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In theoretical physicsFeynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic particles. The scheme is named after its inventor, American physicist Richard Feynmanand was first introduced in The interaction of sub-atomic particles can be complex and difficult to understand intuitively. Feynman diagrams give a simple visualization of what would otherwise be an 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 so “Feynman diagrams have revolutionized nearly every aspect of theoretical physics”. Feynman used Ernst Stueckelberg ‘s interpretation of the positron as if it were an electron moving 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 geynman 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 S -matrix. Alternatively, the path integral siagramme of quantum field theory represents the transition amplitude diagrxmme 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 feynnman matrix element of the S -matrix between the initial and the final states of the quantum system.
When calculating scattering cross-sections in particle physicsthe interaction between particles can riagramme described by starting from a free field that describes the incoming and outgoing particles, and including an interaction Hamiltonian to ds 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 expansionand the time-dependent 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 old-fashioned perturbation theory. The Dyson series can be alternatively rewritten as a sum over Feynman diagrams, where at each vertex both the energy and momentum are conservedbut where the length of the energy-momentum four-vector is not necessarily equal to the mass. 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 diagrammme 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 the Feynman rules, below for any given diagram from a field theory Lagrangian.
Each internal line corresponds to a factor of the 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 fenymanalso invented by Feynman—see path integral formulation. The technique of renormalizationsuggested by Ernst Stueckelberg and Hans Bethe and implemented by DysonFeynman, Schwingerand 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. Murray Gell-Mann always referred to Feynman diagrams as Stueckelberg diagramsafter a Swiss physicist, Ernst Stueckelbergwho 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 diaframme paths, all without the path-integral.
Historically, as a book-keeping device of covariant perturbation theory, the graphs were called Feynman—Dyson diagrams or Dyson graphs because the path integral was unfamiliar when feynmab were introduced, and Freeman Dyson ‘s derivation from old-fashioned perturbation theory was easier to follow for physicists trained in earlier methods.
In their diagramne 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 diahramme the physics of quantum scattering of fundamental particles.
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 theoryuse of graphical methods in the many-body problem shows that this formalism is flexible enough to deal with phenomena of nonperturbative characters … Some modification of the Feynman rules of calculation may well outlive the elaborate mathematical structure of local canonical quantum field theory ….
So far there are no opposing opinions. In quantum field theories the Feynman diagrams are obtained from Lagrangian by Feynman rules. Dimensional regularization is a method for regularizing integrals in the evaluation of Feynman diagrams; it assigns ciagramme to them that are meromorphic functions of an auxiliary complex parameter dcalled the dimension.
Dimensional regularization writes a Feynman integral as an integral depending on the spacetime dimension d and spacetime points. A Feynman diagram is a representation of quantum field theory processes in terms of particle interactions.
The particles 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 a vertexand this is where the particles meet and interact: There are three different types of lines: Traditionally, the bottom of the diagram is the past and the top the future; feynan times, the past is to the left and the future to the right. When calculating correlation functions instead of scattering amplitudesthere 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 interaction of particles rather than the physical position of the particle during 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 to the total amplitude for the fyenman.
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: 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.
The particles in the initial state are depicted by lines sticking out in the direction of the initial state e. In QED there are two types of particles: They are represented in Feynman diagrams as follows:.
In QED a vertex always has three lines attached to it: The vertices might be connected by a bosonic or fermionic propagator. The number of vertices gives the order of the term in the perturbation series expansion of the transition amplitude. In the canonical quantum field theory the S -matrix is represented within the interaction picture by the perturbation series in the powers of the interaction Lagrangian. A Feynman diagram is a graphical representation of a term in the Wick’s expansion of the time-ordered product in the n th order term S n of the S -matrix.
The diagrams are drawn according djagramme the Feynman rules, which depend upon the interaction Lagrangian. For the QED interaction Lagrangian. The second order perturbation term in the S -matrix is. The Wick’s expansion of the integrand gives among others the following term. This term is represented by the Feynman diagram at the right.
This diagram gives contributions to the following processes:. In a path integralthe 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 stateand the integral should be performed a little bit rotated into imaginary time, i. A simple example is diagra,me free relativistic scalar field in d dimensions, whose action integral is:.
This is the field-to-field transition amplitude. The path integral can be thought of as analogous to a probability distribution, and it is convenient to define it so that feyjman 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.
If the final results do not depend on the shape of the lattice or the value of athen the continuum limit exists.
The Particle Adventure
On a lattice, ithe field can be expanded in Fourier modes:. 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 diagrwmme 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 ciagramme of the factors in front of the k -integrals and the momentum-conserving delta functions that will arise.
Now we have the continuum Fourier transform of the original action. The Fourier transform avoids double-counting, so that it can be written:. When diagramem 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. On a finite volume lattice, the determinant is nonzero and independent of the field values.
The factor d d k is the infinitesimal volume of a discrete cell in k -space, in a square lattice box. Diagramm 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 ddiagramme interpreted as a probability distribution.
The expectation value of the field is the statistical expectation value of the feymman when chosen according to the probability distribution:. The path integral defines a probabilistic algorithm to generate a Euclidean scalar field configuration. To find any correlation function, generate a field again and again by this procedure, and find the statistical average:.
The Euclidean correlation function is just the same as the correlation function in statistics or statistical mechanics. The quantum mechanical correlation functions are an analytic continuation of the Euclidean correlation functions. For free fields with a quadratic action, the probability distribution is a high-dimensional Gaussian, and the statistical average is given by an explicit formula. But the Monte Carlo method also works well for bosonic interacting field theories where there is no closed form for the correlation functions.
Each mode is independently Gaussian distributed. The expectation of field modes is easy to calculate:. In the infinite volume limit. The form of the propagator can be more easily diagramme by using the equation of motion for the field.