Projection-valued measure
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
- 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
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
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,
Then
Suppose
for every E ∈ M.
Theorem. If (X, M) is a standard Borel space, then for every projection-valued measure
The measure class[clarification needed] of
A projection-valued measure
Theorem. Any projection-valued measure
where
and
Application in quantum mechanics[edit]
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
where the latter notation is preferred in physics.
We can parse this in two ways.
First, for each fixed E, the projection
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
If X is the real number line, there exists, associated to
which takes the more readable form
if the support of
The above operator A 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.
References[edit]
- 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, V. (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.