In theoretical physics, Feynman diagrams are pictorial representations
of the mathematical expressions describing the behavior of subatomic
particles. The scheme is named after its inventor, American physicist
Richard Feynman, and was first introduced in 1948. 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".[1] While the diagrams are
applied primarily to quantum field theory, they can also be used in
other fields, such as solid-state theory.
Feynman used Ernst Stueckelberg's interpretation of the positron as if
it were an electron moving backward in time.[2] Thus, antiparticles
are represented as moving backward along the time axis in Feynman
diagrams.
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
Quantum field theory Feynman diagram History Background Field theory
Electromagnetism
Weak force
Strong force
Quantum mechanics
Symmetries Symmetry in quantum mechanics C-symmetry P-symmetry T-symmetry Space translation symmetry Time translation symmetry Rotation symmetry Lorentz symmetry Poincaré symmetry Gauge symmetry Explicit symmetry breaking Spontaneous symmetry breaking Yang–Mills theory Noether charge Topological charge Tools Anomaly Crossing Effective field theory Expectation value Faddeev–Popov ghosts Feynman diagram Lattice gauge theory LSZ reduction formula Partition function Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman axioms Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations Standard Model Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism Incomplete theories Topological quantum field theory String theory Superstring theory M-Theory Supersymmetry Supergravity Technicolor Theory of everything Quantum gravity Scientists C. D. Anderson P. W. Anderson Bethe Bjorken Bogoliubov Brout Callan Coleman DeWitt Dirac Dyson Englert Fermi Feynman Fierz Fock Fröhlich Glashow Gell-Mann Gross Guralnik Heisenberg Higgs Haag Hagen 't Hooft Jordan Kendall Kibble Lamb Landau Lee Majorana Mills Nambu Nishijima Parisi Polyakov Salam Schwinger Skyrme Sudarshan Tomonaga Veltman Ward Weinberg Weisskopf Weyl Wilczek Wilson Yang Yukawa v t e Contents 1 Motivation and history 1.1 Alternative names 2 Representation of physical reality 3 Particle-path interpretation 4 Description 4.1 Electron–positron annihilation example 5
5.1 Feynman rules 5.2 Example: second order processes in QED 5.2.1 Scattering of fermions 5.2.2 Compton scattering and annihilation/generation of e− e+ pairs 6 Path integral formulation 6.1 Scalar field Lagrangian 6.2 On a lattice 6.3 Monte Carlo 6.4 Scalar propagator 6.5 Equation of motion 6.5.1 Wick theorem 6.5.2 Higher Gaussian moments — completing Wick's theorem 6.5.3 Interaction 6.5.4 Feynman diagrams 6.5.5 Loop order 6.5.6 Symmetry factors 6.5.7 Connected diagrams: linked-cluster theorem 6.5.8 Vacuum bubbles 6.5.9 Sources 6.6 Spin 1/2; "photons" and "ghosts" 6.6.1 Spin 1/2: Grassmann integrals 6.6.2 Spin 1: photons 6.6.3 Spin 1: non-Abelian ghosts 7 Particle-path representation 7.1 Schwinger representation 7.2 Combining denominators 7.3 Scattering 8 Nonperturbative effects 9 In popular culture 10 See also 11 Notes 12 References 13 External links Motivation and history[edit] In this diagram, a kaon, made of an up and anti-strange quark, decays both weakly and strongly into three pions, with intermediate steps involving a W boson and a gluon (represented by the blue sine wave and green spiral, respectively). 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.
The
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
So far there are no opposing opinions. In quantum field theories the
Feynman diagrams are obtained from Lagrangian by Feynman rules.
General features of the scattering process A + B → C + D: • internal lines (red) for intermediate particles and processes, which has a propagator factor ("prop"), external lines (orange) for incoming/outgoing particles to/from vertices (black), • at each vertex there is 4-momentum conservation using delta functions, 4-momenta entering the vertex are positive while those leaving are negative, the factors at each vertex and internal line are multiplied in the amplitude integral, • space x and time t axes are not always shown, directions of external lines correspond to passage of time. A
In
The electron–positron annihilation interaction: e+ e− → 2γ has a contribution from the second order
S f i = ⟨ f
S
i ⟩ , displaystyle S_ fi =langle fSirangle ;, where S is the S-matrix.
In the canonical quantum field theory the
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! int prod _ j=1 ^ n d^ 4 x_ j Tprod _ j=1 ^ n L_ v left(x_ j right)equiv sum _ n=0 ^ infty S^ (n) ;, where Lv is the interaction Lagrangian and T signifies the
time-ordered product of operators.
A
T ∏ j = 1 n L v ( x j ) = ∑ all possible contractions ( ± ) N ∏ j = 1 n L v ( x j ) , displaystyle Tprod _ j=1 ^ n L_ v left(x_ j right)=sum _ text all possible atop text contractions (pm )Nprod _ j=1 ^ n L_ v left(x_ j right);, where N 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).
Feynman rules[edit]
The diagrams are drawn according to the Feynman rules, which depend
upon the interaction Lagrangian. For the
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
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[edit]
The second order perturbation term in the
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[edit] The
N ψ ¯ ( x ) i e γ μ ψ ( x ) ψ ¯ ( x ′ ) i e γ ν ψ ( x ′ ) A μ ( x ) A ν ( x ′ ) displaystyle N bar psi (x)iegamma ^ mu psi (x) bar psi (x')iegamma ^ nu psi (x')A_ mu (x)A_ nu (x') 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 (2pi )^ 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
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[edit] 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 (2pi )^ 4 frac i gamma p-m+i0 e^ -ip(x-x') is the fermionic contraction (propagator). Path integral formulation[edit] 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, i.e. a Wick rotation. Scalar field Lagrangian[edit] A simple 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 probability amplitude for a process is: ∫ A B e i S D ϕ , displaystyle int _ A ^ B e^ iS ,Dphi ,, 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 initial value of the field, 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 ),Dphi =leftlangle Aleftphi (x_ 1 )cdots phi (x_ n )rightBrightrangle ,, 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 can 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: ∫ 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 ),Dphi displaystyle int e^ iS ,Dphi =leftlangle 0leftphi (x_ 1 )cdots phi (x_ n )right0rightrangle ,. 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 can be thought of as on a discrete square lattice, with lattice spacing a and the limit a → 0 should be taken carefully[clarification needed]. 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[edit] 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 (2pi )^ d phi (k)e^ ikcdot 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. It is important to note that 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,yrangle 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
ϕ ( k )
2 . displaystyle S=int _ k tfrac 1 2 k^ 2 leftphi (k)right^ 2 ,. Now we have the continuum Fourier transform of the original action. In finite volume, the quantity ddk is not infinitesimal, but becomes the volume of a box made by neighboring Fourier modes, or (2π/V)d . The field φ is real-valued, so the Fourier transform obeys: ϕ ( k ) ∗ = ϕ ( − k ) . displaystyle phi (k)^ * =phi (-k),. In terms of real and imaginary parts, the real part of φ(k) is an even function of k, while the imaginary part is odd. The Fourier transform avoids double-counting, so that it can be written: S = ∫ k 1 2 k 2 ϕ ( k ) ϕ ( − k ) displaystyle S=int _ k tfrac 1 2 k^ 2 phi (k)phi (-k) over an integration domain that integrates over each pair (k,−k) exactly once. For a complex scalar field with action 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 Fourier transform is unconstrained: S = ∫ k 1 2 k 2
ϕ ( k )
2 displaystyle S=int _ k tfrac 1 2 k^ 2 leftphi (k)right^ 2 and the integral is over all k. Integrating over all different values of φ(x) is equivalent to integrating over all Fourier modes, because taking a Fourier transform is a unitary linear transformation of field coordinates. When you change coordinates in a multidimensional integral by a linear transformation, the value of the new integral is given by the determinant of the transformation matrix. If y i = A i j x j , displaystyle y_ i =A_ ij x_ j ,, then det ( A ) ∫ d x 1 d x 2 ⋯ d x n = ∫ d y 1 d y 2 ⋯ d y n . displaystyle det(A)int dx_ 1 ,dx_ 2 cdots ,dx_ n =int dy_ 1 ,dy_ 2 cdots ,dy_ n ,. If A is a rotation, then A T A = I displaystyle A^ mathrm T A=I so that det A = ±1, and the sign depends on whether the rotation includes a reflection or not. The matrix that changes coordinates from φ(x) to φ(k) can be read off from the definition of a Fourier transform. A k x = e i k x displaystyle A_ kx =e^ ikx , and the Fourier inversion theorem tells you the inverse: A k x − 1 = e − i k x displaystyle A_ kx ^ -1 =e^ -ikx , which is the complex conjugate-transpose, up to factors of 2π. On a finite volume lattice, the determinant is nonzero and independent of the field values. det A = 1 displaystyle det A=1, and the path integral is a separate factor at each value of k. ∫ exp ( i 2 ∑ k k 2 ϕ ∗ ( k ) ϕ ( k ) ) D ϕ = ∏ k ∫ ϕ k e i 2 k 2
ϕ k
2 d d k displaystyle int exp left( frac i 2 sum _ k k^ 2 phi ^ * (k)phi (k)right),Dphi =prod _ k int _ phi _ k e^ frac i 2 k^ 2 leftphi _ k right^ 2 ,d^ d k , The factor ddk is the infinitesimal volume of a discrete cell in k-space, in a square lattice box d d k = ( 1 L ) d , displaystyle d^ d k=left( frac 1 L right)^ d ,, 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 φk is e ∫ k − 1 2 k 2 ϕ k ∗ ϕ k = ∏ k e − k 2
ϕ k
2 d d k . displaystyle e^ int _ k - tfrac 1 2 k^ 2 phi _ k ^ * phi _ k =prod _ k e^ -k^ 2 leftphi _ k right^ 2 ,d^ d k ,. The expectation value of the field is the statistical expectation value of the field when chosen according to the probability distribution: ⟨ ϕ ( x 1 ) ⋯ ϕ ( x n ) ⟩ = ∫ e − S ϕ ( x 1 ) ⋯ ϕ ( x n ) D ϕ ∫ e − S D ϕ displaystyle leftlangle phi (x_ 1 )cdots phi (x_ n )rightrangle = frac displaystyle int e^ -S phi (x_ 1 )cdots phi (x_ n ),Dphi displaystyle int e^ -S ,Dphi Since the probability of φk is a product, the value of φk at each separate value of k is independently Gaussian distributed. The variance of the Gaussian is 1/k2ddk, 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 V/k2. Monte Carlo[edit] 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 1/k2. This generates a configuration φC(k) at random, and the Fourier transform gives φC(x). For real scalar fields, the algorithm must generate only one of each pair φ(k), φ(−k), 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: ⟨ ϕ ( x 1 ) ⋯ ϕ ( x n ) ⟩ = lim
C
→ ∞ ∑ C ϕ C ( x 1 ) ⋯ ϕ C ( x n )
C
displaystyle leftlangle phi (x_ 1 )cdots phi (x_ n )rightrangle =lim _ Crightarrow infty frac displaystyle sum _ C phi _ C (x_ 1 )cdots phi _ C (x_ n ) C where C 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
⟨ ϕ k ϕ k ′ ⟩ = 0 displaystyle leftlangle phi _ k phi _ k' rightrangle =0, for k ≠ k′, since then the two Gaussian random variables are independent and both have zero mean. ⟨ ϕ k ϕ k ⟩ = V k 2 displaystyle leftlangle phi _ k phi _ k rightrangle = frac V k^ 2 in finite volume V, when the two k-values coincide, since this is the variance of the Gaussian. In the infinite volume limit, ⟨ ϕ ( k ) ϕ ( k ′ ) ⟩ = δ ( k − k ′ ) 1 k 2 displaystyle leftlangle phi (k)phi (k')rightrangle =delta (k-k') frac 1 k^ 2 Strictly speaking, this is an approximation: the lattice propagator is: ⟨ ϕ ( k ) ϕ ( k ′ ) ⟩ = δ ( k − k ′ ) 1 2 ( d − cos ( k 1 ) + cos ( k 2 ) ⋯ + cos ( k d ) ) displaystyle leftlangle phi (k)phi (k')rightrangle =delta (k-k') frac 1 2 big ( d-cos(k_ 1 )+cos(k_ 2 )cdots +cos(k_ d ) big ) 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 2π, so that they cancel out the 2π factors in the measure for k integrals. δ ( k ) = ( 2 π ) d δ D ( k 1 ) δ D ( k 2 ) ⋯ δ D ( k d ) displaystyle delta (k)=(2pi )^ d delta _ D (k_ 1 )delta _ D (k_ 2 )cdots delta _ D (k_ d ), where δD(k) is the ordinary one-dimensional Dirac delta function. This convention for delta-functions is not universal—some authors keep the factors of 2π in the delta functions (and in the k-integration) explicit. Equation of motion[edit] 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: ∂ μ ∂ μ ϕ = 0 displaystyle partial _ mu partial ^ mu phi =0, and in an expectation value, this says: ∂ μ ∂ μ ⟨ ϕ ( x ) ϕ ( y ) ⟩ = 0 displaystyle partial _ mu partial ^ mu leftlangle phi (x)phi (y)rightrangle =0 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 Δ as the Fourier transform of the time-ordered two-point function (the one that comes from the path-integral): ∂ 2 Δ ( x ) = i δ ( x ) displaystyle partial ^ 2 Delta (x)=idelta (x), So that: Δ ( k ) = i k 2 displaystyle Delta (k)= frac i k^ 2 If the equations of motion are linear, the propagator will always be the reciprocal of the quadratic-form matrix that 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[edit] Main article: Wick's theorem Because each field mode is an independent Gaussian, the expectation values for the product of many field modes obeys Wick's theorem: ⟨ ϕ ( k 1 ) ϕ ( k 2 ) ⋯ ϕ ( k n ) ⟩ displaystyle leftlangle phi (k_ 1 )phi (k_ 2 )cdots phi (k_ n )rightrangle is zero unless the field modes coincide in pairs. This means that it is zero for an odd number of φ, and for an even number of φ, it is equal to a contribution from each pair separately, with a delta function. ⟨ ϕ ( k 1 ) ⋯ ϕ ( k 2 n ) ⟩ = ∑ ∏ i , j δ ( k i − k j ) k i 2 displaystyle leftlangle phi (k_ 1 )cdots phi (k_ 2n )rightrangle =sum prod _ i,j frac delta left(k_ i -k_ j right) k_ i ^ 2 where the sum is over each partition of the field modes into pairs, and the product is over the pairs. For example, ⟨ ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) ⟩ = δ ( k 1 − k 2 ) k 1 2 δ ( k 3 − k 4 ) k 3 2 + δ ( k 1 − k 3 ) k 3 2 δ ( k 2 − k 4 ) k 2 2 + δ ( k 1 − k 4 ) k 1 2 δ ( k 2 − k 3 ) k 2 2 displaystyle leftlangle phi (k_ 1 )phi (k_ 2 )phi (k_ 3 )phi (k_ 4 )rightrangle = frac delta (k_ 1 -k_ 2 ) k_ 1 ^ 2 frac delta (k_ 3 -k_ 4 ) k_ 3 ^ 2 + frac delta (k_ 1 -k_ 3 ) k_ 3 ^ 2 frac delta (k_ 2 -k_ 4 ) k_ 2 ^ 2 + frac delta (k_ 1 -k_ 4 ) k_ 1 ^ 2 frac delta (k_ 2 -k_ 3 ) k_ 2 ^ 2 An interpretation of
I = ∫ e − a x 2 / 2 d x = 2 π a displaystyle I=int e^ -ax^ 2 /2 dx= sqrt frac 2pi a ∂ n ∂ a n I = ∫ x 2 n 2 n e − a x 2 / 2 d x = 1 ⋅ 3 ⋅ 5 … ⋅ ( 2 n − 1 ) 2 ⋅ 2 ⋅ 2 … ⋅ 2 2 π a − 2 n + 1 2 displaystyle frac partial ^ n partial a^ n I=int frac x^ 2n 2^ n e^ -ax^ 2 /2 dx= frac 1cdot 3cdot 5ldots cdot (2n-1) 2cdot 2cdot 2ldots ;;;;;cdot 2;;;;;; sqrt 2pi ,a^ - frac 2n+1 2 Dividing by I, ⟨ x 2 n ⟩ = ∫ x 2 n e − a x 2 / 2 ∫ e − a x 2 / 2 = 1 ⋅ 3 ⋅ 5 … ⋅ ( 2 n − 1 ) 1 a n displaystyle leftlangle x^ 2n rightrangle = frac displaystyle int x^ 2n e^ -ax^ 2 /2 displaystyle int e^ -ax^ 2 /2 =1cdot 3cdot 5ldots cdot (2n-1) frac 1 a^ n ⟨ x 2 ⟩ = 1 a displaystyle leftlangle x^ 2 rightrangle = frac 1 a If
⟨ x 1 x 2 x 3 ⋯ x 2 n ⟩ displaystyle leftlangle x_ 1 x_ 2 x_ 3 cdots x_ 2n rightrangle where the x 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 leaving 2n − 4, and so on. This means that Wick's theorem, uncorrected, says that the expectation value of x2n should be: ⟨ x 2 n ⟩ = ( 2 n − 1 ) ⋅ ( 2 n − 3 ) … ⋅ 5 ⋅ 3 ⋅ 1 ⟨ x 2 ⟩ n displaystyle leftlangle x^ 2n rightrangle =(2n-1)cdot (2n-3)ldots cdot 5cdot 3cdot 1leftlangle x^ 2 rightrangle ^ n and this is in fact the correct answer. So
S = ∫ ∂ μ ϕ ∂ μ ϕ + λ 4 ! ϕ 4 . displaystyle S=int partial ^ mu phi partial _ mu phi + frac lambda 4! phi ^ 4 . The reason for the combinatorial factor 4! will be clear soon. Writing the action in terms of the lattice (or continuum) Fourier modes: S = ∫ k k 2
ϕ ( k )
2 + ∫ k 1 k 2 k 3 k 4 ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) δ ( k 1 + k 2 + k 3 + k 4 ) = S F + X . displaystyle S=int _ k k^ 2 leftphi (k)right^ 2 +int _ k_ 1 k_ 2 k_ 3 k_ 4 phi (k_ 1 )phi (k_ 2 )phi (k_ 3 )phi (k_ 4 )delta (k_ 1 +k_ 2 +k_ 3 +k_ 4 )=S_ F +X. Where SF 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 λ, giving a series of corrections to the free action. e − S = e − S F ( 1 + X + 1 2 ! X X + 1 3 ! X X X + ⋯ ) displaystyle e^ -S =e^ -S_ F left(1+X+ frac 1 2! XX+ frac 1 3! XXX+cdots right) 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 φ(k). The half-lines meet at a vertex, which contributes a delta-function that ensures that the sum of the momenta are all equal. To compute a correlation function in the interacting theory, there is a contribution from the X terms now. For example, the path-integral for the four-field correlator: ⟨ ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) ⟩ = ∫ e − S ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) D ϕ Z displaystyle leftlangle phi (k_ 1 )phi (k_ 2 )phi (k_ 3 )phi (k_ 4 )rightrangle = frac displaystyle int e^ -S phi (k_ 1 )phi (k_ 2 )phi (k_ 3 )phi (k_ 4 )Dphi Z which in the free field was only nonzero when the momenta k were equal
in pairs, is now nonzero for all values of k. The momenta of the
insertions φ(ki) can now match up with the momenta of the Xs in the
expansion. The insertions should also be thought of as half-lines,
four in this case, which carry a momentum k, but one that is not
integrated.
The lowest-order contribution comes from the first nontrivial term
e−SFX in the Taylor expansion of the action.
λ 1 k 1 2 1 k 2 2 1 k 3 2 1 k 4 2 . displaystyle lambda frac 1 k_ 1 ^ 2 frac 1 k_ 2 ^ 2 frac 1 k_ 3 ^ 2 frac 1 k_ 4 ^ 2 ,. 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 k1,2,3,4, by Wick's theorem. Feynman diagrams[edit] The expansion of the action in powers of X gives a series of terms with progressively higher number of Xs. The contribution from the term with exactly n Xs is called nth order. The nth order terms has: 4n internal half-lines, which are the factors of φ(k) from the Xs. These all end on a vertex, and are integrated over all possible k. external half-lines, which are the come from the φ(k) 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 δ ( k 1 + k 2 ) k 1 2 displaystyle frac delta (k_ 1 +k_ 2 ) k_ 1 ^ 2 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 that are formed by linking the half-lines in the Xs with
the external half-lines, representing insertions, are the Feynman
diagrams of this theory. Each line carries a factor of 1/k2, 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[edit]
A forest diagram is one where all the internal lines have momentum
that is completely determined by the external lines and the condition
that the incoming and outgoing momentum are equal at each vertex. The
contribution of these diagrams is a product of propagators, without
any integration. A tree diagram is a connected forest diagram.
An example of a tree diagram is the one where each of four external
lines end on an X. Another is when three external lines end on an X,
and the remaining 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
that is not a tree is when eight external lines end on two Xs.
It is easy to verify that in all these cases, the momenta on all the
internal lines is determined by the external momenta and the condition
of momentum conservation in each vertex.
A diagram that is not a forest diagram is called a loop diagram, and
an example is one where two lines of an X are joined to external
lines, while the remaining two lines are joined to each other. The two
lines joined to each other can have any momentum at all, since they
both enter and leave the same vertex. A more complicated example is
one where two Xs 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 that are left undetermined by momentum
conservation is equal to the number of independent closed loops in the
diagram, where independent loops are counted as in homology theory.
The homology is real-valued (actually Rd 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 valid k-values can be arbitrarily redefined whenever there is
a closed loop. A closed loop is a cyclical path of adjacent vertices
that never revisits the same vertex. Such a cycle can be thought of as
the boundary of a hypothetical 2-cell. The k-labellings of a graph
that conserve momentum (i.e. which has zero boundary) up to
redefinitions of k (i.e. up to boundaries of 2-cells) define the first
homology of a graph. The number of independent momenta that are not
determined is then equal to the number of independent homology loops.
For many graphs, this is equal to the number of loops as counted in
the most intuitive way.
Symmetry factors[edit]
The number of ways to form a given
If a line l goes from vertex v to vertex v′, then M(l) goes from N(v) to N(v′). If the line is undirected, as it is for a real scalar field, then M(l) can go from N(v′) to N(v) too. If a line l ends on an external line, M(l) ends on the same external line. If there are different types of lines, M(l) should preserve the type. This theorem has an interpretation in terms of particle-paths: when
identical particles are present, the integral over all intermediate
particles must not double-count states that differ only by
interchanging identical particles.
Proof: To prove this theorem, label all the internal and external
lines of a diagram with a unique name. Then form the diagram by
linking 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 Xs 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.
Connected diagrams: linked-cluster theorem[edit]
Roughly speaking, a
i W [ J ] ≡ ln Z [ J ] . displaystyle iW[J]equiv ln Z[J]. To see this, one should recall that Z [ J ] ∝ ∑ k D k displaystyle Z[J]propto sum _ k D_ k with Dk constructed from some (arbitrary)
∏ i C i n i n i ! displaystyle prod _ i frac C_ i ^ n_ i n_ i ! where i labels the (infinitely) many connected Feynman diagrams possible. A scheme to successively create such contributions from the Dk to Z[J] is obtained by ( 1 0 ! + C 1 1 ! + C 1 2 2 ! + ⋯ ) ( 1 + C 2 + 1 2 C 2 2 + ⋯ ) ⋯ displaystyle left( frac 1 0! + frac C_ 1 1! + frac C_ 1 ^ 2 2! +cdots right)left(1+C_ 2 + frac 1 2 C_ 2 ^ 2 +cdots right)cdots and therefore yields Z [ J ] ∝ ∏ i ∑ n i = 0 ∞ C i n i n i ! = exp ∑ i C i ∝ exp W [ J ] . displaystyle Z[J]propto prod _ i sum _ n_ i =0 ^ infty frac C_ i ^ n_ i n_ i ! =exp sum _ i C_ i propto exp W[J] ,. To establish the normalization Z0 = exp W[0] = 1 one simply calculates all connected vacuum diagrams, i.e., the diagrams without any sources J (sometimes referred to as external legs of a Feynman diagram). Vacuum bubbles[edit] 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: ⟨ ϕ 1 ( x 1 ) ⋯ ϕ n ( x n ) ⟩ = ∫ e − S ϕ 1 ( x 1 ) ⋯ ϕ n ( x n ) D ϕ ∫ e − S D ϕ . displaystyle leftlangle phi _ 1 (x_ 1 )cdots phi _ n (x_ n )rightrangle = frac displaystyle int e^ -S phi _ 1 (x_ 1 )cdots phi _ n (x_ n ),Dphi displaystyle int e^ -S ,Dphi ,. The top is the sum over all Feynman diagrams, including disconnected diagrams that do not link up to external lines at all. In terms of the connected diagrams, the numerator includes the same contributions of vacuum bubbles as the denominator: ∫ e − S ϕ 1 ( x 1 ) ⋯ ϕ n ( x n ) D ϕ = ( ∑ E i ) ( exp ( ∑ i C i ) ) . displaystyle int e^ -S phi _ 1 (x_ 1 )cdots phi _ n (x_ n ),Dphi =left(sum E_ i right)left(exp left(sum _ i C_ i right)right),. Where the sum over E diagrams includes only those diagrams each of whose connected components end on at least one external line. The vacuum bubbles are the same whatever the external lines, and give an overall multiplicative factor. The denominator is the sum over all vacuum bubbles, and dividing gets rid of the second factor. The vacuum bubbles then are only useful for determining Z itself, which from the definition of the path integral is equal to: Z = ∫ e − S D ϕ = e − H T = e − ρ V displaystyle Z=int e^ -S Dphi =e^ -HT =e^ -rho V where ρ is the energy density in the vacuum. Each vacuum bubble
contains a factor of δ(k) zeroing the total k at each vertex, and
when there are no external lines, this contains a factor of δ(0),
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[edit]
∫ h ( x ) ϕ ( x ) d d x = ∫ h ( k ) ϕ ( k ) d d k displaystyle int h(x)phi (x),d^ d x=int h(k)phi (k),d^ d k, In the Feynman expansion, this contributes H terms with one half-line
ending on a vertex. Lines in a
log ( Z [ h ] ) = ∑ n , C h ( k 1 ) h ( k 2 ) ⋯ h ( k n ) C ( k 1 , ⋯ , k n ) displaystyle log big ( Z[h] big ) =sum _ n,C h(k_ 1 )h(k_ 2 )cdots h(k_ n )C(k_ 1 ,cdots ,k_ n ), where C(k1,…,kn) is the connected diagram with n external lines carrying momentum as indicated. The sum is over all connected diagrams, as before. The field h is not dynamical, which means that there is no path integral over h: h is just a parameter in the Lagrangian, which varies from point to point. The path integral for the field is: Z [ h ] = ∫ e i S + i ∫ h ϕ D ϕ displaystyle Z[h]=int e^ iS+iint hphi ,Dphi , 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 Rn, the Fourier transform of the probability density is: ∫ ρ ( y ) e i k y d n y = ⟨ e i k y ⟩ = ⟨ ∏ i = 1 n e i h i y i ⟩ displaystyle int rho (y)e^ iky ,d^ n y=leftlangle e^ iky rightrangle =leftlangle prod _ i=1 ^ n e^ ih_ i y_ i rightrangle , The Fourier transform is the expectation of an oscillatory exponential. The path integral in the presence of a source h(x) is: Z [ h ] = ∫ e i S e i ∫ x h ( x ) ϕ ( x ) D ϕ = ⟨ e i h ϕ ⟩ displaystyle Z[h]=int e^ iS e^ iint _ x h(x)phi (x) ,Dphi =leftlangle e^ ihphi rightrangle which, on a lattice, is the product of an oscillatory exponential for each field value: ⟨ ∏ x e i h x ϕ x ⟩ displaystyle leftlangle prod _ x e^ ih_ x phi _ x rightrangle The fourier transform of a delta-function is a constant, which gives a formal expression for a delta function: δ ( x − y ) = ∫ e i k ( x − y ) d k displaystyle delta (x-y)=int e^ ik(x-y) ,dk This tells you what a field delta function looks like in a path-integral. For two scalar fields φ and η, δ ( ϕ − η ) = ∫ e i h ( x ) ( ϕ ( x ) − η ( x ) ) d d x D h , displaystyle delta (phi -eta )=int e^ ih(x) big ( phi (x)-eta (x) big ) ,d^ d x ,Dh,, which integrates over the Fourier transform coordinate, over h. This expression is useful for formally changing field coordinates in the path integral, much as a delta function is used to change coordinates in an ordinary 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: ⟨ ϕ ( x ) ⟩ = 1 Z ∂ ∂ h ( x ) Z [ h ] = ∂ ∂ h ( x ) log ( Z [ h ] ) . displaystyle leftlangle phi (x)rightrangle = frac 1 Z frac partial partial h(x) Z[h]= frac partial partial h(x) log big ( Z[h] big ) ,. In Euclidean space, source contributions to the action can still appear with a factor of i, so that they still do a Fourier transform. Spin 1/2; "photons" and "ghosts"[edit] Spin 1/2: Grassmann integrals[edit] The field path integral can be extended to the Fermi case, but only if the notion of integration is expanded. A Grassmann integral of a free Fermi field is a high-dimensional determinant or Pfaffian, which defines the new type of Gaussian integration appropriate for Fermi fields. The two fundamental formulas of Grassmann integration are: ∫ e M i j ψ ¯ i ψ j D ψ ¯ D ψ = D e t ( M ) , displaystyle int e^ M_ ij bar psi ^ i psi ^ j ,D bar psi ,Dpsi =mathrm Det (M),, where M is an arbitrary matrix and ψ, ψ are independent Grassmann variables for each index i, and ∫ e 1 2 A i j ψ i ψ j D ψ = P f a f f ( A ) , displaystyle int e^ frac 1 2 A_ ij psi ^ i psi ^ j ,Dpsi =mathrm Pfaff (A),, where A is an antisymmetric matrix, ψ is a collection of Grassmann variables, and the 1/2 is to prevent double-counting (since ψiψj = −ψjψi). In matrix notation, where ψ and η are Grassmann-valued row vectors, η and ψ are Grassmann-valued column vectors, and M is a real-valued matrix: Z = ∫ e ψ ¯ M ψ + η ¯ ψ + ψ ¯ η D ψ ¯ D ψ = ∫ e ( ψ ¯ + η ¯ M − 1 ) M ( ψ + M − 1 η ) − η ¯ M − 1 η D ψ ¯ D ψ = D e t ( M ) e − η ¯ M − 1 η , displaystyle Z=int e^ bar psi Mpsi + bar eta psi + bar psi eta ,D bar psi ,Dpsi =int e^ left( bar psi + bar eta M^ -1 right)Mleft(psi +M^ -1 eta right)- bar eta M^ -1 eta ,D bar psi ,Dpsi =mathrm Det (M)e^ - bar eta M^ -1 eta ,, where the last equality is a consequence of the translation invariance of the Grassmann integral. The Grassmann variables η are external sources for ψ, and differentiating with respect to η pulls down factors of ψ. ⟨ ψ ¯ ψ ⟩ = 1 Z ∂ ∂ η ∂ ∂ η ¯ Z
η = η ¯ = 0 = M − 1 displaystyle leftlangle bar psi psi rightrangle = frac 1 Z frac partial partial eta frac partial partial bar eta Z_ eta = bar eta =0 =M^ -1 again, in a schematic matrix notation. The meaning of the formula above is that the derivative with respect to the appropriate component of η and η gives the matrix element of M−1. This is exactly analogous to the bosonic path integration formula for a Gaussian integral of a complex bosonic field: ∫ e ϕ ∗ M ϕ + h ∗ ϕ + ϕ ∗ h D ϕ ∗ D ϕ = e h ∗ M − 1 h D e t ( M ) displaystyle int e^ phi ^ * Mphi +h^ * phi +phi ^ * h ,Dphi ^ * ,Dphi = frac e^ h^ * M^ -1 h mathrm Det (M) ⟨ ϕ ∗ ϕ ⟩ = 1 Z ∂ ∂ h ∂ ∂ h ∗ Z
h = h ∗ = 0 = M − 1 . displaystyle leftlangle phi ^ * phi rightrangle = frac 1 Z frac partial partial h frac partial partial h^ * Z_ h=h^ * =0 =M^ -1 ,. 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 Grassmann fields, for Majorana fermions, the path integral a
∫ ψ ¯ ( γ μ ∂ μ − m ) ψ displaystyle int bar psi left(gamma ^ mu partial _ mu -mright)psi 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: S = ∫ k ψ ¯ ( i γ μ k μ − m ) ψ . displaystyle S=int _ k bar psi left(igamma ^ mu k_ mu -mright)psi ,. The propagator is the inverse of the matrix M linking ψ(k) and ψ(k), since different values of k do not mix together. ⟨ ψ ¯ ( k ′ ) ψ ( k ) ⟩ = δ ( k + k ′ ) 1 γ ⋅ k − m = δ ( k + k ′ ) γ ⋅ k + m k 2 − m 2 displaystyle leftlangle bar psi (k')psi (k)rightrangle =delta (k+k') frac 1 gamma cdot k-m =delta (k+k') frac gamma cdot k+m k^ 2 -m^ 2 The analog of
⟨ ψ ¯ ( k 1 ) ψ ¯ ( k 2 ) ⋯ ψ ¯ ( k n ) ψ ( k 1 ′ ) ⋯ ψ ( k n ) ⟩ = ∑ p a i r i n g s ( − 1 ) S ∏ p a i r s i , j δ ( k i − k j ) 1 γ ⋅ k i − m displaystyle leftlangle bar psi (k_ 1 ) bar psi (k_ 2 )cdots bar psi (k_ n )psi (k'_ 1 )cdots psi (k_ n )rightrangle =sum _ mathrm pairings (-1)^ S prod _ mathrm pairs ;i,j delta left(k_ i -k_ j right) frac 1 gamma cdot k_ i -m where S is the sign of the permutation that reorders the sequence of
ψ and ψ to put the ones that are paired up to make the
delta-functions next to each other, with the ψ coming right before
the ψ. Since a ψ, ψ pair is a commuting element of the Grassmann
algebra, it doesn't matter what order the pairs are in. If more than
one ψ, ψ 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 Grassmann analog of the
higher Gaussian moments that 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
Grassmann 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 loop 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 nuclear 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 that
together form a bosonic factor of the term in the Lagrangian, and when
entering a vertex by one line, the algorithm should always leave with
the partner line.
To clarify and prove the rule, consider a
S = ∫ 1 4 F μ ν F μ ν = ∫ − 1 2 ( ∂ μ A ν ∂ μ A ν − ∂ μ A μ ∂ ν A ν ) . displaystyle S=int tfrac 1 4 F^ mu nu F_ mu nu =int - tfrac 1 2 left(partial ^ mu A_ nu partial _ mu A^ nu -partial ^ mu A_ mu partial _ nu A^ nu right),. 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: ∫ δ ( ∂ μ A μ − f ) e − f 2 2 D f . displaystyle int delta left(partial _ mu A^ mu -fright)e^ - frac f^ 2 2 ,Df,. The first factor, the delta function, fixes the gauge. The second factor sums over different values of f that are inequivalent gauge fixings. This is simply e − ( ∂ μ A μ ) 2 2 . displaystyle e^ - frac left(partial _ mu A_ mu right)^ 2 2 ,. The additional contribution from gauge-fixing cancels the second half of the free Lagrangian, giving the Feynman Lagrangian: S = ∫ ∂ μ A ν ∂ μ A ν displaystyle S=int partial ^ mu A^ nu partial _ mu A_ nu which is just like four independent free scalar fields, one for each component of A. The Feynman propagator is: ⟨ A μ ( k ) A ν ( k ′ ) ⟩ = δ ( k + k ′ ) g μ ν k 2 . displaystyle leftlangle A_ mu (k)A_ nu (k')rightrangle =delta left(k+k'right) frac g_ mu nu k^ 2 . The one difference is that the sign of one propagator is wrong in the Lorentz case: the timelike component has an opposite sign propagator. This means that these particle states have negative norm—they are not physical states. In the case of photons, it is easy to show by diagram methods that these states are not physical—their contribution cancels with longitudinal photons to only leave two physical photon polarization contributions for any value of k. If the averaging over f is done with a coefficient different from 1/2, the two terms don't cancel completely. This gives a covariant Lagrangian with a coefficient λ displaystyle lambda , which does not affect anything: S = ∫ 1 2 ( ∂ μ A ν ∂ μ A ν − λ ( ∂ μ A μ ) 2 ) displaystyle S=int tfrac 1 2 left(partial ^ mu A^ nu partial _ mu A_ nu -lambda left(partial _ mu A^ mu right)^ 2 right) and the covariant propagator for
⟨ A μ ( k ) A ν ( k ′ ) ⟩ = δ ( k + k ′ ) g μ ν − λ k μ k ν k 2 k 2 . displaystyle leftlangle A_ mu (k)A_ nu (k')rightrangle =delta left(k+k'right) frac g_ mu nu -lambda frac k_ mu k_ nu k^ 2 k^ 2 . Spin 1: non-Abelian ghosts[edit]
To find the
δ ( ∂ μ A μ − f ) e − f 2 2 det M displaystyle delta left(partial _ mu A_ mu -fright)e^ - frac f^ 2 2 det M To find the form of the determinant, consider first a simple two-dimensional integral of a function f that depends only on r, not on the angle θ. Inserting an integral over θ: ∫ f ( r ) d x d y = ∫ f ( r ) ∫ d θ δ ( y )
d y d θ
d x d y displaystyle int f(r),dx,dy=int f(r)int dtheta ,delta (y)left frac dy dtheta right,dx,dy The derivative-factor ensures that popping the delta function in θ removes the integral. Exchanging the order of integration, ∫ f ( r ) d x d y = ∫ d θ ∫ f ( r ) δ ( y )
d y d θ
d x d y displaystyle int f(r),dx,dy=int dtheta ,int f(r)delta (y)left frac dy dtheta right,dx,dy but now the delta-function can be popped in y, ∫ f ( r ) d x d y = ∫ d θ 0 ∫ f ( x )
d y d θ
d x . displaystyle int f(r),dx,dy=int dtheta _ 0 ,int f(x)left frac dy dtheta right,dx,. The integral over θ just gives an overall factor of 2π, while the rate of change of y with a change in θ is just x, so this exercise reproduces the standard formula for polar integration of a radial function: ∫ f ( r ) d x d y = 2 π ∫ f ( x ) x d x displaystyle int f(r),dx,dy=2pi int f(x)x,dx In the path-integral for a nonabelian gauge field, the analogous manipulation is: ∫ D A ∫ δ ( F ( A ) ) det ( ∂ F ∂ G ) D G e i S = ∫ D G ∫ δ ( F ( A ) ) det ( ∂ F ∂ G ) e i S displaystyle int DAint delta big ( F(A) big ) det left( frac partial F partial G right),DGe^ iS =int DGint delta big ( F(A) big ) det left( frac partial F partial G right)e^ 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. ∫ det ( ∂ F ∂ G ) e i S G F D A displaystyle int det left( frac partial F partial G right)e^ iS_ GF ,DA, To get a covariant gauge, the gauge fixing condition is the same as in the Abelian case: ∂ μ A μ = f , displaystyle partial _ mu A^ mu =f,, Whose variation under an infinitesimal gauge transformation is given by: ∂ μ D μ α , displaystyle partial _ mu ,D_ mu alpha ,, where α is the adjoint valued element of the Lie algebra at every point that performs the infinitesimal gauge transformation. This adds the Faddeev Popov determinant to the action: det ( ∂ μ D μ ) displaystyle det left(partial _ mu ,D_ mu right), which can be rewritten as a Grassmann integral by introducing ghost fields: ∫ e η ¯ ∂ μ D μ η D η ¯ D η displaystyle int e^ bar eta partial _ mu ,D^ mu eta ,D bar eta ,Deta , 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: S = ∫ Tr ∂ μ A ν ∂ μ A ν + f j k i ∂ ν A i μ A μ j A ν k + f j r i f k l r A i A j A k A l + Tr ∂ μ η ¯ ∂ μ η + η ¯ A j η displaystyle S=int operatorname Tr partial _ mu A_ nu partial ^ mu A^ nu +f_ jk ^ i partial ^ nu A_ i ^ mu A_ mu ^ j A_ nu ^ k +f_ jr ^ i f_ kl ^ r A_ i A_ j A^ k A^ l +operatorname Tr partial _ mu bar eta partial ^ mu eta + bar eta A_ j eta , 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 longitudinal 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[edit]
Feynman diagrams were originally discovered by Feynman, by trial and
error, as a way to represent the contribution to the
1 p 2 + m 2 = ∫ 0 ∞ e − τ ( p 2 + m 2 ) d τ displaystyle frac 1 p^ 2 +m^ 2 =int _ 0 ^ infty e^ -tau left(p^ 2 +m^ 2 right) ,dtau The meaning of this identity (which is an elementary integration) is made clearer by Fourier transforming to real space. Δ ( x ) = ∫ 0 ∞ d τ e − m 2 τ 1 ( 4 π τ ) d / 2 e − x 2 4 τ displaystyle Delta (x)=int _ 0 ^ infty dtau e^ -m^ 2 tau frac 1 ( 4pi tau )^ d/2 e^ frac -x^ 2 4tau The contribution at any one value of τ to the propagator is a Gaussian of width √τ. The total propagation function from 0 to x is a weighted sum over all proper times τ of a normalized Gaussian, the probability of ending up at x after a random walk of time τ. The path-integral representation for the propagator is then: Δ ( x ) = ∫ 0 ∞ d τ ∫ D X e − ∫ 0 τ ( x ˙ 2 2 + m 2 ) d τ ′ displaystyle Delta (x)=int _ 0 ^ infty dtau int DX,e^ -int limits _ 0 ^ tau left( frac dot x ^ 2 2 +m^ 2 right)dtau ' 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[edit] The Schwinger representation has an immediate practical application to loop diagrams. For example, for the diagram in the φ4 theory formed by joining two xs together in two half-lines, and making the remaining lines external, the integral over the internal propagators in the loop is: ∫ k 1 k 2 + m 2 1 ( k + p ) 2 + m 2 . displaystyle int _ k frac 1 k^ 2 +m^ 2 frac 1 (k+p)^ 2 +m^ 2 ,. Here one line carries momentum k and the other k + p. The asymmetry can be fixed by putting everything in the Schwinger representation. ∫ t , t ′ e − t ( k 2 + m 2 ) − t ′ ( ( k + p ) 2 + m 2 ) d t d t ′ . displaystyle int _ t,t' e^ -t(k^ 2 +m^ 2 )-t'left((k+p)^ 2 +m^ 2 right) ,dt,dt',. Now the exponent mostly depends on t + t′, ∫ t , t ′ e − ( t + t ′ ) ( k 2 + m 2 ) − t ′ 2 p ⋅ k − t ′ p 2 , displaystyle int _ t,t' e^ -(t+t')(k^ 2 +m^ 2 )-t'2pcdot k-t'p^ 2 ,, except for the asymmetrical little bit. Defining the variable u = t + t′ and v = t′/u, the variable u goes from 0 to ∞, while v goes from 0 to 1. The variable u is the total proper time for the loop, while v parametrizes the fraction of the proper time on the top of the loop versus the bottom. The Jacobian for this transformation of variables is easy to work out from the identities: d ( u v ) = d t ′ d u = d t + d t ′ , displaystyle d(uv)=dt'quad du=dt+dt',, and "wedging" gives u d u ∧ d v = d t ∧ d t ′ displaystyle u,duwedge dv=dtwedge dt', . This allows the u integral to be evaluated explicitly: ∫ u , v u e − u ( k 2 + m 2 + v 2 p ⋅ k + v p 2 ) = ∫ 1 ( k 2 + m 2 + v 2 p ⋅ k − v p 2 ) 2 d v displaystyle int _ u,v ue^ -uleft(k^ 2 +m^ 2 +v2pcdot k+vp^ 2 right) =int frac 1 left(k^ 2 +m^ 2 +v2pcdot k-vp^ 2 right)^ 2 ,dv leaving only the v-integral. This method, invented by Schwinger but usually attributed to Feynman, is called combining denominator. Abstractly, it is the elementary identity: 1 A B = ∫ 0 1 1 ( v A + ( 1 − v ) B ) 2 d v displaystyle frac 1 AB =int _ 0 ^ 1 frac 1 big ( vA+(1-v)B big ) ^ 2 ,dv But this form does not provide the physical motivation for introducing v; v is the proportion of proper time on one of the legs of the loop. Once the denominators are combined, a shift in k to k′ = k + vp symmetrizes everything: ∫ 0 1 ∫ 1 ( k 2 + m 2 + 2 v p ⋅ k + v p 2 ) 2 d k d v = ∫ 0 1 ∫ 1 ( k ′ 2 + m 2 + v ( 1 − v ) p 2 ) 2 d k ′ d v displaystyle int _ 0 ^ 1 int frac 1 left(k^ 2 +m^ 2 +2vpcdot k+vp^ 2 right)^ 2 ,dk,dv=int _ 0 ^ 1 int frac 1 left(k'^ 2 +m^ 2 +v(1-v)p^ 2 right)^ 2 ,dk',dv This form shows that the moment that p2 is more negative than four 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: ∫ d k 1 k 2 + m 2 1 ( k + p 1 ) 2 + m 2 ⋯ 1 ( k + p n ) 2 + m 2 displaystyle int dk, frac 1 k^ 2 +m^ 2 frac 1 (k+p_ 1 )^ 2 +m^ 2 cdots frac 1 (k+p_ n )^ 2 +m^ 2 The general rule follows from the Schwinger prescription for n + 1 denominators: 1 D 0 D 1 ⋯ D n = ∫ 0 ∞ ⋯ ∫ 0 ∞ e − u 0 D 0 ⋯ − u n D n d u 0 ⋯ d u n . displaystyle frac 1 D_ 0 D_ 1 cdots D_ n =int _ 0 ^ infty cdots int _ 0 ^ infty e^ -u_ 0 D_ 0 cdots -u_ n D_ n ,du_ 0 cdots du_ n ,. The integral over the Schwinger parameters ui can be split up as before into an integral over the total proper time u = u0 + u1 … + un and an integral over the fraction of the proper time in all but the first segment of the loop vi = ui/u for i ∈ 1,2,…,n . The vi 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: d u = d u 0 + d u 1 ⋯ + d u n displaystyle du=du_ 0 +du_ 1 cdots +du_ n , d ( u v i ) = d u i . displaystyle d(uv_ i )=du_ i ,. Wedging all these equations together, one obtains u n d u ∧ d v 1 ∧ d v 2 ⋯ ∧ d v n = d u 0 ∧ d u 1 ⋯ ∧ d u n . displaystyle u^ n ,duwedge dv_ 1 wedge dv_ 2 cdots wedge dv_ n =du_ 0 wedge du_ 1 cdots wedge du_ n ,. This gives the integral: ∫ 0 ∞ ∫ s i m p l e x u n e − u ( v 0 D 0 + v 1 D 1 + v 2 D 2 ⋯ + v n D n ) d v 1 ⋯ d v n d u , displaystyle int _ 0 ^ infty int _ mathrm simplex u^ n e^ -uleft(v_ 0 D_ 0 +v_ 1 D_ 1 +v_ 2 D_ 2 cdots +v_ n D_ n right) ,dv_ 1 cdots dv_ n ,du,, where the simplex is the region defined by the conditions v i > 0 and ∑ i = 1 n v i < 1 displaystyle v_ i >0quad mbox and quad sum _ i=1 ^ n v_ i <1 as well as v 0 = 1 − ∑ i = 1 n v i . displaystyle v_ 0 =1-sum _ i=1 ^ n v_ i ,. Performing the u integral gives the general prescription for combining denominators: 1 D 0 ⋯ D n = n ! ∫ s i m p l e x 1 ( v 0 D 0 + v 1 D 1 ⋯ + v n D n ) n + 1 d v 1 d v 2 ⋯ d v n displaystyle frac 1 D_ 0 cdots D_ n =n!int _ mathrm simplex frac 1 left(v_ 0 D_ 0 +v_ 1 D_ 1 cdots +v_ n D_ n right)^ n+1 ,dv_ 1 ,dv_ 2 cdots dv_ n Since the numerator of the integrand is not involved, the same prescription works for any loop, no matter what the spins are carried by the legs. The interpretation of the parameters vi is that they are the fraction of the total proper time spent on each leg. Scattering[edit] 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, 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 that form an irreducible representation of the Poincaré group. Single particle states describe an object with a finite mass, a well defined momentum, and a spin. This definition is fine for protons and neutrons, electrons and photons, but it excludes quarks, which are permanently confined, so the modern point of view is more accommodating: a particle is anything whose interaction can be described in terms of Feynman diagrams, which have an interpretation as a sum over particle trajectories. A field operator can act to produce a one-particle state from the vacuum, which means that the field operator φ(x) produces a superposition of Wigner particle states. In the free field theory, the field produces one particle states only. But when there are interactions, the field operator can also produce 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 λφ4 interaction of the previous section, the order λ contribution to the (Lorentz) correlation function is: ⟨ ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) ⟩ = i k 1 2 i k 2 2 i k 3 2 i k 4 2 i λ displaystyle leftlangle phi (k_ 1 )phi (k_ 2 )phi (k_ 3 )phi (k_ 4 )rightrangle = frac i k_ 1 ^ 2 frac i k_ 2 ^ 2 frac i k_ 3 ^ 2 frac i k_ 4 ^ 2 ilambda , Stripping off the external propagators, that is, removing the factors of i/k2, gives the invariant scattering amplitude M: M = i λ displaystyle M=ilambda , which is a constant, independent of the incoming and outgoing momentum. The interpretation of the scattering amplitude is that the sum of M2 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: ∫ d k
k ⟩ ⟨ k
. displaystyle int dk,krangle langle k,. 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: E 2 − k 2 = m 2 . displaystyle E^ 2 -k^ 2 =m^ 2 ,. 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: ∫ δ ( E 2 − k 2 − m 2 )
E , k ⟩ ⟨ E , k
d E d k = ∫ d k 2 E
k ⟩ ⟨ k
. displaystyle int delta (E^ 2 -k^ 2 -m^ 2 )E,krangle langle E,k,dE,dk=int dk over 2E krangle langle k,. So the normalized k-states are different from the relativistically normalized k-states by a factor of E = ( k 2 − m 2 ) 1 4 . displaystyle sqrt E =left(k^ 2 -m^ 2 right)^ frac 1 4 ,. The invariant amplitude M is then the probability amplitude for
relativistically normalized incoming states to become relativistically
normalized outgoing states.
For nonrelativistic values of k, the relativistic normalization is the
same as the nonrelativistic normalization (up to a constant factor
√m). In this limit, the φ4 invariant scattering amplitude is still
constant. The particles created by the field φ scatter in all
directions with equal amplitude.
The nonrelativistic potential, which scatters in all directions with
an equal amplitude (in the Born approximation), is one whose Fourier
transform is constant—a 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[edit]
Thinking of Feynman diagrams as a perturbation series, nonperturbative
effects like tunneling do not show up, because any effect that goes to
zero faster than any polynomial does not affect the Taylor series.
Even bound states are absent, since at any finite order particles are
only exchanged a finite number of times, and to make a bound state,
the binding force must last forever.
But this point of view is misleading, because the diagrams not only
describe scattering, but they also are a representation of the
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.
Nonperturbative tunneling processes involve field configurations that
on average get big when the coupling constant gets small, but each
configuration is a coherent superposition of particles whose local
interactions are described by Feynman diagrams. When the coupling is
small, these become collective processes that involve large numbers of
particles, but where the interactions between each of the particles is
simple.[citation needed]
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
The use of the above diagram of the virtual particle producing a
quark–antiquark pair was featured in the television sit-com The Big
Bang Theory, in the episode "The Bat Jar Conjecture".
See also[edit] Julian Schwinger#Schwinger and Feynman Stueckelberg–Feynman interpretation Penguin diagram Path integral formulation Propagator List of Feynman diagrams Angular momentum diagrams (quantum mechanics) Notes[edit] ^ Kaiser, David (2005). "Physics and Feynman's Diagrams" (PDF).
American Scientist. 93: 156.
^ Feynman, Richard (1949). "The Theory of Positrons". Physical Review.
76 (76): 749. Bibcode:1949PhRv...76..749F. doi:10.1103/PhysRev.76.749.
In this solution, the "negative energy states" appear in a form which
may be pictured (as by Stückelberg) in space-time as waves traveling
away from the external potential backwards in time. Experimentally,
such a wave corresponds to a positron approaching the potential and
annihilating the electron.
^ R. Penco, D. Mauro (2006). "
References[edit] Gerardus 't Hooft, Martinus Veltman, Diagrammar, CERN Yellow Report
1973, online
David Kaiser, Drawing Theories Apart: The Dispersion of Feynman
Diagrams in Postwar Physics, Chicago: University of Chicago Press,
2005. ISBN 0-226-42266-6
Martinus Veltman, Diagrammatica: The Path to Feynman Diagrams,
Cambridge Lecture Notes in Physics, ISBN 0-521-45692-4 (expanded,
updated version of above)
Mark Srednicki, Quantum Field Theory, online Script (2006)
Schweber, S. S. (1994).
External links[edit] Wikimedia Commons has media related to Feynman diagrams. AMS article: "What's New in Mathematics: Finite-dimensional Feynman
Diagrams"
Draw Feynman diagrams explained by Flip Tanedo at Quantumdiaries.com
Drawing Feynman diagrams with FeynDiagram C++ library that produces
PostScript output.
Feynman Diagram Examples using Thorsten Ohl's Feynmf LaTeX package.
Online Diagram Tool A graphical application for creating publication
ready diagrams.
JaxoDraw A Java program for drawing Feynman diagrams.
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