Observables in quantum mechanics are generally treated on the premise that
the values they take (their eigenvalues) are quantities of such a kind as may be
scaled and added: that is, tensor, vector or scalar quantities. However, General Relativity requires us to describe the universe
as a smooth manifold, in which context
position is not a vector quantity: at best, one can represent
positions (in modest-sized chunks of space-time) by vectors (using a chart), but
then the notions of addition
and scaling
this induces depend on
your choice of representation.
The need for an observable's eigenvalues to be addable and scalable arises
from the observable being a linear map from S = {system states} to V⊗S
where V is the space of nominally possible
values for the observable,
among which the eigenvalues are the feasible
(or, indeed, observable)
possibilities. [The observable, (V⊗S: Q |S), then has to satisfy a
structural relationship with a given symmetric metric, an antilinear mapping
(dual(S): g |S) for which, for any s, t in S, t·g(s) and s·g(t)
are mutually conjugate; the constraint says that, for any s, t in S,
Q(s)·g(t) and Q(t)·g(s) must also be mutually conjugate; Q is then
described as hermitian
.] When the observable is position on our smooth
manifold, M, V gets replaced by M, which is not a linear space, and the ⊗
tensor space combinator is not available to us: we cannot ask which mappings (M:
|dual(S)) are linear. None the less, the position observable is clearly
meaningful, so how may we describe it ?
The natural way to approach this is to look at the conventional description and seek the extent to which it is free of the requirement that the observable is a vector quantity. To this end, note that the orthodox treatment in terms of diagonalisation of an observable is alternatively described as decomposition of the observable into a sum: each term in which is the (if necessary tensor) product of an eigenvalue of the observable with a projection operator (projector) which selects the eigenspace for that eigenvalue.
The sum of just the projectors, without multiplication by their eigenvalues, delivers the identity linear operator and these projectors commute with one another (the product of any distinct pair is zero). For any set of values in the space within which the observable's values lie, there is a projector equal to the sum of the projectors for those eigenvalues which lie in the set given.
The probability, when the system being described is in some given state, of finding the observable to have a value in some set turns out to be the result of contracting this projector for the set with the state's bra (on the left; its image, in dual(S), under g) and ket (on the right; the member of S representing the state). One may obtain a hermitian operator (actually a projector), with trace 1, from this bra and ket: their (tensor) product the other way round. The probability just cited is then the trace of the product of this hermitian operator with the projector for the given set of values for the observable.
If we do not know the state of the system but, instead, know a real
probability measure over the possible states
(loosely a probability density
for what state the system is in) then we
can integrate this ket tensor bra product using that measure: the result will
also be a hermitian operator with trace 1 (but not, as far as I can see,
necessarily a projector). Consequently, it is more natural to describe the
state of the system in terms of a hermitian with trace 1 rather than in terms of
definite state vectors: the probability of finding the system (for which we had
a probability measure over possible states) to have its value for an observable
in some set is still the trace of the product of the projector for the
observable to be in that set times the hermitian operator with trace 1
associated with the system's state.
The decomposition of the identity into a sum of commuting projectors commuting with our observable can, alternatively, be regarded as a (generalised) probability measure on the space within which the observable's values lie. The values taken by the measure lie in a commuting algebra of hermitian projectors. If the values taken by the observable do lie in some vector space (which will, naturally, be real given a hermitian observable) then the expected value of this probability measure comes to the observable itself. However, all other aspects of its use are liberated from dependence on the vectorial nature of the values taken by the observable.
Consequently, we can describe the position observable on a smooth manifold
as a probability measure on that manifold, taking values in the space of
hermitian projectors on the Hilbert space via which we describe our quantum
mechanical states. You can, roughly, think of this as a probability density for
the position, albeit the probabilities
delivered are not real numbers
between 0 and 1. Such numbers may be obtained, however, by multiplying the
projector delivered with the unit-trace hermitian operator describing the state
of the system under study and taking the trace of the product. For any
measurable subset of the smooth manifold, the measure yields a projector; the
whole manifold is measurable and its projector is the identity on our space S of
states. Any subset of the manifold in which the particle definitely isn't
yields the zero projector. Any measurable (e.g. smooth and bounded) mapping, f,
from the manifold to some fixed linear space U may be integrated
using
the measure to produce a linear map (U⊗S: |S) which may be contracted
with the hermitian operator describing the system state to yield an expected
value, in U, of f.
Written by Eddy.