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(9-48)
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we find (9-49) and this relation extends by linearity to an arbitrary operator. Starting from Eqs. (9-44) and (9-49) we may compute the kernel corresponding to the evolution operator for a quantum mechanical problem. We assume the hamiltonian to be given in normal form in terms of the operators at and a. Its normal kernel is simply obtained by substituting complex numbers for the creation and annihilation operators. We denote this function by h(z, ~). For an infinitesimal time interval !J..t we have approximately
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(9-50)
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This reduces correctly to the kernel of the identity when !J..t -+ O. Repeated application ofEq. (9-44) over a finite interval leads to the path integral
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(9-51) We use for the limit the symbolic notation
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_ U(Zf,tf;z;,t;)=
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_ ZfZf+ZiZi. E0(z,z)exp 2 +I
{- -
It dt [..zz-zz : 2i -.
h(i,
Z)]}
(9-52)
In this expression the integration variables z(tf) and i(ti) remain independent from i(tf) and z(t i), which are fixed by the boundary conditions. We recognize once again the classical action in the exponent of (9-52). Indeed the form p dq or, rather, 1(P dq - q dp) may be written according to (9-37) as
1 1(P dq - q dp) = 2i (z di - i dz)
(9-53)
which requires to treat Z and i as independent variables. Clearly, all that is said above immediately generalizes to time-dependent hamiltonians and to several degrees of freedom.
Let us compute the evolution operator for an harmonic oscillator driven by an external timedependent force in such a way that
(9-54)
Here 1 is the complex conjugate of f As illustrated by Eqs. (9-19) and (9-21) the result should be proportional to eiI(f.') where I(f, i) is the action computed on the extremal trajectory satisfying the classical equations of motion
z + i[wz The solution is
Z(t)
f(t)]
Z(ti) z(tf)
Zi zf
i-i[wz-J(t)]=O
Zi e-iw(t-t,)
r dt' e-iw(t-t') f(t') Jt,
z(t)
zf eiW(t-t f ) + i 1'f dt' eiW(t-t') J(t')
These quantities are obviously not conjugate due to the dissymmetry of boundary conditions, The corresponding exponent of the path integral along this trajectory is
if = i(zfzf
+ ZiZi) + ~
rtf dt [iz -
zz -
2ih(z, z, t)]
= zf e-iW(tf-t,) Zi
rtf dt [if e-iw(tf-t) f(t)
+ J(t) e-iW(t-t,) Z;]
(9-55a)
rtf _ - Jt {tf dt dt' f(t) e-iW(t-t')f(t')(}(t t
FUNCTIONAL METHODS
Explicit calculation of the path integral gives simply
(9-55b)
Thus the case of the driven osciIIator is very simple in this formalism and the kernel is regular everywhere.
The gaussian integration formula used repeatedly in these evaluations is worth quoting here as it occurs as a cornerstone of the application of path integrals. If A stands for the matrix of a nonsingular quadratic form the hermitian part of which is positive and z and u stand for column vectors of complex numbers, (9-56) Note that the exponent on the right-hand side is the saddle-point value of the exponent of the integrand.
9-1-3 Fermion Systems
Since path integrals exhibit the close relationship between classical and quantum mechanics it would seem a priori that we would encounter some difficulties when extending the treatment to fermions. Fortunately the relevant construction in terms of an anticommuting algebra has been devised by Berezin and we have already made some use of it in Chap. 4. Let us start from a two-level system with the two operators a and at fulfilling (9-57) We shall try to represent them as acting on a Hilbert space of "analytic functions." The analogy with the infinite series in z and z used previously for bosons suggests the following. Let us consider series with complex coefficients in two anticommuting variables 1] and ii, that is, such that
1]ii + ii1] =
1]2 =
ii2 =
(9-58)
These series reduce to polynomials of the form
P(ii, 1]) = Po + Plii + Pl1] + P121]ii
(9-59)
The set of these polynomials of dimension 22 = 4 may be identified with the exterior algebra on a two-dimensional vector space (generated by homogeneous polynomials of degree one). The associative multiplication is defined in agreement with the rules (9-58). We also introduce the linear derivation (9-60) On each monomial they act by suppressing the corresponding 1] (for 8) or (for 8) once the latter has been brought to the left; otherwise they give zero.