Editing Feynman diagram (section)

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

:<math> \partial_\mu \partial^\mu \phi = 0\,</math>

and in an expectation value, this says:

:<math>\partial_\mu\partial^\mu \left\langle \phi(x) \phi(y)\right\rangle =0</math>

Where the derivatives act on {{mvar|x}}, and the identity is true everywhere except when {{mvar|x}} and {{mvar|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'' {{math|Δ}} as the Fourier transform of the time-ordered two-point function (the one that comes from the path-integral):

:<math> \partial^2 \Delta (x) = i\delta(x)\,</math>

So that:

:<math> \Delta(k) = \frac{i}{k^2}</math>

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 {{mvar|i}} disappears in the Euclidean theory.

==== Wick theorem ====
{{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'':

:<math> \left\langle \phi(k_1) \phi(k_2) \cdots \phi(k_n)\right\rangle</math>

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

:<math>\left\langle \phi(k_1) \cdots \phi(k_{2n})\right\rangle = \sum \prod_{i,j} \frac{\delta\left(k_i - k_j\right) }{k_i^2 } </math>

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

:<math> \left\langle \phi(k_1) \phi(k_2) \phi(k_3) \phi(k_4) \right\rangle = \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}</math>

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

==== Higher Gaussian moments — completing Wick's theorem ====
There is a subtle point left before Wick's theorem is proved—what if more than two of the <math>\phi</math>s have the same momentum? If it's an odd number, the integral is zero; negative values cancel with the positive values. But if the number is even, the integral is positive. The previous demonstration assumed that the <math>\phi</math>s would only match up in pairs.

But the theorem is correct even when arbitrarily many of the <math>\phi</math> are equal, and this is a notable property of Gaussian integration:
:<math> I = \int e^{-ax^2/2}dx = \sqrt\frac{2\pi}{a} </math>
:<math> \frac{\partial^n}{\partial a^n } I = \int \frac{x^{2n}}{2^n} e^{-ax^2/2}dx = \frac{1\cdot 3 \cdot 5 \ldots \cdot (2n-1) }{ 2 \cdot 2 \cdot 2 \ldots \;\;\;\;\;\cdot 2\;\;\;\;\;\;} \sqrt{2\pi}\, a^{-\frac{2n+1}{2}}</math>

Dividing by {{mvar|I}},

:<math> \left\langle x^{2n}\right\rangle=\frac{\displaystyle\int x^{2n} e^{-a x^2/2} }{\displaystyle \int e^{-a x^2/2} } = 1 \cdot 3 \cdot 5 \ldots \cdot (2n-1) \frac{1}{a^n} </math>
:<math> \left\langle x^2 \right\rangle = \frac{1}{a} </math>

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

:<math> \left\langle x_1 x_2 x_3 \cdots x_{2n} \right\rangle</math>

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

:<math> \left\langle x^{2n} \right\rangle = (2n-1)\cdot(2n-3)\ldots \cdot5 \cdot 3 \cdot 1 \left\langle x^2\right\rangle^n </math>

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

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

:<math> S = \int \partial^\mu \phi \partial_\mu\phi +\frac {\lambda}{ 4!} \phi^4. </math>

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

:<math> S = \int_k k^2 \left|\phi(k)\right|^2 + \frac{\lambda}{4!}\int_{k_1k_2k_3k_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. </math>

Where {{mvar|S<sub>F</sub>}} is the free action, whose correlation functions are given by Wick's theorem. The exponential of {{mvar|S}} in the path integral can be expanded in powers of {{mvar|λ}}, giving a series of corrections to the free action.

:<math> e^{-S} = e^{-S_F} \left( 1 + X + \frac{1}{2!} X X + \frac{1}{3!} X X X + \cdots \right) </math>

The path integral for the interacting action is then a [[power series]] of corrections to the free action. The term represented by {{mvar|X}} should be thought of as four half-lines, one for each factor of {{math|''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 {{mvar|X}} terms now. For example, the path-integral for the four-field correlator:

:<math>\left\langle \phi(k_1) \phi(k_2) \phi(k_3) \phi(k_4) \right\rangle = \frac{\displaystyle\int e^{-S} \phi(k_1)\phi(k_2)\phi(k_3)\phi(k_4) D\phi }{ Z}</math>

which in the free field was only nonzero when the momenta {{mvar|k}} were equal in pairs, is now nonzero for all values of {{mvar|k}}. The momenta of the insertions {{math|''φ''(''k<sub>i</sub>'')}} can now match up with the momenta of the {{mvar|X}}s in the expansion. The insertions should also be thought of as half-lines, four in this case, which carry a momentum {{mvar|k}}, but one that is not integrated.

The lowest-order contribution comes from the first nontrivial term {{math|''e''<sup>−''S<sub>F</sub>''</sup>''X''}} in the Taylor expansion of the action. Wick's theorem requires that the momenta in the {{mvar|X}} half-lines, the {{math|''φ''(''k'')}} factors in {{math|X}}, should match up with the momenta of the external half-lines in pairs. The new contribution is equal to:

:<math> \lambda \frac{1}{ k_1^2} \frac{1}{ k_2^2} \frac{1}{ k_3^2} \frac{1}{ k_4^2}\,. </math>

The 4! inside {{math|X}} is canceled because there are exactly 4! ways to match the half-lines in {{mvar|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 {{math|''k''<sub>1,2,3,4</sub>}}, by Wick's theorem.

==== Feynman diagrams ====
The expansion of the action in powers of {{mvar|X}} gives a series of terms with progressively higher number of {{mvar|X}}s. The contribution from the term with exactly {{mvar|n}} {{mvar|X}}s is called {{mvar|n}}th order.

The {{mvar|n}}th order terms has:
# {{math|4''n''}} internal half-lines, which are the factors of {{math|''φ''(''k'')}} from the {{mvar|X}}s. These all end on a vertex, and are integrated over all possible {{mvar|k}}.
# external half-lines, which are the come from the {{math|''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

:<math> \frac{\delta(k_1 + k_2)}{k_1^2} </math>

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 {{mvar|k}}. The half-line at the tail end of the arrow carries momentum {{mvar|k}}, while the half-line at the head-end carries momentum {{mvar|−''k''}}. If one of the two half-lines is external, this kills the integral over the internal {{mvar|k}}, since it forces the internal {{mvar|k}} to be equal to the external {{mvar|k}}. If both are internal, the integral over {{mvar|k}} remains.

The diagrams that are formed by linking the half-lines in the {{mvar|X}}s with the external half-lines, representing insertions, are the Feynman diagrams of this theory. Each line carries a factor of {{math|{{sfrac|1|''k''<sup>2</sup>}}}}, 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 {{mvar|k}} is equal to the total outgoing {{mvar|k}}.

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

==== Loop order ====
A forest diagram is one where all the internal lines have momentum 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 {{mvar|X}}. Another is when three external lines end on an {{mvar|X}}, and the remaining half-line joins up with another {{mvar|X}}, and the remaining half-lines of this {{mvar|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 {{mvar|X}}s.

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

A diagram that is not a forest diagram is called a ''loop'' diagram, and an example is one where two lines of an {{mvar|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 {{mvar|X}}s are joined to each other by matching the legs one to the other. This diagram has no external lines at all.

The reason loop diagrams are called loop diagrams is because the number of {{mvar|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 {{math|'''R'''<sup>''d''</sup>}} 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 {{mvar|k}}-valued weighted graph is zero.

A set of valid {{mvar|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 {{mvar|k}}-labellings of a graph that conserve momentum (i.e. which has zero boundary) up to redefinitions of {{mvar|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 ====
The number of ways to form a given Feynman diagram by joining half-lines is large, and by Wick's theorem, each way of pairing up the half-lines contributes equally. Often, this completely cancels the factorials in the denominator of each term, but the cancellation is sometimes incomplete.

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

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

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

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

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

An [[automorphism]] of a Feynman graph is a permutation {{mvar|M}} of the lines and a permutation {{mvar|N}} of the vertices with the following properties:

# If a line {{mvar|l}} goes from vertex {{mvar|v}} to vertex {{mvar|v′}}, then {{math|''M''(''l'')}} goes from {{math|''N''(''v'')}} to {{math|''N''(''v′'')}}. If the line is undirected, as it is for a real scalar field, then {{math|''M''(''l'')}} can go from {{math|''N''(''v′'')}} to {{math|''N''(''v'')}} too.
# If a line {{mvar|l}} ends on an external line, {{math|''M''(''l'')}} ends on the same external line.
# If there are different types of lines, {{math|''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 {{mvar|X}}s gives a different pattern of linking names to half-lines, and this is a factor of {{math|''n''!}}. Each permutation of the half-lines in a single {{mvar|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.

{{anchor|Linked-cluster theorem}}

==== Connected diagrams: ''linked-cluster theorem'' ====
Roughly speaking, a Feynman diagram is called ''connected'' if all vertices and propagator lines are linked by a sequence of vertices and propagators of the diagram itself. If one views it as an [[Graph (discrete mathematics)|undirected graph]] it is connected. The remarkable relevance of such diagrams in QFTs is due to the fact that they are sufficient to determine the [[Partition function (quantum field theory)|quantum partition function]] {{math|''Z''[''J'']}}. More precisely, connected Feynman diagrams determine

:<math>i W[J]\equiv \ln Z[J].</math>

To see this, one should recall that

:<math> Z[J]\propto\sum_k{D_k}</math>

with {{mvar|D<sub>k</sub>}} constructed from some (arbitrary) Feynman diagram that can be thought to consist of several connected components {{mvar|C<sub>i</sub>}}. If one encounters {{mvar|n<sub>i</sub>}} (identical) copies of a component {{mvar|C<sub>i</sub>}} within the Feynman diagram {{mvar|D<sub>k</sub>}} one has to include a ''symmetry factor'' {{mvar|''n<sub>i</sub>''!}}. However, in the end each contribution of a Feynman diagram {{mvar|D<sub>k</sub>}} to the partition function has the generic form

:<math>\prod_i \frac{C_{i}^{n_i} }{ n_i!} </math>

where {{mvar|i}} labels the (infinitely) many connected Feynman diagrams possible.

A scheme to successively create such contributions from the {{mvar|D<sub>k</sub>}} to {{math|''Z''[''J'']}} is obtained by

:<math>\left(\frac{1}{0!}+\frac{C_1}{1!}+\frac{C^2_1}{2!}+\cdots\right)\left(1+C_2+\frac{1}{2}C^2_2+\cdots\right)\cdots </math>

and therefore yields

:<math>Z[J]\propto\prod_i{\sum^\infty_{n_i=0}{\frac{C_i^{n_i}}{n_i!}}}=\exp{\sum_i{C_i}}\propto \exp{W[J]}\,.</math>

To establish the ''normalization'' {{math|''Z''<sub>0</sub> {{=}} exp ''W''[0] {{=}} 1}} one simply calculates all connected ''vacuum diagrams'', i.e., the diagrams without any ''sources'' {{mvar|J}} (sometimes referred to as ''external legs'' of a Feynman diagram).

The linked-cluster theorem was first proved to order four by [[Keith Brueckner]] in 1955, and for infinite orders by [[Jeffrey Goldstone]] in 1957.<ref>{{Cite book |last1=Fetter |first1=Alexander L. |url=https://books.google.com/books?id=0wekf1s83b0C |title=Quantum Theory of Many-particle Systems |last2=Walecka |first2=John Dirk |date=2003-06-20 |publisher=Courier Corporation |isbn=978-0-486-42827-7 |language=en}}</ref>

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

:<math> \left\langle \phi_1(x_1) \cdots \phi_n(x_n)\right\rangle = \frac{\displaystyle\int e^{-S} \phi_1(x_1) \cdots\phi_n(x_n)\, D\phi }{\displaystyle \int e^{-S}\, D\phi}\,.</math>

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:

:<math> \int e^{-S}\phi_1(x_1)\cdots\phi_n(x_n)\, D\phi = \left(\sum E_i\right)\left( \exp\left(\sum_i C_i\right) \right)\,.</math>

Where the sum over {{mvar|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 {{mvar|Z}} itself, which from the definition of the path integral is equal to:

:<math> Z= \int e^{-S} D\phi = e^{-HT} = e^{-\rho V} </math>

where {{mvar|ρ}} is the energy density in the vacuum. Each vacuum bubble contains a factor of {{math|''δ''(''k'')}} zeroing the total {{mvar|k}} at each vertex, and when there are no external lines, this contains a factor of {{math|''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 ====
Correlation functions are the sum of the connected Feynman diagrams, but the formalism treats the connected and disconnected diagrams differently. Internal lines end on vertices, while external lines go off to insertions. Introducing ''sources'' unifies the formalism, by making new vertices where one line can end.

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

:<math> \int h(x) \phi(x)\, d^dx = \int h(k) \phi(k)\, d^dk \,</math>

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

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

:<math> \log\big(Z[h]\big) = \sum_{n,C} h(k_1) h(k_2) \cdots h(k_n) C(k_1,\cdots,k_n)\,</math>

where {{math|''C''(''k''<sub>1</sub>,...,''k<sub>n</sub>'')}} is the connected diagram with {{mvar|n}} external lines carrying momentum as indicated. The sum is over all connected diagrams, as before.

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

:<math> Z[h] = \int e^{iS + i\int h\phi}\, D\phi \,</math>

and it is a function of the values of {{mvar|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 {{math|'''R'''<sup>''n''</sup>}}, the Fourier transform of the probability density is:

:<math> \int \rho(y) e^{i k y}\, d^n y = \left\langle e^{i k y} \right\rangle = \left\langle \prod_{i=1}^{n} e^{ih_i y_i}\right\rangle \,</math>

The Fourier transform is the expectation of an oscillatory exponential. The path integral in the presence of a source {{mvar|''h''(''x'')}} is:

:<math> Z[h] = \int e^{iS} e^{i\int_x h(x)\phi(x)}\, D\phi = \left\langle e^{i h \phi }\right\rangle</math>

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

:<math> \left\langle \prod_x e^{i h_x \phi_x}\right\rangle </math>

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

:<math> \delta(x-y) = \int e^{ik(x-y)}\, dk </math>

This tells you what a field delta function looks like in a path-integral. For two scalar fields {{mvar|φ}} and {{mvar|η}},

:<math> \delta(\phi - \eta) = \int e^{ i h(x)\big(\phi(x) -\eta(x)\big)\,d^dx}\, Dh\,, </math>

which integrates over the Fourier transform coordinate, over {{mvar|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 {{mvar|h}}, and the physical partition function is the value when {{mvar|h}} is the zero function:

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

:<math> \left\langle\phi(x)\right\rangle = \frac{1}{Z} \frac{\partial}{\partial h(x)} Z[h] = \frac{\partial}{\partial h(x)} \log\big(Z[h]\big)\,.</math>

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