PERTURBATION THEORY in .NET

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CHAPTER
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PERTURBATION THEORY
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We develop the relativistic perturbation theory and present the technique of Feynman diagrams. Some attention is paid to the case where the interaction contains derivatives. We identify the expansion as a series in Planck's constant and study some elementary topological properties. The parametric representation of Feynman integrals is introduced and used to define the euclidean continuation. Analytic properties and discontinuity formulas are indicated.
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6-1 INTERACTION REPRESENTATION AND FEYNMAN RULES
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The basic problem is the calculation of Green functions, i.e., vacuum expectation values of time-ordered products of interacting fields
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(6-1)
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For brevity's sake we shall consider a generic scalar field (or collection of fields) <p, and shall focus our attention on quantum electrodynamics in the next section. It is convenient to collect all the Green functions in a generating functional
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Z(j) =
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in! fd4Xl ... d4xnj(Xl) j(Xn)G(Xlo . .. , Xn) d4x j(X) <p(X)
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<01 T exp [i f
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JiO)
(6-2)
QUANTUM FIELD THEORY
The S matrix may be expressed in the compact form we saw in Eq. (5-38): S = : exp
(f d y {CPin(Y{(D
+ m 2) Z!/2
t5j~y)J}): Z(j)li=O
(6-3)
This relation is itself a generating functional for reduction formulas. The field cp(x) satisfies dynamical equations derived from a lagrangian
(6-4)
where fi'o is a quadratic free lagrangian and fi'int is an interaction lagrangian. For simplicity 2"int is assumed at first to contain no field derivative. Since we are generally unable to find an exact expression of Green functions, we rely on a perturbative method, with fi'int considered as a small perturbation to fi'o. Generally fi'int depends on one or several coupling constants and the perturbative expansion will be a power series in this or these coupling constants. Even when the coupling constant is small, as in electrodynamics where a = e 2 /4nhc = 1/137, we do not expect it to converge. As we shall see later, there are some indications that it is, rather, a divergent asymptotic series. In any case, the perturbative expansion may be considered as a formal mathematical series, from which much information may be extracted. When the theory contains several coupling constants, the natural "small" parameter may be identified as h. The corresponding series has also a topological characterization. It is a loopwise expansion, according to the increasing number of independent loops of the associated diagrams. Therefore, the expansion may simultaneously be regarded as a small coupling series, a semiclassical expansion about the free-field configuration, or a topologically well-defined procedure. As in the previous chapters, we proceed to a heuristic construction and neglect at first various difficulties such as the infinite volume limit, the possible occurrence of ultraviolet divergences, or the proper definition of the lagrangian. These points will be reconsidered in the following.
6-1-1 Self-Interacting Scalar Field
As explained above, we first confine ourselves to a single scalar field cp. We have shown in Chap. 4 that there exists formally a unitary operator U(t) which transforms the free field CPin into the interacting one cp(t): with t = where Since the coupling terms contain no field derivative,
Hint(t)
= X O
(6-5)
(6-6)
Lo=t d x fi'int[CPin(X)]
(6-7)
More generally, let U(t2' t1) stand for
PERTURBATION THEORY
U(t2' t1)
= Texp
[i 1: dt' fd x .Pint(X, t') ]
U(t,t) = I
(6-8)
where .Pint is expressed as before in terms of </Jin' This operator enjoys the following properties:
U(t, - 00) = U(t)
We shall now derive the fundamental relation
<01 Texp
d xj(x)</J(x) ]10) <01 Texp
d x {.Pint [</Jin(X)]
+ j(X)</Jin(X)} ]
(6-9)
<01 Texp
{i J x .Pint [</Jin(X)] } 10) d
or, equivalently, after identification of the term of degree n in j:
<01 T</Jin(X1)'"
G(Xr, ... ,Xn)
</Jin(Xn) exp
<OITexp i
0 {i fd x .Pint [</Jin(X)] } 1)
d x.Pint [</Jin(X)]
(6-10)
In the right-hand side of Eqs. (6-9) and (6-10), the T symbol acts on the whole expression. This means that after an expansion of the exponential in powers of .Pint all the fields </Jin have to be time ordered. The vacuum state 10) is to be understood as the "in" vacuum state 10 in). We assume that the interaction is adiabatically switched off in the remote past; therefore limt~ - 00 U (t) = I. Let us consider a set of time-ordered points Xr, ... , X n , that is, satisfying x > xg > ... > x~. We then have that
0 <01 T</J(xr) </J(xn) 1) =
<01 </J(X1)'" </J(xn) 10) <01 U- 1(t1)</Jin(X1)U(tr, t2)</Jin(X2)'" x U(t n-1, tn)</Jin(X n) U(t n) 10)
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