Mathematical Quantum Mechanics Glossary

Mathematical quantum mechanics reuses familiar words in several technical senses. “Normal,” “pure,” “complete,” and “continuous” are particularly easy to misread. This glossary explains the intended distinctions in prose; use the Definition Index for exact predicates and the linked articles for full developments.

Operator and domain vocabulary

Adjoint. For a densely defined operator, the adjoint is determined by which vectors make the associated pairing bounded. Its domain is part of the construction. A formal conjugate differential expression does not specify that domain. See Adjoints of Unbounded Operators.

Closure and closability. Closing an operator means closing its graph. A closable operator has a graph closure that is still the graph of a single-valued operator. These terms do not refer to closing the physical system to environmental interactions. See Closed and Closable Operators.

Core. A smaller domain that is dense in the full operator domain for the graph norm. Ordinary Hilbert-space density is weaker: it does not guarantee that the operator’s action is approximated. See Dense Domains and Tracking Domains.

Essential self-adjointness. A symmetric operator has a self-adjoint closure, so its initial action determines a unique self-adjoint realization. This is not the assertion that every symmetric operator has a unique extension. See Essential Self-Adjointness.

Form domain. The vectors on which a quadratic form is finite. For a nonnegative self-adjoint $A$, it is $D(A^{1/2})$, usually larger than $D(A)$. A finite energy form does not automatically mean that $A\psi$ exists as a vector. See Quadratic Forms.

Normal operator. A bounded operator satisfying $A^\dagger A=AA^\dagger$. This operator classification has a different meaning from a normal state or normal channel. The unbounded definition must also handle equality of the relevant operators and domains. See The Bounded Spectral Theorem.

Spectral and representation vocabulary

Continuous spectrum. In the point/continuous/residual classification, spectral values where $A-\lambda I$ is injective with dense range but has no bounded everywhere-defined inverse. “Absolutely continuous spectral measure” is a different, more refined statement. See Point, Continuous, and Residual Spectrum.

Generalized eigenvector. An eigenfunctional in a specified test-space dual, not necessarily a Hilbert-space vector. Its admissibility depends on the chosen rigging and operator action there. See Distributions as Generalized Vectors.

Multiplicity. The number of independent spectral channels at a spectral value. An eigenvalue’s degeneracy is its eigenspace dimension; continuous spectral multiplicity is encoded by fibers in a spectral representation. See Complete Sets of Commuting Observables.

Regular representation of the CCR. A Weyl representation with the continuity needed to recover self-adjoint generators for the one-parameter unitary groups. “Regular” here does not merely mean that a displayed wavefunction looks smooth. See Regularity Assumptions.

Representation. A structure-preserving map from an algebra or group into operators. A representation may be reducible, and inequivalent representations need not be connected by a unitary on the chosen spaces. See Representations.

Strong commutation. Commutation of the spectral projections of self-adjoint observables. A commutator that vanishes only on a selected common domain is not automatically this statement. See Compatible Observables.

States, maps, and dynamics vocabulary

Faithful state. A state that assigns strictly positive expectation to every nonzero positive algebra element. For a normal state on $B(\mathcal H)$, its density operator has zero kernel. In infinite dimension that density operator need not have a bounded inverse. See Normal States.

Normal state. A positive normalized functional that respects increasing bounded operator limits on a von Neumann algebra. On $B(\mathcal H)$ it is represented by a density operator. Normality is a continuity condition, not purity. See Normal and Singular States.

Pure state. An extreme point of the state space of the specified observable algebra. For normal states on $B(\mathcal H)$ it means rank one. A restriction to a smaller algebra can cease to be pure. See Pure and Mixed States.

Separable state. A state in the trace-norm closed convex hull of product states in the normal bipartite setting. The word does not refer to separability of the underlying Hilbert space, which means existence of a countable dense subset. See Separable and Entangled Mixed States and Separability Assumptions.

Complete positivity. Positivity that survives adjoining every finite-dimensional untouched ancilla. “Complete” here does not refer to metric completeness or to completeness of scattering wave operators. See Completely Positive Maps.

Conservative semigroup. A CP evolution preserving trace on states, equivalently preserving the identity on observables. Formal infinitesimal trace cancellation on an unproved domain is weaker. See Bounded and Unbounded Generators.

Strong continuity. Continuity of each orbit in the stated space. For a trace-class semigroup it means trace-norm continuity of every fixed input, not convergence uniformly over all inputs. See Strong Continuity.

Exercise: purity depends on the observable algebra

For a qubit in $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$, restrict its state functional to the diagonal algebra. Is the restricted state pure? Is it normal?

Solution

On a diagonal observable $\operatorname{diag}(a,b)$, the expectation is $(a+b)/2$. This is the equal mixture of the two evaluation states that select $a$ or $b$, so the restriction is not pure on the diagonal algebra. It is normal, as all states on this finite-dimensional algebra are. On the full matrix algebra the original rank-one state is pure. This links algebraic vocabulary to superselection: changing the accessible algebra changes which distinctions are operational.

References

• Bratteli, Ola, and Derek W. Robinson. Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed. Springer, 1987. doi:10.1007/978-3-662-02520-8. Algebraic states, representations, and normality.
• Hall, Brian C. Quantum Theory for Mathematicians. Graduate Texts in Mathematics 267, Springer, 2013. doi:10.1007/978-1-4614-7116-5. Operator, spectral, and quantum terminology.