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-adjoint projections on a fixed Hilbert space.[1] A projection-valued measure (PVM) is formally similar to a real-valued measure, except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.
Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators, in which case the PVM is sometimes referred to as the spectral measure. The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements.[clarification needed] They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix generalizes the notion of a pure state.
Definition
[edit]Let denote a separable complex Hilbert space and a measurable space consisting of a set and a Borel σ-algebra on . A projection-valued measure is a map from to the set of bounded self-adjoint operators on satisfying the following properties:[2][3]
- is an orthogonal projection for all
- and , where is the empty set and the identity operator.
- If in are disjoint, then for all ,
- for all
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
with total variation at most .[6] It reduces to a real-valued measure when
and a probability measure when is a unit vector.
Example Let be a σ-finite measure space and, for all , let
be 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
[edit]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 is a finite Borel measure given by
Hence, is a finite measure space.
The theorem is also correct for unbounded measurable functions but then will be an unbounded linear operator on the Hilbert space .
This allows to define the Borel functional calculus for such operators and then pass to measurable functions via the Riesz–Markov–Kakutani representation theorem. That is, if is a measurable function, then a unique measure exists such that
Spectral theorem
[edit]Let be a separable complex Hilbert space, be a bounded self-adjoint operator and the spectrum of . Then the spectral theorem says that there exists a unique projection-valued measure , defined on a Borel subset , such that[9]
where the integral extends to an unbounded function when the spectrum of is unbounded.[10]
Direct integrals
[edit]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 equivalent if 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 multiplicity n 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:
where
and
Application in quantum mechanics
[edit]In quantum mechanics, given a projection-valued measure of a measurable space to the space of continuous endomorphisms upon a Hilbert space ,
- the projective space of the Hilbert space is interpreted as the set of possible (normalizable) states of a quantum system,[11]
- the measurable space is the value space for some quantum property of the system (an "observable"),
- the projection-valued measure expresses the probability that the observable takes on various values.
A common choice for is the real line, but it may also be
- (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 .
Let be a measurable subset of and a normalized vector quantum state in , so that its Hilbert norm is unitary, . The probability that the observable takes its value in , given the system in state , is
We can parse this in two ways. First, for each fixed , the projection is a self-adjoint operator on whose 1-eigenspace are the states for which the value of the observable always lies in , and whose 0-eigenspace are the states for which the value of the observable never lies in .
Second, for each fixed normalized vector state , the association
is a probability measure on making the values of the observable into a random variable.
A measurement that can be performed by a projection-valued measure is called a projective measurement.
If is the real number line, there exists, associated to , a self-adjoint operator defined on by
which reduces to
if the support of is a discrete subset of .
The above operator is called the observable associated with the spectral measure.
Generalizations
[edit]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.
See also
[edit]Notes
[edit]- ^ Conway 2000, p. 41.
- ^ Hall 2013, p. 138.
- ^ Reed & Simon 1980, p. 234.
- ^ Rudin 1991, p. 308.
- ^ Hall 2013, p. 541.
- ^ a b Conway 2000, p. 42.
- ^ Kowalski, Emmanuel (2009), Spectral theory in Hilbert spaces (PDF), ETH Zürich lecture notes, p. 50
- ^ Reed & Simon 1980, p. 227,235.
- ^ Reed & Simon 1980, p. 235.
- ^ Hall 2013, p. 205.
- ^ Ashtekar & Schilling 1999, pp. 23–65.
References
[edit]- Ashtekar, Abhay; Schilling, Troy A. (1999). "Geometrical Formulation of Quantum Mechanics". On Einstein's Path. New York, NY: Springer New York. arXiv:gr-qc/9706069. doi:10.1007/978-1-4612-1422-9_3. ISBN 978-1-4612-7137-6.* Conway, John B. (2000). A course in operator theory. Providence (R.I.): American mathematical society. ISBN 978-0-8218-2065-0.
- Hall, Brian C. (2013). Quantum Theory for Mathematicians. New York: Springer Science & Business Media. ISBN 978-1-4614-7116-5.
- Mackey, G. W., The Theory of Unitary Group Representations, The University of Chicago Press, 1976
- Moretti, Valter (2017), Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, vol. 110, Springer, Bibcode:2017stqm.book.....M, ISBN 978-3-319-70705-1
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6.
- Rudin, Walter (1991). Functional Analysis. Boston, Mass.: McGraw-Hill Science, Engineering & Mathematics. ISBN 978-0-07-054236-5.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
- Varadarajan, V. S., Geometry of Quantum Theory V2, Springer Verlag, 1970.