Projection-valued measure#Application in quantum mechanics

Let $H$ denote a separable complex Hilbert space and $(X,\; M)$ a measurable space consisting of a set $X$ and a Borel σ-algebra $M$ on $X$. A projection-valued measure $\backslash pi$ is a map from $M$ to the set of bounded self-adjoint operators on $H$ satisfying the following properties:{{sfn | Hall | 2013 | p=138}}{{sfn | Reed | Simon | 1980 | p=234}}

• $\backslash pi(E)$ is an orthogonal projection for all $E\; \backslash in\; M.$
• $\backslash pi(\backslash emptyset)\; =\; 0$ and $\backslash pi(X)\; =\; I$, where $\backslash emptyset$ is the empty set and $I$ the identity operator.
• If $E\_1,\; E\_2,\; E\_3,\backslash dotsc$ in $M$ are disjoint, then for all $v\; \backslash in\; H$,

::$\backslash pi\backslash left(\backslash bigcup\_\{j=1\}^\{\backslash infty\}\; E\_j\; \backslash right)v\; =\; \backslash sum\_\{j=1\}^\{\backslash infty\}\; \backslash pi(E\_j)\; v.$

• $\backslash pi(E\_1\; \backslash cap\; E\_2)=\; \backslash pi(E\_1)\backslash pi(E\_2)$ for all $E\_1,\; E\_2\; \backslash in\; M.$

The second and fourth property show that if $$

E_1 and $E\_2$ are disjoint, i.e., $E\_1\; \backslash cap\; E\_2\; =\; \backslash emptyset$, the images $\backslash pi(E\_1)$ and $\backslash pi(E\_2)$ are orthogonal to each other.

Let $V\_E\; =\; \backslash operatorname\{im\}(\backslash pi(E))$ and its orthogonal complement $V^\backslash perp\_E=\backslash ker(\backslash pi(E))$ denote the image and kernel, respectively, of $\backslash pi(E)$. If $V\_E$ is a closed subspace of $H$ then $H$ can be wrtitten as the orthogonal decomposition $H=V\_E\; \backslash oplus\; V^\backslash perp\_E$ and $\backslash pi(E)=I\_E$ is the unique identity operator on $V\_E$ satisfying all four properties.{{sfn | Rudin | 1991 | p=308}}{{sfn | Hall | 2013 | p=541}}

For every $\backslash xi,\backslash eta\backslash in\; H$ and $E\backslash in\; M$ the projection-valued measure forms a complex-valued measure on $H$ defined as

:$\backslash mu\_\{\backslash xi,\backslash eta\}(E)\; :=\; \backslash langle\; \backslash pi(E)\backslash xi\; \backslash mid\; \backslash eta\; \backslash rangle$

with total variation at most $\backslash |\backslash xi\backslash |\backslash |\backslash eta\backslash |$.{{sfn | Conway | 2000 | p=42}} It reduces to a real-valued measure when

:$\backslash mu\_\{\backslash xi\}(E)\; :=\; \backslash langle\; \backslash pi(E)\backslash xi\; \backslash mid\; \backslash xi\; \backslash rangle$

and a probability measure when $\backslash xi$ is a unit vector.

Example Let $(X,\; M,\; \backslash mu)$ be a Measure space#Important classes of measure spaces and, for all $E\; \backslash in\; M$, let

:$$

\pi(E) : L^2(X) \to L^2 (X)

be defined as

:$\backslash psi\; \backslash mapsto\; \backslash pi(E)\backslash psi=1\_E\; \backslash psi,$

i.e., as multiplication by the indicator function $1\_E$ on L²(X). Then $\backslash pi(E)=1\_E$ defines a projection-valued measure.{{sfn | Conway | 2000 | p=42}} For example, if $X\; =\; \backslash mathbb\{R\}$, $E\; =\; (0,1)$, and $\backslash varphi,\backslash psi\; \backslash in\; L^2(\backslash mathbb\{R\})$ there is then the associated complex measure $\backslash mu\_\{\backslash varphi,\backslash psi\}$ which takes a measurable function $f:\; \backslash mathbb\{R\}\; \backslash to\; \backslash mathbb\{R\}$ and gives the integral

:$\backslash int\_E\; f\backslash ,d\backslash mu\_\{\backslash varphi,\backslash psi\}\; =\; \backslash int\_0^1\; f(x)\backslash psi(x)\backslash overline\{\backslash varphi\}(x)\backslash ,dx$

If {{pi}} is a projection-valued measure on a measurable space (X, M), then the map

: $$

\chi_E \mapsto \pi(E)

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.

{{math theorem|Theorem|For any bounded Borel function $f$ on $X$, there exists a unique bounded operator $T\; :\; H\; \backslash to\; H$ such that

{{Citation |last=Kowalski|first=Emmanuel| year=2009|title=Spectral theory in Hilbert spaces| series = ETH Zürich lecture notes | url=https://people.math.ethz.ch/~kowalski/spectral-theory.pdf|page = 50}}{{sfn | Reed | Simon | 1980 | p=227,235}}

:$\backslash langle\; T\; \backslash xi\; \backslash mid\; \backslash xi\; \backslash rangle\; =\; \backslash int\_X\; f(\backslash lambda)\; \backslash ,d\backslash mu\_\{\backslash xi\}(\backslash lambda),\; \backslash quad\; \backslash forall\; \backslash xi\; \backslash in\; H.$

where $\backslash mu\_\{\backslash xi\}$ is a finite Borel measure given by

:$\backslash mu\_\{\backslash xi\}(E)\; :=\; \backslash langle\; \backslash pi(E)\backslash xi\; \backslash mid\; \backslash xi\; \backslash rangle,\; \backslash quad\; \backslash forall\; E\; \backslash in\; M.$

Hence, $(X,M,\backslash mu)$ is a finite measure space.}}

The theorem is also correct for unbounded measurable functions $f$ but then $T$ will be an unbounded linear operator on the Hilbert space $H$.

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 $g:\backslash mathbb\{R\}\backslash to\backslash mathbb\{C\}$ is a measurable function, then a unique measure exists such that

:$g(T)\; :=\backslash int\_\backslash mathbb\{R\}\; g(x)\; \backslash ,\; d\backslash pi(x).$

{{see also|Self-adjoint operator#Spectral theorem}}

Let $H$ be a separable complex Hilbert space, $A:H\backslash to\; H$ be a bounded self-adjoint operator and $\backslash sigma(A)$ the spectrum of $A$. Then the spectral theorem says that there exists a unique projection-valued measure $\backslash pi^A$, defined on a Borel subset $E\; \backslash subset\; \backslash sigma(A)$, such that{{sfn | Reed | Simon | 1980 | p=235}}

:$A\; =\backslash int\_\{\backslash sigma(A)\}\; \backslash lambda\; \backslash ,\; d\backslash pi^A(\backslash lambda),$

where the integral extends to an unbounded function $\backslash lambda$ when the spectrum of $A$ is unbounded.{{sfn | Hall | 2013 | p=205}}

First we provide a general example of projection-valued measure based on direct integrals. Suppose (X, M, μ) is a measure space and let {H_x}_{x ∈ X} be a μ-measurable family of separable Hilbert spaces. For every E ∈ M, let {{pi}}(E) be the operator of multiplication by 1_E on the Hilbert space

:$\backslash int\_X^\backslash oplus\; H\_x\; \backslash \; d\; \backslash mu(x).$

Then {{pi}} is a projection-valued measure on (X, M).

Suppose {{pi}}, ρ are projection-valued measures on (X, M) with values in the projections of H, K. {{pi}}, ρ are unitarily equivalent if and only if there is a unitary operator U:H → K such that

:$\backslash pi(E)\; =\; U^*\; \backslash rho(E)\; U\; \backslash quad$

for every E ∈ M.

Theorem. If (X, M) is a standard Borel space, then for every projection-valued measure {{pi}} 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 {H_x}_{x ∈ X} , such that {{pi}} is unitarily equivalent to multiplication by 1_E on the Hilbert space

:$\backslash int\_X^\backslash oplus\; H\_x\; \backslash \; d\; \backslash mu(x).$

The measure class{{clarify|reason=What is a measure class? A measure up to measure-preserving equivalence? Should the measure be completed?|date=May 2015}} of μ and the measure equivalence class of the multiplicity function x → dim H_x completely characterize the projection-valued measure up to unitary equivalence.

A projection-valued measure {{pi}} is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly,

Theorem. Any projection-valued measure {{pi}} taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures:

:$\backslash pi\; =\; \backslash bigoplus\_\{1\; \backslash leq\; n\; \backslash leq\; \backslash omega\}\; (\backslash pi\; \backslash mid\; H\_n)$

where

:$H\_n\; =\; \backslash int\_\{X\_n\}^\backslash oplus\; H\_x\; \backslash \; d\; (\backslash mu\; \backslash mid\; X\_n)\; (x)$

and

:$X\_n\; =\; \backslash \{x\; \backslash in\; X:\; \backslash dim\; H\_x\; =\; n\backslash \}.$

{{see also|Expectation value (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 projective space $\backslash mathbf\{P\}(H)$ of the Hilbert space $H$ is interpreted as the set of possible (normalizable) states $\backslash varphi$ of a quantum system,{{sfn | Ashtekar | Schilling | 1999 | pp=23–65}}
• the measurable space $X$ is the value space for some quantum property of the system (an "observable"),
• the projection-valued measure $\backslash pi$ expresses the probability that the observable takes on various values.

A common choice for $X$ is the real line, but it may also be

• $\backslash mathbb\{R\}^3$ (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 $\backslash varphi$.

Let $E$ be a measurable subset of $X$ and $\backslash varphi$ a normalized vector quantum state in $H$, so that its Hilbert norm is unitary, $\backslash |\backslash varphi\backslash |=1$. The probability that the observable takes its value in $E$, given the system in state $\backslash varphi$, is

:$$

P_\pi(\varphi)(E) = \langle \varphi\mid\pi(E)(\varphi)\rangle = \langle \varphi\mid\pi(E)\mid\varphi\rangle.

We can parse this in two ways. First, for each fixed $E$, the projection $\backslash pi(E)$ is a self-adjoint operator on $H$ whose 1-eigenspace are the states $\backslash varphi$ for which the value of the observable always lies in $E$, and whose 0-eigenspace are the states $\backslash varphi$ for which the value of the observable never lies in $E$.

Second, for each fixed normalized vector state $\backslash varphi$, the association

:$$

P_\pi(\varphi) :

E \mapsto \langle\varphi\mid\pi(E)\varphi\rangle

is a probability measure on $X$ making the values of the observable into a random variable.

{{Anchor|Projective measurement}}A measurement that can be performed by a projection-valued measure $\backslash pi$ is called a projective measurement.

If $X$ is the real number line, there exists, associated to $\backslash pi$, a self-adjoint operator $A$ defined on $H$ by

:$A(\backslash varphi)\; =\; \backslash int\_\{\backslash mathbb\{R\}\}\; \backslash lambda\; \backslash ,d\backslash pi(\backslash lambda)(\backslash varphi),$

which reduces to

:$A(\backslash varphi)\; =\; \backslash sum\_i\; \backslash lambda\_i\; \backslash pi(\{\backslash lambda\_i\})(\backslash varphi)$

if the support of $\backslash pi$ is a discrete subset of $X$.

The above operator $A$ is called the observable associated with the spectral measure.

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", i.e. a set of positive semi-definite Hermitian operators that sum to the identity. This generalization is motivated by applications to quantum information theory.

• Spectral theorem
• Spectral theory of compact operators
• Spectral theory of normal C*-algebras

{{reflist}}

• {{cite book | last1=Ashtekar | first1=Abhay | last2=Schilling | first2=Troy A. | title=On Einstein's Path | chapter=Geometrical Formulation of Quantum Mechanics | publisher=Springer New York | publication-place=New York, NY | year=1999 | isbn=978-1-4612-7137-6 | doi=10.1007/978-1-4612-1422-9_3 | arxiv=gr-qc/9706069 }}* {{cite book | last=Conway | first=John B. | title=A course in operator theory | publisher=American mathematical society | publication-place=Providence (R.I.) | date=2000 | isbn=978-0-8218-2065-0}}
• {{cite book | last=Hall | first=Brian C. | title=Quantum Theory for Mathematicians | publisher=Springer Science & Business Media | publication-place=New York | date=2013 | isbn=978-1-4614-7116-5}}
• Mackey, G. W., The Theory of Unitary Group Representations, The University of Chicago Press, 1976
• {{citation |last=Moretti |first=Valter |title=Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation |volume=110 |year=2017 |publisher=Springer |bibcode=2017stqm.book.....M |isbn=978-3-319-70705-1}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{cite book | last1=Reed | first1=M. | author-link=Michael C. Reed | last2=Simon | first2=B. | author-link2=Barry Simon| title=Methods of Modern Mathematical Physics: Vol 1: Functional analysis | publisher=Academic Press | year=1980 | isbn=978-0-12-585050-6}}
• {{cite book | last=Rudin | first=Walter | title=Functional Analysis | publisher=McGraw-Hill Science, Engineering & Mathematics | publication-place=Boston, Mass. | date=1991 | isbn=978-0-07-054236-5}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• 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 Topological vector spaces, distributions and kernels}}
• Varadarajan, V. S., Geometry of Quantum Theory V2, Springer Verlag, 1970.

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