Observables \(\cdot\) Part II. November 10, 2026.
Earlier in this series. I. The basic idea.
Part I described a real observable through its bounded tests, recovered its self-adjoint coordinate operator, and identified a natural notion of convergence. But a single real number is rarely all one measures. The position of a particle in space has three coordinates, the multiplication operators \(X_1, X_2, X_3\) on \(L^2 (\mathbb{R}^3)\). Each is an unbounded self-adjoint operator of the kind studied in Part I, and none has a single eigenvector. Yet physicists speak without hesitation of the position, a point of \(\mathbb{R}^3\) whose probability law in a unit vector \(u\) is \(| u (x) |^2 \mathrm{d} x\). What does it mean to measure three unbounded operators together, and when is it possible at all? The same question returns when the outcome is a complex number, a direction on a sphere, or a point of a manifold.
This part answers it in the language of Part I. Section 1 constructs joint observables from compatible coordinates; Sections 2–3 extend the outcome space to the complex plane and to manifolds. Section 4 adds probability laws by choosing a state, and shows what remains when observables are not compatible. Sections 5–6 are an outlook on distances and quantization; they assume more background than the rest. The optional subsections can be skipped on a first reading.
We continue to work with sharp observables: each yes-or-no question about the outcome is represented by an orthogonal projection, and the calculus preserves products. Functions of one observable therefore always commute with one another. Two different observables need not, and this is where the new questions begin. The proofs are collected in this part's companion and use the foundational results in Part I.
Contents.
Consider two diagonal Hermitian matrices on \(\mathbb{C}^2\). Let \(P_1, P_2\) be the projections onto the two coordinate axes, and write
On the first axis the two values are \((1, 2)\); on the second they are \((- 1, 3)\). A joint test is therefore
This is one observable with two possible outcomes in the plane. Its spectral measure assigns the projections \(P_1\) and \(P_2\) to those two points. For general commuting Hermitian matrices, simultaneous diagonalization gives the same construction: each common eigenvector carries a tuple of eigenvalues.
Compare the Pauli matrices
Each is Hermitian with eigenvalues \(\pm 1\), so each is a perfectly good observable with two outcomes. But the eigenvectors of \(\sigma_1\) are \((1, \pm 1) / \sqrt{2}\) and those of \(\sigma_2\) are \((1, \pm i) / \sqrt{2}\): no vector is an eigenvector of both, so there is no list of joint outcomes. Nor is there a consistent joint calculus: it would make \(\sigma_1\) and \(\sigma_2\) functions of one observable, hence commuting, whereas \(\sigma_1 \sigma_2 = - \sigma_2 \sigma_1 \neq 0\). Already for \(f (x, y) = xy\) the candidates \(\sigma_1 \sigma_2\), \(\sigma_2 \sigma_1\) and their average are three different matrices; the average is zero. The first pair commutes and the second does not, and this is the property to be formulated for unbounded operators. The Pauli matrices return in Section 4, where a state still assigns them ordered expectations.
We want to keep this picture when the coordinates are unbounded and there may be no common eigenvectors. Suppose that self-adjoint operators \(A_1, \ldots, A_d\) on one Hilbert space have one-dimensional spectral measures \(E_j\). We seek an observable with outcomes in \(\mathbb{R}^d\) whose spectral measure combines their individual spectral events consistently.
For bounded self-adjoint operators, \(AB = BA\) implies that all their spectral projections commute. In finite dimensions this is simultaneous diagonalization. For unbounded operators the identity \(AB = BA\) is no longer a usable test: \(AB\) is defined only on the vectors \(u \in D (B)\) with \(Bu \in D (A)\), and this set, or its counterpart for \(BA\), can be very small. We therefore formulate compatibility directly through the bounded spectral projections.
Strong commutation is the statement about the full observables: \(E_A (\Delta)\) commutes with \(E_B (\Gamma)\) for every pair of Borel sets. Equivalently, any bounded Borel function of \(A\) commutes with any bounded Borel function of \(B\). The meaning is operational: two commuting projections \(P, Q\) are yes-or-no questions that can be asked together, since \(PQ\) is again a projection, the event that both answers are yes. Strong commutation can also be tested on their resolvents \(R_A (z) = (A - z)^{- 1}\) or on their unitary groups:
or, equivalently,
All the operators in these tests are bounded. The equivalences follow because the resolvents generate the continuous calculus and bounded spectral approximation gives the projections; the unitary groups determine the same spectral measures by uniqueness of Fourier transforms. The companion proves these implications.
For a finite family, we require this strong commutation for every pair. No further compatibility hypothesis is needed: pairwise commuting spectral projections can be multiplied to impose all coordinate conditions simultaneously.
Assume the finite family is pairwise strongly commuting. If \(R = \Delta_1 \times \cdots \times \Delta_d\) is a Borel rectangle, the only possible joint projection is
Commutation makes this product an orthogonal projection. Its range is the intersection of the ranges of the factors: it imposes all coordinate conditions at once. Projections belonging to disjoint rectangles are orthogonal. Refining a finite rectangular grid therefore gives consistent sums of these projections.
Joint spectral theorem for a finite family. Pairwise strongly commuting self-adjoint operators \(A_1, \ldots, A_d\) determine a unique normalized projection-valued measure (PVM) \(E\) on \(\mathbb{R}^d\) satisfying \(E (R) = P (R)\) on rectangles. Its coordinate marginals are the \(E_j\). Conversely, the coordinate operators \(\int x_j \mathrm{d} E (x)\) of any normalized PVM \(E\) on \(\mathbb{R}^d\) are pairwise strongly commuting.
A coordinate marginal simply ignores all the other coordinates. If \(\pi_j (x) = x_j\), the marginal identity is
Thus asking whether the outcome's \(j\)-th coordinate lies in \(\Delta\) reproduces the original one-dimensional spectral event.
How to pass from rectangles to all outcomes (a sketch). Approximate a compactly supported continuous \(f\) by a step function \(s = \sum_{\alpha} c_{\alpha} 1_{R_{\alpha}}\) on a finite rectangular grid, and set \(J (s) = \sum_{\alpha} c_{\alpha} P (R_{\alpha})\). Orthogonality gives \(\| J (s) \| \leqslant \| s \|_{\infty}\). Refinement makes the definition independent of the grid, so uniform approximation defines a contractive homomorphism
It preserves adjoints and is nondegenerate: taking larger cutoffs in every coordinate makes their product converge strongly to \(I\). In other words, these bounded tests collectively see the whole Hilbert space. This is the nondegeneracy condition from Part I. Riesz–Markov now turns \(f \mapsto \langle u, J (f) u \rangle\) into a finite Radon measure. Polarization and multiplicativity recover the PVM, exactly as in Part I. Its marginals identify the rectangle projections above, and rectangles determine the measure uniquely. The companion proves each step.
This constructs a multidimensional observable from its compatible one-dimensional coordinates. Every real Borel function \(h\) of the outcomes then gives a self-adjoint operator, with the domain supplied by the same joint measure:
where \(\mu_u (F) = \langle u, E (F) u \rangle\). Here and below \(h\) is finite at every point; an almost-everywhere version requires specifying the spectral null sets. Bounded joint tests act on all of \(H\), even when the coordinate operators have different domains. Their common domain consists of the vectors for which \(\int | x |^2 \mathrm{d} \mu_u (x) < \infty\).
On \(L^2 (\mathbb{R}^d, \mathrm{d} x)\), take the coordinate multiplication operators \((X_j u) (x) = x_j u (x)\) on their maximal domains. Their spectral projections multiply by coordinate indicators, so the construction gives \(E (F) u = 1_F u\). For \(d = 3\) this is the position observable of the introduction. A unit vector has joint outcome law \(| u (x) |^2 \mathrm{d} x\). This law need not factor into its coordinate marginals: strong commutation permits a joint sharp measurement, while independence is a further property of that law in the chosen state.
An even simpler example is the pair \((A, A)\). It is strongly commuting, but its joint measure is concentrated on the diagonal \(\{ (x, x) : x \in \mathbb{R} \}\): it is the pushforward of \(E_A\) under \(x \mapsto (x, x)\). The two outcomes agree exactly. Thus the rectangle formula is a product of projections, not a claim that the scalar joint probabilities are products.
More generally a Borel map \(F : \mathbb{R}^d \rightarrow X\) transports the constructed observable by \(E_F (B) = E (F^{- 1} (B))\). Taking \(F (x, y) = x + iy\) will produce the normal operators in Section 2. Taking coordinates constrained to a sphere or an embedded manifold leads to Section 3. The geometry of the outcome space now enters through a concrete joint measure.
We can now add two compatible observables by applying an ordinary scalar function to their joint outcome. For \(d = 2\), write the coordinates as \(A, B\) and take \(h (x, y) = x + y\). The result is
This operator agrees with \(A + B\) on \(D (A) \cap D (B)\), but its domain may be larger. For instance, with \(B = - A\) the algebraic sum is zero on \(D (A)\), while \(C\) is the zero operator on all of \(H\): the joint measure lives on the line \(y = - x\).
Sum of compatible observables. If \(A\) and \(B\) strongly commute, the algebraic sum \(A + B\), defined on \(D (A) \cap D (B)\), is essentially self-adjoint, and its graph closure is the joint spectral integral:
To see why its closure is all of \(C\), take \(u \in D (C)\) and set \(u_n = E ([- n, n]^2) u\). These vectors lie in both coordinate domains. Moreover
Thus the intersection is a core for \(C\). The joint measure supplies both the value of the sum and the domain of its self-adjoint closure. The companion gives the full argument.
When strong commutation is unavailable, no joint spectral integral supplies the sum. A separate domain or form construction is needed.
The joint measure can also be identified by its bounded Fourier tests:
Strong commutation makes this a strongly continuous unitary representation of the additive group \(\mathbb{R}^d\). Conversely, a strongly continuous unitary representation of that group gives commuting strongly continuous one-parameter unitary groups on the coordinate axes. Stone's theorem recovers their self-adjoint generators and the joint construction applies. Fourier uniqueness identifies the joint spectral measure. One may therefore construct the coordinate operators first, or construct their compatible bounded evolutions first: both routes lead to the same multidimensional observable.
The joint construction uses the full spectral projections. We end this section by explaining why a formal commutator on convenient test vectors cannot replace that hypothesis. This caution can be read independently of the construction.
Commutation on a test domain means \(ABu = BAu\) for \(u\) in a specified common domain on which both products exist, often a dense subspace invariant under both operators. It is an assertion about those restrictions. A weaker, form version on a specified subspace \(\mathcal{D} \subseteq D (A) \cap D (B)\) is
This makes sense without applying either operator twice. On a common invariant domain, symmetry and algebraic commutation imply this identity. Neither assertion on an arbitrary dense test space determines how the full self-adjoint observables interact.
The distinction is visible in momentum on \(L^2 (0, 1)\). The realizations of \(- i \frac{\mathrm{d}}{\mathrm{d} x}\) with periodic boundary condition \(u (1) = u (0)\) and antiperiodic condition \(u (1) = - u (0)\) both act on \(C_c^{\infty} (0, 1)\) by the same formula; there they are the same operator, so they commute trivially. Nevertheless, their spectral projections onto the constant function \(1\) and \(e^{i \pi x}\) do not commute: these normalized eigenfunctions are neither parallel nor orthogonal. The test space is not an operator core for either realization; it does not retain their boundary conditions. A formal commutator on this space cannot even tell the two boundary conditions apart.
Being a common invariant core would not, by itself, make algebraic commutation a sufficient criterion either. A core determines each closed operator separately; strong commutation concerns their interaction through all bounded spectral tests. Nelson's Analytic vectors (1959) gives a counterexample even with essentially self-adjoint restrictions on a common invariant domain. We therefore use the resolvent or unitary-group identities as the criteria.
The real line entered because a Hermitian matrix has real eigenvalues. A normal matrix can have complex eigenvalues, but it still has an orthonormal eigenbasis. Its calculus applies a function to each of those eigenvalues, now using functions on \(\mathbb{C}\). The same change of outcome space works for unbounded operators.
A densely defined closed operator \(N\) is normal when \(N^{\ast} N = NN^{\ast}\), including equality of the product domains. Equivalently, \(D (N) = D (N^{\ast})\) and \(\| Nu \| = \| N^{\ast} u \|\) on this domain. Self-adjoint operators are the case in which the spectral values lie on the real axis.
Spectral theorem for normal operators. Normal operators on \(H\) correspond bijectively to nondegenerate \(\ast\)-homomorphisms
Equivalently, they correspond to projection-valued measures \(E\) on \(\mathbb{C}\) with \(E (\mathbb{C}) = I\). Writing \(\mu_u (\Delta) = \langle u, E (\Delta) u \rangle\), the coordinate and its domain are
The adjoint is the integral of \(\bar{z}\) on the same domain. Riesz–Markov and polarization work exactly as before; the spectral integral is built by cutting off large \(| z |\). See van Neerven, Chapter 10 and the proof companion.
The real and imaginary coordinate functions give self-adjoint operators
Complex observables are compatible pairs. The real and imaginary parts \(A, B\) of a normal operator \(N\) strongly commute, and \(N = A + iB\) on \(D (N) = D (A) \cap D (B)\). Conversely, every strongly commuting pair of self-adjoint operators arises in this way from exactly one normal operator.
Thus a complex observable is a pair of compatible real observables, with a joint calculus. The first half holds because the spectral projections of \(A\) and \(B\) are spectral projections of \(N\). The converse follows from Section 1: construct the joint PVM of \(A, B\) on \(\mathbb{R}^2\) and push it forward by \((x, y) \mapsto x + iy\). The identity \(| x + iy |^2 = x^2 + y^2\) gives precisely the stated domain. The companion gives the details.
There is a small domain point worth retaining. The algebraic expression \((N + N^{\ast}) / 2\) is initially defined only on \(D (N)\); its closure is \(A\), whose domain can be larger. For instance, if \(N = iB\) with \(B\) unbounded and self-adjoint, this expression is zero on \(D (B)\), while its closure is zero on all of \(H\). Even the words real part deserve a domain convention.
The Fourier description now uses two real parameters:
This is a strongly continuous unitary representation of \(\mathbb{R}^2\) and determines \(E\) by Fourier uniqueness. Conversely, its two commuting one-parameter subgroups recover \(A\) and \(B\) by Stone's theorem. The expression \(e^{itN}\) generally has neither boundedness nor unitarity: at \(z = x + iy\) its modulus is \(e^{- ty}\). The correct bounded tests are the characters of the additive plane.
The compatible-pair description already gives the complex observable. There is also a useful bounded encoding of it. The purpose of this optional construction is the same as that of the Cayley transform in Part I: to retain the entire observable, including its domain, inside a bounded operator. The complex plane requires a different scalar map.
A normal operator can have spectrum equal to all of \(\mathbb{C}\). Multiplication by \(z\) on \(L^2 (\mathbb{C}, \mathrm{d}^2 z)\) is an example. Every small disc around a proposed resolvent point supplies unit vectors on which \(N - z_0\) is arbitrarily small. Thus there need not be even one bounded resolvent, and the Cayley transform of Part I, Section 7 cannot be copied unchanged.
Instead use the homeomorphism from the plane to the open unit disc
Its calculus gives a bounded normal contraction \(T = N (I + N^{\ast} N)^{- 1 / 2}\). It satisfies \(\ker (I - T^{\ast} T) = \{ 0 \}\), meaning that its spectral measure puts no mass on the boundary circle. The boundary may still belong to its spectrum. Conversely, every normal contraction with this kernel condition gives a unique normal operator by
This also suggests a proof of the normal spectral theorem: construct \(T\) using the polar decomposition of \(N\), apply the bounded normal spectral theorem to \(T\), and transport its measure back from the disc to the plane. The companion checks the domains and uniqueness. Bounded functions still carry the full observable, even when the particular bounded functions called resolvents are unavailable.
The passage from \(\mathbb{R}\) to \(\mathbb{C}\) suggests a further question: why should the outcomes of an observable live in a vector space at all? A particle may have a position on a sphere, an orientation, or a phase on a circle. The pullback description of random variables already supplied the appropriate language in Part I, Section 3. We now develop that construction and ask which parts of the outcome space's structure it retains.
Let \(M\) be a second-countable Hausdorff manifold. A sharp observable with outcomes in \(M\) is a projection-valued measure \(E\) on its Borel sets, normalized by \(E (M) = I\).
Outcome-space dictionary. Normalized PVMs on \(M\) correspond bijectively to nondegenerate representations
The proof is the Riesz–Markov and polarization argument of Part I, Section 4 on a different locally compact space; see the companion. There is no distinguished coordinate function whose integral should be called the operator. The observable is the whole calculus. Each real Borel function \(h : M \rightarrow \mathbb{R}\) yields a self-adjoint observable \(h (E)\) with its own domain
A measurable map \(F : M \rightarrow L\) transports the observable by \(E_F (B) = E (F^{- 1} (B))\). Its bounded tests are \(g (F (E)) = (g \circ F) (E)\). For a continuous map that is not proper, the pullback of a \(C_0 (L)\) function need only be in \(C_b (M)\); the bounded extension of the calculus supplies exactly what is needed. Changes of coordinates therefore have a precise meaning without manipulating unbounded expressions by hand.
A useful model is position on \(M\). Choose a Radon measure \(\nu\) and take \(H = L^2 (M, \nu)\). Then \(j (f) u = fu\) and \(E (B) u = 1_B u\). A unit vector \(u\) has position law \(| u (m) |^2 \nu (\mathrm{d} m)\). This is the manifold version of the multiplication example in Part I: bounded tests act by multiplication, without a choice of global coordinates.
The same model works for square-integrable sections of a Hermitian vector bundle, replacing \(| u (m) |^2\) by the squared norm in its fibre. This allows internal degrees of freedom without changing the position outcome space. A connection or a Hamiltonian supplies further information about how those degrees of freedom evolve.
On the circle this is the spectral theorem for a unitary. On the sphere \(S^2 \subset \mathbb{R}^3\), it is equivalent to three commuting bounded self-adjoint operators satisfying
Section 1 constructs their joint measure on \(\mathbb{R}^3\). The displayed relation forces it to be concentrated on the sphere: the joint integral of \((x_1^2 + x_2^2 + x_3^2 - 1)^2\) is zero, so the scalar measures give no mass where this function is positive. Restricting the measure to \(S^2\) constructs the desired observable. Coordinate polynomials are dense in \(C (S^2)\), giving another way to see its uniqueness. The companion proves both statements.
Smoothness was not used to construct the spectral measure. Any second-countable locally compact Hausdorff space \(X\) admits the same dictionary between nondegenerate representations of \(C_0 (X)\) and normalized PVMs; Kustermans and Vaes explain this locally compact dictionary in operator-algebraic language. This includes spaces with singularities, as well as manifolds.
For a countable discrete space, an observable is simply a family of mutually orthogonal projections \(P_x\) with \(\sum_x P_x = I\) strongly, and \(j (f) = \sum_x f (x) P_x\). A graph-valued observable has this description when its outcomes are vertices. Adjacency or a graph Laplacian is additional structure; neither is determined by that collection of projections.
For a second-countable locally compact group \(G\), the group law suggests how to combine outcomes. Two \(G\)-valued sharp observables can be combined by multiplication if their spectral projections commute: they then have a joint PVM on \(G \times G\), which can be pushed forward by \((g, h) \mapsto gh\); see the companion. The order matters for a nonabelian group. Compatibility of the two calculi is what makes this an operation on jointly observed outcomes. For example, two commuting unitaries admit a joint calculus on the torus \(S^1 \times S^1\), and their product is its pushforward under \((z, w) \mapsto zw\). The product of two noncommuting unitaries is still a unitary, but it is an ordered operator construction, not the pushforward of a joint sharp law. A representation \(G \rightarrow U (H)\), which implements symmetries, should also be distinguished from a PVM with outcomes in \(G\).
A metric, smooth structure or symplectic form is extra data. The algebra \(C_0 (M)\) describes the topology of \(M\); multiplication by its elements does not by itself distinguish two Riemannian metrics or two symplectic forms on the same manifold. One must specify how the calculus interacts with the additional structure. Covariance is one such requirement.
Symmetry provides an established way to relate an observable to geometry. A particularly natural formulation is Mackey's system of imprimitivity: a group \(G\) acting on \(M\), a strongly continuous unitary representation \(U\), and a PVM satisfying
For a unit vector \(u\), the covariance identity implies \(\langle U (g) u, E (gB) U (g) u \rangle = \langle u, E (B) u \rangle\): transforming both the state and the outcome region leaves the probability unchanged. This is extra information relating the group action to the observable; the representation of \(C_0 (M)\) alone does not supply it.
The basic example is position on the line. Take \(M = G =\mathbb{R}\) acting by translation, \(H = L^2 (\mathbb{R})\), the position PVM \(E (B) u = 1_B u\), and \((U (a) u) (x) = u (x - a)\). Then \(U (a) E (B) U (a)^{\ast} = E (B + a)\). By Stone's theorem \(U (a) = e^{- iaP}\), where \(P\) is the momentum \(- i \frac{\mathrm{d}}{\mathrm{d} x}\), so this covariance is the Weyl form of the canonical commutation relation between position and momentum. In this case Mackey's theorem reduces to the Stone–von Neumann uniqueness theorem: on a separable Hilbert space, every such covariant pair is a direct sum of copies of this one.
On homogeneous spaces, Mackey's theory relates such covariant position observables to induced representations; see his 1949 paper. Thus the question has been studied, though usually under the names spectral measures, observables on an outcome space, and systems of imprimitivity rather than manifold-valued operators.
The sphere construction generalizes. Fix a proper embedding \(\iota : M \rightarrow \mathbb{R}^k\). Start with a strongly commuting self-adjoint tuple, construct its joint PVM, and check that it is concentrated on \(\iota (M)\). Transporting this measure back by \(\iota^{- 1}\) produces the manifold-valued observable. The support condition is additional geometric information, not a consequence of commutation. Conversely every sharp observable on \(M\) supplies such a tuple by integrating the embedded coordinates. The individual coordinates can be unbounded; their domains are supplied by the joint measure.
For a standard Borel outcome space without a chosen locally compact topology, the PVM formulation remains available. Equivalently, one can use a unital homomorphism on bounded Borel functions with the bounded pointwise continuity of Part I, Section 5. A particular continuous calculus and Radon regularity require a topology. Infinite-dimensional targets require this distinction: on an infinite-dimensional Hilbert space with its norm topology, \(C_0 (X) = \{ 0 \}\). Here vanishing at infinity means that each set \(\{ x : | f (x) | \geqslant \varepsilon \}\) is compact for \(\varepsilon > 0\); decay as the norm of \(x\) grows is a weaker condition. A nonzero function with compact superlevel sets would provide a compact neighbourhood, which that space does not have. Thus the measurable formulation can remain useful even when the locally compact \(C_0\) formulation cannot be copied.
So far we have extended the outcome space of a single observable. We now place several observables in one algebra and specify how their expectations are taken. In the matrix example of Part I, choosing a unit vector \(u\) turns an operator \(T\) into its expectation \(\langle u, Tu \rangle\). More generally, a state \(\omega\) on a unital algebra of bounded operators is a positive linear functional with \(\omega (I) = 1\). It plays the role of expectation.
The setting is a pair \((\mathcal{M}, \omega)\), where \(\mathcal{M} \subseteq B (H)\) is a von Neumann algebra and \(\omega\) is a normal state; such a pair is called a non-commutative probability space. A von Neumann algebra is a unital \(\ast\)-subalgebra of \(B (H)\) closed in the weak operator topology; normality means that the state preserves suprema of increasing bounded nets of positive operators.
Two examples explain this choice. The classical case is \(\mathcal{M}= L^{\infty} (\Omega, \mathbb{P})\), acting by multiplication on \(L^2 (\Omega, \mathbb{P})\), with \(\omega (f) = \int f \mathrm{d} \mathbb{P}\). This is ordinary probability, and normality of \(\omega\) is the monotone convergence theorem. In the other direction, \(\mathcal{M}\) can be smaller than \(B (H)\). For a composite system \(H = H_1 \otimes H_2\), the observables of the first part form \(\mathcal{M}= B (H_1) \otimes I\). A unit vector of \(H\) then gives \(\omega (T \otimes I) = \text{Tr} (\rho T)\) with a density matrix \(\rho\) on \(H_1\), which for an entangled vector is not a vector state of \(H_1\). Vector and density-matrix states are normal. Normality is what lets probability pass through the monotone limits used below: it makes spectral probabilities countably additive.
A real observable is now a nondegenerate homomorphism \(j : C_0 (\mathbb{R}) \rightarrow \mathcal{M}\). Its expectation functional \(\omega \circ j\) gives its law. The Riesz–Markov construction of Part I, Section 4 gives a unique Radon probability measure \(\mu\), of total mass one because \(\omega\) is normal and \(j\) nondegenerate (see below), such that
The spectral description makes this law explicit. If the calculus \(j\) takes values in \(\mathcal{M}\), its spectral projections \(E (\Delta)\) lie there as well. For the normal state \(\omega\),
Normality gives countable additivity, and the second identity now holds for every bounded Borel function.
Thus each individual observable has an ordinary probability law. Its image algebra is commutative because its scalar test functions commute. The quantum feature appears in comparisons between different observables: the images of two homomorphisms \(j_1\) and \(j_2\) need not commute with one another. The matrix example below will make this difference explicit.
Knowing the two separate laws does not determine mixed expectations. A joint non-commutative distribution records all finite products of bounded tests, keeping track of their order:
Here each index \(k_r\) selects one of the observables, and the homomorphisms have been extended to their bounded Borel calculi. All these products are bounded operators, so their expectations make sense before any moment or domain assumptions.
The Pauli matrices of Section 1 show what the individual laws leave out. We add a unit vector:
In the vector state \(\omega (T) = \langle u, Tu \rangle\), both observables have law \((\delta_{- 1} + \delta_1) / 2\). Nevertheless \(\omega (\sigma_1 \sigma_2) = i\) and \(\omega (\sigma_2 \sigma_1) = - i\). Replacing \(\sigma_2\) by \(- \sigma_2\) leaves the two individual laws unchanged and reverses the mixed expectation. Thus marginal laws do not determine even these simplest ordered tests. For real tests, reversing two factors gives the complex conjugate expectation; the order still carries information absent from the marginals.
No classical pair of random variables reproduces these numbers: for real random variables \(X, Y\) the expectation \(\mathbb{E} [XY]\) is real, while \(\omega (\sigma_1 \sigma_2) = i\). The symmetrized product, \(\omega ((\sigma_1 \sigma_2 + \sigma_2 \sigma_1) / 2) = 0\), is what two independent fair signs would give; the obstruction lies in the order. The companion verifies the example.
A joint sharp measurement of two real observables would be a PVM on \(\mathbb{R}^2\), and by the converse half of the joint spectral theorem its marginals strongly commute. Noncommuting observables therefore cannot be marginals of such a joint sharp measurement, although their ordered mixed expectations still make sense. This is a compatibility condition on two observables, not an additional restriction on either one alone.
The same construction describes an \(M\)-valued quantum observable by a nondegenerate representation \(j : C_0 (M) \rightarrow \mathcal{M}\) on \(H\). The normal state \(\omega\) gives its Radon probability law through \(\omega \circ j\). The same distinction between a single observable and a family applies. The outcome-space constructions of Section 3 therefore fit this same probability language.
Recall the pullback \(f \mapsto f \circ X\) for random variables in Part I, Section 3. It suggests a geometric interpretation: a map of spaces acts on functions in the opposite direction. Non-commutative geometry retains this algebraic description when multiplication in the target algebra no longer commutes. A represented algebra plays the role of the functions on a space; a homomorphism describes a map in the opposite direction. Thus \(j\) is naturally read as the pullback \(f \mapsto f \circ X\) along a real-valued observable \(X\) on that non-commutative space, although \(X\) exists only through \(j\). A state supplies integration, and \(\omega \circ j\) is its pushforward law. This interpretation uses the nondegeneracy of Part I, Section 2.
For an early algebraic formulation of quantum mechanics, see I. E. Segal's Postulates for General Quantum Mechanics (1947). It is useful historical background for the language of observables, algebras and states.
The normal-state setting above is enough for the main argument. The following qualification explains what can fail without it. Normality is a substantive assumption. Vector states and density-matrix states are normal, and give countably additive laws through the spectral projections. For a general state there are two different questions.
First, restricting \(\omega\) to the continuous calculus gives a positive functional on \(C_0 (\mathbb{R})\). Riesz–Markov always gives a finite measure, but it is a probability measure only when that functional has norm one. Equivalently, the state must give full mass to the cutoffs \(\chi_n\) of Part I, Section 2: \(\omega (j (\chi_n)) \rightarrow 1\). Normality and nondegeneracy ensure this condition.
Second, does this Riesz measure agree with \(\Delta \mapsto \omega (E (\Delta))\) on every Borel set? Normality ensures this too, by preserving spectral limits. Full mass on the continuous calculus alone does not.
For a concrete warning, let \(A\) multiply by \(t\) on \(L^2 (0, 1)\). The companion constructs a state that evaluates every continuous spectral test at 0, so its Riesz law is \(\delta_0\). Yet \(E_A (\{ 0 \}) = 0\), since the singleton has Lebesgue measure zero. Hence \(\omega (E_A (\{ 0 \})) = 0\) while the Riesz law assigns that set mass one. This state is not normal. The example separates having a probability law for continuous tests from evaluating all spectral events by that same law.
The Part II companion proves the relation between the algebra, its spectral projections and the law in a normal state, and proves the counterexample above.
An outcome space tells us where measurements take their values. What additional data describe distances, symmetries or Hamiltonian dynamics? Symmetry has already appeared through covariance. This section turns to distance; Section 6 considers Hamiltonian structure. These two perspectives use data beyond the spectral measure.
The algebra \(C (M)\) remembers the topology of a compact manifold. Smooth and metric structure require further data. In Connes's compact spectral triple \((\mathcal{A}, H, D)\), a unital \(\ast\)-algebra \(\mathcal{A} \subseteq B (H)\) is accompanied by a self-adjoint \(D\) with compact resolvent. Each \(a \in \mathcal{A}\) preserves the domain \(\text{Dom} D\), and \([D, a]\) on that domain extends to a bounded operator. This domain requirement is part of the geometry.
The circle shows the idea with nothing beyond Section 1. Take \(M =\mathbb{R}/ 2 \pi \mathbb{Z}\), \(H = L^2 (M)\), the algebra \(\mathcal{A}= C^{\infty} (M)\) acting by multiplication, and the periodic momentum \(D = - i \frac{\mathrm{d}}{\mathrm{d} \theta}\). Its eigenvalues are the integers, with eigenfunctions \(e^{ik \theta}\), so its resolvent is compact. For smooth \(f\), the Leibniz rule gives \([D, f] u = - if' u\), hence \(\| [D, f] \| = \sup | f' |\). A function with \(| f' | \leqslant 1\) changes by at most the length of the shorter arc between two points, and the distance to a fixed point, suitably smoothed, nearly attains this bound. Thus the supremum of \(| f (p) - f (q) |\) over such \(f\) is the arc-length distance from \(p\) to \(q\).
The same holds in higher dimensions. On a compact connected Riemannian spin manifold without boundary, take \(\mathcal{A}= C^{\infty} (M)\), the Hilbert space of square-integrable spinors, and the self-adjoint Dirac operator \(D\). For real \(f\), the commutator \([D, f]\) is Clifford multiplication by \(\mathrm{d} f\), up to the conventional factor of \(i\). Its norm is the supremum norm of the gradient. A bound on the commutator is therefore a bound on how rapidly a function can change along the manifold. Taking the largest possible change between two points under that bound recovers their distance:
Integrating the gradient along paths proves one inequality; smooth approximation of distance functions gives the reverse; the companion gives the argument. See Connes, Definition 1 and Proposition 2.
The present viewpoint describes \(D\) by its bounded calculus, including its resolvents and unitary group; compact resolvent implies \(f (D)\) is compact for \(f \in C_0 (\mathbb{R})\). Geometry also uses how \(D\) interacts with the represented coordinate algebra. Knowing its calculus alone does not specify that algebra or those commutators. Indeed, spectral data alone do not fix the geometry: Milnor (1964) exhibited two flat 16-dimensional tori that are not isometric although their Laplacians have the same eigenvalues with the same multiplicities. As observables, the two Laplacians are therefore unitarily equivalent; the difference between the manifolds is visible only through the represented functions. Allowing \(\mathcal{A}\) to be noncommutative extends this geometric framework beyond ordinary manifolds, while retaining the same care with self-adjointness and domains.
The convergence notion in Part I, Section 8 extends directly to observables with outcomes in \(M\): on a fixed Hilbert space, require \(j_n (f) u \rightarrow j (f) u\) for every \(f \in C_0 (M)\) and every \(u\). There need not be a distinguished resolvent on \(M\); the bounded tests still define the convergence of the observables.
One can also ask whether such convergence retains a specified metric, symmetry, or Poisson structure. Which additional estimates express that structure? Convergence of the bounded tests alone does not control derivatives or unbounded coordinate functions.
A symplectic manifold provides a useful test of what this means. Its symplectic form determines a Poisson bracket \(\{ f, g \}\) on smooth functions, expressing Hamiltonian dynamics. For bounded smooth tests, a sharp observable on this manifold still has \([j (f), j (g)] = 0\): all its scalar tests belong to a commutative algebra. Thus specifying a phase-space outcome does not by itself turn its Poisson bracket into a quantum commutator.
Quantization asks for different data. Choose an algebra \(\mathcal{A}\) of smooth functions in \(C_0 (M)\), closed under products, conjugation and Poisson brackets; \(C_c^{\infty} (M)\) is a natural choice. A family of linear maps
assigns bounded operators to these tests. Here \(\hbar > 0\) is the parameter tending to zero in the classical limit. A positive quantization map additionally sends nonnegative tests to positive operators. Positivity alone does not make it a homomorphism.
First, multiplying two quantized tests should approach quantizing their product. For fixed \(f, g \in \mathcal{A}\), this asks for
Second, the commutator, rescaled by the classical-limit parameter, should approach the quantization of the Poisson bracket. This is the Dirac condition:
These conditions recover multiplication and relate quantum commutators to Hamiltonian dynamics. We use \([X, Y] = XY - YX\) and the Poisson convention \(\{ q_j, p_k \} = \delta_{jk}\) in canonical coordinates. One also asks for \(\| Q_{\hbar} (f) \| \rightarrow \| f \|_{\infty}\), so the classical information is retained. These are standard ingredients of strict quantization, whose full formulation also specifies how the algebras at different \(\hbar\) fit together. They are not an existence theorem for arbitrary manifolds or arbitrary positive maps. See Landsman's strict quantization of coadjoint orbits for a concrete construction with positive maps. For a finite-dimensional family, the fuzzy sphere is the basic example: spin-\(j\) matrices rescaled by \(1 / \sqrt{j (j + 1)}\) satisfy \(X_1^2 + X_2^2 + X_3^2 = I\), like the sphere coordinates of Section 3, but their commutators are of order \(\hbar_j = 1 / \sqrt{j (j + 1)}\) instead of zero. For \(j = 1 / 2\) they are the Pauli matrices of Section 1 divided by \(\sqrt{3}\).
This explains a limit of the sharp-observable picture. If every \(Q_{\hbar}\) were an exact homomorphism, all its commutators would vanish. The Dirac condition and the norm limit would then force \(\{ f, g \} = 0\) for every pair of tests. A nontrivial Poisson structure therefore calls for a different notion of quantization, rather than an additional identity imposed on a single sharp calculus. The companion proves this elementary obstruction.
I would like to understand how quantitative control of these defects relates to convergence of states, outcome laws and dynamics. That is a direction for further exploration; the spectral-triple discussion in Section 5 supplies an established way to retain metric information.
For applications it helps to specify three things: the outcome space, the bounded tests describing the observable, and the extra structure those tests are required to respect. For a real sharp observable one coordinate encodes the whole observable. On a manifold, and especially in the symplectic setting, keeping those three choices explicit prevents several quite different constructions from being conflated.
Proofs and complements for this part.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, revised and enlarged edition, Academic Press, 1980, especially Chapters VII–VIII. Publisher. A systematic account of spectral theory and unbounded operators.
G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, second edition, Graduate Studies in Mathematics 157, AMS, 2014. Chapters 2–3 cover domains, adjoints and the spectral theorem. Section 4.2 explains commutation through the bounded functional calculus. Author's book page and freely available text.
E. Nelson, Analytic vectors, Annals of Mathematics (2) 70 (1959), 572–615. doi:10.2307/1970331. Analytic vectors, integration of infinitesimal representations and the distinction between algebraic and strong commutation.
J. van Neerven, Functional Analysis, Cambridge University Press, 2022. Chapters 9–10 treat bounded and unbounded normal operators; Chapter 15 connects states and observables. Chapter 10.
G. W. Mackey, Imprimitivity for Representations of Locally Compact Groups I, Proceedings of the National Academy of Sciences 35 (1949), 537–545. doi:10.1073/pnas.35.9.537. Covariant spectral measures on homogeneous outcome spaces and their relation to induced representations.
N. P. Landsman, Strict quantization of coadjoint orbits (1998), arXiv:math-ph/9807027. Definition 2, Proposition 1 and Theorem 2 give the classical-limit conditions and a construction using positive quantization maps on integral coadjoint orbits of compact connected Lie groups.
A. Connes, Noncommutative Geometry, Academic Press, 1994. Author's freely available book. Chapter II.1 develops the algebra–space dictionary. See also Noncommutative geometry and reality, Journal of Mathematical Physics 36 (1995), 6194–6231, author's text: Definition 1 and Proposition 2 introduce spectral triples and recover geodesic distance from the Dirac commutator.
J. Kustermans and S. Vaes, The operator algebra approach to quantum groups, Proceedings of the National Academy of Sciences 97 (2000), 547–552. doi:10.1073/pnas.97.2.547. Section 1 explains the locally compact dictionary and morphisms into multiplier algebras.
J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proceedings of the National Academy of Sciences 51 (1964), 542. doi:10.1073/pnas.51.4.542. Two non-isometric flat tori with the same Laplace spectrum.
I. E. Segal, Postulates for General Quantum Mechanics, Annals of Mathematics (2) 48 (1947), 930–948. Original paper. An early algebraic formulation of quantum mechanics.