Resolution of Identity

A resolution of identity expresses every vector as the sum or integral of components selected by a complete family of events. In a discrete basis it is an orthogonal series; in continuous spectral theory it is a normalized projection-valued measure. The identity is meaningful only together with the topology or pairing in which the sum or integral is understood.

Helpful background. Projection-Valued Measures supplies strong countable additivity. The Unbounded Spectral Theorem explains why spectral resolution does not require an eigenbasis.

Orthogonal completeness is a strong limit

Let $\{e_n\}$ be an orthonormal basis of a separable Hilbert space. The finite-rank projectors

$$P_N=\sum_{n=1}^N|e_n\rangle\langle e_n|$$

satisfy

$$\|P_N\psi-\psi\|^2 =\sum_{n>N}|\langle e_n,\psi\rangle|^2\longrightarrow0.$$

Thus the notation $\sum_n|e_n\rangle\langle e_n|=I$ means strong convergence: each vector’s expansion converges in Hilbert norm. Parseval’s equality, $\|\psi\|^2=\sum_n|\langle e_n,\psi\rangle|^2$, expresses the same completeness quantitatively.

For an infinite-dimensional space the convergence is not in operator norm, since $\|I-P_N\|=1$ for every finite $N$. The omitted subspace always contains a unit vector.

If the orthonormal family is incomplete, the strong sum is the projection onto its closed span, not the identity. Orthonormality and completeness are separate statements.

Degeneracy requires whole eigenspaces

Suppose $A$ has pure point spectral type with mutually orthogonal eigenspace projections $P_\lambda$. Completeness reads

$$\sum_\lambda P_\lambda=I$$

strongly. Each $P_\lambda$ can be expanded in an orthonormal basis of its eigenspace, but it need not be rank one.

In a spin-independent Hamiltonian acting on spatial wavefunctions tensored with $\mathbb C^2$, every spatial eigenfunction gives two spin states. Summing only one spin representative per energy omits half of the corresponding state space. The full projector onto that energy includes both spin directions.

A spectrum containing discrete eigenvalues does not imply that their projections exhaust the identity. In a model with bound and scattering states,

$$I=P_{\mathrm{bound}}+P_{\mathrm{continuous}},$$

where the first term is a strong eigenprojection sum and the second can be nonzero. A completeness claim must account for both subspaces, and the continuous part can itself contain different spectral types.

Spectral normalization and partitions

For a normalized PVM $E$ on $(\Omega,\Sigma)$,

$$I=E(\Omega)=\int_\Omega 1\,dE.$$

Any finite or countable measurable partition $\Omega=\bigsqcup_j\Delta_j$ gives $I=\sum_jE(\Delta_j)$ with strong convergence. Components belonging to distinct bins are orthogonal, so

$$\|\psi\|^2=\sum_j\|E(\Delta_j)\psi\|^2.$$

This is a spectral version of Parseval’s equality. A partition of a continuous spectrum resolves the identity without introducing any point eigenvectors.

For position on $\mathbb R$, partition the line into unit intervals. The projections multiply by their indicator functions, splitting a wavefunction into pieces of disjoint support. The countable sum reconstructs the wavefunction in $L^2$, even though every exact-position singleton projection is zero.

What a continuous ket integral means

The notation

$$I=\int_{\mathbb R}|x\rangle\langle x|\,dx$$

is a useful shorthand in a representation with suitable generalized position vectors. It is not an ordinary integral of bounded rank-one projectors on $L^2(\mathbb R)$, because $|x\rangle$ is not a vector in that Hilbert space.

For suitable test functions $\phi,\psi$, the weak meaning is

$$\langle\phi,\psi\rangle =\int_{\mathbb R}\overline{\phi(x)}\psi(x)\,dx.$$

Equivalently, the identity kernel is the distribution $\delta(x-y)$, acting on test functions. Values of a general $L^2$ vector at individual points are not intrinsically defined, so pointwise evaluation requires an additional test-space structure.

The invariant operator statement remains $E_Q(\mathbb R)=I$. Rigged Hilbert Spaces develops the extra spaces that make generalized-ket pairings precise.

For momentum, normalization conventions matter. With $\langle x|p\rangle=(2\pi\hbar)^{-1/2}e^{ipx/\hbar}$, the formal completeness integral uses $dp$. Dropping the normalization changes the measure factor; one must not mix the two conventions.

A resolution need not be projection-valued

An overcomplete family such as coherent states can also resolve the identity, but its individual rank-one operators are not mutually orthogonal. Such an integral defines a POVM structure rather than the spectral PVM of a single real self-adjoint observable.

The distinction is visible in probability addition: orthogonal spectral components satisfy a Pythagorean norm sum, while overlapping state vectors do not form an orthogonal decomposition. “Resolution of identity” describes normalization; it does not by itself establish orthogonality or sharpness.

Exercises

Let $\{e_n\}$ be an orthonormal basis and retain only the even-indexed vectors. Find the strong sum of their rank-one projections and the missing norm of a vector $\psi$.

Solution

The sum is the projection $P_{\mathrm{even}}$ onto the closed span of the even-indexed vectors. The missing squared norm is $\|(I-P_{\mathrm{even}})\psi\|^2=\sum_{n\text{ odd}}|\langle e_n,\psi\rangle|^2$. The retained family remains orthonormal but is not complete.

For position, show that the projections onto $[-N,N]$ converge strongly to $I$ and compute their operator-norm error.

Solution

The squared vector error is $\int_{|x|>N}|\psi(x)|^2\,dx\to0$. The complement contains normalized functions supported outside $[-N,N]$, so $\|I-E_Q([-N,N])\|=1$ for every finite $N$.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — spectral representations and generalized physics notation.
• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012, doi:10.1007/978-94-007-4753-1 — spectral integrals and multiplicity.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — completeness and spectral subspaces.