As we have seen in the previous chapter
Dϕ(x) where relevant includes ground-state factors
Z( J , gn)≡<0|U(T)|0>=
Dϕ(x) exp
i
d4xL(ϕ)
(8.1)
will play the role of a generating functional for calculating expectation values of products of field operators, which will now be studied in more detail. In general the Lagrange density for a scalar field theory is given by
L(ϕ)=L2(ϕ)−V(ϕ)−J (x)ϕ(x), (8.2)
whereL2(ϕ) is quadratic in the fields, hence for a scalar field L2(ϕ)= 12
∂µϕ(x)∂µϕ(x)−m2ϕ2(x) ,
(8.3) V(ϕ)= g3
3!ϕ3(x)+g4
4!ϕ4(x)+ · · · .
As mentioned before, it is customary to not include the mass term in the poten-tial V, such that V describes the interactions. We can add the interaction as an operator, when evaluating the path integral for the quadratic approximation L(ϕ)=L2(ϕ)−ϕ(x) J (x),
Z( J , gn) =
Dϕ(x) exp
i
d4x {L2(ϕ)−ϕ(x) J (x)}
exp
−i
d4x V(ϕ)
=<0|U2(T)Texp
−i
d4x V( ˆϕ)
|0>.
(8.4) 51
52 A Course in Field Theory We can now use the fact that
to find a somewhat formal, but in an expansion with respect to the coupling constants gn, well-defined expression for the fully interacting path integral
Z( J , gn)=exp
We have assumed the vacuum energy to be normalised to zero, in absence of interactions, such that Z( J =gn=0) =1. Equivalently, Z( J , gn) is synony-mous with Z( J , gn)/Z( J =gn=0). We now define GJ as
GJ ≡log Z( J , gn), (8.7)
where the dependence on the coupling constants in GJ is implicit. We will show that GJ can be seen as the sum of all connected diagrams. A diagram is connected if it cannot be decomposed in the product of two diagrams that are not connected. Note that at J =0, i GJ/T equals the energy of the ground state as a function of the coupling constants, normalised so as to vanish at zero couplings.
The different diagrams arise from the expansion of exp
will represent an-point vertex, with coordinate x, which is to be integrated over. As we saw in the derivation of the classical equations of motion, the integral over x in the Fourier representation gives rise to conservation of momentum at the vertex. Using
ϕ(x)= 1
Perturbative Expansion in Field Theory 53
we have for each vertex in the Fourier representation
−ig Note that factors of 2πare dropping out in the identities
exp In the quantum theory we have to keep track of the factors i. Compared to the Feynman rules of Chapter 4, the propagator will come with an extra factor
−i. A vertex will now carry a factor ig
(2π)2/i2−
(in a finite volume this be-comes ig√
2πV/i)2−
); see Table 8.1. To compute the vacuum energy, there is an overall factor i since E0T=i GJ=0=i log Z( J =0, gn). The same factor of i applies for using the tree-level diagrams to solve the classical equations of motion. It is easy to see that these Feynman rules give identical results for these tree-level diagrams, as compared to the Feynman rules introduced in Chapter 4. The factors of i exactly cancel each other.
We note that the propagator connected to a source comes down whenever a derivative in the source acts on Z2( J ); see Equation (8.7). When this derivative acts on terms that have already come down from previous derivatives, one of the sources connected to a propagator is removed and this connects that propagator to the vertex associated toδ/δJ (x). As any derivative is connected to a vertex, the propagator either runs between a vertex and a source HHq ×, between two vertices HHq qHH , or it connects two legs of the same vertex
j
q . The possibility of closed loops did not occur in solving the classical equations of motion and is specific to the quantum theory.
54 A Course in Field Theory To prove that GJ only contains connected diagrams we write
GJ =log quan-tum mechanics, however generalised to infinite dimensions), also X and Y are elements from the algebra, and we can express GJ in a sum of multiple com-mutators using the Campbell–Baker–Hausdorff formula [see Equation (6.44)].
GJ =
Due to the multiple commutators, all components that do not commute are connected. However, if the components would commute they would not con-tribute to the commutators. This is even true if we do not put the derivatives with respect to the source to zero, once they have been moved to the right (this is why we consider the action on the identity).
The only thing that remains to be discussed is with which combinatorial factor each diagram should contribute. This is, as in Chapter 4, with the inverse of the order of the permutation group that leaves the topology of the diagram unchanged. These combinatorial factors are clearly independent of the space-time integrations and possible contractions of vector or other indices. We can check them by reducing the path integral to zero dimensions, orϕ(x) → ϕ andDϕ(x)→dϕ. In other words, we replace the path integral by an ordinary integral. As an example consider
Z( J , g)=C
Perturbative Expansion in Field Theory 55 The constant C is simply to normalise Z( J =gn=0) =1. Expanding the exponents we get in lowest nontrivial order
Z( J =0) =1− g2
In the last term, the numerical factors in front of the diagrams indicate the combinatorial factors (for the first diagram a factor 2 from interchanging the two vertices and a factor 3! from interchanging the three propagators; for the second diagram the latter factor is replaced by 4 as we can only interchange for each vertex the two legs that do not interconnect the two vertices). The Feynman rules for this simple case are that each vertex gets a factor−g (in zero dimensions there are no factors 2π) and each propagator gets a factor
−i/M. In Problem 13 the exponentiation is checked for Z( J ) to O(g2) and O( J5) (giving the simplest nontrivial check).
We will now show how the number of loops in a diagram is related to the expansion in ¯h. We can expect such a relation, as we have shown at the end of Chapter 6 that the ¯h→0 limit is related to the classical equations of motion, whereas we have shown in Chapter 4 that these classical equations are solved by tree diagrams. If we call L the number of loops of a diagram, we will show that where GL , J is the sum of all connected diagrams with exactly L loops. This means that a loop expansion is equivalent with an expansion in ¯h. To prove this we first note that due to reinstating ¯h the source term will get an extra factor 1/¯h, the propagator a factor ¯h, and the coupling constants gnare replaced by gn/¯h. A diagram with Vnn-point vertices, E external lines (connected to a source), and P propagators has therefore an extra overall factor of
¯h−E¯hP
n=3
¯h−Vn. (8.18)
56 A Course in Field Theory We can relate this to the number of loops by noting that the number of mo-mentum integrations (i.e., the number of independent momenta) in a diagram equals the number of loops plus the number of external lines, minus one for the overall conservation of energy and momentum, i.e., L+E −1. On the other hand, the number of momentum integrations is also the number of propagators minus the number of delta functions coming from the vertices, i.e., P−
n=3Vn. Hence
L =1+P−E−
n=3
Vn, (8.19)
which implies that the total number of ¯h factors in a diagram is given by L−1.
In the next chapters we will often consider so-called amputated diagrams, where the external propagators connected to a source are taken off from the expressions for the diagram. If we do not count these external propagators, Equation (8.19) has to be replaced by L =1+P−
n=3Vn, as there are exactly E such external propagators.
DOI: 10.1201/b15364-9