Functional Derivative Approach
Let us now leave the path integral formalism temporarily and reformulate operatorial quantum mechanics in a way which will make it easy later on to establish the formal connection between operator and path integral formalism. Our objective is to introduce
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Let us now leave the path integral formalism temporarily and reformulate operatorial quantum mechanics in a way which will make it easy later on to establish the formal connection between operator and path integral formalism. Our objective is to introduce the generating functional into quantum mechanics . Naturally we want to generate transistion amplitudes . The problem confronting us is how to transcribe operator quantum mechanics as expressed in Heisenberg's equation of motion into a theory formulated solely in terms of c-numbers. This can be achieved either by Schwinger's action principle or with the aid of a generation functional defined as follows : (17.1) Here Q(t) and P(t) stand for arbitrary c-number functions ("sources") and T denotes the time-ordering operation with respect to the Heisenberg operators q(t) and p(t) . When acting on a string of operators with different time arguments it orders the operators in such a way that the time arguments increase from the right to the left. The exponential in (17.1) should be interpreted as a power series, and in each term T acts according to the above rule: dtA(t) Te j'Z q
1 == ~ ~ ,n=O
n.
l
TZ
Tl
di, .. .
l
TZ
dtnTA(td . . . A(tn).
(17 .2)
Tj
The advantage offered by the introduction of external sources is that any number of operators between brackets can be generated by a simple functional differentiation, e.g., (17.3) (17.4) or, somewhat more generally :
W. Dittrich, Classical and Quantum Dynamics © Springer-Verlag Berlin Heidelberg 2001
188
17. Functional Derivative Approach
where F[q , p] is any functional of q(t) and pet) . To be more specific, let us consider the two-point function 1
I T(q(t)q(t)) I qv, tJl
(q2, tz
Ii 0 Ii 0 QP = -;- 0P( ) -;- 0P( I) (% tz I qi , t\) , IQ=P=O . ] o t ] o t (17.6)
The LHS contains the time-ordering operation, which is given by
I T(q(t)q(t')) I qi , tJl =
(q2, ti
I q(t)q(t l) I qi. t] )8(t - t l) (q2, ti I q(tl)q(t) I qi , tl )8(t ' - t),
(q2' ti +
(17.7)
where the step function 8(t) is defined by 8(t) = { I, t > 0 0, t < O.
The functional derivative referred to above is the limiting process °oF[(Q)] = lim ~(F[Q(t') uQ t £---> 0 e
+ £o(t -
t l)] - F[Q(t')J).
(17 .8)
Here we list a few examples: (a)
F[Q] = f Q(t')f(t/)dt',
(17 .9)
of[Q] oQ(t) = f(t). oQ(t') = O(tl - t) . oQ(t)
Note (b)
(17.10)
F[Q] = eJ Q(t')f(t')dt',
(17.11)
of[Q] = +(t) J Q(t')f(t')dt' oQ(t) J e . (c)
F[Q] = eJ dt'dt"Q(t')A(t',t")Q(t")
(17.12)
of[Q] = [fdtIlA(t tll)Q(tll)+fdt'Q(t')A(t' t)]e J QAQ. oQ(t) , ,
Our main goal is to rewrite Heisenberg's operator quantum mechanics equations into a coupled set of c-number functional differential equations. For this reason we start with the Hamiltonian operator H = H(q(t), p(t)) and write Heisenberg's equations of motion for the operators q(t) and p(t) as . t _ aH(q(t), pet)) q() ap(t) , pet)
= _ aH(q(t) , pet)) aq(t)
.
(17 .13)
17. Functional Derivative Approach
189
In the following we want to keep the c-number sources, i.e., we do not set them equal to zero after f
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