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S5f. Why normal ordering?
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Field theory often deals with polynomial expressions in annihilation
operators a(p) and their adjoint creation operators a^*(p).
While a(p) is a linear operator on a dense subspace H of the
corresponding Fock space, its adjoint isn't. But both are densely
defined sesquilinear forms on Fock space.
A sesquilinear form is a linear mapping f from a space H (the domain;
a dense subspace of the Hilbert space, in the present case of Fock
space) to its dual space H^* (which properly contains H), while
an operator maps H into H. Thus the latter can be iterated
while the former usually cannot.
is always defined when phi,psi in H (since f|psi> is in H^*,
the inner product is defined). Thus Hermitian sesquilinear forms are
satisfying candidates for 'observables'. However, matrix elements
of products fg make sense only for
operators f,g, since fg|psi> is not defined if g|psi> is outside H.
In particular, a(p)a(p)^* is a meaningless construct, while
:a(p)a(p)^*: = +-a^(p)*a(p)
makes sense as a Hermitian sesquilinear forms. But f(p)=a^(p)*a(p) is
no longer an operator in any sense (though good 1-particle
operators can be made by integration with suitable test functions).
That's why f(p)f(q) is meaningless while the permuted form
:f(p)f(q): = +-a^*(p)a^*(q)a(q)a(p)
(+ for Bosons, - for Fermions) is well defined (again as sesquilinear
form only).
More generally, any product O of creation and annihilation operators
which has all its creation terms to the left of all its annihilation
terms (these are called normally ordered products) defines a
sesquilinear form. The reason is that such an O can be written as
O=A^*B where A and B are products of annihilation operators only,
hence = can be interpreted as the inner
product of the two vectors A|phi> and B|psi> obtained from phi and psi
by applying annihilation operators only, which produces vectors in H
for which the inner product is always defined.
Normal ordering just permutes arbitrary products to put them into the
normally ordered and hence well-defined form (and adds a minus sign
if an odd number of transpositions of Fermion operators is needed
to order the product). This is extended by linearity to polynomials
and infinite series in power products. Note that normal ordering is
defined for formal expressions (i.e. strings of letters),
not for operators or forms; only _after_ nornal ordering an
expression O one gets a sesquilinear form :O:.
In Fock spaces over finite-dimensional Hilbert spaces, the situation is
different; there a(p) and a^*(p) are indeed operators on Fock space
(and the index p ranges over finitely many items only). Thus all
products make sense, and the normally ordered version of a product
differs from the original product by terms involving fewer operators.
Normal ordering is usually motivated by starting with a
finite-dimensional discretization where integrals become finite sums;
then one can do all the formal manipulations rigorously. Upon passing
to the continuum limit, most expressions become infinite and hence
meaningless, but the normally ordered expressions happen to have a
well-definedlimit and hence are meaningful. So these are the relevant
'operators' or rather sesquilinear forms. Presenting things as above
avoids any infinities.