3 · spectral theory and generalized quantum states

the need for generalized states

Let $A$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$. The spectral theorem asserts the existence of a unique projection-valued measure $E_A$ such that $$ A=\int_{\sigma(A)}\lambda\,dE_A(\lambda). $$ When $\sigma(A)$ is discrete the projections $E_A(\{\lambda\})$ are finite-rank (or at least supported on eigenspaces inside $\mathcal{H}$) and the corresponding eigenvectors lie in $\mathcal{H}$. When $\sigma(A)$ contains a continuous part the spectral projections $E_A(\Delta)$ for Borel sets $\Delta$ of positive spectral measure are nontrivial, yet there are typically no eigenvectors in $\mathcal{H}$ belonging to individual points of the continuum.

Nevertheless, Dirac's bra-ket notation and the everyday practice of quantum mechanics freely employ "eigenvectors" $|x\rangle$, $|p\rangle$, $|E\rangle$ satisfying $$ Q|x\rangle=x|x\rangle,\qquad P|p\rangle=p|p\rangle, $$ together with continuum normalization $$ \langle x|x'\rangle=\delta(x-x'),\qquad\langle p|p'\rangle=\delta(p-p'). $$ These objects cannot be vectors in $\mathcal{H}$. They are generalized eigenvectors living in a larger dual space.

remark (what the pvm already gives). the spectral theorem never requires continuum eigenvectors as Hilbert-space vectors. spectral projections $E_A(\Delta)$ are the rigorous objects. generalized eigenvectors refine that picture into a pointwise eigenfunction expansion useful for calculations and for matching Dirac notation.

test-function spaces and distributions

Let $\Phi$ be a nuclear Frechet space densely and continuously embedded in $\mathcal{H}$. Typical choices relevant to quantum mechanics are the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ of smooth functions of rapid decrease; spaces $\mathcal{D}(K)$ of smooth functions supported in a compact set $K$; and spaces of smooth vectors for a given Lie-group representation (for instance, the analytic vectors of the harmonic-oscillator Hamiltonian).

The dual $\Phi'$ consists of continuous linear functionals on $\Phi$: the generalized functions or distributions. The Hilbert space sits between them via the Riesz identification of $\mathcal{H}$ with its antidual: $$ \Phi\subset\mathcal{H}\simeq\mathcal{H}'\subset\Phi'. $$ Every vector $\psi\in\mathcal{H}$ defines a regular distribution $\phi\mapsto\langle\psi|\phi\rangle$ for $\phi\in\Phi$. The embedding is continuous and dense.

definition (gelfand triple). a rigged Hilbert space is a triple $\Phi\subset\mathcal{H}\subset\Phi'$ as above, with $\Phi$ nuclear, densely and continuously embedded in $\mathcal{H}$, and $\Phi'$ its continuous dual (with the strong dual topology).

generalized eigenvectors

Assume $A$ maps $\Phi$ continuously into itself, so that the dual operator $A'$ is well-defined on $\Phi'$. A generalized eigenvector of $A$ belonging to $\lambda\in\sigma(A)$ is a nonzero functional $F_\lambda\in\Phi'$ such that $$ \langle F_\lambda|A\phi\rangle=\lambda\langle F_\lambda|\phi\rangle\qquad\text{for all }\phi\in\Phi. $$ In Dirac notation one writes $A|F_\lambda\rangle=\lambda|F_\lambda\rangle$ with $|F_\lambda\rangle\in\Phi'$.

example (position). on $L^2(\mathbb{R})$ with $\Phi=\mathcal{S}(\mathbb{R})$, the generalized eigenvectors of $Q$ are the Dirac deltas: $|x\rangle=\delta_x\in\mathcal{S}'(\mathbb{R})$ with $\langle x|\phi\rangle=\phi(x)$.

example (momentum). the generalized eigenvectors of $P=-i\hbar d/dx$ are the plane waves $$ |p\rangle=\frac{1}{\sqrt{2\pi\hbar}}e^{ipx/\hbar}, $$ understood as tempered distributions. both families lie in $\mathcal{S}'$ but not in $L^2$.

the nuclear spectral theorem

When $\Phi$ is nuclear, the spectral theorem admits a far sharper form (the Gelfand-Maurin theorem). There exists a measure $\mu$ on $\sigma(A)$ and a family of generalized eigenvectors $\{|F_\lambda\rangle\}_{\lambda\in\sigma(A)}$ such that every $\phi\in\Phi$ possesses the generalized eigenfunction expansion $$ \phi=\int_{\sigma(A)}|F_\lambda\rangle\langle F_\lambda|\phi\rangle\,d\mu(\lambda). $$ In other words, the identity admits the resolution $$ I=\int_{\sigma(A)}|F_\lambda\rangle\langle F_\lambda|\,d\mu(\lambda), $$ where the integral converges in the weak sense appropriate to $\Phi'$. For the position operator one recovers $$ I=\int_{-\infty}^{\infty}|x\rangle\langle x|\,dx. $$ The same statement holds for any complete set of strongly commuting self-adjoint operators, with a joint generalized eigenbasis labelled by points of the joint spectrum.

remark (relation to the pvm). the nuclear expansion refines $A=\int\lambda\,dE_A(\lambda)$: formally $dE_A(\lambda)=|F_\lambda\rangle\langle F_\lambda|\,d\mu(\lambda)$. the projection-valued measure remains the primary Hilbert-space object; the Gelfand-Maurin data make the continuum "eigenvectors" precise.

fourier transformation in the distributional setting

The Fourier transform $$ (\mathcal{F}\phi)(p)=\frac{1}{(2\pi\hbar)^{n/2}}\int e^{-ip\cdot x/\hbar}\phi(x)\,dx $$ is an automorphism of the Schwartz space $\mathcal{S}(\mathbb{R}^n)$. By duality it extends to an automorphism of the tempered distributions $\mathcal{S}'$: $$ \langle\mathcal{F}T|\phi\rangle=\langle T|\mathcal{F}\phi\rangle. $$ In particular the Fourier transform interchanges the generalized eigenbases of position and momentum (up to the conventional factors of $2\pi\hbar$ fixed by the normalization of $|p\rangle$).

Plancherel's theorem and the Fourier inversion formula remain valid on $\mathcal{S}$ and extend by continuity to $L^2$, giving a rigorous justification of the unitarity of the Fourier transform on the physical Hilbert space.

example (free evolution). the free propagator on the line is multiplication by $e^{-itp^2/(2m\hbar)}$ in the momentum representation. transporting that multiplier back by $\mathcal{F}^{-1}$ yields the distributional kernel of $e^{-itH/\hbar}$ on configuration space.

kernels and the schwartz kernel theorem

Many operators of quantum mechanics are integral operators. The Schwartz kernel theorem states that every continuous linear map $K:\Phi\to\Phi'$ is represented by a unique distributional kernel $K(x,y)\in\Phi'_x\,\widehat{\otimes}\,\Phi'_y$: $$ \langle K\phi|\psi\rangle=\langle K(\,\cdot\,,\,\cdot\,)|\psi\otimes\phi\rangle. $$ In the familiar $L^2$ setting one recovers ordinary integral kernels when they happen to be locally integrable functions; in general the kernel is a distribution. Green's functions, propagators, and the kernels of spectral projections all find their natural home in this framework.

example (position projector density). the formal kernel $\langle x|\psi\rangle=\psi(x)$ is the evaluation functional, while $\langle x|A|y\rangle$ for a differential operator $A$ is a distributional kernel supported on the diagonal and involving derivatives of $\delta(x-y)$.

continuous spectra in concrete systems

free particle. the Hamiltonian $H=P^2/(2m)$ has purely continuous spectrum $[0,\infty)$. its generalized eigenfunctions are plane waves, and the spectral measure is Lebesgue measure on the positive axis (with multiplicity two in one dimension for the two directions of motion).

scattering states. for a short-range potential the continuous spectrum of the Schrodinger operator remains $[0,\infty)$ under standard hypotheses. generalized eigenfunctions are the distorted plane waves (Jost solutions); their asymptotic form defines the scattering amplitude.

bloch waves. on a periodic lattice the generalized eigenfunctions are labelled by quasi-momentum in the Brillouin zone; the spectral measure is supported on the union of energy bands.

In each case the rigged-Hilbert-space formalism converts formal Dirac manipulations into theorems about nuclear spaces and duals.

preparation for quantum fields

A quantum field is not an operator-valued function but an operator-valued distribution. For a real scalar field one has a linear map $$ f\mapsto\phi(f),\qquad f\in\mathcal{S}(\mathbb{R}^{1,3}), $$ taking values in operators on a Hilbert space (or, more precisely, in operators affiliated with a von Neumann algebra). The same dual-pairing language developed for generalized eigenvectors applies verbatim: $$ \phi(f)=\int\phi(x)f(x)\,d^4x, $$ where $\phi(x)$ itself is only a symbolic expression for the distribution. The Wightman axioms, the reconstruction of fields from vacuum expectation values, and the treatment of local algebras all rest on the theory of distributions and nuclear spaces introduced here.

summary

The simple Hilbert-space picture is enlarged to a Gelfand triple $\Phi\subset\mathcal{H}\subset\Phi'$. Self-adjoint operators acquire complete systems of generalized eigenvectors lying in $\Phi'$. Continuous spectra, Dirac delta normalization, Fourier transformation, and distributional kernels receive precise analytic meaning. The nuclear spectral theorem supplies the rigorous continuum expansion that complements the projection-valued spectral theorem $A=\int_{\sigma(A)}\lambda\,dE_A(\lambda)$.

With this machinery in place one is prepared to treat quantum fields as operator-valued distributions and to pass from the kinematics of a fixed number of particles to the full local structure of relativistic quantum field theory.