In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced the diagrams in 1948. The calculation of probability amplitudes in theoretical particle physics requires the use of large, complicated integrals over a large number of variables. Feynman diagrams instead represent these integrals graphically. Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula. According to David Kaiser, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations. Feynman diagrams have revolutionized nearly every aspect of theoretical physics." While the diagrams apply primarily to quantum field theory, they can be used in other areas of physics, such as solid-state theory. Frank Wilczek wrote that the calculations that won him the 2004 Nobel Prize in Physics "would have been literally unthinkable without Feynman diagrams, as would [Wilczek's] calculations that established a route to production and observation of the Higgs particle." A Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory. Within the canonical formulation of quantum field theory, a Feynman diagram represents a term in the Wick's expansion of the perturbative S-matrix. Alternatively, the path integral formulation of quantum field theory represents the transition amplitude as a weighted sum of all possible histories of the system from the initial to the final state, in terms of either particles or fields. The transition amplitude is then given as the matrix element of the S-matrix between the initial and final states of the quantum system. Feynman used Ernst Stueckelberg's interpretation of the positron as if it were an electron moving backward in time. Thus, antiparticles are represented as moving backward along the time axis in Feynman diagrams.
Motivation and history
When calculating scattering cross-sections in particle physics, the interaction between particles can be described by starting from a free field that describes the incoming and outgoing particles, and including an interaction Hamiltonian to describe how the particles deflect one another. The amplitude for scattering is the sum of each possible interaction history over all possible intermediate particle states. The number of times the interaction Hamiltonian acts is the order of the perturbation expansion, and the 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 (or time-dependent/time-ordered 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 conserved, but where the length of the energy-momentum four-vector is not necessarily equal to the mass, i.e. the intermediate particles are so-called off-shell. 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 classical 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 mechanics, also invented by Feynman—see path integral formulation. The naïve application of such calculations often produces diagrams whose amplitudes are infinite, because the short-distance particle interactions require a careful limiting procedure, to include particle self-interactions. The technique of renormalization, suggested by Ernst Stueckelberg and Hans Bethe and implemented by Dyson, Feynman, Schwinger, and Tomonaga compensates for this effect and eliminates the troublesome infinities. After renormalization, calculations using Feynman diagrams match experimental results with very high accuracy. Feynman diagram and path integral methods are also used in statistical mechanics and can even be applied to classical mechanics.
Alternate names Murray Gell-Mann always referred to Feynman diagrams as Stueckelberg diagrams, after Swiss physicist Ernst Stueckelberg, who devised a similar notation many years earlier. Stueckelberg was motivated by the need for a manifestly covariant formalism for quantum field theory, but did not provide as automated a way to handle symmetry factors and loops, although he was first to find the correct physical interpretation in terms of forward and backward in time particle paths, all without the path-integral. 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 they were introduced, and Freeman Dyson's derivation from old-fashioned perturbation theory borrowed from the perturbative expansions in statistical mechanics was easier to follow for physicists trained in earlier methods. Feynman had to lobby hard for the diagrams, which confused physicists trained in equations and graphs.
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 the physics of quantum scattering of fundamental particles. Their motivations are consistent with the convictions of James Daniel Bjorken and Sidney Drell:
The Feynman graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand. Although the statement of the theory in terms of graphs may imply perturbation theory, use of graphical methods in the many-body problem shows that this formalism is flexible enough to deal with phenomena of nonperturbative characters ... Some modification of the Feynman rules of calculation may well outlive the elaborate mathematical structure of local canonical quantum field theory ... In quantum field theories, Feynman diagrams are obtained from a Lagrangian by Feynman rules. Dimensional regularization is a method for regularizing integrals in the evaluation of Feynman diagrams; it assigns values to them that are meromorphic functions of an auxiliary complex parameter d, called the dimension. Dimensional regularization writes a Feynman integral as an integral depending on the spacetime dimension d and spacetime points.
Particle-path interpretation A Feynman diagram is a representation of quantum field theory processes in terms of particle interactions. The particles are represented by the diagram lines. The lines 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 vertex, and this is where the particles meet and interact. The interactions are: emit/absorb particles, deflect particles, or change particle type. The three different types of lines are: internal lines, connecting vertices; incoming lines, extending from "the past" to a vertex, representing an initial state; and outgoing lines, extending from a vertex to "the future", representing the end state (the latter two are also known as external lines). Traditionally, the bottom of the diagram is the past and the top the future; alternatively, the past is to the left and the future to the right. When calculating correlation functions instead of scattering amplitudes, past and future are not relevant and all lines are internal. The particles then begin and end on small x's, which represent the positions of the operators whose correlation is calculated. Feynman diagrams are a pictorial representation of a contribution to the total amplitude for a process that can happen in different ways. When a group of incoming particles 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 graphs that represent the interaction of particles rather than the physical position of the particle during a scattering process. They are not the same as spacetime diagrams and bubble chamber images even though they all describe particle scattering. Unlike a bubble chamber picture, only the sum of all relevant 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 process's amplitude.
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, while the final state is two photons. Conventionally, the initial state is at the left of the diagram and the final state at the right (although other layouts are also used). The particles in the initial state are depicted by lines pointing in the direction of the initial state (e.g., to the left). The particles in the final state are represented by lines pointing in the direction of the final state (e.g., to the right). QED involves two types of particles: matter particles such as electrons or positrons (called fermions) and exchange particles (called gauge bosons). They are represented in Feynman diagrams as follows:
Electron in the initial state is represented by a solid line, with an arrow indicating the spin of the particle e.g. pointing toward the vertex (→•). Electron in the final state is represented by a line, with an arrow indicating the spin of the particle e.g. pointing away from the vertex: (•→). Positron in the initial state is represented by a solid line, with an arrow indicating the spin of the particle e.g. pointing away from the vertex: (←•). Positron in the final state is represented by a line, with an arrow indicating the spin of the particle e.g. pointing toward the vertex: (•←). Virtual Photon in the initial and the final states is represented by a wavy line (~• and •~). In QED each vertex 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. Vertices can be connected by a bosonic or fermionic propagator. A bosonic propagator is represented by a wavy line connecting two vertices (•~•). A fermionic propagator is represented by a solid line with an arrow connecting two vertices, (•←•). 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:
e+ + e− → 2γ has a contribution from the second order Feynman diagram: In the initial state (at the bottom; early time) there is one electron (e−) and one positron (e+) and in the final state (at the top; late time) there are two photons (γ).
Canonical quantization formulation The probability amplitude for a transition of a quantum system (between asymptotically free states) from the initial state |i⟩ to the final state | f ⟩ is given by the matrix element
S f i = ⟨ f | S | i ⟩ , {\displaystyle S_{\rm {fi}}=\langle \mathrm {f} |S|\mathrm {i} \rangle \;,}
where S is the S-matrix. In terms of the time-evolution operator U, it is simply
S = lim t 2 → + ∞ [ lim t 1 → − ∞ U ( t 2 , t 1 ) ] . {\displaystyle S=\lim _{t_{2}\rightarrow +\infty }\left[\lim _{t_{1}\rightarrow -\infty }U(t_{2},t_{1})\right]\;.}
In the interaction picture, this expands to
S = T exp ( − i ∫ − ∞ + ∞ d τ H V ( τ ) ) . {\displaystyle S={\mathcal {T}}\exp \left(-i\int _{-\infty }^{+\infty }d\tau H_{V}(\tau )\right).}
where HV is the interaction Hamiltonian and T {\displaystyle {\mathcal {T}}} signifies the time-ordered product of operators. Dyson's formula expands the time-ordered matrix exponential into a perturbation series in the powers of the interaction Hamiltonian density,
S = ∑ n = 0 ∞ ( − i ) n n ! ( ∏ j = 1 n ∫ d 4 x j ) T { ∏ j = 1 n H V ( x j ) } ≡ ∑ n = 0 ∞ S ( n ) . {\displaystyle S=\sum _{n=0}^{\infty }{\frac {(-i)^{n}}{n!}}\left(\prod _{j=1}^{n}\int d^{4}x_{j}\right){\mathcal {T}}\left\{\prod _{j=1}^{n}{\mathcal {H}}_{V}\left(x_{j}\right)\right\}\equiv \sum _{n=0}^{\infty }S^{(n)}\;.}
Equivalently, with the interaction Lagrangian LV, it is
S = ∑ n = 0 ∞ i n n ! ( ∏ j = 1 n ∫ d 4 x j ) T { ∏ j = 1 n L V ( x j ) } ≡ ∑ n = 0 ∞ S ( n ) . {\displaystyle S=\sum _{n=0}^{\infty }{\frac {i^{n}}{n!}}\left(\prod _{j=1}^{n}\int d^{4}x_{j}\right){\mathcal {T}}\left\{\prod _{j=1}^{n}{\mathcal {L}}_{V}\left(x_{j}\right)\right\}\equiv \sum _{n=0}^{\infty }S^{(n)}\;.}
A Feynman diagram is a graphical representation of a single summand in the Wick's expansion of the time-ordered product in the nth-order term S(n) of the Dyson series of the S-matrix,
T ∏ j = 1 n L V ( x j ) = ∑ A ( ± ) N ∏ j = 1 n L V ( x j ) , {\displaystyle {\mathcal {T}}\prod _{j=1}^{n}{\mathcal {L}}_{V}\left(x_{j}\right)=\sum _{\text{A}}(\pm ){\mathcal {N}}\prod _{j=1}^{n}{\mathcal {L}}_{V}\left(x_{j}\right)\;,}
where N signifies the normal-ordered 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) and A represents all possible contractions.
Feynman rules The diagrams are drawn according to the Feynman rules, which depend upon the interaction Lagrangian. For the QED interaction Lagrangian
L v = − g ψ ¯ γ μ ψ A μ {\displaystyle L_{v}=-g{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu }}
describing the interaction of a fermionic field ψ with a bosonic gauge field Aμ, the Feynman rules can be formulated in coordinate space as follows:
Each integration coordinate xj is represented by a point (sometimes called a vertex); A bosonic propagator is represented by a wiggly line connecting two points; A fermionic propagator is represented by a solid line connecting two points; A bosonic field A μ ( x i ) {\displaystyle A_{\mu }(x_{i})} is represented by a wiggly line attached to the point xi; A fermionic field ψ(xi) is represented by a solid line attached to the point xi with an arrow toward the point; An anti-fermionic field ψ(xi) is represented by a solid line attached to the point xi with an arrow away from the point.
Example: second order processes in QED The second order perturbation term in the S-matrix is
S ( 2 ) = ( i e ) 2 2 ! ∫ d 4 x d 4 x ′ T ψ ¯ ( x ) γ μ ψ ( x ) A μ ( x ) ψ ¯ ( x ′ ) γ ν ψ ( x ′ ) A ν ( x ′ ) . {\displaystyle S^{(2)}={\frac {(ie)^{2}}{2!}}\int d^{4}x\,d^{4}x'\,T{\bar {\psi }}(x)\,\gamma ^{\mu }\,\psi (x)\,A_{\mu }(x)\,{\bar {\psi }}(x')\,\gamma ^{\nu }\,\psi (x')\,A_{\nu }(x').\;}
Scattering of fermions
The Wick's expansion of the integrand gives (among others) the following term
N ψ ¯ ( x ) γ μ ψ ( x ) ψ ¯ ( x ′ ) γ ν ψ ( x ′ ) A μ ( x ) A ν ( x ′ ) _ , {\displaystyle N{\bar {\psi }}(x)\gamma ^{\mu }\psi (x){\bar {\psi }}(x')\gamma ^{\nu }\psi (x'){\underline {A_{\mu }(x)A_{\nu }(x')}}\;,}
where
A μ ( x ) A ν ( x ′ ) _ = ∫ d 4 k ( 2 π ) 4 − i g μ ν k 2 + i 0 e − i k ( x − x ′ ) {\displaystyle {\underline {A_{\mu }(x)A_{\nu }(x')}}=\int {\frac {d^{4}k}{(2\pi )^{4}}}{\frac {-ig_{\mu \nu }}{k^{2}+i0}}e^{-ik(x-x')}}
is the electromagnetic contraction (propagator) in the Feynman gauge. This term is represented by the Feynman diagram at the right. This diagram gives contributions to the following processes:
e− e− scattering (initial state at the right, final state at the left of the diagram); e+ e+ scattering (initial state at the left, final state at the right of the diagram); e− e+ scattering (initial state at the bottom/top, final state at the top/bottom of the diagram).
Compton scattering and annihilation/generation of e− e+ pairs Another interesting term in the expansion is
N ψ ¯ ( x ) γ μ ψ ( x ) ψ ¯ ( x ′ ) _ γ ν ψ ( x ′ ) A μ ( x ) A ν ( x ′ ) , {\displaystyle N{\bar {\psi }}(x)\,\gamma ^{\mu }\,{\underline {\psi (x)\,{\bar {\psi }}(x')}}\,\gamma ^{\nu }\,\psi (x')\,A_{\mu }(x)\,A_{\nu }(x')\;,}
where
ψ ( x ) ψ ¯ ( x ′ ) _ = ∫ d 4 p ( 2 π ) 4 i γ p − m + i 0 e − i p ( x − x ′ ) {\displaystyle {\underline {\psi (x){\bar {\psi }}(x')}}=\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i}{\gamma p-m+i0}}e^{-ip(x-x')}}
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 must have a well-defined ground state, and the integral must be performed a little bit rotated into imaginary time, i.e. a Wick rotation. The path integral formalism is completely equivalent to the canonical operator formalism above.
Scalar field Lagrangian An example is the free relativistic scalar field in d dimensions, whose action integral is:
S = ∫ 1 2 ∂ μ ϕ ∂ μ ϕ d d x . {\displaystyle S=\int {\tfrac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi \,d^{d}x\,.}
The unnormalized probability amplitude for a process is:
∫ A B e i S D ϕ , {\displaystyle \int _{A}^{B}e^{iS}\,D\phi \,,}
where A and B are space-like hypersurfaces that define the boundary conditions. The collection of all the φ(A) on the starting hypersurface give the field's initial value, analogous to the starting position for a point particle, and the field values φ(B) at each point of the final hypersurface defines the final field value, which is allowed to vary, giving a different amplitude to end up at different values. This is the field-to-field transition amplitude. The path integral gives the expectation value of operators between the initial and final state:
∫ A B e i S ϕ ( x 1 ) ⋯ ϕ ( x n ) D ϕ = ⟨ A | ϕ ( x 1 ) ⋯ ϕ ( x n ) | B ⟩ , {\displaystyle \int _{A}^{B}e^{iS}\phi (x_{1})\cdots \phi (x_{n})\,D\phi =\left\langle A\left|\phi (x_{1})\cdots \phi (x_{n})\right|B\right\rangle \,,}
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 is normalized in the following way:
∫ e i S ϕ ( x 1 ) ⋯ ϕ ( x n ) D ϕ ∫ e i S D ϕ = ⟨ 0 | ϕ ( x 1 ) ⋯ ϕ ( x n ) | 0 ⟩ . {\displaystyle {\frac {\displaystyle \int e^{iS}\phi (x_{1})\cdots \phi (x_{n})\,D\phi }{\displaystyle \int e^{iS}\,D\phi }}=\left\langle 0\left|\phi (x_{1})\cdots \phi (x_{n})\right|0\right\rangle \,.}
The field's partition function is the normalization factor on the bottom, which 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 can be thought of as on a discrete square lattice, with lattice spacing a and the limit a → 0 should be taken carefully. If the final results do not depend on the shape of the lattice or the value of a, then the continuum limit exists.
On a lattice On a lattice, (i), the field can be expanded in Fourier modes:
ϕ ( x ) = ∫ d k ( 2 π ) d ϕ ( k ) e i k ⋅ x = ∫ k ϕ ( k ) e i k x . {\displaystyle \phi (x)=\int {\frac {dk}{(2\pi )^{d}}}\phi (k)e^{ik\cdot x}=\int _{k}\phi (k)e^{ikx}\,.}
Here the integration domain is over k restricted to a cube of side length 2π/a, so that large values of k are not allowed. The k-measure contains the factors of 2π 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 a → 0. Sometimes, instead of a lattice, the field modes are just cut off at high values of k instead. It is also convenient from time to time to consider the 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 that will arise. On a lattice, (ii), the action needs to be discretized:
S = ∑ ⟨ x , y ⟩ 1 2 ( ϕ ( x ) − ϕ ( y ) ) 2 , {\displaystyle S=\sum _{\langle x,y\rangle }{\tfrac {1}{2}}{\big (}\phi (x)-\phi (y){\big )}^{2}\,,}
where ⟨x,y⟩ is a pair of nearest lattice neighbors x and y. The discretization should be thought of as defining what the derivative ∂μφ means. In terms of the lattice Fourier modes, the action can be written:
S = ∫ k ( ( 1 − cos ( k 1 ) ) + ( 1 − cos ( k 2 ) ) + ⋯ + ( 1 − cos ( k d ) ) ) ϕ k ∗ ϕ k . {\displaystyle S=\int _{k}{\Big (}{\big (}1-\cos(k_{1}){\big )}+{\big (}1-\cos(k_{2}){\big )}+\cdots +{\big (}1-\cos(k_{d}){\big )}{\Big )}\phi _{k}^{*}\phi ^{k}\,.}
For k near zero this is:
S = ∫ k 1 2 k 2