diagrams close onto themselves, no propagator lines end up on the external states,

and they appear as unlinked diagrams. According to the multiplication rule, the

two amplitudes represented by the two sub-parts of the total diagram are multiplied

together to get the total amplitude represented by the diagram. The ¬rst and second

diagrams on the right in Figure 9.4 are of the connected and disconnected type,

respectively. These diagrams, according to the general rule of diagram construction,

can also be accompanied by any vacuum ¬‚uctuations constructable. The symbol + · · ·

in the ¬gure summarizes envisioning all diagrams constructable with the vertices and

propagators de¬ning the theory. The total class of diagrams is thus an in¬nite myriad

with in¬nite repetitions.

The totality of all diagrams can thus (with the help of our most developed sense)

feature that the factorial provided by the expansion of the exponential function is canceled by this

redundancy.

9.1. Diagrammatics 261

be envisioned perturbatively. However, this is of little use unless only trivial lowest

order perturbation theory needs to be invoked. One approach to a more powerful

diagrammatic representation is by using topological arguments to partially re-sum

the diagrammatic perturbation expansion in terms of e¬ective vertices and the full 2-

state propagators, i.e. in terms of so-called skeleton diagrams.12 In the next section,

we shall ¬rst pursue the hierarchal option on our way to this goal, expressing any

N -state amplitude in terms of amplitudes with di¬erent numbers of external states.

Before embarking on deriving the fundamental diagrammatic equation, we intro-

duce the inverse propagator. The inverse of the free or bare propagator is speci¬ed

by the (partial di¬erential) equation satis¬ed by the free propagator

(G’1 )1¯ G(0) = δ12 (9.5)

0 1 ¯

12

or since the propagator is symmetric in its labels

(G’1 )1¯ G(0) = δ12 = G1¯ (G’1 )¯ .

(0)

(9.6)

0 0

1

¯

1 12

12

We have written the equation satis¬ed by the free propagator in matrix notation,

in terms of an integral operator as summation over repeated indices is implied.13

For later use we introduce diagrammatic notation for the inverse free propagators as

depicted in Figure 9.6.

1

1

(G’1 )11 =

0

Figure 9.6 Diagrammatic notation for the inverse free propagator.

Using the basic diagrammatic rule: to interact or not, we shall start obtaining dia-

grammatic identities relating amplitudes, and eventually express these diagrammatic

relations in terms of di¬erential equations.

9.1.3 Fundamental dynamic relation

To get started on a systematic categorization of the plethora of diagrams, let us ¬rst

consider the case where one particle is not allowed to interact and let us separate out

its state to appear on the left in the diagram specifying the amplitude in question as

depicted in Figure 9.7. Since not interacting is an option even for a particle capable of

interacting, this seemingly irrelevant case of a completely non-interacting particle is a

¬rst step in the general deconstruction of an N -state amplitude into amplitudes with

less external states, and allows furthermore a comment on the quantum statistics of

identical particles.

12 This was performed in Section 4.5.2, starting with the canonical formalism.

13 The inverse free contour-ordered Green™s function encountered in Section 4.4.1, or the inverse

free matrix Green™s function of Section 5.2.1, stipulating the additional matrix structure in the

dynamical indices, had integral kernels typically consisting of di¬erential operators operating on the

delta function.

262 9. Diagrammatics and generating functionals

2

1

N

Figure 9.7 General N -state diagram.

Since the particle in state 1 is assumed not to interact, its only option is to

propagate directly to a ¬nal state, and the amplitude A1234...N can in this case be

expressed in terms of the amplitude which has two external states less according to

the basic rule: everything can happen on the way between the (N ’ 2) other ¬nal

states, and the diagrammatic equation displayed in Figure 9.8 is obtained.

2 2

1 2

1

3

=

+···+

N ’1

N N

1 N

Figure 9.8 Diagrams for the non-interacting particle labeled by 1.

The N -state amplitude is in this case represented by the amplitudes speci¬ed by

(N ’ 2) external states, i.e. A23...N without the index M labeling the state where the

propagator starting in state 1 ends up. If sM = s1 , the process is not allowed since

a non-interacting particle can not propagate to a di¬erent species state, and this

feature is faithfully respected by the diagrammatics, since then the corresponding

(0)

propagator according to Eq. (9.3) vanishes, G1M = 0, and the contribution from the

corresponding diagram vanishes since by the multiplication rule the bare propagator

amplitude multiplies the adjacent (N ’ 2)-state amplitude.

The quantum statistics of identical particles introduces minus signs when two

identical fermions interchange states and the amplitudes are symmetric upon inter-

change of bosons, say

A213...N = ± A123...N (9.7)

where the upper (lower) sign is for bosons (fermions), respectively.

For the case of non-interacting identical particles, only free propagation and e¬ects

of the quantum statistics of the particles are involved as displayed in Figure 9.9.

9.1. Diagrammatics 263

3

2

3 3

2 2

3

2

± +

=

1 1

4 4

1 4

1 4

Figure 9.9 The 4-state diagrams for two non-interacting identical particles.

In the following we consider ¬rst bosons, in which case the amplitude functions

are symmetric upon interchange of pairs of external state labels. The features of

antisymmetry for fermions are then added.14 The symmetry property of amplitudes

forces the vertices to be symmetric in their indices, e.g. for the 3-vertex g213 = g123 ,

etc.

Returning to the diagram for the general N -state amplitude and respecting the

other option for the particle in state 1, to interact, gives the additional ¬rst two

diagrams as depicted on the right in Figure 9.10 for the case of a theory with three-

and four-line connector vertices. The equation relating amplitudes as depicted in

Figure 9.10 is the fundamental dynamic equation of motion in the diagrammatic

language (for the case of three- and four-line vertices but trivially generalized).

2 2 2

1 1

1 1 1

= + 3!

2!

N N N

2

1 2

3

+···+

+

N ’1

N

1 N

Figure 9.10 Fundamental dynamic equation for three- and four-vertex interactions.

The option of interaction through the 3-state vertex is for the N -external-state

(N +1)

amplitude expressed in terms of the amplitude A¯¯ with (N + 1) external

233...N +1

states, where two internal propagators are contracted at the vertex. This leads to

the ¬rst diagram on the right in Figure 9.10, representing according to the Feynman

14 In diagrammatics the essential is the topology of a diagram, and the interpretation of diagrams

for the case of fermions is by the end of the day the same as for bosons except for the rule that a

relative minus sign must be assigned to a diagram for each closed loop of fermion propagators.

264 9. Diagrammatics and generating functionals

rules the amplitude as speci¬ed in Figure 9.11.

2

1

(0)

= G1¯ g¯¯¯ A¯¯

1 123 232...N

N

Figure 9.11 Diagram and corresponding analytical expression.

Repeated state labels are summed over in accordance with the superposition

principle. Similarly for diagrams with higher-order vertices in Figure 9.10 displayed

for a theory with an additional 4-attachment vertex.

Although combinatorial prefactors are an abomination in diagrammatics we have

in accordance with custom introduced them in Figure 9.10 by hand, the convention

being: an N -line vertex carries an explicit prefactor 1/(N ’ 1)!, the reason being to

be relieved at a di¬erent junction as immediately to be revealed. Consider a theory

with only a 3-attachment vertex, and follow the further adventures of one of the

particles emanating from the interaction vertex according to its two options, interact

or not, as depicted in Figure 9.12.

2

2

2 2

1

1 1

= + +

1

3 3

3

3

2

1

1

+2

3

2

1

+ disconnected diagram

= 2—

+ higher-order contributions.

3

Figure 9.12 Further adventures of a particle line emanating at a vertex.

The upper row of diagrams on the right in Figure 9.12 corresponds to the option

of not interacting. In lowest order in the interaction, the second and third diagrams

on the right give the same contribution. The inserted combinatorial factor in Figure

9.10 is thus the device to make the bare vertex diagram (here a 3-vertex) appear

9.1. Diagrammatics 265

with no combinatorial factor. In a theory with only a 3-attachment vertex, the

inserted combinatorial factor appearing with the vertex in Figure 9.10, thus makes

the diagrammatic expansion of the 3-state amplitude start out with the lowest-order

connected diagram, the bare vertex 3-state amplitude, carrying no additional factor

as depicted in Figure 9.13.

2 2

1

1

···

=

+

3 3

Figure 9.13 Lowest-order connected 3-state diagram for a 3-vertex theory.

A similar function has the combinatorial factor inserted in front of the 4-vertex

diagram in Figure 9.10.

9.1.4 Low order diagrams

Let us now familiarize ourselves with the Feynman rules and derive the expressions of

lowest-order diagrammatic perturbation theory. The reader not interested in entering

into this in¬nite forest of diagrams can skip the next few pages and go straight to

the next sections where more powerful methods are developed. These will allow us

systematically to generate the jungle of diagrams. However, for the adventurous

reader let us see what kind of diagrams will emerge when we apply the simple law

of dynamics, to interact or not to interact! A lesson to be learned from this is

that although the basic rule is as simple as it possibly can be, in this brute force

generation of diagrams one can easily miss a diagram, something history has proved

over and again. The functional methods we shall consider shortly are able to capture

the complete diagammatics in a simple way and in this way are able to help us in

ensuring against mistakes.

We can now in any diagram follow the further possible options of any particle

line emanating at a vertex, interact or not, and in this way unfold order by order

the in¬nite total canopy in the jungle of diagrams constituting perturbation theory.

For example, consider the 2-state amplitude (or two-point or 2-state propagator

or Green™s function) and a theory with the option of interaction only through the

3-attachment vertex. The two options for dynamics then generate the diagrams

depicted in Figure 9.14.

1

+

= 2

Figure 9.14 Interaction or not option for the 2-state amplitude.

266 9. Diagrammatics and generating functionals

A new diagrammatic entity enters in the ¬rst diagram on the right in Figure

9.14, the sum of all vacuum diagrams. The ¬rst diagram on the right in Figure

9.14 represents the product of two quantities, the bare 2-state amplitude, the bare

propagator, times the amplitude resulting from the sum of all vacuum diagrams:

free propagation accompanied by vacuum ¬‚uctuations, and nothing further is to be

revealed diagrammatically in this part. The second diagram on the right corresponds

to the option of interaction (in QED it could represent photon absorption or emission

by electrons and positrons or pair creation). We note the general structure emerging

in this way for the 2-state amplitude: the appearance of the bare 2-state amplitude

and the appearance of a higher-order amplitude, here the 3-state amplitude.

Next we concentrate on the second diagram on the right in Figure 9.14, and

explore the options, interact or not, of one of the lines emanating from the vertex

and obtain the diagrams depicted in Figure 9.15.

1

= + 2

1

+ 2

Figure 9.15 Diagrams generated by particle emanating at the vertex.

The ¬rst two diagrams on the right in Figure 9.15 correspond to the option of not

interacting, viz. either propagating freely back to the vertex or freely to the external

state. The last diagram encompasses the option of interacting, exposing one more

vertex in our 3-state vertex theory.

The 1-state amplitude appearing in the ¬rst and second diagram on the right

in Figure 9.15 (as a disconnected and connected piece, respectively), the tadpole

diagram, can in a 3-vertex theory be expressed in terms of the 2-state amplitude

contracted at the vertex as depicted in Figure 9.16, since the only option for the

line is to interact (the option of not interacting was already exhausted in the ¬rst

diagram in Figure 9.14).

1

=

2

Figure 9.16 Tadpole or 1-state amplitude in a 3-vertex theory is expressable in

terms of the vertex and the 2-state amplitude contracted at the vertex.

Inserting into the second diagram on the right in Figure 9.16 the expression for

9.1. Diagrammatics 267

the 2-state amplitude speci¬ed by the expression in Figure 9.14 gives in a three-line-

vertex theory the diagrammatic equation for the tadpole depicted in Figure 9.17.

1 1

+

=

2 4

Figure 9.17 Tadpole equation for a three-line-vertex theory.

The 1-state diagram, the tadpole, has thus been expressed in terms of the bare

tadpole times the amplitude representing the sum of all the vacuum diagrams plus

a higher correlation amplitude, here the 3-state amplitude contracted at vertices

according to the second diagram on the right in Figure 9.17.

Exercise 9.1. Obtain the diagrammatic equation for the tadpole if a 4-line vertex

is also included in the theory.

Let us now further expose interactions in the 2-state amplitude in Figure 9.14.

Insert the diagrammatic expansion of the tadpole in Figure 9.17 into the ¬rst diagram

on the right in Figure 9.15, and then substitute the resulting expression for the second

diagram on the right in Figure 9.14, and further explore the options for particle lines

emanating from vertices, interaction or not. This gives the diagrammatic expansion

of the 2-state amplitude depicted in Figure 9.18.