On a complex vector space, V, in the presence of an antilinear morphism, g, from V to its dual, we say that an automorphism of V is g-Hermitian precisely if its product with $\sigma $ is symmetric under Hermitian conjugation. Given that all symmetric antilinear forms are diagonalisable, and the ensuing diagonal entries must necessarily be real, it is constructive to think of g-Hermitian automorphisms as being `real' with respect to g. 0 is Hermitian with respect to any antilinear morphism: 1 is Hermitian only with respect to symmetric ones.

Under the same circumstances, we call a projection a g-projector precisely if it is g-Hermitian. With both Hermitian and projector, we elide the prefix when there is a standard antilinear morphism, usually an isomorphism, given in our context as the metric or in some similar rôle: this antilinear morphism is assumed to be the elided prefix.

Let P be a g-projector and v any vector in their domain. Consider g(Pv) = (gP)v. Now, gP is symmetric, so equal to its transpose: thus, for any vector u, (gP)vu = (gP)uv. In particular, taking (1-P)w for u, we obtain g(Pv)((1-P)w) = (gP(1-P))wv = 0 since P(1-P) = 0. Thus, viewing g as an inner product, Pv and (1-P)w are always orthogonal: that is, the range of a g-projector is necessarily g-orthogonal to its kernel.

Now, if g is not symmetric this doesn't imply that the kernel is g-orthogonal to the range: however, if g is symmetric then 1 is a g-projector and so is our 1-P; thus P's kernel ((1-P)'s range) is orthogonal to P's range. In particular, any v in range(P) and u in kernel(P) yield g(u+v)(u+v) = guu + guv + gvu + gvv = guu + gvv since guv=0=gvu. Applying this to P's range/kernel decomposition of a vector, v, as v= Pv+ (1-P)v we obtain gvv= g(Pv)(Pv)+ g((1-P)v)((1-P)v).

If our antilinear g is symmetric and positive-definite then it defines a metric in which the square of the `length' of any vector, v, is gvv. The length of any non-zero vector is then positive, with the zero vector having zero length. If v, Pv and (1-P)v have lengths r, s and t respectively, the foregoing decomposition of gvv yields rr= ss+ tt, whence rr is at least ss and at least tt, so r is at least as big as s and t (with equality to either only if the other is zero). (We also obtain rr= (s+t)(s+t) - 2st, whence r is no more than s+t, with equality only if one of s,t is zero.) Thus the length of v is no less than that of Pv: g-projectors never make vectors longer in terms of g's distance.

If our base anti-isomorphism from V to its dual is itself symmetric and positive definite, we are guaranteed that we can diagonalise it: furthermore that, for any second symmetric linear map from V to its dual, we can chose a basis which diagonalise both while making the positive definite one's diagonal entries all 1. When we apply this to g and some g-Hermitian automorphism of V (or, rather, to the latter's product with the former) we discover that the latter's diagonal entries are either 1 or 0: these are the only self-square real values.

Thus, for any g-projector P, we can chose a basis (dim| i
-> b_{i} :V) (with dim the dimension of V) and some n ≤ dim for
which: $\sigma \; =\; \sum $_{dim} i -> b^{i} x
b^{*i} [where x denotes tensor product; (dim| i ->
b^{i}) is the dual of our basis (dim| i -> b_{i}: V); and
$b*$ is the result of composing complex conjugation with
this dual, so (V| v ->b^{*i}v = (b^{i}v)^{*})] and
$P\; =\; \sum $_{n} i-> b_{i} x b^{i}.

For any g-projectors P, Q diagonalised, as above by bases p, q respectively: we can use the resulting representations of P and Q to explore the relationships between P and Q.

In our standard model of quantum mechanics, the Universe being in some
definite state, v, is described by a vector |v> in our Hilbert space of
length 1 wrt our (symmetric antilinear) metric. This can be used to define a
trace-1 projector, $|v><v|\; =\; |v>\; x\; \sigma |v>$ which
suffices to say everything we know about the observable properties of the system
under study when in state v. When the Universe's state is less well known, we
describe it in terms of a probability distribution, p, on possible states: from
this we can obtain an automorphism of our Hilbert space's dual,
$\int $_{v} p(v) |v><v|, which serves correctly in
place of the corresponding definite-state projector. This Hermitian morphism
describes everything we know about the system described: I shall call it the
state morphism. It always has trace 1 and it is always Hermitian: however, it
need not be a projector. Indeed, a trace 1 projector always defines a
one-dimensional space (its range) in which, up to phase, there is precisely one
unit vector; so whenever the state morphism is a projector, it describes a
definite state.

In this same model, projectors on the Hilbert space correspond to statements: the probability associated with any statement is the result of applying the state morphism to the projector for the statement. (Applying, here, is equivalent to composing the projector with the transpose of the state morphism and taking the trace of the result.) Any experiment is expressed in terms of

- a presumed model
- H, which expresses our real world in terms of a collection of probability distributions describing correlations between real phenomena
- some prior facts
- D, usually expressed as probability distributions whose meaning is mediated by the model, and
- a test
- which samples a variate, A, from a distribution expressed in terms of those of H and D.

Localisation of A (*ie* saying that it only depends on matters relating
to some bounded region of space) amounts to saying that P_{A} depends
only on channels *via* the conjunction of H and D; now, P_{H∧D}
is a projector-measure; so what we're saying is that P_{A} commutes with
the supremum of P_{H∧D} {whose values all commute; their supremum is
their global disjunction (`or') - *ie* it is the product (`and') of all
projectors that commute with every value taken by P_{H∧D}} [Note the
potential for a `blur at infinity' with a gap of definite dimension between an
infinite-dimensional collection of projectors and their higher-infinite
dimensional supremum - a countable basis is at best dense.]

Conjunction of statements (`and') needs to be related to some way of combining their associated projectors to obtain the projector associated with their conjunction. For commuting projectors, the answer is simply their product: in general, the resulting projector is subsumed by each of the conjoined operators and subsumes any other having the same property. One projector subsumes another if they commute with product the subsumed projector. Disjunction (`or') is similarly the minimal projector subsuming both: for commuting projectors, this is the sum minus the product.

Let Q be an observable: that is, there is some range of values on which
there is a projector measure, P_{Q}, whose values all commute with one
another: their supremum is the projector corresponding to the collection of
states on which Q is meaningful. Let B be a P_{Q}-measurable subset of
the range of Q. Let the state automorphism be S: then our probability for
observing Q with a valie in B is P(Q in B given S) = (S)(P_{Q}(B)).
This, in turn, can be viewed as the probability of finding the system's state in
the collection of states associated with P_{Q}(B), thus declaring this
collection of states to be measurable: it's measure is this probability.

Let R be another observable and C a P_{R}-measurable subset of its
range. We can only ask whether Q lies in B and R lies in C (a conjunction) if
we can measure the intersection of the collections of states associated with the
projectors P_{Q}(B) and P_{R}(C). The inevitable question is:
when do the collections of states associated with two (not necessarily
commuting) projectors have measurable intersection ?

If Q and R commute, or at least if P_{Q}(B) and P_{R}(C)
commute, we have P(Q in B and R in C given S) = S(P_{Q}(B)
P_{R}(C)). We say Q and R are independent if, for any measurable B and
C, this is equal to the product of P(Q in B given S) and P(R in C given S),
*ie* to S(P_{Q}(B)) S(P_{R}(C)). It seems to be a known
truth of Quantum mechanics that commuting observables are independent: this
seems to say a great deal about S.

With such a model, I wish to examine what we may say about Bayes' theorem, which enables us to refine our prior facts in the light of our model and our test's result.

Transcribed Eddy.$Id: projector.html,v 1.4 2009-08-09 14:42:29 eddy Exp $