =\; 0\; quad,quadleft\; [phi(mathbf\{r\})\; ,\; phi^dagger(mathbf\{r\text{'}\})\; ight]\; =\; delta^3(mathbf\{r\}\; -\; mathbf\{r\text{'}\})$

where $delta(x)$ stands for the

Dirac delta function . As before, the fermionic relations are the same, with the commutators replaced by anticommutators.It should be emphasized that the field operator is "not" the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say

:$H\; =\; -\; frac\{hbar^2\}\{2m\}\; sum\_i\; abla\_i^2\; +\; sum\_\{i\; <\; j\}\; U(|mathbf\{r\}\_i\; -\; mathbf\{r\}\_j|)$

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

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

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

**Implications of quantum field theory****Unification of fields and particles**The "second quantization" procedure that we have outlined in the previous section takes a set of single-particle quantum states as a starting point. Sometimes, it is impossible to define such single-particle states, and one must proceed directly to quantum field theory. For example, a quantum theory of the

electromagnetic field "must" be a quantum field theory, because it is impossible (for various reasons) to define awavefunction for a singlephoton . In such situations, the quantum field theory can be constructed by examining the mechanical properties of the classical field and guessing the corresponding quantum theory. The quantum field theories obtained in this way have the same properties as those obtained using second quantization, such as well-defined creation and annihilation operators obeying commutation or anticommutation relations.Quantum field theory thus provides a unified framework for describing "field-like" objects (such as the electromagnetic field, whose excitations are photons) and "particle-like" objects (such as electrons, which are treated as excitations of an underlying electron field).

**Physical meaning of particle indistinguishability**The second quantization procedure relies crucially on the particles being identical. We would not have been able to construct a quantum field theory from a distinguishable many-particle system, because there would have been no way of separating and indexing the degrees of freedom.

Many physicists prefer to take the converse interpretation, which is that "quantum field theory explains what identical particles are". In ordinary quantum mechanics, there is not much theoretical motivation for using symmetric (bosonic) or antisymmetric (fermionic) states, and the need for such states is simply regarded as an empirical fact. From the point of view of quantum field theory, particles are identical

if and only if they are excitations of the same underlying quantum field. Thus, the question "why are all electrons identical?" arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental.**Particle conservation and non-conservation**During second quantization, we started with a Hamiltonian and state space describing a fixed number of particles ($N$), and ended with a Hamiltonian and state space for an arbitrary number of particles. Of course, in many common situations $N$ is an important and perfectly well-defined quantity, e.g. if we are describing a gas of atoms sealed in a box. From the point of view of quantum field theory, such situations are described by quantum states that are eigenstates of the number operator $hat\{N\}$, which measures the total number of particles present. As with any quantum mechanical observable, $hat\{N\}$ is conserved if it commutes with the Hamiltonian. In that case, the quantum state is trapped in the $N$-particle subspace of the total Fock space, and the situation could equally well be described by ordinary $N$-particle quantum mechanics.

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

On the other hand, it is possible, and indeed common, to encounter quantum states that are "not" eigenstates of $hat\{N\}$, which do not have well-defined particle numbers. Such states are difficult or impossible to handle using ordinary quantum mechanics, but they can be easily described in quantum field theory as

quantum superposition s of states having different values of $N$. For example, suppose we have a bosonic field whose particles can be created or destroyed by interactions with a fermionic field. The Hamiltonian of the combined system would be given by the Hamiltonians of the free boson and free fermion fields, plus a "potential energy" term such as:$H\_I\; =\; sum\_\{k,q\}\; V\_q\; (a\_q\; +\; a\_\{-q\}^dagger)\; c\_\{k+q\}^dagger\; c\_k$,

where $a\_k^dagger$ and $a\_k$ denotes the bosonic creation and annihilation operators, $c\_k^dagger$ and $c\_k$ denotes the fermionic creation and annihilation operators, and $V\_q$ is a parameter that describes the strength of the interaction. This "interaction term" describes processes in which a fermion in state $k$ either absorbs or emits a boson, thereby being kicked into a different eigenstate $k+q$. (In fact, this type of Hamiltonian is used to describe interaction between

conduction electron s andphonon s inmetal s. The interaction between electrons andphoton s is treated in a similar way, but is a little more complicated because the role of spin must be taken into account.) One thing to notice here is that even if we start out with a fixed number of bosons, we will typically end up with a superposition of states with different numbers of bosons at later times. The number of fermions, however, is conserved in this case.In

condensed matter physics , states with ill-defined particle numbers are particularly important for describing the varioussuperfluid s. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers.**Axiomatic approaches**The preceding description of quantum field theory follows the spirit in which most

physicist s approach the subject. However, it is not mathematically rigorous. Over the past several decades, there have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set ofaxiom s for it. These attempts fall into two broad classes.The first class of axioms, first proposed during the 1950s, include the Wightman, Osterwalder-Schrader, and

Haag-Kastler systems. They attempted to formalize the physicists' notion of an "operator-valued field" within the context offunctional analysis , and enjoyed limited success. It was possible to prove that any quantum field theory satisfying these axioms satisfied certain general theorems, such as thespin-statistics theorem and the CPT theorem. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory, including theStandard Model , satisfied these axioms. Most of the theories that could be treated with these analytic axioms were physically trivial, being restricted to low-dimensions and lacking interesting dynamics. The construction of theories satisfying one of these sets of axioms falls in the field ofconstructive quantum field theory . Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others.During the 1980s, a second set of axioms based on geometric ideas was proposed. This line of investigation, which restricts its attention to a particular class of quantum field theories known as topological quantum field theories, is associated most closely with

Michael Atiyah andGraeme Segal , and was notably expanded upon byEdward Witten ,Richard Borcherds , andMaxim Kontsevich . However, most physically-relevant quantum field theories, such as theStandard Model , are not topological quantum field theories; the quantum field theory of the fractional quantum Hall effect is a notable exception. The main impact of axiomatic topological quantum field theory has been on mathematics, with important applications inrepresentation theory ,algebraic topology , anddifferential geometry .Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the

Millennium Prize Problems —proving the existence of a mass gap in Yang-Mills theory—is linked to this issue.**Phenomena associated with quantum field theory**In the previous part of the article, we described the most general properties of quantum field theories. Some of the quantum field theories studied in various fields of theoretical physics possess additional special properties, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections.

**Renormalization**Early in the history of quantum field theory, it was found that manyseemingly innocuous calculations, such as the perturbative shift in the energy of an electron due to the presence of the electromagnetic field, give infinite results. The reason is that the perturbation theory for the shift in an energy involves a sum over all other energy levels, and there are infinitely many levels at short distances which each give a finite contribution.

Many of these problems are related to failures in

classical electrodynamics that were identified but unsolved in the 19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field which it carries around with it. The energy carried by a single electron—itsself energy —is not simply the bare value, but also includes the energy contained in its electromagnetic field, its attendant cloud of photons. The energy in a field of a spherical source diverges in both classical and quantum mechanics, but as discovered by Weisskopf, in quantum mechanics the divergence is much milder, going only as the logarithm of the radius of the sphere.The solution to the problem, presciently suggested by Stueckelberg, independently by Bethe after the crucial experiment by Lamb, implemented at one loop by Schwinger, and systematically extended to all loops by Feynman and Dyson, with converging work by Tomonaga in isolated postwar Japan, is called

renormalization . The technique of renormalization recognizes that the problem is essentially purely mathematical, that extremely short distances are at fault. In order to define a theory on a continuum, first place acutoff on the fields, by postulating that quanta cannot have energies above some extremely high value. This has the effect of replacing continuous space by a structure where very short wavelengths do not exist, as on a lattice. Lattices break rotational symmetry, and one of the crucial contributions made by Feynman, Pauli and Villars, and modernized by 't Hooft and Veltman, is a symmetry preserving cutoff for perturbation theory. There is no known symmetrical cutoff outside of perturbation theory, so for rigorous or numerical work people often use an actual lattice.On a lattice, every quantity is finite but depends on the spacing. When taking the limit of zero spacing, we make sure that the physically-observable quantities like the observed electron mass stay fixed, which means that the constants in the Lagrangian defining the theory depend on the spacing. Hopefully, by allowing the constants to vary with the lattice spacing, all the results at long distances become insensitive to the lattice, defining a continuum limit.

The renormalization procedure only works for a certain class of quantum field theories, called

**renormalizable quantum field theories**. A theory is**perturbatively renormalizable**when the constants in the Lagrangian only diverge at worst as logarithms of the lattice spacing for very short spacings. The continuum limit is then well defined in perturbation theory, and even if it is not fully well defined non-perturbatively, the problems only show up at distance scales which are exponentially small in the inverse coupling for weak couplings. TheStandard Model ofparticle physics is perturbatively renormalizable, and so are its component theories (quantum electrodynamics /electroweak theory andquantum chromodynamics ). Of the three components, quantum electrodynamics is believed to not have a continuum limit, while the asymptotically free SU(2) and SU(3) weak hypercharge and strong color interactions are nonperturbatively well defined.The

renormalization group describes how renormalizable theories emerge as the long distance low-energyeffective field theory for any given high-energy theory. Because of this, renormalizable theories are insensitive to the precise nature of the underlying high-energy short-distance phenomena. This is a blessing because it allows physicists to formulate low energy theories without knowing the details of high energy phenomenon. It is also a curse, because once a renormalizable theory like the standard model is found to work, it gives very few clues to higher energy processes. The only way high energy processes can be seen in the standard model is when they allow otherwise forbidden events, or if they predict quantitative relations between the coupling constants.**Gauge freedom**A

gauge theory is a theory that admits a symmetry with a local parameter. For example, in every quantum theory the global phase of thewave function is arbitrary and does not represent something physical. Consequently, the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is aglobal symmetry . Inquantum electrodynamics , the theory is also invariant under a "local" change of phase, that is - one may shift the phase of allwave function s so that the shift may be different at every point inspace-time . This is alocal symmetry . However, in order for a well-definedderivative operator to exist, one must introduce a new field, thegauge field , which also transforms in order for the local change of variables (the phase in our example) not to affect the derivative. Inquantum electrodynamics thisgauge field is theelectromagnetic field . The change of local gauge of variables is termedgauge transformation .In quantum field theory the excitations of fields represent particles. The particle associated with excitations of the

gauge field is thegauge boson , which is thephoton in the case ofquantum electrodynamics .The degrees of freedom in quantum field theory are local fluctuations of the fields. The existence of a

gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero bygauge transformation s, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or "gauge artifacts"; usually some of them have a negative norm, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum anomaly. If a gauge symmetry is anomalous (i.e. not kept in the quantum theory) then the theory is non-consistent: for example, inquantum electrodynamics , had there been agauge anomaly , this would require the appearance ofphoton s with longitudinalpolarization andpolarization in the time direction, the latter having a negative norm, rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any interaction, making the theory non unitary and again inconsistent (seeoptical theorem ).In general, the

gauge transformation s of a theory consist several different transformations, which may not becommutative . These transformations are together described by a mathematical object known as agauge group .Infinitesimal gauge transformation s are thegauge group generators. Therefore the number ofgauge boson s is the group dimension (i.e. number of generators forming a basis).All the

fundamental interaction s in nature are described by gauge theories. These are:

*Quantum electrodynamics , whosegauge transformation is a local change of phase, so that thegauge group isU(1) . Thegauge boson is thephoton .

*Quantum chromodynamics , whosegauge group isSU(3) . Thegauge boson s are eightgluon s.

* The electroweak Theory, whosegauge group is $U(1)\; imes\; SU(2)$ (adirect product ofU(1) andSU(2) ).

*Gravity , whose classical theory isgeneral relativity , admits the equivalence principle which is a form of gauge symmetry.**Supersymmetry**Supersymmetry assumes that every fundamentalfermion has a superpartner that is aboson and vice versa. It was introduced in order to solve the so-called Hierarchy Problem, that is, to explain why particles not protected by any symmetry (like theHiggs boson ) do not receive radiative corrections to its mass driving it to the larger scales (GUT, Planck...). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity (Supergravity ), and it is a key ingredient for the consistency ofstring theory .The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite.

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

Large Hadron Collider .**See also***

Abraham-Lorentz force

*Dangerously irrelevant

*Feynman path integral

*Green–Kubo relations

*Green's function (many-body theory)

*Invariance mechanics

*List of quantum field theories

*Quantum chromodynamics

*Quantum electrodynamics

*Quantum flavordynamics

*Quantum geometrodynamics

*Quantum magnetodynamics

*Photon polarization

*Relationship between string theory and quantum field theory

*Schwinger-Dyson equation

*Theoretical and experimental justification for the Schrödinger equation **Notes****Suggested reading for the layman*** Gribbin, John ; "Q is for Quantum: Particle Physics from A to Z", Weidenfeld & Nicolson (1998) [ISBN 0297817523|] Dictionary of all things quantum.

* Feynman, Richard ; The Character of Physical Law

* Feynman, Richard ; QED

**Suggested reading*** Wilczek, Frank ; "Quantum Field Theory", Review of Modern Physics 71 (1999) S85-S95. Review article written by a master of Q.C.D., [

*http://nobelprize.org/physics/laureates/2004/wilczek-autobio.html "Nobel laureate 2003"*] . Full text available at : [*http://fr.arxiv.org/abs/hep-th/9803075 "hep-th/9803075"*]* Ryder, Lewis H. ; "Quantum Field Theory " (Cambridge University Press, 1985), [ISBN 0-521-33859-X] . Introduction to relativistic Q.F.T. for particle physics.

* Yndurain, Francisco Jose; "The Theory of Quark and Gluon Interactions",(Springer,2006), Fourth Edition. [ISBN 978-3-540-33209-I]

* Yndurain, Francisco Jose; "Quantum Chromodynamics". Springler-Verlag 1983. [ISBN 3-540-11752-0]

* Zee, Anthony ; "Quantum Field Theory in a Nutshell", Princeton University Press (2003) [ISBN 0-691-01019-6] .

* Peskin, M and Schroeder, D. ;"An Introduction to Quantum Field Theory" (Westview Press, 1995) [ISBN 0-201-50397-2]* Weinberg, Steven ; "The Quantum Theory of Fields" (3 volumes) Cambridge University Press (1995). A monumental treatise on Q.F.T. written by a leading expert, [

*http://nobelprize.org/physics/laureates/1979/weinberg-lecture.html "Nobel laureate 1979"*] .* Loudon, Rodney ; "The Quantum Theory of Light" (Oxford University Press, 1983), [ISBN 0-19-851155-8]

*cite book | author=Greiner, Walter and Müller, Berndt | title=Gauge Theory of Weak Interactions | publisher=Springer | year=2000 | id=ISBN 3-540-67672-4

* Paul H. Frampton , "Gauge Field Theories", Frontiers in Physics, Addison-Wesley (1986), Second Edition, Wiley (2000).

*cite book | author=Gordon L. Kane | title=Modern Elementary Particle Physics | publisher=Perseus Books | year=1987 | id=ISBN 0-201-11749-5

*Kleinert, Hagen, "Multivalued Fields in in Condensed Matter, Electrodynamics, and Gravitation", [

*http://www.worldscibooks.com/physics/6742.html World Scientific (Singapore, 2008)*] (also available [*http://www.physik.fu-berlin.de/~kleinert/re.html#B9 online*] )**External links*** Siegel, Warren ; [

*http://insti.physics.sunysb.edu/%7Esiegel/errata.html "Fields"*] (also available from )

* 't Hooft, Gerard ; "The Conceptual Basis of Quantum Field Theory", Handbook of the Philosophy of Science, Elsevier (to be published). Review article written by a master of gauge theories, [*http://nobelprize.org/physics/laureates/1999/thooft-autobio.html"Nobel laureate 1999"*] . Full text available in [*http://www.phys.uu.nl/~thooft/lectures/basisqft.pdf"pdf"*] .

* Srednicki, Mark ; [*http://gabriel.physics.ucsb.edu/~mark/qft.html "Quantum Field Theory"*]

* Kuhlmann, Meinard ; [*http://plato.stanford.edu/entries/quantum-field-theory/ "Quantum Field Theory"*] , Stanford Encyclopedia of Philosophy

* Quantum field theory textbooks: [*http://motls.blogspot.com/2006/01/qft-didactics.html a list with links to amazon.com*]

* [*http://quantumfieldtheory.info Pedagogic Aids to Quantum Field Theory*] . Click on the link "Introduction" for a simplified introduction to QFT suitable for someone familiar with quantum mechanics.

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