A projection-valued measure on a measurable space , where is a σ-algebra of subsets of, is a mapping from to the set of self-adjoint projections on a Hilbert space such that and for every, the following function is a complex measure on '. We denote this measure by Note that is a real-valued measure, and a probability measure when has length one. If is a projection-valued measure and then the images, are orthogonal to each other. From this follows that in general, and they commute. Example'. Suppose is a measure space. Let, for every measurable subset in, be the operator of multiplication by the indicator function on L''2. Then is a projection-valued measure.
Extensions of projection-valued measures, integrals and the spectral theorem
If π is a projection-valued measure on a measurable space, 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 boundedM-measurable function f on X, there existsa unique bounded linear operator such that for all. Where,denotes the complex measure from the definition of. The map is a homomorphism of rings. An integral notation is often used for, as in The theorem is also correct for unbounded measurable functions f, but then will be an unbounded linear operator on the Hilbert space H''. The spectral theorem says that every self-adjoint operator has an associated projection-valued measure defined on the real axis, such that This allows to define the Borel functional calculus for such operators: if is a measurable function, we set
Structure of projection-valued measures
First we provide a general example of projection-valued measure based on direct integrals. Suppose is a measure space and let x ∈ X be a μ-measurable family of separable Hilbert spaces. For every E ∈ M, let π be the operator of multiplication by 1E on the Hilbert space Then π is a projection-valued measure on. Suppose π, ρ are projection-valued measures on 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 is a standard Borel space, then for every projection-valued measure π on taking values in the projections of a separable Hilbert space, there is a Borel measure μ and a μ-measurable family of Hilbert spaces x ∈ X, such that π is unitarily equivalent to multiplication by 1E on the Hilbert space The measure class 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 valuen. 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: where and
Application in quantum mechanics
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 unit sphere 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, - 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, - a discrete set, - 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 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. Any operator so obtained is called an observable, in quantum mechanics.
