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In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions governing the behavior of subatomic particles. The scheme is named for its inventor, Nobel Prizewinning American physicist Richard Feynman, and was first introduced in 1948. The interaction of subatomic 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".^{[1]} While the diagrams are applied primarily to quantum field theory, they can also be used in other fields, such as solidstate theory.
Feynman proposed an interpretation of the positron as if it were an electron moving backward in time.^{[2]} Thus Feynman diagrams contain both a space axis and a time axis, and antiparticles are interpreted as moving forward in space but backward in time.
The calculation of probability amplitudes 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 formulation of quantum field theory, a Feynman diagram represents a term in the Wick's expansion of the perturbative Smatrix. 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 Smatrix between the initial and the final states of the quantum system.
Quantum field theory 

Feynman diagram 
History 








Motivation and history
When calculating scattering crosssections in particle physics, the interaction between particles can be described by starting from a free field that describes the incoming and outgoing particles, and including an interaction 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 expansion, and the timedependent perturbation theory for fields is known as the Dyson series. When the intermediate states at intermediate times are energy eigenstates (collections of particles with a definite momentum) the series is called oldfashioned perturbation theory.
The Dyson series can be alternatively 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 oldfashioned terms, because the oldfashioned way treats the particle and antiparticle contributions as separate. Each Feynman diagram is the sum of exponentially many oldfashioned terms, because each internal line can separately represent either a particle or an antiparticle. In a nonrelativistic theory, there are no antiparticles and there is no doubling, so each Feynman diagram includes only one term.
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.
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.
The naïve application of such calculations often produces diagrams whose amplitudes are infinite, because the shortdistance particle interactions require a careful limiting procedure, to include particle selfinteractions. The technique of renormalization, suggested by Ernst Stueckelberg and Hans Bethe and implemented by Dyson, Feynman, Schwinger, and Tomonaga compensates for this effect and eliminates the troublesome infinities. After renormalization, calculations using Feynman diagrams match experimental results with very high accuracy.
Feynman diagram and path integral methods are also used in statistical mechanics and can even be applied to classical mechanics.^{[3]}
Alternative names
Murray GellMann 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 pathintegral.^{[4]} Historically they were sometimes called FeynmanDyson diagrams or Dyson graphs,^{[5]} because when they were introduced the path integral was unfamiliar, and Freeman Dyson's derivation from oldfashioned 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". This reflects historical fact: Feynman had to lobby hard for the diagrams which confused the establishment physicists trained in equations and graphs.^{[6]}
Representation of physical reality
In their presentations of fundamental interactions,^{[7]}^{[8]} written from the particle physics perspective, Gerard 't Hooft and Martinus Veltman gave good arguments for taking the original, nonregularized 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:^{[9]} "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 manybody 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.
Particlepath interpretation
A Feynman diagram is a representation of quantum field theory processes in terms of 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.
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 amplitudes, 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.
Feynman diagrams are a pictorial representation of a contribution to the total amplitude for a process that 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.
Feynman diagrams are often confused with spacetime diagrams and bubble chamber images because they all describe particle scattering. Feynman diagrams are graphs 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.
Description
A Feynman diagram represents a perturbative contribution to the amplitude of a quantum transition from some initial quantum state to some final quantum state.
For example, in the process of electronpositron annihilation the initial state is one electron and one positron, the final state: two photons.
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).
A Feynman diagram consists of points, called vertices, and lines attached to the vertices.
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).
In QED there are two types of particles: electrons/positrons (called fermions) and photons (called gauge bosons). They are represented in Feynman diagrams as follows:
 Electron in the initial state is represented by a solid line with an arrow pointing toward the vertex (→•).
 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 (~• and •~).
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.
The vertices might be connected by a bosonic or fermionic propagator. A bosonic propagator is represented by a wavy line connecting two vertices (•~•). A fermionic propagator is represented by a solid line (with an arrow in one or another direction) connecting two vertices, (•←•).
The number of vertices gives the order of the term in the perturbation series expansion of the transition amplitude.
Electronpositron annihilation example
The electronpositron annihilation interaction:
$e^+e^\backslash to2\backslash gamma$
has a contribution from the second order Feynman diagram shown adjacent:
In the initial state (at the bottom; early time) there is one electron (e^{−}) and one positron (e^{+}) and in the final state (at the top; late time) there are two photons (γ).
Canonical quantization formulation
The probability amplitude for a transition of a quantum system from the initial state $i\backslash rangle$ to the final state $f\backslash rangle$ is given by the matrix element
 $S\_\{fi\}=\backslash langle\; fSi\backslash rangle\backslash ;,$
where $S$ is the Smatrix.
In the canonical quantum field theory the Smatrix is represented within the interaction picture by the perturbation series in the powers of the interaction Lagrangian,
 $S=\backslash sum\_\{n=0\}^\{\backslash infty\}\{i^n\backslash over\; n!\}\backslash int\backslash prod\_\{j=1\}^n\; d^4\; x\_j\; T\backslash prod\_\{j=1\}^n\; L\_v(x\_j)\backslash equiv\backslash sum\_\{n=0\}^\{\backslash infty\}S^\{(n)\}\backslash ;,$
where $L\_v$ is the interaction Lagrangian and $T$ signifies the timeordered product of operators.
A Feynman diagram is a graphical representation of a term in the Wick's expansion of the timeordered product in the $n$th order term $S^\{(n)\}$ of the Smatrix,
 $T\backslash prod\_\{j=1\}^nL\_v(x\_j)=\backslash sum\_\{\backslash mathrm\{all\backslash ;possible\backslash ;contractions\}\}(\backslash pm)N\backslash prod\_\{j=1\}^nL\_v(x\_j)\backslash ;,$
where $N$ signifies the normalproduct of the operators and $(\backslash pm)$ 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 QED interaction Lagrangian, $L\_v=g\backslash bar\backslash psi\backslash gamma^\backslash mu\backslash psi\; A\_\backslash mu$, describing the interaction of a fermionic field $\backslash psi$ with a bosonic gauge field $A\_\backslash mu$, the Feynman rules can be formulated in coordinate space as follows:
 Each integration coordinate $x\_j$ 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 $A\_\backslash mu(x\_i)$ is represented by a wiggly line attached to the point $x\_i$;
 A fermionic field $\backslash psi(x\_i)$ is represented by a solid line attached to the point $x\_i$ with an arrow toward the point;
 A fermionic field $\backslash bar\backslash psi(x\_i)$ is represented by a solid line attached to the point $x\_i$ with an arrow from the point;
Example: second order processes in QED
The second order perturbation term in the Smatrix is
 $S^\{(2)\}=\{(ie)^2\backslash over\; 2!\}\backslash int\; d^4x\backslash ,\; d^4x\text{'}\backslash ,\; T\backslash bar\backslash psi(x)\backslash ,\backslash gamma^\backslash mu\backslash ,\backslash psi(x)\backslash ,A\_\backslash mu(x)\backslash ,\backslash bar\backslash psi(x\text{'})\backslash ,\backslash gamma^\backslash nu\backslash ,\backslash psi(x\text{'})\backslash ,A\_\backslash nu(x\text{'}).\backslash ;$
Scattering of fermions
The Wick's expansion of the integrand gives (among others) the following term
$N\backslash bar\backslash psi(x)\backslash gamma^\backslash mu\backslash psi(x)\backslash bar\backslash psi(x\text{'})\backslash gamma^\backslash nu\backslash psi(x\text{'})\backslash underline\{A\_\backslash mu(x)A\_\backslash nu(x\text{'})\}\backslash ;,$
where
$\backslash underline\{A\_\backslash mu(x)A\_\backslash nu(x\text{'})\}=\backslash int\{d^4k\backslash over(2\backslash pi)^4\}\{ig\_\{\backslash mu\backslash nu\}\backslash over\; k^2+i0\}e^\{ik(xx\text{'})\}$
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:
 $e^e^$ scattering (initial state at the right, final state at the left of the diagram);
 $e^+e^+$ scattering (initial state at the left, final state at the right of the diagram);
 $e^e^+$ scattering (initial state at the bottom/top, final state at the top/bottom of the diagram).
Compton scattering and annihilation/generation of $e^e^+$ pairs
Another interesting term in the expansion is
 $N\backslash bar\backslash psi(x)\backslash ,\backslash gamma^\backslash mu\backslash ,\backslash underline\{\backslash psi(x)\backslash ,\backslash bar\backslash psi(x\text{'})\}\backslash ,\backslash gamma^\backslash nu\backslash ,\backslash psi(x\text{'})\backslash ,A\_\backslash mu(x)\backslash ,A\_\backslash nu(x\text{'})\backslash ;,$
where
 $\backslash underline\{\backslash psi(x)\backslash bar\backslash psi(x\text{'})\}=\backslash int\{d^4p\backslash over(2\backslash pi)^4\}\{i\backslash over\; \backslash gamma\; pm+i0\}e^\{ip(xx\text{'})\}$
is the fermionic contraction (propagator).
Path integral formulation
In a pathintegral, 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 welldefined 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 ddimensions, whose action integral is:
 $S\; =\; \backslash int\; \{1\backslash over\; 2\}\; \backslash partial\_\backslash mu\; \backslash phi\; \backslash partial^\backslash mu\; \backslash phi\; d^dx\; \backslash ,.$
The probability amplitude for a process is:
 $\backslash int\_A^B\; e^\{iS\}\; D\backslash phi\backslash ,,$
where A and B are spacelike hypersurfaces that define the boundary conditions. The collection of all the $\backslash ,\backslash phi(A)$ on the starting hypersurface give the initial value of the field, analogous to the starting position for a point particle, and the field values $\backslash ,\backslash phi(B)$ 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 fieldtofield transition amplitude.
The path integral gives the expectation value of operators between the initial and final state:
 $\backslash int\_A^B\; e^\{iS\}\; \backslash phi(x\_1)\; ...\; \backslash phi(x\_n)\; D\backslash phi\; =\; \backslash langle\; A\; \backslash phi(x\_1)\; ...\; \backslash phi(x\_n)\; B\; \backslash rangle\backslash ,,$
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 pathintegral 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:
 $\{\backslash int\; e^\{iS\}\; \backslash phi(x\_1)\; ...\; \backslash phi(x\_n)\; D\backslash phi\; \backslash over\; \backslash int\; e^\{iS\}\; D\backslash phi\; \}\; =\; \backslash langle\; 0\; \; \backslash phi(x\_1)\; ....\; \backslash phi(x\_n)\; 0\backslash rangle\; \backslash ,.$
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 initialtofinal amplitudes are illdefined if one thinks of the continuum limit right from the beginning, because the fluctuations in the field can become unbounded. So the pathintegral should be thought of as on a discrete square lattice, with lattice spacing $a$ and the limit $a\backslash rightarrow\; 0$ 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:
 $$
\phi(x) = \int {dk\over (2\pi)^d} \phi(k) e^{ik\cdot x} = \int_k \phi(k) e^{ikx}\,.
Here the integration domain is over k restricted to a cube of side length $2\backslash pi/a$, so that large values of k are not allowed. It is important to note that the kmeasure contains the factors of $2\backslash pi$ from Fourier transforms, this is the best standard convention for kintegrals in QFT. The lattice means that fluctuations at large k are not allowed to contribute right away, they only start to contribute in the limit $a\backslash rightarrow\; 0$. Sometimes, instead of a lattice, the field modes are just cut off at high values of k instead.
It is also convenient from time to time to consider the spacetime volume to be finite, so that the k modes are also a lattice. This is not strictly as necessary as the spacelattice limit, because interactions in k are not localized, but it is convenient for keeping track of the factors in front of the kintegrals and the momentumconserving delta functions that will arise.
On a lattice, (ii), the action needs to be discretized:
 $S=\; \backslash sum\_\{,y>\}\; \{1\backslash over\; 2\}\; (\backslash phi(x)\; \; \backslash phi(y)\; )^2\backslash ,,$
where $,y>$ is a pair of nearest lattice neighbors $x$ and $y$. The discretization should be thought of as defining what the derivative $\backslash partial\_\backslash mu\; \backslash phi$ means.
In terms of the lattice Fourier modes, the action can be written:
 $$
S= \int_k ( (1\cos(k_1)) +(1\cos(k_2)) + ... + (1\cos(k_d)) )\phi^*_k \phi^k\,.
For k near zero this is:
 $$
S = \int_k {1\over 2} k^2 \phi(k)^2\,.
Now we have the continuum Fourier transform of the original action. In finite volume, the quantity $d^dk$ is not infinitesimal, but becomes the volume of a box made by neighboring Fourier modes, or $(2\backslash pi/V)^d$.
The field $\backslash ,\backslash phi$ is realvalued, so the Fourier transform obeys:
 $\backslash phi(k)^*\; =\; \backslash phi(k)\backslash ,.$
In terms of real and imaginary parts, the real part of $\backslash ,\backslash phi(k)$ is an even function of k, while the imaginary part is odd. The Fourier transform avoids doublecounting, so that it can be written:
 $S\; =\; \backslash int\_k\; \{1\backslash over\; 2\}\; k^2\; \backslash phi(k)\; \backslash phi(k)$
over an integration domain that integrates over each pair (k,−k) exactly once.
For a complex scalar field with action
 $S\; =\; \backslash int\; \{1\backslash over\; 2\}\; \backslash partial\_\backslash mu\backslash phi^*\; \backslash partial^\backslash mu\backslash phi\; d^dx$
the Fourier transform is unconstrained:
 $S\; =\; \backslash int\_k\; \{1\backslash over\; 2\}\; k^2\; \backslash phi(k)^2$
and the integral is over all k.
Integrating over all different values of $\backslash ,\backslash phi(x)$ is equivalent to integrating over all Fourier modes, because taking a Fourier transform is a unitary linear transformation of field coordinates. When you change coordinates in a multidimensional integral by a linear transformation, the value of the new integral is given by the determinant of the transformation matrix. If
 $y\_i\; =\; A\_\{ij\}\; x\_j\backslash ,,$
then
 $$
\det(A) \int dx_1 dx_2 ... dx_n = \int dy_1 dy_2 ... dy_n\,.
If A is a rotation, then
 $$
A^T A = I
\,
so that $\backslash det\; A\; =\; \backslash pm\; 1$, and the sign depends on whether the rotation includes a reflection or not.
The matrix that changes coordinates from $\backslash ,\backslash phi(x)$ to $\backslash ,\backslash phi(k)$ can be read off from the definition of a Fourier transform.
 $A\_\{kx\}\; =\; e^\{ikx\}\; \backslash ,$
and the Fourier inversion theorem tells you the inverse:
 $A^\{1\}\_\{kx\}\; =\; e^\{ikx\}\; \backslash ,$
which is the complex conjugatetranspose, up to factors of $2\backslash pi$. On a finite volume lattice, the determinant is nonzero and independent of the field values.
 $\backslash det\; A\; =\; 1\; \backslash ,$
and the path integral is a separate factor at each value of k.
 $\backslash int$
\exp \biggl({i \over 2} \sum_k k^2 \phi^*(k) \phi(k) \biggr) D\phi = \prod_k \int_{\phi_k} e^
Dividing by I,
 $\backslash langle\; x^\{2n\}\backslash rangle=\{\backslash int\; x^\{2n\}\; e^\{a\; x^2/2\}\; \backslash over\; \backslash int\; e^\{a\; x^2/2\}\; \}\; =\; 1\; \backslash cdot\; 3\; \backslash cdot\; 5\; ...\; \backslash cdot\; (2n1)\; \{1\backslash over\; a^n\}$
 $\backslash langle\; x^2\; \backslash rangle\; =\; \{1\backslash over\; a\}$
If Wick's theorem were correct, the higher moments would be given by all possible pairings of a list of 2n x's:
 $\backslash langle\; x\_1\; x\_2\; x\_3\; ...\; x\_\{2n\}\; \backslash rangle$
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 2n3 different x's leaving 2n−4, and so on. This means that Wick's theorem, uncorrected, says that the expectation value of $x^\{2n\}$ should be:
 $\backslash langle\; x^\{2n\}\; \backslash rangle\; =\; (2n1)\backslash cdot(2n3)....\; \backslash cdot5\; \backslash cdot\; 3\; \backslash cdot\; 1\; (\backslash langle\; x^2\backslash rangle)^n$
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 selfinteraction, with an action:
 $S\; =\; \backslash int\; \backslash partial^\backslash mu\; \backslash phi\; \backslash partial\_\backslash mu\backslash phi\; +\; \{\backslash lambda\; \backslash over\; 4!\}\; \backslash phi^4.$
The reason for the combinatorial factor 4! will be clear soon. Writing the action in terms of the lattice (or continuum) Fourier modes:
 $S\; =\; \backslash int\_k\; k^2\; \backslash phi(k)^2\; +\; \backslash int\_\{k\_1k\_2k\_3k\_4\}\; \backslash phi(k\_1)\; \backslash phi(k\_2)\; \backslash phi(k\_3)\backslash phi(k\_4)\; \backslash delta(k\_1+k\_2+k\_3\; +\; k\_4)\; =\; S\_F\; +\; X.$
Where $S\_F$ 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 $\backslash lambda$, giving a series of corrections to the free action.
 $e^\{S\}\; =\; e^\{S\_F\}\; (\; 1\; +\; X\; +\; \{1\backslash over\; 2!\}\; X\; X\; +\; \{1\backslash over\; 3!\}\; X\; X\; X\; +\; ...\; )$
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 halflines, one for each factor of $\backslash phi(k)$. The halflines meet at a vertex, which contributes a deltafunction that 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 pathintegral for the fourfield correlator:
 $\backslash langle\; \backslash phi(k\_1)\; \backslash phi(k\_2)\; \backslash phi(k\_3)\; \backslash phi(k\_4)\; \backslash rangle\; =\; \{\backslash int\; e^\{S\}\; \backslash phi(k\_1)\backslash phi(k\_2)\backslash phi(k\_3)\backslash phi(k\_4)\; D\backslash phi\; \backslash over\; Z\}$
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 $\backslash phi(k\_i)$ can now match up with the momenta of the X's in the expansion. The insertions should also be thought of as halflines, four in this case, which carry a momentum k, but one that is not integrated.
The lowest order contribution comes from the first nontrivial term $e^\{S\_F\}\; X$ in the Taylor expansion of the action. Wick's theorem requires that the momenta in the X halflines, the $\backslash phi(k)$ factors in X, should match up with the momenta of the external halflines in pairs. The new contribution is equal to:
 $\backslash lambda\; \{1\backslash over\; k\_1^2\}\; \{1\backslash over\; k\_2^2\}\; \{1\backslash over\; k\_3^2\}\; \{1\backslash over\; k\_4^2\}.$
The 4! inside X is canceled because there are exactly 4! ways to match the halflines in X to the external halflines. Each of these different ways of matching the halflines 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 nth order.
The nth order terms has:
 4n internal halflines, which are the factors of $\backslash phi(k)$ from the X's. These all end on a vertex, and are integrated over all possible k.
 external halflines, which are the come from the $\backslash phi(k)$ insertions in the integral.
By Wick's theorem, each pair of halflines must be paired together to make a line, and this line gives a factor of
 $\backslash delta(k\_1\; +\; k\_2)\; \backslash over\; k\_1^2$
which multiplies the contribution. This means that the two halflines 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 halfline at the tail end of the arrow carries momentum k, while the halfline at the headend carries momentum −k. If one of the two halflines 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 that are formed by linking the halflines in the X's with the external halflines, representing insertions, are the Feynman diagrams of this theory. Each line carries a factor of $1\backslash over\; k^2$, 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 halflines 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 that 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 halfline joins up with another X, and the remaining halflines 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 that 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 that 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 kintegrals that 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 realvalued (actually R^d valued), the value associated with each line is the momentum. The boundary operator takes each line to the sum of the endvertices 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 kvalued weighted graph is zero.
A set of kvalues 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 2cell. The klabelings of a graph that conserves momentum (which has zero boundary) up to redefinitions of k (up to boundaries of 2cells) define the first homology of a graph. The number of independent momenta that 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 halflines is large, and by Wick's theorem, each way of pairing up the halflines 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 halflines in the X joined to each other. There are 4×3 ways to join the external halflines 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 halflines of one X to all the halflines 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 halflines 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 halflines 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 2.
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 particlepaths: when identical particles are present, the integral over all intermediate particles must not doublecount states that differ only 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 halfline 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 halflines, and this is a factor of n!. Each permutation of the halflines 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.
Connected diagrams: linkedcluster theorem
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 (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 quantum partition function $Z[J]$. More precisely, connected Feynman diagrams determine
 $i\; W[J]\backslash equiv\; \backslash ln\; Z[J].$
To see this, one should recall that
 $Z[J]\backslash propto\backslash sum\_k\{D\_k\}$
with $D\_k$ constructed from some (arbitrary) Feynman diagram that can be thought to consist of several connected components $C\_i$. If one encounters $n\_i$ (identical) copies of a component $C\_i$ within the Feynman diagram $D\_k$ one has to include a symmetry factor $n\_i!$. However, in the end each contribution of a Feynman diagram $D\_k$ to the partition function has the generic form
 $\backslash prod\_i\; \{C\_\{i\}^\{n\_i\}\; \backslash over\; n\_i!\}$
where $i$ labels the (infinite) many connected Feynman diagrams possible.
A scheme to successively create such contributions from the $D\_k$ to $Z[J]$ is obtained by
 $\backslash left(\backslash frac\{1\}\{0!\}+\backslash frac\{C\_1\}\{1!\}+\backslash frac\{C^2\_1\}\{2!\}+\backslash dots\backslash right)\backslash left(1+C\_2+\backslash frac\{1\}\{2\}C^2\_2+\backslash dots\backslash right)\backslash dots$
and therefore yields
 $Z[J]\backslash propto\backslash prod\_i\{\backslash sum^\backslash infty\_\{n\_i=0\}\{\backslash frac\{C\_i^\{n\_i\}\}\{n\_i!\}\}\}=\backslash exp\{\backslash sum\_i\{C\_i\}\}\backslash propto\; \backslash exp\{W[J]\}.$
To establish the normalization $Z\_0=\backslash exp\{W[0]\}=1$ one simply calculates all connected vacuum diagrams, i.e., the diagrams without any sources $J$ (sometimes referred to as external legs of a Feynman diagram).
Vacuum bubbles
An immediate consequence of the linkedcluster theorem is that all vacuum bubbles, diagrams without external lines, cancel when calculating correlation functions. A correlation function is given by a ratio of pathintegrals:
 $\backslash langle\; \backslash phi\_1(x\_1)\; ...\; \backslash phi\_n(x\_n)\backslash rangle\; =\; \{\backslash int\; e^\{S\}\; \backslash phi\_1(x\_1)\; ...\backslash phi\_n(x\_n)\; D\backslash phi\; \backslash over\; \backslash int\; e^\{S\}\; D\backslash phi\}.$
The top is the sum over all Feynman diagrams, including disconnected diagrams that 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:
 $\backslash int\; e^\{S\}\backslash phi\_1(x\_1)...\backslash phi\_n(x\_n)\; D\backslash phi\; =\; (\backslash sum\; E\_i)(\; \backslash exp(\backslash sum\_i\; C\_i)\; ).$
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:
 $Z=\; \backslash int\; e^\{S\}\; D\backslash phi\; =\; e^\{HT\}\; =\; e^\{\backslash rho\; V\}$
where $\backslash rho$ is the energy density in the vacuum. Each vacuum bubble contains a factor of $\backslash delta(k)$ zeroing the total k at each vertex, and when there are no external lines, this contains a factor of $\backslash delta(0)$, because the momentum conservation is overenforced. 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 that contribute to the action, but are not dynamical variables. A scalar field source is another scalar field h that contributes a term to the (Lorentz) Lagrangian:
 $\backslash int\; h(x)\; \backslash phi(x)\; d^dx\; =\; \backslash int\; h(k)\; \backslash phi(k)\; d^dk\; \backslash ,$
In the Feynman expansion, this contributes H terms with one halfline ending on a vertex. Lines in a Feynman diagram can now end either on an X vertex, or on an Hvertex, and only one line enters an H vertex. The Feynman rule for an Hvertex 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.
 $\backslash log(Z[h])\; =\; \backslash sum\_\{n,C\}\; h(k\_1)\; h(k\_2)\; ...\; h(k\_n)\; C(k\_1,...,k\_n)\backslash ,$
where $C(k\_1,....,k\_n)$ 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:
 $Z[h]\; =\; \backslash int\; e^\{iS\; +\; i\backslash int\; h\backslash phi\}\; D\backslash phi\; \backslash ,$
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:
 $\backslash int\; \backslash rho(y)\; e^\{i\; k\; y\}\; d^n\; y\; =\; \backslash langle\; e^\{i\; k\; y\}\; \backslash rangle\; =\; \backslash langle\; \backslash prod\_\{i=1\}^\{n\}\; e^\{ih\_i\; y\_i\}\backslash rangle\; \backslash ,$
The fourier transform is the expectation of an oscillatory exponential. The path integral in the presence of a source h(x) is:
 $Z[h]\; =\; \backslash int\; e^\{iS\}\; e^\{i\backslash int\_x\; h(x)\backslash phi(x)\}\; D\backslash phi\; =\; \backslash langle\; e^\{i\; h\; \backslash phi\; \}\backslash rangle$
which, on a lattice, is the product of an oscillatory exponential for each field value:
 $\backslash langle\; \backslash prod\_x\; e^\{i\; h\_x\; \backslash phi\_x\}\backslash rangle$
The fourier transform of a deltafunction is a constant, which gives a formal expression for a delta function:
 $\backslash delta(xy)\; =\; \backslash int\; e^\{ik(xy)\}\; dk$
This tells you what a field delta function looks like in a pathintegral. For two scalar fields $\backslash phi$ and $\backslash eta$,
 $\backslash delta(\backslash phi\; \; \backslash eta)\; =\; \backslash int\; e^\{\; i\; h(x)(\backslash phi(x)\; \backslash eta(x)d^dx\}\; Dh$
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 multidimensional 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:
 $\backslash langle\backslash phi(x)\backslash rangle\; =\; \{1\backslash over\; Z\}\; \{\backslash partial\; \backslash over\; \backslash partial\; h(x)\}\; Z[h]\; =\; \{\backslash partial\backslash over\backslash partial\; h(x)\}\; \backslash log(Z[h]).$
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 pathintegral can be extended to the Fermi case, but only if the notion of integration is expanded. A Grassman integral of a free Fermi field is a highdimensional determinant or Pfaffian, which defines the new type of Gaussian integration appropriate for Fermi fields.
The two fundamental formulas of Grassmann integration are:
 $\backslash int\; e^\{M\_\{ij\}\{\backslash bar\backslash psi\}^i\; \backslash psi^j\}\; D\backslash bar\backslash psi\; D\backslash psi=\; \backslash mathrm\{Det\}(M)$
where M is an arbitrary matrix and $\backslash scriptstyle\; \backslash psi,\backslash bar\backslash psi$ are independent Grassmann variables for each index i, and
 $\backslash int\; e^\; (1)^S\; \backslash prod\_\{\backslash mathrm\{pairs\}\backslash ;\; i,j\}\; \backslash delta(k\_i\; k\_j)\; \{1\backslash over\; \backslash gamma\backslash cdot\; k\_i\; \; m\}$
where S is the sign of the permutation that reorders the sequence of psibars and psis to put the ones that are paired up to make the deltafunctions next to each other, with the psibar coming right before the psi. Since a psipsibar pair is a commuting element of the Grassman algebra, it doesn't matter what order the pairs are in. If more than one psi/psibar 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 that completed the Bosonic Wick's theorem earlier.
The rules for spin1/2 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 2color 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 fourFermi interactions (like in the Fermi effective theory of the Weak interactions) there are more kintegrals 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 that 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 halfline must be joined with the unlinked first halfline. This requires one permutation to move the last psibar 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 Afield is:
 $S\; =\; \backslash int\; \{1\backslash over\; 4\}\; F^\{\backslash mu\backslash nu\}\; F\_\{\backslash mu\backslash nu\}\; =\; \backslash int\; \; \{1\backslash over\; 2\}(\backslash partial^\backslash mu\; A\_\backslash nu\; \backslash partial\_\backslash mu\; A^\backslash nu\; \; \backslash partial^\backslash mu\; A\_\backslash mu\; \backslash partial\_\backslash nu\; A^\backslash nu\; ).\; \backslash ,$
The quadratic form defining the propagator is noninvertible. 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:
 $\backslash int\; \backslash delta(\backslash partial\_\backslash mu\; A^\backslash mu\; \; f)\; e^\{\{f^2\backslash over\; 2\}\; \}\; Df.$
The first factor, the delta function, fixes the gauge. The second factor sums over different values of f that are inequivalent gauge fixings. This is simply
 $e^\{\; \{(\backslash partial\_\backslash mu\; A\_\backslash mu)^2\backslash over\; 2\}\}.$
The additional contribution from gaugefixing cancels the second half of the free Lagrangian, giving the Feynman Lagrangian:
 $S=\; \backslash int\; \backslash partial^\backslash mu\; A^\backslash nu\; \backslash partial\_\backslash mu\; A\_\backslash nu$
which is just like four independent free scalar fields, one for each component of A. The Feynman propagator is:
 $\backslash langle\; A\_\backslash mu(k)\; A\_\backslash nu(k\text{'})\; \backslash rangle\; =\; \backslash delta(k+k\text{'})\; \{g\_\{\backslash mu\backslash nu\}\; \backslash over\; k^2\; \}.$
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 $\backslash lambda$, which does not affect anything:
 $S=\; \backslash int\; \{1\backslash over\; 2\}(\backslash partial^\backslash mu\; A^\backslash nu\; \backslash partial\_\backslash mu\; A\_\backslash nu\; \; \backslash lambda\; (\backslash partial\_\backslash mu\; A^\backslash mu)^2)$
and the covariant propagator for QED is:
 $\backslash langle\; A\_\backslash mu(k)\; A\_\backslash nu(k\text{'})\; \backslash rangle\; =\backslash delta(k+k\text{'})\{g\_\{\backslash mu\backslash nu\}\; \; \backslash lambda\{k\_\backslash mu\; k\_\backslash nu\; \backslash over\; k^2\}\; \backslash over\; k^2\}.$
Spin 1: nonabelian ghosts
To find the Feynman rules for nonabelian Gauge fields, the procedure that performs the Gauge fixing must be carefully corrected to account for a change of variables in the pathintegral.
The gauge fixing factor has an extra determinant from popping the delta function:
 $\backslash delta(\backslash partial\_\backslash mu\; A\_\backslash mu\; \; f)\; e^\{\{f^2\backslash over\; 2\}\}\; \backslash mathrm\{Det\}\{M\}$
To find the form of the determinant, consider first a simple twodimensional integral of a function f that depends only on r, not on the angle $\backslash scriptstyle\; \backslash theta$. Inserting an integral over theta:
 $\backslash int\; f(r)\; dx\; dy\; =\; \backslash int\; f(r)\; \backslash int\; d\backslash theta\; \backslash delta(y)\; \{dy\; \backslash over\; d\backslash theta\}\; dx\; dy$
The derivativefactor ensures that popping the delta function in $\backslash scriptstyle\; \backslash theta$ removes the integral. Exchanging the order of integration,
 $\backslash int\; f(r)\; dx\; dy\; =\; \backslash int\; d\backslash theta\; \backslash int\; f(r)\; \backslash delta(y)\; \{dy\backslash over\; d\backslash theta\}\; dx\; dy$
but now the deltafunction can be popped in y,
 $\backslash int\; f(r)\; dx\; dy\; =\; \backslash int\; d\backslash theta\_0\; \backslash int\; f(x)\; \{dy\backslash over\; d\backslash theta\}\; dx\backslash ,.$
The integral over $\backslash scriptstyle\backslash theta$ just gives an overall factor of $\backslash scriptstyle\; 2\backslash pi$, while the rate of change of $\backslash scriptstyle\; y$ with a change in $\backslash theta$ is just x, so this exercise reproduces the standard formula for polar integration of a radial function:
 $\backslash int\; f(r)\; dx\; dy\; =\; 2\backslash pi\; \backslash int\; f(x)\; x\; dx$
In the pathintegral for a nonabelian gauge field, the analogous manipulation is:
 $\backslash int\; DA\; \backslash int\; \backslash delta(F(A))\; \backslash mathrm\{Det\}(\{\backslash partial\; F\backslash over\; \backslash partial\; G\})\; DG\; e^\{iS\}\; =\; \backslash int\; DG\; \backslash int\; \backslash delta(F(A))\backslash mathrm\{Det\}(\{\backslash partial\; F\backslash over\; \backslash partial\; G\})\; e^\{iS\}\; \backslash ,$
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.
 $\backslash int\; \backslash mathrm\{Det\}(\{\backslash partial\; F\backslash over\; \backslash partial\; G\})e^\{iS\_\{GF\}\}\; DA\; \backslash ,$
To get a covariant gauge, the gauge fixing condition is the same as in the Abelian case:
 $\backslash partial\_\backslash mu\; A^\backslash mu\; =\; f\; \backslash ,$
Whose variation under an infinitesimal gauge transformation is given by:
 $\backslash partial\_\backslash mu\; D\_\backslash mu\; \backslash alpha\; \backslash ,$
where $\backslash scriptstyle\; \backslash alpha$ is the adjoint valued element of the Lie algebra at every point that performs the infinitesimal gauge transformation. This adds the Faddeev Popov determinant to the action:
 $Det(\backslash partial\_\backslash mu\; D\_\backslash mu)\; \backslash ,$
which can be rewritten as a Grassman integral by introducing ghost fields:
 $\backslash int\; e^\{\backslash bar\backslash eta\; \backslash partial\_\backslash mu\; D^\backslash mu\; \backslash eta\}\; D\backslash bar\backslash eta\; D\backslash eta\; \backslash ,$
The determinant is independent of f, so the pathintegral 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:
 $S=\; \backslash int\; Tr\; \backslash partial\_\backslash mu\; A\_\backslash nu\; \backslash partial^\backslash mu\; A^\backslash nu\; +\; f^i\_\{jk\}\; \backslash partial^\backslash nu\; A\_i^\backslash mu\; A^j\_\backslash mu\; A^k\_\backslash nu\; +\; f^i\_\{jr\}\; f^r\_\{kl\}\; A\_i\; A\_j\; A^k\; A^l\; +\; Tr\; \backslash partial\_\backslash mu\; \backslash bar\backslash eta\; \backslash partial^\backslash mu\; \backslash eta\; +\; \backslash bar\backslash eta\; A\_j\; \backslash eta\; \backslash ,$
The diagrams are derived from this action. The propagator for the spin1 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.
Particlepath representation
Feynman diagrams were originally discovered by Feynman, by trial and error, as a way to represent the contribution to the Smatrix from different classes of particle trajectories.
Schwinger representation
The Euclidean scalar propagator has a suggestive representation:
 $\{1\backslash over\; p^2+m^2\}\; =\; \backslash int\_0^\backslash infty\; e^\{\backslash tau(p^2\; +\; m^2)\}\; d\backslash tau$
The meaning of this identity (which is an elementary integration) is made clearer by Fourier transforming to real space.
 $\backslash Delta(x)\; =\; \backslash int\_0^\backslash infty\; d\backslash tau\; e^\{m^2\backslash tau\}\; \{1\backslash over\; (\{4\backslash pi\backslash tau\})^\{d/2\}\}e^\{x^2\backslash over\; 4\backslash tau\}$
The contribution at any one value of $\backslash tau$ to the propagator is a Gaussian of width $\backslash scriptstyle\; \backslash sqrt\{\backslash tau\}$. The total propagation function from 0 to x is a weighted sum over all proper times $\backslash tau$ of a normalized Gaussian, the probability of ending up at x after a random walk of time $\backslash tau$.
The pathintegral representation for the propagator is then:
 $\backslash Delta(x)\; =\; \backslash int\_0^\backslash infty\; d\backslash tau\; \backslash int\; DX\; e^\{\; \backslash int\_0^\{\backslash tau\}\; (\backslash dot\{x\}^2/2\; +\; m^2)\; d\backslash tau\text{'}\}$
which is a pathintegral 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 phi4 theory formed by joining two x's together in two halflines, and making the remaining lines external, the integral over the internal propagators in the loop is:
 $\backslash int\_k\; \{1\backslash over\; (k^2\; +\; m^2)\}\; \{1\backslash over\; ((k+p)^2\; +\; m^2)\}\; \backslash ,.$
Here one line carries momentum k and the other k+p. The asymmetry can be fixed by putting everything in the Schwinger representation.
 $\backslash int\_\{t,t\text{'}\}\; e^\{t(k^2+m^2)\; \; t\text{'}((k+p)^2\; +m^2)\; \}\; dt\; dt\text{'}\backslash ,.$
Now the exponent mostly depends on t+t',
 $\backslash int\_\{t,t\text{'}\}\; e^\{(t+t\text{'})(k^2+m^2)\; \; t\text{'}\; 2p\backslash cdot\; k\; t\text{'}\; p^2\}\backslash ,,$
except for the asymmetrical little bit. Defining the variable u=(t+t') and $\backslash scriptstyle\; v$= t'/u, the variable u goes from 0 to infinity, while $\backslash scriptstyle\; v$ goes from 0 to 1. The variable u is the total proper time for the loop, while $\backslash scriptstyle\; v$ 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:
 $d(uv)=\; dt\text{'}\backslash ;\backslash ;\backslash ;du\; =\; dt+dt\text{'}\backslash ,,$
and "wedging" gives
 $u\; du\; \backslash wedge\; dv\; =\; dt\; \backslash wedge\; dt\text{'}\backslash ,$.
This allows the u integral to be evaluated explicitly:
 $\backslash int\_\{u,v\}\; u\; e^\{u\; (\; k^2+m^2\; +\; v\; 2p\backslash cdot\; k\; +\; v\; p^2)\}\; =\; \backslash int\; \{1\backslash over\; (k^2\; +\; m^2\; +\; v\; 2p\backslash cdot\; k\; \; v\; p^2)^2\}\; dv$
leaving only the $\backslash ,v$integral. This method, invented by Schwinger but usually attributed to Feynman, is called combining denominator. Abstractly, it is the elementary identity:
 $\{1\backslash over\; AB\}=\; \backslash int\_0^1\; \{1\backslash over(\; vA+\; (1v)B)^2\}\; dv$
But this form does not provide the physical motivation for introducing $\backslash ,v$—$\backslash ,v$ is the proportion of proper time on one of the legs of the loop.
Once the denominators are combined, a shift in k to $k\text{'}=k+vp$ symmetrizes everything:
 $\backslash int\_0^1\; \backslash int\{1\backslash over\; (k^2\; +\; m^2\; +\; v\; 2p\; \backslash cdot\; k\; +\; v\; p^2)^2\}\; dk\; dv\; =\; \backslash int\_0^1\; \backslash int\; \{1\backslash over\; (k\text{'}^2\; +\; m^2\; +\; v(1v)p^2)^2\}\; dk\text{'}\; dv$
This form shows that the moment that p^{2} 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:
 $\backslash int\; dk\; \{1\backslash over\; (k^2\; +\; m^2)\}\; \{1\backslash over\; ((k+p\_1)^2\; +\; m^2)\}\; ...\; \{1\backslash over\; ((k+p\_n)^2\; +\; m^2)\}$
The general rule follows from the Schwinger prescription for n+1 denominators:
 $\{1\backslash over\; D\_0\; D\_1\; ...\; D\_n\}\; =\; \backslash int\_0^\backslash infty\; ...\backslash int\_0^\backslash infty\; e^\{u\_0\; D\_0\; ...\; u\_n\; D\_n\}\; du\_0\; ...\; du\_n\; \backslash ,.$
The integral over the Schwinger parameters $\backslash scriptstyle\; u\_i$ can be split up as before into an integral over the total proper time $\backslash scriptstyle\; u\; =\; u\_0\; +\; u\_1\; ...\; +\; u\_n$ and an integral over the fraction of the proper time in all but the first segment of the loop $\backslash scriptstyle\; v\_i\; =\; u\_i/u$ for $\backslash scriptstyle\; i\backslash in\; \backslash \{\; 1,2,...,n\backslash \}$. 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:
 $du\; =\; du\_0\; +\; du\_1\; ...\; +\; du\_n\; \backslash ,$
 $d(uv\_i)\; =\; d\; u\_i\; \backslash ,.$
"Wedging" all these equation together, one obtains
 $u^n\; du\; \backslash wedge\; dv\_1\; \backslash wedge\; dv\_2\; ...\; \backslash wedge\; dv\_n\; =\; du\_0\; \backslash wedge\; du\_1\; ...\; \backslash wedge\; du\_n\; \backslash ,.$
This gives the integral:
 $\backslash int\_0^\backslash infty\; \backslash int\_\{\backslash mathrm\{simplex\}\}\; u^n\; e^\{u(v\_0\; D\_0\; +\; v\_1\; D\_1\; +\; v\_2\; D\_2\; ...\; +\; v\_n\; D\_n)\}\; dv\_1\; ...dv\_n\; du\backslash ,,$
where the simplex is the region defined by the conditions $\backslash scriptstyle\; v\_i>0$ and $\backslash scriptstyle\; \backslash sum\_\{i=1\}^n\; v\_i\; <\; 1$ as well as $\backslash scriptstyle\; v\_0\; =\; 1\backslash sum\_\{i=1\}^n\; v\_i$. Performing the u integral gives the general prescription for combining denominators:
 $\{1\backslash over\; D\_0\; ...\; D\_n\; \}\; =\; n!\; \backslash int\_\{\backslash mathrm\{simplex\}\}\; \{1\backslash over\; (v\_0\; D\_0\; +v\_1\; D\_1\; ...\; +\; v\_n\; D\_n)^\{n+1\}\}\; dv\_1\; dv\_2\; ...\; dv\_n$
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 $\backslash scriptstyle\; v\_i$ 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 selfevident, 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 singleparticle state is to some extent incompatible with the notion of an object localized in space.
In the 1930s, Wigner gave a mathematical definition for singleparticle states: they are a collection of states that 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 oneparticle state from the vacuum, which means that the field operator $\backslash phi(x)$ 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 3particle, 5particle (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 higherorder corrections.
The relation between scattering and correlation functions is the LSZtheorem: The scattering amplitude for n particles to go to mparticles 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 $\backslash lambda\; \backslash phi^4$ interaction of the previous section, the order $\backslash lambda$ contribution to the (Lorentz) correlation function is:
 $\backslash langle\; \backslash phi(k\_1)\backslash phi(k\_2)\backslash phi(k\_3)\backslash phi(k\_4)\backslash rangle\; =\; \{i\backslash over\; k\_1^2\}\{i\backslash over\; k\_2^2\}\; \{i\backslash over\; k\_3^2\}\; \{i\backslash over\; k\_4^2\}\; i\backslash lambda\; \backslash ,$
Stripping off the external propagators, that is, removing the factors of $i/k^2$, gives the invariant scattering amplitude M:
 $M\; =\; i\backslash lambda\; \backslash ,$
which is a constant, independent of the incoming and outgoing momentum. The interpretation of the scattering amplitude is that the sum of $M^2$ over all possible final states is the probability for the scattering event. The normalization of the singleparticle states must be chosen carefully, however, to ensure that M is a relativistic invariant.
Nonrelativistic 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:
 $\backslash int\; dk\; k\backslash rangle\backslash langle\; k\backslash ,$
In relativity, the integral over the kstates for a particle of mass m integrates over a hyperbola in E,k space defined by the energymomentum relation:
 $E^2\; \; k^2\; =\; m^2\; \backslash ,$
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:
 $\backslash int\; \backslash delta(E^2k^2\; \; m^2)\; E,k\backslash rangle\backslash langle\; E,k\; dE\; dk\; =\; \backslash int\; \{dk\; \backslash over\; 2\; E\}\; k\backslash rangle\backslash langle\; k$
So the normalized kstates are different from the relativistically normalized kstates by a factor of $\backslash sqrt\{E\}\; =\; (k^2m^2)^\{1\backslash over\; 4\}$
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 $\backslash sqrt\{m\}$ ). In this limit, the $\backslash phi^4$ 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 deltafunction potential. The lowest order scattering of the theory reveals the nonrelativistic interpretation of this theory—it describes a collection of particles with a deltafunction 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 that 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 shortdistance 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 that 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 that 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 largescale configurations describe nonperturbative 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 BetheSalpeter equation describes the class of diagrams to include to describe a relativistic atom. For quantum chromodynamics, the Shifman Vainshtein Zakharov sum rules describe nonperturbatively excited longwavelength 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 nonperturbative 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 quarkantiquark pair was featured in the television sitcom The Big Bang Theory, in the 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^{[10]}
See also
Notes
References
 Gerardus 't Hooft, Martinus Veltman, Diagrammar, CERN Yellow Report 1973, online
 David Kaiser, Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics, Chicago: University of Chicago Press, 2005. ISBN 0226422666
 Martinus Veltman, Diagrammatica: The Path to Feynman Diagrams, Cambridge Lecture Notes in Physics, ISBN 0521456924 (expanded, updated version of above)
 Mark Srednicki, Quantum Field Theory, online Script (2006)
External links
 AMS article: "What's New in Mathematics: Finitedimensional Feynman Diagrams"
 WikiTeX supports editing Feynman diagrams directly in Wiki articles.
 Drawing Feynman diagrams with FeynDiagram C++ library that produces PostScript output.
 Feynman Diagram Examples using Thorsten Ohl's Feynmf LaTeX package.
 JaxoDraw A Java program for drawing Feynman diagrams.

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