Feynman diagram
Encyclopedia
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

, and first introduced in 1948. The interaction of sub-atomic particles can be complex and difficult to understand intuitively, and the Feynman diagrams allow for a simple visualization of what would otherwise be a rather arcane and abstract formula. As David Kaiser writes, "since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations," and as such "Feynman diagrams have revolutionized nearly every aspect of theoretical physics". However, while the diagrams are applied primarily to quantum field theory, they can also be used in other fields, such as solid-state theory
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...

.

The calculation of probability amplitudes
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...

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
Canonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...

formulation of quantum field theory, a Feynman diagram represents a term in the Wick's expansion
Wick's theorem
Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem . It is named after Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to sums of products of pairs of these operators...

of the perturbative S-matrix. Alternatively, the path integral formulation
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...

of quantum field theory represents the transition amplitude as a weighted sum of all possible histories of the system from the initial to the final state, in terms of either particles or fields. The transition amplitude is then given as the matrix element of the S-matrix between the initial and the final states of the quantum system.

## Motivation and history

When calculating scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

cross section
Cross section (physics)
A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....

s in particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, the interaction between particles can be described by starting from a free field which describes the incoming and outgoing particles, and including an interaction Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

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
Perturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...

, and the time-dependent perturbation theory for fields is known as the Dyson series
Dyson series
In scattering theory, the Dyson series, formulated by British-born American physicist Freeman Dyson, is a perturbative series, and each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics at the second order the difference from...

. When the intermediate states at intermediate times are energy eigenstates (collections of particles with a definite momentum) the series is called old-fashioned perturbation theory.

The Dyson series can be alternately rewritten as a sum over Feynman diagrams, where at each interaction vertex both the energy and momentum are conserved, but where the length of the energy momentum four vector is not equal to the mass. The Feynman diagrams are much easier to keep track of than old-fashioned terms, because the old-fashioned way treats the particle and antiparticle contributions as separate. Each Feynman diagram is the sum of exponentially many old-fashioned terms, because each internal line can separately represent either a particle or an antiparticle. In a non-relativistic theory, there are no antiparticles and there is no doubling, so each Feynman diagram includes only one term.

Feynman gave a prescription for calculating the amplitude for any given diagram from a field theory Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

- the Feynman rules. Each internal line corresponds to a factor of the corresponding virtual particle
Virtual particle
In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...

's propagator
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...

; each vertex where lines meet gives a factor derived from an interaction term in the Lagrangian, and incoming and outgoing lines carry an energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

, momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

, and spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

.

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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, also invented by Feynman–see path integral formulation
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...

.

The naïve application of such calculations often produces diagrams whose amplitudes are infinite
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

, because the short-distance particle interactions require a careful limiting procedure, to include particle self-interactions. The technique of renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....

, suggested by Ernst Stueckelberg
Ernst Stueckelberg
Ernst Carl Gerlach Stueckelberg was a Swiss mathematician and physicist.- Career :In 1927 Stueckelberg got his Ph. D. at the University of Basel under August Hagenbach...

and Hans Bethe
Hans Bethe
Hans Albrecht Bethe was a German-American nuclear physicist, and Nobel laureate in physics for his work on the theory of stellar nucleosynthesis. A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and...

and implemented by Dyson
Freeman Dyson
Freeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...

, Feynman, Schwinger
Julian Schwinger
Julian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...

, and Tomonaga
Sin-Itiro Tomonaga
was a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with Richard Feynman and Julian Schwinger.-Biography:...

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
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

.

### Alternative names

Murray Gell-Mann
Murray Gell-Mann
Murray Gell-Mann is an American physicist and linguist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles...

always referred to Feynman diagrams as Stueckelberg diagrams, after a Swiss physicist, Ernst Stueckelberg
Ernst Stueckelberg
Ernst Carl Gerlach Stueckelberg was a Swiss mathematician and physicist.- Career :In 1927 Stueckelberg got his Ph. D. at the University of Basel under August Hagenbach...

, who devised a similar notation many years earlier. Stueckelberg was motivated by the need for a manifestly covariant formalism for quantum field theory, but did not provide as automated a way to handle symmetry factors and loops, although he was first to find the correct physical interpretation in terms of forward and backward in time particle paths, all without the path-integral. Historically they were sometimes called Feynman-Dyson diagrams or Dyson graphs, because when they were introduced the path integral was unfamiliar, and Freeman Dyson
Freeman Dyson
Freeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...

's derivation from old-fashioned perturbation theory was easier to follow for physicists trained in earlier methods. However, in 2006 Dyson himself confirmed that the diagrams should be called Feynman diagrams because "he taught us how to use them".

## Representation of physical reality

In their presentations of fundamental interactions, written from the particle physics perspective, Gerard ’t Hooft and Martinus Veltman gave good arguments for taking the original, non-regularized Feynman diagrams as the most succinct representation of our present knowledge about the physics of quantum scattering of fundamental particles. Their motivations are consistent with the convictions of James Daniel Bjorken and Sidney Drell
Sidney Drell
Sidney David Drell is an American theoretical physicist and arms control expert. He is a professor emeritus at the Stanford Linear Accelerator Center and a senior fellow at Stanford University's Hoover Institution. Drell is a noted contributor in the field of quantum electrodynamics and particle...

: ”The Feynman graphs and rules of calculation summarize quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

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
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

, use of graphical methods in the many-body problem
Many-body problem
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of a large number of interacting particles. Microscopic here implies that quantum mechanics has to be used to provide an accurate description of the system...

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
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

by Feynman Rules.

## Particle-path interpretation

A Feynman diagram is a representation of quantum field theory processes in terms of particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

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 amplitude
Scattering amplitude
In quantum physics, the scattering amplitude is the amplitude of the outgoing spherical wave relative to the incoming plane wave in the stationary-state scattering process...

s, there is no past and future and all the lines are internal. The particles then begin and end on little x's, which represent the positions of the operators whose correlation is being calculated.

Feynman diagrams are a pictorial representation of a contribution to the total amplitude for a process which can happen in several different ways. When a group of incoming particles are to scatter off each other, the process can be thought of as one where the particles travel over all possible paths, including paths that go backward in time.

Feynman diagrams are often confused with spacetime diagrams and bubble chamber
Bubble chamber
A bubble chamber is a vessel filled with a superheated transparent liquid used to detect electrically charged particles moving through it. It was invented in 1952 by Donald A. Glaser, for which he was awarded the 1960 Nobel Prize in Physics...

images because they all describe particle scattering. Feynman diagrams are graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

s that represent the trajectories of particles in intermediate stages of a scattering process. Unlike a bubble chamber picture, only the sum of all the Feynman diagrams represent any given particle interaction; particles do not choose a particular diagram each time they interact. The law of summation is in accord with the principle of superposition--- every diagram contributes a factor to the total amplitude for the process.

## 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 electron-positron 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
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

there are two types of particles: electrons/positrons (called fermions) and photons (called gauge bosons). They are represented in Feynman diagrams as follows:
1. Electron in the initial state is represented by a solid line with an arrow pointing toward the vertex (→•).
2. Electron in the final state is represented by a line with an arrow pointing away from the vertex: (•→).
3. Positron in the initial state is represented by a solid line with an arrow pointing away from the vertex: (←•).
4. Positron in the final state is represented by a line with an arrow pointing toward the vertex: (•←).
5. 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
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...

. A bosonic propagator is represented by a wavy line connecting two vertexes (•~•). A fermionic propagator is represented by a solid line (with an arrow in one or another direction) connecting two vertexes, (•←•).

The number of vertices gives the order of the term in the perturbation series expansion of the transition amplitude.

### Electron-positron annihilation example

The electron-positron annihilation interaction:

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).

### Perturbative S-matrix

The probability amplitude for a transition of a quantum system from the initial state to the final state is given by the matrix element

where is the S-matrix.

In the canonical quantum field theory the S-matrix is represented within the interaction picture
Interaction picture
In quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...

by the perturbation series in the powers of the interaction Lagrangian,

where is the interaction Lagrangian and signifies the time-ordered product of operators.

A Feynman diagram is a graphical representation of a term in the Wick's expansion of the time-ordered product in the -th order term of the S-matrix,

where signifies the normal-product of the operators and takes care of the possible sign change when commuting the fermionic operators to bring them together for a contraction (a propagator
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...

).

### Feynman rules

The diagrams are drawn according to the Feynman rules which depend upon the interaction Lagrangian. For the QED
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

interaction Lagrangian, , describing the interaction of a fermionic field with a bosonic gauge field , the Feynman rules can be formulated in coordinate space as follows:
1. Each integration coordinate is represented by a point (sometimes called a vertex);
2. A bosonic propagator
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...

is represented by a wiggly line connecting two points;
3. A fermionic propagator is represented by a solid line connecting two points;
4. A bosonic field is represented by a wiggly line attached to the point ;
5. A fermionic field is represented by a solid line attached to the point with an arrow toward the point;
6. A fermionic field is represented by a solid line attached to the point with an arrow from the point;

### Example: second order processes in QED

The second order perturbation term in the S-matrix is

#### Scattering of fermions

{|align="right"
|
|}
The Wick's expansion of the integrand gives (among others) the following term

where

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:
1. scattering (initial state at the right, final state at the left of the diagram);
2. scattering (initial state at the left, final state at the right of the diagram);
3. scattering (initial state at the bottom/top, final state at the top/bottom of the diagram).

#### Compton scattering and annihilation/generation of pairs

Another interesting term in the expansion is

where

is the fermionic contraction (propagator).

## Path integral formulation

In a path-integral, the field Lagrangian, integrated over all possible field histories, defines the probability amplitude to go from one field configuration to another. In order to make sense, the field theory should have a well-defined ground state, and the integral should be performed a little bit rotated into imaginary time.

### Scalar Field Lagrangian

A simple example is the free relativistic scalar field in d-dimensions, whose action integral is:

The probability amplitude for a process is:

where A and B are space-like hypersurfaces which define the boundary conditions. The collection of all the on the starting hypersurface give the initial value of the field, analogous to the starting position for a point particle, and the field values at each point of the final hypersurface defines the final field value, which is allowed to vary, giving a different amplitude to end up at different values. This is the field-to-field transition amplitude.

The path integral gives the expectation value of operators between the initial and final state:

and in the limit that A and B recede to the infinite past and the infinite future, the only contribution that matters is from the ground state (this is only rigorously true if the path-integral is defined slightly rotated into imaginary time). The path integral should be thought of as analogous to a probability distribution, and it is convenient to define it so that multiplying by a constant doesn't change anything:

The normalization factor on the bottom is called the partition function for the field, and it coincides with the statistical mechanical partition function at zero temperature when rotated into imaginary time.

The initial-to-final amplitudes are ill-defined if one thinks of the continuum limit right from the beginning, because the fluctuations in the field can become unbounded. So the path-integral should be thought of as on a discrete square lattice, with lattice spacing and the limit should be taken carefully. If the final results do not depend on the shape of the lattice or the value of a, then the continuum limit exists.

### On a lattice ...

On a lattice, (i), the field can be expanded in Fourier modes:

Here the integration domain is over k restricted to a cube of side length , so that large values of k are not allowed. It is important to note that the k-measure contains the factors of from Fourier transforms, this is the best standard convention for k-integrals in QFT. The lattice means that fluctuations at large k are not allowed to contribute right away, they only start to contribute in the limit . Sometimes, instead of a lattice, the field modes are just cut off at high values of k instead.

It is also convenient from time to time to consider the space-time volume to be finite, so that the k modes are also a lattice. This is not strictly as necessary as the space-lattice limit, because interactions in k are not localized, but it is convenient for keeping track of the factors in front of the k-integrals and the momentum-conserving delta functions which will arise.

On a lattice, (ii), the action needs to be discretized:

where means that x and y are nearest lattice neighbors. The discretization should be thought of as defining what the derivative means.

In terms of the lattice Fourier modes, the action can be written:

For k near zero this is:

Now we have the continuum Fourier transform of the original action. In finite volume, the quantity is not infinitesimal, but becomes the volume of a box made by neighboring Fourier modes, or .

The field is real-valued, so the Fourier transform obeys:

In terms of real and imaginary parts, the real part of is an even function of k, while the imaginary part is odd. The Fourier transform avoids double-counting, so that it can be written:

over an integration domain which integrates over each pair (k,-k) exactly once.

For a complex scalar field with action

the Fourier transform is unconstrained:

and the integral is over all k.

Integrating over all different values of 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

then

If A is a rotation, then

so that , and the sign depends on whether the rotation includes a reflection or not.

The matrix which changes coordinates from to can be read off from the definition of a Fourier transform.

and the Fourier inversion theorem tells you the inverse:

which is the complex conjugate-transpose, up to factors of . On a finite volume lattice, the determinant is nonzero and independent of the field values.

and the path integral is a separate factor at each value of k.

The factor is the infinitesimal volume of a discrete cell in k-space, in a square lattice box , where L is the side-length of the box. Each separate factor is an oscillatory Gaussian, and the width of the Gaussian diverges as the volume goes to infinity.

In imaginary time, the Euclidean action becomes positive definite, and can be interpreted as a probability distribution. The probability of a field having values is

The expectation value of the field is the statistical expectation value of the field when chosen according to the probability distribution:

Since the probability of is a product, the value of at each separate value of k is independently Gaussian distributed. The variance of the Gaussian is 1/ (k^2 d^dk), which is formally infinite, but that just means that the fluctuations are unbounded in infinite volume. In any finite volume, the integral is replaced by a discrete sum, and the variance of the integral is .

### Monte-Carlo

The path integral defines a probabilistic algorithm to generate a Euclidean scalar field configuration. Randomly pick the real and imaginary parts of each Fourier mode at wavenumber k to be a gaussian random variable with variance . This generates a configuration at random, and the Fourier transform gives . For real scalar fields, the algorithm must generate only one of each pair , and make the second the complex conjugate of the first.

To find any correlation function, generate a field again and again by this procedure, and find the statistical average:

where is the number of configurations, and the sum is of the product of the field values on each configuration. The Euclidean correlation function is just the same as the correlation function in statistics or statistical mechanics. The quantum mechanical correlation functions are an analytic continuation of the Euclidean correlation functions.

For free fields with a quadratic action, the probability distribution is a high-dimensional Gaussian, and the statistical average is given by an explicit formula. But the Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

also works well for bosonic interacting field theories where there is no closed form for the correlation functions.

### Scalar Propagator

Each mode is independently Gaussian distributed. The expectation of field modes is easy to calculate:

for , since then the two Gaussian random variables are independent and both have zero mean.

in finite volume V, when the two k-values coincide, since this is the variance of the Gaussian. In the infinite volume limit,

Strictly speaking, this is an approximation: the lattice propagator is:

But near k=0, for field fluctuations long compared to the lattice spacing, the two forms coincide.

It is important to emphasize that the delta functions contain factors of , so that they cancel out the factors in the measure for k integrals.

where is the ordinary one-dimensional Dirac delta function. This convention for delta-functions is not universal--- some authors keep the factors of in the delta functions (and in the k-integration) explicit.

### Equation of Motion

The form of the propagator can be more easily found by using the equation of motion for the field. From the Lagrangian, the equation of motion is:

and in an expectation value, this says:

Where the derivatives act on x, and the identity is true everywhere except when x and y coincide, and the operator order matters. The form of the singularity can be understood from the canonical commutation relations to be a delta-function. Defining the (euclidean) Feynman propagator as the Fourier transform of the time-ordered two-point function (the one that comes from the path-integral):

So that:

If the equations of motion are linear, the propagator will always be the reciprocal of the quadratic-form matrix which defines the free Lagrangian, since this gives the equations of motion. This is also easy to see directly from the Path integral. The factor of i disappears in the Euclidean theory.

#### Wick Theorem

Because each field mode is an independent Gaussian, the expectation values for the product of many field modes obeys Wick's theorem:

is zero unless the field modes coincide in pairs. This means that it is zero for an odd number of 's, and for an even number of phi's, it is equal to a contribution from each pair separately, with a delta function.

where the sum is over each partition of the field modes into pairs, and the product is over the pairs. For example,

An interpretation of Wick's theorem is that each field insertion can be thought of as a dangling line, and the expectation value is calculated by linking up the lines in pairs, putting a delta function factor that ensures that the momentum of each partner in the pair is equal, and dividing by the propagator.

#### Higher Gaussian moments--- completing Wick's theorem

There is a subtle point left before Wick's theorem is proved--- what if more than two of the phi's have the same momentum? If its an odd number, the integral is zero, negative values cancel with the positive values, But if the number is even, the integral is positive. The previous demonstration assumed that the phi's would only match up in pairs.

But the theorem is correct even when arbitrarily many of the phis are equal, and this is a notable property of Gaussian integration:

Dividing by I,

If Wick's theorem were correct, the higher moments would be given by all possible pairings of a list of 2n x's:

where the x-s are all the same variable, the index is just to keep track of the number of ways to pair them. The first x can be paired with 2n-1 others, leaving 2n-2. The next unpaired x can be paired with 2n-3 different x's leaving 2n-4, and so on. This means that Wick's theorem, uncorrected, says that the expectation value of should be:

and this is in fact the correct answer. So Wick's theorem holds no matter how many of the momenta of the internal variables coincide.

#### Interaction

Interactions are represented by higher order contributions, since quadratic contributions are always Gaussian. The simplest interaction is the quartic self-interaction, with an action:

The reason for the combinatorial factor 4! will be clear soon. Writing the action in terms of the lattice (or continuum) Fourier modes:

Where 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 , giving a series of corrections to the free action.

The path integral for the interacting action is then a power series of corrections to the free action. The term represented by X should be thought of as four half-lines, one for each factor of . The half-lines meet at a vertex, which contributes a delta-function which ensures that the sum of the momenta are all equal.

To compute a correlation function in the interacting theory, there is a contribution from the X terms now. For example, the path-integral for the four-field correlator:

which in the free field was only nonzero when the momenta k were equal in pairs, is now nonzero for all values of the k. The momenta of the insertions can now match up with the momenta of the X's in the expansion. The insertions should also be thought of as half-lines, four in this case, which carry a momentum k, but one which is not integrated.

The lowest order contribution comes from the first nontrivial term in the Taylor expansion of the action. Wick's theorem requires that the momenta in the X half-lines, the factors in X, should match up with the momenta of the external half-lines in pairs. The new contribution is equal to:

The 4! inside X is canceled because there are exactly 4! ways to match the half-lines in X to the external half-lines. Each of these different ways of matching the half-lines together in pairs contributes exactly once, regardless of the values of the k's, by Wick's theorem.

#### Feynman Diagrams

The expansion of the action in powers of X gives a series of terms with progressively higher number of X's. The contribution from the term with exactly n X's are called n-th order.

The n-th order terms has:
1. 4n internal half-lines, which are the factors of from the X's. These all end on a vertex, and are integrated over all possible k.
2. external half-lines, which are the come from the insertions in the integral.

By Wick's theorem, each pair of half-lines must be paired together to make a line, and this line gives a factor of

which multiplies the contribution. This means that the two half-lines that make a line are forced to have equal and opposite momentum. The line itself should be labelled by an arrow, drawn parallel to the line, and labeled by the momentum in the line k. The half-line at the tail end of the arrow carries momentum k, while the half-line at the head-end carries momentum -k. If one of the two half-lines is external, this kills the integral over the internal k, since it forces the internal k to be equal to the external k. If both are internal, the integral over k remains.

The diagrams which are formed by linking the half-lines in the X's with the external half-lines, representing insertions, are the Feynman diagrams of this theory. Each line carries a factor of , the propagator, and either goes from vertex to vertex, or ends at an insertion. If it is internal, it is integrated over. At each vertex, the total incoming k is equal to the total outgoing k.

The number of ways of making a diagram by joining half-lines into lines almost completely cancels the factorial factors coming from the Taylor series of the exponential and the 4! at each vertex.

#### Loop Order

A forest diagram is one where all the internal lines have momentum which is completely determined by the external lines and the condition that the incoming and outgoing momentum are equal at each vertex. The contribution of these diagrams is a product of propagators, without any integration. A tree diagram is a connected forest diagram.

An example of a tree diagram is the one where each of four external lines end on an X. Another is when three external lines end on an X, and the remaining half-line joins up with another X, and the remaining half-lines of this
X run off to external lines. These are all also forest diagrams (as every tree is a forest); an example of a forest which is not a tree is when eight external lines end on two X's.

It is easy to verify that in all these cases, the momenta on all the internal lines is determined by the external momenta and the condition of momentum conservation in each vertex.

A diagram which is not a forest diagram is called a loop diagram, and an example is one where two lines of an X are joined to external lines, while the remaining two lines are joined to each other. The two lines joined to each other can have any momentum at all, since they both enter and leave the same vertex. A more complicated example is one where two X's are joined to each other by matching the legs one to the other. This diagram has no external lines at all.

The reason loop diagrams are called loop diagrams is because the number of k-integrals which are left undetermined by momentum conservation is equal to the number of independent closed loops in the diagram, where independent loops are counted as in homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...

. The homology is real-valued (actually R^d valued), the value associated with each line is the momentum. The boundary operator takes each line to the sum of the end-vertices with a positive sign at the head and a negative sign at the tail. The condition that the momentum is conserved is exactly the condition that the boundary of the k-valued weighted graph is zero.

A set of k-values can be relabeled whenever there is a closed loop going from vertex to vertex, never revisiting the same vertex. Such a cycle can be thought of as the boundary of a 2-cell. The k-labelings of a graph which conserve momentum (which have zero boundary) up to redefinitions of k (up to boundaries of 2-cells) define the first homology of a graph. The number of independent momenta which are not determined is then equal to the number of independent homology loops. For many graphs, this is equal to the number of loops as counted in the most intuitive way.

#### Symmetry factors

The number of ways to form a given Feynman diagram by joining together half-lines is large, and by Wick's theorem, each way of pairing up the half-lines contributes equally. Often, this completely cancels the factorials in the denominator of each term, but the cancellation is sometimes incomplete.

The uncancelled denominator is called the symmetry factor of the diagram. The contribution of each diagram to the correlation function must be divided by its symmetry factor.

For example, consider the Feynman diagram formed from two external lines joined to one X, and the remaining two half-lines in the X joined to each other. There are 4*3 ways to join the external half-lines to the X, and then there is only one way to join the two remaining lines to each other. The X comes divided by 4!=4*3*2, but the number of ways to link up the X half lines to make the diagram is only 4*3, so the contribution of this diagram is divided by two.

For another example, consider the diagram formed by joining all the half-lines of one X to all the half-lines of another X. This diagram is called a vacuum bubble, because it does not link up to any external lines. There are 4! ways to form this diagram, but the denominator includes a 2! (from the expansion of the exponential, there are two X's) and two factors of 4!. The contribution is multiplied by 4!/(2*4!*4!) = 1/48.

Another example is the Feynman diagram formed from two X's where each X links up to two external lines, and the remaining two half-lines of each X are joined to each other. The number of ways to link an X to two external lines is 4*3, and either X could link up to either pair, giving an additional factor of 2. The remaining two half-lines in the two X's can be linked to each other in two ways, so that the total number of ways to form the diagram is 4*3*4*3*2*2, while the denominator is 4!4!2!. The total symmetry factor is 2, and the contribution of this diagram is divided by two.

The symmetry factor theorem gives the symmetry factor for a general diagram: the contribution of each Feynman diagram must be divided by the order of its group of automorphisms, the number of symmetries that it has.

An automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...

of a Feynman graph is a permutation M of the lines and a permutation N of the vertices with the following properties:
1. 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.
2. If a line l ends on an external line, M(l) ends on the same external line.
3. If there are different types of lines, M(l) should preserve the type.

This theorem has an interpretation in terms of particle-paths: when identical particles are present, the integral over all intermediate particles must not double-count states which only differ by interchanging identical particles.

Proof: To prove this theorem, label all the internal and external lines of a diagram with a unique name. Then form the diagram by linking the a half-line to a name and then to the other half line.

Now count the number of ways to form the named diagram. Each permutation of the X's gives a different pattern of linking names to half-lines, and this is a factor of n!. Each permutation of the half-lines in a single X gives a factor of 4!. So a named diagram can be formed in exactly as many ways as the denominator of the Feynman expansion.

But the number of unnamed diagrams is smaller than the number of named diagram by the order of the automorphism group of the graph.

Roughly speaking, a Feynman diagram is called connected if all vertices and propagator lines are linked by a sequence of vertices and propagators of the diagram itself. If one views it as a (undirected) graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

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
Partition function (quantum field theory)
In quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral:...

. More precisely, connected Feynman diagrams determine
.

To see this, one should recall that

with constructed from some (arbitrary) Feynman diagram which can be thought to consist of several connected components . If one encounters (identical) copies of a component within the Feynman diagram one has to include a symmetry factor . However, in the end each contribution of a Feynman diagram to the partition function has the generic form

where labels the (infinite) many connected Feynman diagrams possible.

A scheme to successively create such contributions from the to is obtained by

and therefore yields
.

To establish the normalization one simply calculates all connected vacuum diagrams, i.e., the diagrams without any sources (sometimes referred to as external legs of a Feynman diagram).

#### Vacuum Bubbles

An immediate consequence of the linked-cluster theorem is that all vacuum bubbles, diagrams without external lines cancel when calculating correlation functions. A correlation function is given by a ratio of path-integrals:

The top is the sum over all Feynman diagrams, including disconnected diagrams which do not link up to external lines at all. In terms of the connected diagrams, the numerator includes the same contributions of vacuum bubbles as the denominator:

Where the sum over E diagrams includes only those diagrams each of whose connected components end on at least one external line. The vacuum bubbles are the same whatever the external lines, and give an overall multiplicative factor. The denominator is the sum over all vacuum bubbles, and dividing gets rid of the second factor.

The vacuum bubbles then are only useful for determining Z itself, which from the definition of the path integral is equal to:

where is the energy density in the vacuum. Each vacuum bubble contains a factor of zeroing the total k at each vertex, and when there are no external lines, this contains a factor of , because the momentum conservation is over-enforced. In finite volume, this factor can be identified as the total volume of space time. Dividing by the volume, the remaining integral for the vacuum bubble has an interpretation: it is a contribution to the energy density of the vacuum.

#### Sources

Correlation functions are the sum of the connected Feynman diagrams, but the formalism treats the connected and disconnected diagrams differently. Internal lines end on vertices, while external lines go off to insertions. Introducing sources unifies the formalism, by making new vertices where one line can end.

Sources are external fields, fields which contribute to the action, but are not dynamical variables. A scalar field source is another scalar field h which contributes a term to the (Lorentz) Lagrangian:

In the Feynman expansion, this contributes H terms with one half-line ending on a vertex. Lines in a Feynman diagram can now end either on an X vertex, or on an H-vertex, and only one line enters an H vertex. The Feynman rule for an H-vertex is that a line from an H with momentum k gets a factor of h(k).

The sum of the connected diagrams in the presence of sources includes a term for each connected diagram in the absence of sources, except now the diagrams can end on the source. Traditionally, a source is represented by a little "x" with one line extending out, exactly as an insertion.

where is the connected diagram with n external lines carrying momentum as indicated. The sum is over all connected diagrams, as before.

The field h is not dynamical, which means that there is no path integral over h: h is just a parameter in the Lagrangian which varies from point to point. The path integral for the field is:

and it is a function of the values of h at every point. One way to interpret this expression is that it is taking the Fourier transform in field space. If there is a probability density on R^n, the Fourier transform of the probability density is:

The fourier transform is the expectation of an oscillatory exponential. The path integral in the presence of a source h(x) is:

which, on a lattice, is the product of an oscillatory exponential for each field value:

The fourier transform of a delta-function is a constant, which gives a formal expression for a delta function:

This tells you what a field delta function looks like in a path-integral. For two scalar fields and ,

Which integrates over the Fourier transform coordinate, over h. This expression is useful for formally changing field coordinates in the path integral, much as a delta function is used to change coordinates in an ordinary multi-dimensional integral.

The partition function is now a function of the field h, and the physical partition function is the value when h is the zero function:

The correlation functions are derivatives of the path integral with respect to the source:

In Euclidean space, source contributions to the action can still appear with a factor of "i", so that they still do a Fourier transform.

#### Spin 1/2: Grassman integrals

The field path-integral can be extended to the Fermi case, but only if the notion of integration is expanded. A Grassman integral
Berezin integral
In mathematical physics, a Grassmann integral, or, more correctly, Berezin integral, is a way to define integration for functions of Grassmann variables. It is not an integral in the Lebesgue sense; it is called integration because it has analogous properties and since it is used in physics as a...

of a free Fermi field is a high-dimensional determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

or Pfaffian
Pfaffian
In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries. This polynomial is called the Pfaffian of the matrix, The term Pfaffian was introduced by who named them after Johann Friedrich Pfaff...

which defines the new type of Gaussian integration appropriate for Fermi fields.

The two fundamental formulas of Grassman integration are:

where M is an arbitrary matrix and are independent Grassmann variables for each index i, and

Where A is an antisymmetric matrix, is a collection of Grassmann variables, and the 1/2 is to prevent double-counting (since ). In matrix notation, where and are Grassman valued row vectors, and are Grassman valued column vectors, and M is a real valued matrix:

Where the last equality is a consequence of the translation invariance of the Grassman integral. The Grassman variables are external sources for , and differentiating with respect to pulls down factors of .

again, in a schematic matrix notation. The meaning of the formula above is that the derivative with respect to the appropriate component of and gives the matrix element of . This is exactly analogous to the Bosonic path integration formula for a Gaussian integral of a complex Bosonic field:

So that the propagator is the inverse of the matrix in the quadratic part of the action in both the Bose and Fermi case.

For real Grassman fields, for Majorana fermion
Majorana fermion
In physics, a Majorana fermion is a fermion which is its own anti-particle. The term is used in opposition to Dirac fermion, which describes particles that differ from their antiparticles...

s, the path integral is a Pfaffian times a source quadratic form, and the formulas give the square root of the determinant, just as they do for real Bosonic fields. The propagator is still the inverse of the quadratic part.

The free Dirac Lagrangian:

formally gives the equations of motion and the anticommutation relations of the Dirac field, just as the Klein Gordon Lagrangian in an ordinary path integral gives the equations of motion and commutation relations of the scalar field. By using the spatial fourier-transform of the Dirac field as a new basis for the Grassmann algebra, the quadratic part of the Dirac action becomes simple to invert:

The propagator is the inverse of the matrix M linking and , since different values of k do not mix together.

The analog of Wick's theorem matches psi and psi-bars in pairs:

where S is the sign of the permutation which reorders the sequence of psi-bars and psis to put the ones which are paired up to make the delta-functions next to each other, with the psi-bar coming right before the psi. Since a psi-psi-bar pair is a commuting element of the Grassman algebra, it doesn't matter what order the pairs are in. If more than one psi/psi-bar pair have the same k, the integral is zero, and it is easy to check that the sum over pairings gives zero in this case (there are always an even number of them). This is the Grassman analog of the higher Gaussian moments which completed the Bosonic Wick's theorem earlier.

The rules for spin-1/2 Dirac particles are as follows: The propagator is the inverse of the Dirac operator, the lines have arrows just as for a complex scalar field, and the diagram acquires an overall factor of -1 for each closed Fermi loop. If there are an odd number of Fermi loops, the diagram changes sign. Historically, the -1 rule was very difficult for Feynman to discover. He discovered it after a long process of trial and error, since he lacked a proper theory of Grassman integration.

The rule follows from the observation that the number of Fermi lines at a vertex is always even. Each term in the Lagrangian must always be Bosonic. A Fermi loops is counted by following Fermionic lines until one comes back to the starting point, then removing those lines from the diagram. Repeating this process eventually erases all the Fermionic lines: this is the Euler algorithm to 2-color a graph, which works whenever each vertex has even degree. Note that the number of steps in the Euler algorithm is only equal to the number of independent Fermionic homology cycles in the common special case that all terms in the Lagrangian are exactly quadratic in the Fermi fields, so that each vertex has exactly two Fermionic lines. When there are four-Fermi interactions (like in the Fermi effective theory of the Weak interactions) there are more k-integrals than Fermi loops. In this case, the counting rule should apply the Euler algorithm by pairing up the Fermi lines at each vertex into pairs which together form a bosonic factor of the term in the Lagrangian, and when entering a vertex by one line, the algorithm should always leave with the partner line.

To clarify and prove the rule, consider a Feynman diagram formed from vertices, terms in the Lagrangian, with Fermion fields. The full term is Bosonic, it is a commuting element of the Grassman algebra, so the order in which the vertices appear is not important. The Fermi lines are linked into loops, and when traversing the loop, one can reorder the vertex terms one after the other as one goes around without any sign cost. The exception is when you return to the starting point, and the final half-line must be joined with the unlinked first half-line. This requires one permutation to move the last psi-bar to go in front of the first psi, and this gives the sign.

This rule is the only visible effect of the exclusion principle in internal lines. When there are external lines, the amplitudes are antisymmetric when two Fermi insertions for identical particles are interchanged. This is automatic in the source formalism, because the sources for Fermi fields are themselves Grassman valued.

#### Spin 1: Photons

The naive propagator for photons is infinite, since the Lagrangian for the A-field is:

The quadratic form defining the propagator is non-invertible. The reason is the gauge invariance of the field, adding a gradient to A does not change the physics.

To fix this problem, one needs to fix a gauge. The most convenient way is to demand that the divergence of A is some function f, whose value is random from point to point. It does no harm to integrate over the values of f, since it only determines the choice of gauge. This procedure inserts the following factor into the path integral for A:

The first factor, the delta function, fixes the gauge. The second factor sums over different values of f which are inequivalent gauge fixings. This is simply

The additional contribution from gauge-fixing cancels the second half of the free Lagrangian, giving the Feynman Lagrangian:

which is just like four independent free scalar fields, one for each component of A. The Feynman propagator is:

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 which does not affect anything:

and the covariant propagator for QED is:

#### Spin 1: Nonabelian Ghosts

To find the Feynman rules for nonabelian Gauge fields, the procedure which performs the Gauge fixing must be carefully corrected to account for a change of variables in the path-integral.

The gauge fixing factor has an extra determinant from popping the delta function:

To find the form of the determinant, consider first a simple two-dimensional integral, of a function f which depends only on r, not on the angle . Inserting an integral over theta:

The derivative-factor ensures that popping the delta function in removes the integral. Exchanging the order of integration,

but now the delta-function can be popped in y,

The integral over just gives an overall factor of , while the rate of change of with a change in is just x, so this exercise reproduces the standard formula for polar integration of a radial function:

In the path-integral for a nonabelian gauge field, the analogous manipulation is:

The factor in front is the volume of the gauge group, and it contributes a constant which can be discarded. The remaining integral is over the gauge fixed action.

To get a covariant gauge, the gauge fixing condition is the same as in the Abelian case:

Whose variation under an infinitesimal gauge transformation is given by:

where is the adjoint valued element of the Lie algebra at every point which performs the infinitesimal gauge transformation. This adds the Faddeev Popov determinant to the action:

which can be rewritten as a Grassman integral by introducing ghost fields:

The determinant is independent of f, so the path-integral over f can give the Feynman propagator (or a covariant propagator) by choosing the measure for f as in the abelian case. The full gauge fixed action is then the Yang Mills action in Feynman gauge with an additional ghost action:

The diagrams are derived from this action. The propagator for the spin-1 fields has the usual Feynman form. There are vertices of degree 3 with momentum factors whose couplings are the structure constants, and vertices of degree 4 whose couplings are products of structure constants. There are additional ghost loops, which cancel out timelike and logitudinal states in A loops.

In the Abelian case, the determinant for covariant gauges does not depend on A, so the ghosts do not contribute to the connected diagrams.

## Particle-Path representation

Feynman diagrams were originally discovered by Feynman, by trial and error, as a way to represent the contribution to the S-matrix from different classes of particle trajectories.

### Schwinger representation

The Euclidean scalar propagator has a suggestive representation:

The meaning of this identity (which is an elementary integration) is made clearer by Fourier transforming to real space.

The contribution at any one value of to the propagator is a Gaussian of width . The total propagation function from 0 to x is a weighted sum over all proper times of a normalized Gaussian, the probability of ending up at x after a random walk of time .

The path-integral representation for the propagator is then:

which is a path-integral rewrite of the Schwinger representation.

The Schwinger representation is both useful for making manifest the particle aspect of the propagator, and for symmetrizing denominators of loop diagrams.

### Combining Denominators

The Schwinger representation has an immediate practical application to loop diagrams. For example, For the diagram in the phi-4 theory formed by joining two X-s together in two half-lines, and making the remaining lines external, the integral over the internal propagators in the loop is:

Here one line carries momentum k and the other k+p. The asymmetry can be fixed by putting everything in the Schwinger representation.

Now the exponent mostly depends on t+t',

except for the asymmetrical little bit. Defining the variable u=(t+t') and = t'/u, the variable u goes from 0 to infinity, while goes from 0 to 1. The variable u is the total proper time for the loop, while 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:

and "wedging" gives
.

This allows the u integral to be evaluated explicitly:

leaving only the -integral. This method, invented by Schwinger but usually attributed to Feynman, is called combining denominator. Abstractly, it is the elementary identity:

But this form does not provide the physical motivation for introducing --- is the proportion of proper time on one of the legs of the loop.

Once the denominators are combined, a shift in k to symmetrizes everything:

This form shows that the moment that p2 is more negative than 4 times the mass of the particle in the loop, which happens in a physical region of Lorentz space, the integral has a cut. This is exactly when the external momentum can create physical particles.

When the loop has more vertices, there are more denominators to combine:

The general rule follows from the Schwinger prescription for n+1 denominators:

The integral over the Schwinger parameters can be split up as before into an integral over the total proper time and an integral over the fraction of the proper time in all but the first segment of the loop for . 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:

"Wedging" all these equation together, one obtains

This gives the integral:

where the simplex is the region defined by the conditions and as well as . Performing the u integral gives the general prescription for combining denominators:

Since the numerator of the integrand is not involved, the same prescription works for any loop, no matter what the spins are carried by the legs. The interpretation of the parameters is that they are the fraction of the total proper time spent on each leg.

### Scattering

The correlation functions of a quantum field theory describe the scattering of particles. The definition of "particle" in relativistic field theory is not self-evident, because if you try to determine the position so that the uncertainty is less than the compton wavelength
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons...

, the uncertainty in energy is large enough to produce more particles and antiparticles of the same type from the vacuum. This means that the notion of a single-particle state is to some extent incompatible with the notion of an object localized in space.

In the 1930s, Wigner gave a mathematical definition for single-particle states: they are a collection of states which form an irreducible representation of the Poincaré group. Single particle states describe an object with a finite mass, a well defined momentum, and a spin. This definition is fine for protons and neutrons, electrons and photons, but it excludes quarks, which are permanently confined, so the modern point of view is more accommodating: a particle is anything whose interaction can be described in terms of Feynman diagrams, which have an interpretation as a sum over particle trajectories.

A field operator can act to produce a one-particle state from the vacuum, which means that the field operator produces a superposition of Wigner particle states. In the free field theory, the field produces one particle states only. But when there are interactions, the field operator can also produce 3-particle,5-particle (if there is no +/- symmetry also 2,4,6 particle) states too. To compute the scattering amplitude for single particle states only requires a careful limit, sending the fields to infinity and integrating over space to get rid of the higher-order corrections.

The relation between scattering and correlation functions is the LSZ-theorem: The scattering amplitude for n particles to go to m-particles in a scattering event is the given by the sum of the Feynman diagrams that go into the correlation function for n+m field insertions, leaving out the propagators for the external legs.

For example, for the interaction of the previous section, the order contribution to the (Lorentz) correlation function is:

Stripping off the external propagators, that is, removing the factors of , gives the invariant scattering amplitude M:

which is a constant, independent of the incoming and outgoing momentum. The interpretation of the scattering amplitude is that the sum of over all possible final states is the probability for the scattering event. The normalization of the single-particle states must be chosen carefully, however, to ensure that M is a relativistic invariant.

Non-relativistic single particle states are labeled by the momentum k, and they are chosen to have the same norm at every value of k. This is because the nonrelativistic unit operator on single particle states is:

In relativity, the integral over the k-states for a particle of mass m integrates over a hyperbola in E,k space defined by the energy-momentum relation:

If the integral weighs each k point equally, the measure is not Lorentz invariant. The invariant measure integrates over all values of k and E, restricting to the hyperbola with a Lorentz invariant delta function:

So the normalized k-states are different from the relativistically normalized k-states by a factor of

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 ). In this limit, the 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
Born approximation
In scattering theory and, in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. Born approximation is named after Max Born, winner of the 1954 Nobel Prize for physics.It is...

) is one whose Fourier transform is constant--- a delta-function potential. The lowest order scattering of the theory reveals the non-relativistic interpretation of this theory--- it describes a collection of particles with a delta-function repulsion. Two such particles have an aversion to occupying the same point at the same time.

## Nonperturbative effects

Thinking of Feynman diagrams as a perturbation series, nonperturbative effects like tunneling do not show up, because any effect which goes to zero faster than any polynomial does not affect the Taylor series. Even bound states are absent, since at any finite order particles are only exchanged a finite number of times, and to make a bound state, the binding force must last forever.

But this point of view is misleading, because the diagrams not only describe scattering, but they also are a representation of the short-distance field theory correlations. They encode not only asymptotic processes like particle scattering, they also describe the multiplication rules for fields, the operator product expansion
Operator product expansion
- 2D Euclidean quantum field theory :In quantum field theory, the operator product expansion is a Laurent series expansion of two operators...

. Nonperturbative tunneling processes involve field configurations which on average get big when the coupling constant gets small, but each configuration is a coherent superposition of particles whose local interactions are described by Feynman diagrams. When the coupling is small, these become collective processes which involve large numbers of particles, but where the interactions between each of the particles is simple.

This means that nonperturbative effects show up asymptotically in resummations of infinite classes of diagrams, and these diagrams can be locally simple. The graphs determine the local equations of motion, while the allowed large-scale configurations describe non-perturbative physics. But because Feynman propagators are nonlocal in time, translating a field process to a coherent particle language is not completely intuitive, and has only been explicitly worked out in certain special cases. In the case of nonrelativistic bound states, the Bethe-Salpeter equation
Bethe-Salpeter equation
The Bethe–Salpeter equation, named after Hans Bethe and Edwin Salpeter, describes the bound states of a two-body quantum field theoretical system in a relativistically covariant formalism...

describes the class of diagrams to include to describe a relativistic atom. For quantum chromodynamics, the Shifman Vainshtein Zakharov sum rules describe non-perturbatively excited long-wavelength field modes in particle language, but only in a phenomenological way.

The number of Feynman diagrams at high orders of perturbation theory is very large, because there are as many diagrams as there are graphs with a given number of nodes. Nonperturbative effects leave a signature on the way in which the number of diagrams and resummations diverge at high order. It is only because non-perturbative effects appear in hidden form in diagrams that it was possible to analyze nonperturbative effects in string theory, where in many cases a Feynman description is the only one available.

## Mathematical details

A Feynman diagram can be considered a graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

. When considering a field composed of particles, the edges will represent (sections of) particle world line
World line
In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...

s; the vertices
Vertex (graph theory)
In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...

represent virtual interaction
Interaction
Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect...

s. Since only certain interactions are permitted, the graph is constrained to have only certain types of vertices. The type of field of an edge is its field label; the permitted types of interaction are interaction labels.

The value of a given diagram can be derived from the graph; the value of the interaction as a whole is obtained by summing over all diagrams.

## Mathematical interpretation

Feynman diagrams are really a graphical way of keeping track of deWitt indices, much like Penrose's graphical notation for indices in multilinear algebra
Multilinear algebra
In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...

. There are several different types for the indices, one for each field (this does not depend on how the fields are grouped; for instance, if the up quark field and down quark field are treated as different fields, then there would be the same type assigned to both of them but if they are treated as a single multicomponent field with "flavors", then there would be a problem). The edges, (i.e., propagator
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...

s) are tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s of rank (2,0) in deWitt's notation (i.e., with two contravariant indices
Einstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...

and no covariant indices
Einstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...

), while the vertices of degree n are rank n covariant tensors which are totally symmetric among all bosonic indices of the same type and totally antisymmetric among all fermionic indices of the same type and the contraction
Tensor contraction
In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...

of a propagator with a rank n covariant tensor is indicated by an edge incident to a vertex (there is no ambiguity in which "slot" to contract with because the vertices correspond to totally symmetric tensors). The external vertices correspond to the uncontracted contravariant indices.

A derivation of the Feynman rules using Gaussian functional integrals is given in the functional integral article.

Each Feynman diagram on its own does not have a physical significance. It's only the infinite sum over all possible (bubble-free) Feynman diagrams which gives physical results. This infinite sum is usually only asymptotically convergent
Asymptotic expansion
In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular,...

.

• Stueckelberg-Feynman interpretation
• Invariance mechanics
Invariance mechanics
In physics, invariance mechanics, in its simplest form, is the rewriting of the laws of quantum field theory in terms of invariant quantities only. For example, the positions of a set of particles in a particular coordinate system is not invariant under translations of the system...

• Penguin diagram
Penguin diagram
In quantum field theory, penguin diagrams are a class of Feynman diagrams which are important for understanding CP violating processes in the standard model. They refer to one-loop processes in which a quark temporarily changes flavor , and the flavor-changed quark engages in some tree interaction,...

• Path integral formulation
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...

• Propagator
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...

s