Mathematical operator-value measure of interest in quantum mechanics and functional analysis
In
mathematics, particularly in
functional analysis, a projection-valued measure (or spectral measure) is a function defined on certain subsets of a fixed set and whose values are
self-adjointprojections on a fixed
Hilbert space.[1] A projection-valued measure (PVM) is formally similar to a
real-valuedmeasure, except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to
integratecomplex-valued functions with respect to a PVM; the result of such an integration is a
linear operator on the given Hilbert space.
The second and fourth property show that if and are disjoint, i.e., , the images and are
orthogonal to each other.
Let and its
orthogonal complement denote the
image and
kernel, respectively, of . If is a closed subspace of then can be wrtitten as the orthogonal decomposition and is the unique identity operator on satisfying all four properties.[4][5]
For every and the projection-valued measure forms a
complex-valued measure on defined as
i.e., as multiplication by the
indicator function on
L2(X). Then defines a projection-valued measure.[6] For example, if , , and there is then the associated complex measure which takes a measurable function and gives the integral
Extensions of projection-valued measures
If π is a projection-valued measure on a measurable space (X, M), then the map
extends to a linear map on the vector space of
step functions on X. In fact, it is easy to check that this map is a
ring homomorphism. This map extends in a canonical way to all bounded complex-valued
measurable functions on X, and we have the following.
Theorem — For any bounded Borel function on , there exists a unique
bounded operator such that
[7][8]
where the integral extends to an unbounded function when the spectrum of is unbounded.[10]
Direct integrals
First we provide a general example of projection-valued measure based on
direct integrals. Suppose (X, M, μ) is a measure space and let {Hx}x ∈ X be a μ-measurable family of separable Hilbert spaces. For every E ∈ M, let π(E) be the operator of multiplication by 1E on the Hilbert space
Then π is a projection-valued measure on (X, M).
Suppose π, ρ are projection-valued measures on (X, M) with values in the projections of H, K. π, ρ are unitarily equivalentif and only if there is a unitary operator U:H → K such that
for every E ∈ M.
Theorem. If (X, M) is a
standard Borel space, then for every projection-valued measure π on (X, M) taking values in the projections of a separable Hilbert space, there is a Borel measure μ and a μ-measurable family of Hilbert spaces {Hx}x ∈ X, such that π is unitarily equivalent to multiplication by 1E on the Hilbert space
The measure class[clarification needed] of μ and the measure equivalence class of the multiplicity function x → dim Hx completely characterize the projection-valued measure up to unitary equivalence.
A projection-valued measure π is homogeneous of multiplicityn if and only if the multiplicity function has constant value n. Clearly,
Theorem. Any projection-valued measure π taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures:
In quantum mechanics, given a projection valued measure of a measurable space X to the space of continuous endomorphisms upon a Hilbert space H,
the projective space of the Hilbert space H is interpreted as the set of possible states Φ of a quantum system,
the measurable space X is the value space for some quantum property of the system (an "observable"),
the projection-valued measure π expresses the probability that the
observable takes on various values.
A common choice for X is the real line, but it may also be
R3 (for position or momentum in three dimensions ),
a discrete set (for angular momentum, energy of a bound state, etc.),
the 2-point set "true" and "false" for the truth-value of an arbitrary proposition about Φ.
Let E be a measurable subset of the measurable space X and Φ a normalized vector-state in H, so that its Hilbert norm is unitary, ||Φ|| = 1. The probability that the observable takes its value in the subset E, given the system in state Φ, is
where the latter notation is preferred in physics.
We can parse this in two ways.
First, for each fixed E, the projection π(E) is a self-adjoint operator on H whose 1-eigenspace is the states Φ for which the value of the observable always lies in E, and whose 0-eigenspace is the states Φ for which the value of the observable never lies in E.
Second, for each fixed normalized vector state , the association
is a probability measure on X making the values of the observable into a random variable.
A measurement that can be performed by a projection-valued measure π is called a projective measurement.
If X is the real number line, there exists, associated to π, a Hermitian operator A defined on H by
which takes the more readable form
if the support of π is a discrete subset of R.
The above operator A is called the observable associated with the spectral measure.
Generalizations
The idea of a projection-valued measure is generalized by the
positive operator-valued measure (POVM), where the need for the orthogonality implied by projection operators is replaced by the idea of a set of operators that are a non-orthogonal partition of unity[clarification needed]. This generalization is motivated by applications to
quantum information theory.
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