FUNCTIONAL METHODS in .NET framework

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FUNCTIONAL METHODS
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9-2 RELATIVISTIC FORMULATION
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We generalize the previous approach to the infinite systems of interacting fields.
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9-2-1 S Matrix and Green's Functions in Terms of Path Integrals
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We start with an examination of the familiar case of a neutral scalar field coupled to an external real c-number source j(x). The classical action is
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(9-79)
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and the quantum hamiltonian
f 3 [~TC~p
+ ~(V%p + ~2
o/~p -
j%pJ
(9-80)
It describes an assembly of quantum oscillators coupled to varying external
forces (Chaps. 3 and 4). At a given time the Fourier decomposition of the field is
%p(x) =
dk [a(k) e ik x
+ at(k) e- ik X]
(9-81)
TCop(X) = -i
dk w(k)[a(k)e ik ' x
at(k)e- ik ' x ]
in terms of which
f(t, k) =
dk [w(k)at(k)a(k) - f(t,k)at(k) - f(t,k)a(k)]
(9-82)
e- ik ' x j(x,
In this form the hamiltonian is diagonal and we can apply formula (9-55) which gives the integral kernel of the evolution operator
U(zf' tf; Zi, ti) = exp
(f dk {Zf(k) e-iW(k)(tf-t,) zi(k)
dt[Zf(k) e-iw(k)(tf-t) f(t, k) + f(t, k) e-iW(k)(t-t.l Zi(k)]
(9-83)
~ ff 1:f dt dt' f(t, k) e-iW(k)lt-t'l f(t', k)})
With the source switched off at 1 1-+ 00, the S matrix is defined as the limit t of the operator eitfHo U(tf' ti) e-it,H o, where Ho is obtained from H by setting j = O. For coherent states, such as those used here, the action of e- itHo amounts
QUANTUM FIELD THEORY
simply to the shift Z -+ Z e-iwt, where w is the frequency associated to the oscillator described by z. Consequently, the integral kernel of the S matrix reads
Y'(zj, Zi)
- tl,t f-+OO
exp {fdk [Zf(k)Zi(k)
-~ ff
<Pas (x, t)
rtf dt [Zf(k) eiw(k)tf(t, k) ff dt dt' 1(t,
+ 1(t, k) e-iw(k)t zi(k)]
(9-84)
k) e-iW(k)!t-t'! f(t', k)]}
From Eq. (9-49) the normal kernel will simply follow if we drop the first factor in the exponential. The remaining part is interpreted in terms of the classical asymptotic field
= f dk[zi(k)e- ik . x + zf(k) eik . x ]
(9-85)
solution of the homogeneous Klein-Gordon equation. Since Zf(k) is not the complex conjugate of zi(k), <Pas is given in terms of boundary conditions on positive frequencies for t -+ - 00 and negative ones for t -+ + 00. We recognize Feynman's mixed boundary condition. With these notations
f dk f:: dt [Zf(k) eiw(k)t f(t, k)
+ 1(t, k) e-iw(k)t zi(k)]
= f d4 x f dkj(x)[zf(k)eiw(k)t-ik'x
+ zi(k)e-iW(k)t+ik'x]
= f d4 x j(x) <Pas (x)
Furthermore,
f dk f f dt dt' 1(t, k) e-iw(k)lt-t'1 f(t', k)
= f f d4 x d4 x' j(x)j(x') f dk e-iw(k)lt-t'l+ik'(x-x')
The integral over k is the Feynman propagator
dk e-iw(k)lt-t'l+ik'(x-x') = i
_ _ ,-,;;-e_--.__
d4 k
-ik'(x-x')
(2n)4 k
+ is
= - iGF(x - x') =
<01 T<pop(x)<pop(x') 0 )
where <Pop is the quantized scalar free field. Finally,
FUNCTIONAL METHODS
yN(Zf' Zi) Ii = exp
[i fd4x j(x) <Pas (X) + ~ ffd4x d4x: j(X)GF(X -
X')j(X') ]
(9-86)
This was indeed our result (4-63) and was used as a starting point in the discussion of Wick's theorem. The relativistic covariance as well as Feynman's ie prescription for propagators have followed naturally from the path integral formalism. To obtain the S operator which will be denoted So(j), we substitute <Pop to <Pas and normal order:
So(j) Zo(j)
[i Jd4x j(x) <Pop(x)J: Zo(j)
(9-87) d4x d4x' j(x)GF(x ---.: x')j(x') ]
Gf f
Since
+ m2 ) i
(jj(x) Zo(j)
j(x)Zo(j)
formula (9-87) may be reinterpreted as
So(j) = : exp
{f d4x [<Pop(X)(O + m (jj~X)J}: Zo(j)
(9-88)
where (0 + m 2 )((jj(jj) acts only on Zo(j). We consider now more complex interactions. Let us introduce, for instance, self-coupling through a potential V(<p) in such a way that the action takes the form
J( <p, j)
d4x j<p
(9-89)
As indicated by the notation, we take for simplicity V(<p) to depend only on <p and not on its gradient. For instance, V(<p) may be A,<p4j4!. To derive the S matrix we use the same type of reasoning leading to Eqs. (9-23) and (9-24). In other words, we have the following relation between normal kernels:
= exp [ -
f vG (jj~X))
4 d x
JYN(j) li=o
(9-90)
with yN (j) given by the expression (9-86). The perturbative series follows from the expansion of the exponential operator in (9-90). The S matrix itself may be written S = : exp
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