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L ~<nl f> = L fd~ d.r: r[( zn~n in VS .NET
L ~<nl f> = L fd~ d.r: r[( zn~n Decoding PDF417 In .NET Framework Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications. Painting PDF417 In Visual Studio .NET Using Barcode generator for VS .NET Control to generate, create PDF417 image in Visual Studio .NET applications. f([) PDF417 Reader In .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET applications. Barcode Generation In .NET Using Barcode creator for .NET Control to generate, create bar code image in .NET framework applications. = fd~ ~[ e~~+z~ f([) Scanning Bar Code In .NET Framework Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. PDF417 2d Barcode Creation In C# Using Barcode encoder for VS .NET Control to generate, create PDF417 image in .NET framework applications. (948) Draw PDF417 In .NET Framework Using Barcode creator for ASP.NET Control to generate, create PDF417 image in ASP.NET applications. PDF 417 Maker In VB.NET Using Barcode drawer for .NET Control to generate, create PDF417 2d barcode image in VS .NET applications. we find (949) and this relation extends by linearity to an arbitrary operator. Starting from Eqs. (944) and (949) 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 Paint UPC Code In .NET Framework Using Barcode printer for VS .NET Control to generate, create UPC Code image in .NET applications. Data Matrix Maker In VS .NET Using Barcode encoder for Visual Studio .NET Control to generate, create Data Matrix ECC200 image in .NET framework applications. (950) Draw Matrix 2D Barcode In .NET Using Barcode encoder for .NET framework Control to generate, create Matrix 2D Barcode image in .NET framework applications. International Standard Serial Number Generation In Visual Studio .NET Using Barcode generator for Visual Studio .NET Control to generate, create ISSN  13 image in Visual Studio .NET applications. This reduces correctly to the kernel of the identity when !J..t + O. Repeated application ofEq. (944) over a finite interval leads to the path integral Creating Code 128 Code Set B In Java Using Barcode maker for Android Control to generate, create Code 128 Code Set A image in Android applications. Draw Code 128 Code Set C In Visual Basic .NET Using Barcode generation for VS .NET Control to generate, create USS Code 128 image in Visual Studio .NET applications. QUANTUM FIELD THEORY
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{  It dt [..zzzz : 2i . h(i, Z)]} (952) 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 (952). Indeed the form p dq or, rather, 1(P dq  q dp) may be written according to (937) as 1 1(P dq  q dp) = 2i (z di  i dz) (953) which requires to treat Z and i as independent variables. Clearly, all that is said above immediately generalizes to timedependent 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 (954) Here 1 is the complex conjugate of f As illustrated by Eqs. (919) and (921) 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
ii[wzJ(t)]=O
Zi eiw(tt,) r dt' eiw(tt') f(t') Jt, z(t) zf eiW(tt f ) + i 1'f dt' eiW(tt') 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 eiW(tft,) Zi
rtf dt [if eiw(tft) f(t) + J(t) eiW(tt,) Z;] (955a) rtf _  Jt {tf dt dt' f(t) eiW(tt')f(t')(}(t t
FUNCTIONAL METHODS
Explicit calculation of the path integral gives simply
(955b) 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, (956) Note that the exponent on the righthand side is the saddlepoint value of the exponent of the integrand. 913 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 twolevel system with the two operators a and at fulfilling (957) 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 =
(958) These series reduce to polynomials of the form
P(ii, 1]) = Po + Plii + Pl1] + P121]ii
(959) The set of these polynomials of dimension 22 = 4 may be identified with the exterior algebra on a twodimensional vector space (generated by homogeneous polynomials of degree one). The associative multiplication is defined in agreement with the rules (958). We also introduce the linear derivation (960) 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.

