QUANTUM FIELD THEORY in Visual Studio .NET

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QUANTUM FIELD THEORY
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Define the subset of analytic functions by the condition
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which implies that / depends only on ij.
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Note that and
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which means that polynomials in the derivative operators have the same structure as the original anticommuting algebra. The reader will observe that 8(P 1 P 2 ) is not equal to 8P 1 P 2 + P 18P2 and will easily find the correct version of this identity. The construction may easily be generalized to several degrees of freedom. With 2n degrees of freedom, the exterior algebra will be of dimension 22 " and the space of analytic functions of dimension 2".
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Returning to the case n = 1 we write an analytic function as
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/=/0 + /lij
and define a scalar product such that
(g,f) =
+ gt/l
(9-62)
Here a bar on scalars means complex conjugation. Is it possible to represent this scalar product as an integral as in the boson case The answer is "yes," provided we identify derivation and integration as follows. The integral symbol is defined by linearity starting from the requirements
dij ij = 1
dl1l1 = 1
dij 1 =
dl1 1 = 0
(9-63)
If we also agree that dl1 and dij anticommute and that the rules (9-63) apply when dij and ij or dl1 and 11 are brought next to each other, we indeed see that integrals and derivatives are identical. Consequently,
dl1 P = oP
dij dl1 P =
(9-64)
As a consequence the integral of a derivative vanishes (0 2 = 82 = 0) and the procedure is easily extended to several degrees of freedom. We have the possibility to make changes of integration variables under the integral sign. If we limit ourselves for the time being to linear transformations which automatically respect the structure (9-58), i.e., of the form
where A is a nonsingular c-number matrix, a substitution in any polynomial P yields
P(I1, ij)
Q(~, ~
FUNCTIONAL METHODS
and in particular
As a result
dii drt P(rt, ii)
d~(det A)-!Q(~, ~
(9-65)
This implies a rule differing from the usual one in the sense that the standard jacobian appears inverted since det A = J(rt'
~,~1
A basis being chosen to allow for a definition of analytic functions, let us define complex conjugation as
(9-66)
We then verify that
(g,J) =
dii drt
e-'i'l g(rt)f(ii)
(9-67)
analogous to formula (9-38) for bosons. We have now the following representation of a and at in terms of a pair of adjoint operators:
a-+o
(9-68)
Obviously a2
= a t2 =
O. Furthermore,
a(atf)
As a result
(g, af) = gof!
More generally, we can assign an integral kernel to a linear operator on the space yt of analytic functions. Consider the orthonormal states 10) and 11) such that a 10) = 0, at 10) = 11), corresponding to the functions 1 and ii. Let us write
A= (Af)(ii) = A(ii, ~) =
L In)An,m<ml
d~ e-~~ A(ii, ~)f(~
(9-69)
L iinAn,m~m
(9-70)
As in the case of bosons we represent the projector on the ground state as
1 )<01 =: e- ata : = 1 - ata 0
QUANTUM FIELD THEORY
and rewrite A in normal form: (9-71)
n,m n,m
The associated normal kernel
(9-72)
is related to the previous one through
A(ij, '1)
e'i~ AN(ij, '1)
(9-73)
For instance, the integral kernel of the identity is e'i~ so that
f(ij)
while the product of operators is given by
A 1 A 2(ij, '1) =
f d~ e-~~+'i~ f(~ fifd~ e-~~ ~)A2([,
A 1 (ij,
(9-74)
(9-75)
The analog of formula (9-56) for gaussian integrals is
f fI
dr/k d'1k exp [-
L ijk Akl'11 + L (ijk~k + ~k'1k)] = det A k,l k
(9-76)
The close parallelism with the boson case allows an immediate transcription to obtain a path integral for transition amplitudes. Let H(a t , a, t) be the normal ordered hamiltonian of a fermionic system. The corresponding normal kernel is obtained by substituting if, '1 for at, a in this order. Consequently, the kernel of the evolution operator is given by
U('1f' tf; '1i, ti)
'1f'1f + '1i'1i f ('1, '1) exp 2
{- -
It dt [_..: - h('1, '1, t)]} '1'1 - '1'1 2i
(9-77)
As an exercise consider the motion of a quantum spin ~ submitted to the action of a constant field B along the z axis. The ground state is defined as corresponding to the value Sz = ~ of the spin. The hamiltonian reads H = /lB(2a t a - 1) with /l the gyro magnetic ratio. Equation (9-77) shows that (9-78) Note the similarity with the harmonic oscillator. The treatment can be extended to a time-dependent field, including in particular a transverse field rotating at frequency w. What is striking when looking at Eq. (9-76) is that the integral of a quadratic form is also given, up to a factor, by the value on the "extremal trajectory," i.e., giving a stationary value to its exponent.
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