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S5h. Creation operators and rigged Hilbert space
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Physicists regard Fock space as the Hilbert space containing the
basis states
|x_1:N> = |x_1,...,x_N>
and their linear combinations. However, there is no Hilbert space
containing these states. The state |x_1:N> = |x_1,...,x_N>
is not in the Hilbert Fock space, for the same reason for which
|x> is not in the 1-particle Hilbert space. It is only a
distribution.
The Hilbert Fock space is made instead of all wave functions
psi = sum_N integral dx_1:n psi_N(x_1:N) |x_1,...,x_N>
with finite
= sum_N |psi_N|^2/N!
Physicists also define annihilation operators a(x) and
their adjoints, creation operators a^*(x). However, these are
not operators, but operator-valued distributions. For example,
a^*(x) maps the vacuum state |vac> (with psi_0=1, other psi_N=0)
into a^*(x)|vac> = |x>, which is not in the Hilbert Fock space.
More generally, for every nonzero Hilbert Fock space vector psi,
the vector
psi' = a^*(x) psi
lies outside the Hilbert Fock space state.
Thus the domain of a^*(x) is just {0}.
However, the states |x_1:N> = |x_1,...,x_N> lie in the top
layer H^* of the right Gelfand triple = rigged Hilbert space.
This is the name for a triple H in Hbar in H^* of vector spaces,
where Hbar is a Hilbert space, H a dense 'nuclear' subspace
(containing very smooth states with very good behavior at infintity)
and H^* its dual space (containing among others very singular states
and states with very poor behavior at infintity). Observables (in the
weak sense) are bilinear forms, or, which is the same, linear mappings
from H to H^*. The adjoint of such a linear mapping is again an
observable in the weak sense. Annihilation operators a(x) (and their
adjoints a^*(x)) are observables in this weak sense, although they are
not Hermitian (and a fortiori not self-adjoint).
Most physicists take it lightly since the times of Dirac.
They don't bother about self-adjointness or any other functional
analytic concept, unless ignoring it brings them into trouble.
Almost everything they do in the nonrelativistic regime
can be made rigorous in the rigged Hilbert space, so they fare right
even when they imagine wrongly that they work in a Hilbert
space. Thus they get away with their bad practices.
What they call 'Hilbert space' _is_ in fact always a
rigged Hilbert space; although most of them just don't know and
don't care.