Continuous Basis Notation

Continuous-basis notation compresses spectral transforms and operator kernels into symbols resembling matrix elements. To read it correctly, identify whether a bracket is a test-functional pairing, an almost-everywhere spectral coefficient, or a distribution kernel. These objects permit different operations; matrix multiplication and diagonal traces cannot simply be copied from finite dimension.

Required background. Position Kets and Momentum Kets construct the coefficients. Delta Normalization fixes their integration measure. This page uses the Schwartz rigging of $L^2(\mathbb R,dx)$.

Identify the type of each bracket before calculating

For $\psi\in\mathcal S$, $\langle x|\psi\rangle$ is evaluation at $x$. For a general $L^2$ state it denotes an almost-everywhere position representative, whose individually chosen point values are not invariant data.

The expression $\langle x|p\rangle$ denotes a Fourier transform kernel. It is not the inner product of two Hilbert vectors. Similarly, $\langle x|A|y\rangle$ may denote a distribution on the pair of variables, even when $A$ is a bounded Hilbert-space operator.

An integral such as $\int|x\rangle\langle x|dx$ is a weak resolution: its meaning is obtained after applying it between suitable vectors. The measure, multiplicity labels, and convergence must accompany the formula. Generalized Eigenfunction Expansions gives the general spectral conditions.

The Schwartz kernel theorem explains operator matrix elements

Every continuous linear map $A:\mathcal S(\mathbb R)\to\mathcal S'(\mathbb R)$ has a unique tempered distribution kernel on $\mathbb R^2$. In the Hilbert pairing convention, its action can be expressed as

$$\langle\phi,A\psi\rangle =K_A\bigl(\overline\phi\otimes\psi\bigr),$$

with the pairing interpreted distributionally if the output is not a Hilbert vector. A bounded operator on $L^2$ meets this continuity condition by composing the continuous inclusions $\mathcal S\to L^2\to\mathcal S'$, with the regular-distribution convention on the last inclusion.

The kernel theorem is an imported result about nuclear test spaces. It supplies a distribution kernel, not necessarily an ordinary function, a pointwise evaluation $A\delta_y$, or a justified iterated integral for arbitrary inputs.

For the identity, position, and momentum actions on Schwartz functions, the kernels are

$$\begin{aligned} K_I(x,y)&=\delta(x-y),\\ K_X(x,y)&=x\delta(x-y),\\ K_P(x,y)&=-i\hbar\,\partial_x\delta(x-y). \end{aligned}$$

These are identities after testing. For example, the derivative kernel gives $i\hbar\int\overline{\phi'(x)}\psi(x)dx$, which equals $\langle\phi,P\psi\rangle$ after integration by parts. Its sign follows from the derivative variable and the stated pairing convention.

Operator composition need not be a product of distributions

Finite matrices satisfy $(AB)_{ij}=\sum_kA_{ik}B_{kj}$. The continuous analogue is often written

$$K_{AB}(x,y)=\int K_A(x,z)K_B(z,y)\,dz.$$

This requires a definition of the product and a justification of the integration. Two bounded operators have a well-defined bounded composition, so that composition has its own distribution kernel. It does not follow that arbitrary representatives of their kernels can be multiplied pointwise and integrated.

For sufficiently regular integral kernels, appropriate integrability estimates can justify the familiar formula. For differential operators on a common invariant test space, composing their actions yields the corresponding derivative kernel. For example, $P^2$ has kernel $-\hbar^2\partial_x^2\delta(x-y)$; it is not obtained by squaring the distribution $\partial_x\delta(x-y)$ at the same arguments.

For unbounded operators, check the product domain as well. A formally plausible kernel cannot supply an input domain missing from the operator definition.

A trace is not an arbitrary diagonal value

Trace-class operators on $L^2$ are Hilbert–Schmidt and have square-integrable kernels, but such a kernel is only an almost-everywhere class on $\mathbb R^2$. The diagonal $x=y$ has two-dimensional measure zero. Its assigned values therefore do not follow from that class, so $\operatorname{Tr}A=\int K_A(x,x)dx$ needs additional interpretation or regularity.

For a positive density operator with spectral decomposition $\rho=\sum_nr_n|u_n\rangle\langle u_n|$, the position density is rigorously defined almost everywhere by

$$w_\rho(x)=\sum_nr_n|u_n(x)|^2.$$

Monotone integration gives $\int w_\rho(x)dx=\sum_nr_n=1$. Moreover, $\operatorname{Tr}(\rho E_X(\Delta))=\int_\Delta w_\rho(x)dx$. This defines the probability density without choosing an arbitrary diagonal representative of a two-variable kernel.

Writing it as $\rho(x,x)$ is useful shorthand when that meaning has been fixed. It does not authorize setting $\operatorname{Tr}I=\int\delta(0)dx$ or extracting traces from an arbitrary numerical kernel diagonal. Trace-Class Operators supplies the basis-independent trace definition.

Exercise: changing only the diagonal

Let a positive rank-one operator have kernel $K(x,y)=u(x)\overline{u(y)}$, with $u\in\mathcal S$ normalized. Replace the kernel’s values on the diagonal by zero and leave all other values unchanged. Has the Hilbert–Schmidt operator changed? Does the integral of the assigned new diagonal give its trace?

Solution

The operator has not changed because the kernels agree almost everywhere on $\mathbb R^2$. Its trace remains $\|u\|^2=1$. The new assigned diagonal integrates to zero, demonstrating that a raw diagonal is not an invariant of an arbitrary $L^2$ kernel representative. The original continuous representative and the positive spectral density both recover the correct trace under their justified meanings.

References

• Gel’fand, I. M., and N. Ya. Vilenkin, Generalized Functions, Vol. 4: Applications of Harmonic Analysis, translated by Amiel Feinstein, Academic Press, 1964, Chapter I, publisher edition — nuclear spaces and distribution kernels.
• Hall, Brian C., Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — continuous representations and differential operators.
• Simon, Barry, Trace Ideals and Their Applications, 2nd ed., Mathematical Surveys and Monographs 120, American Mathematical Society, 2005, doi:10.1090/surv/120 — trace-class operators and integral-kernel qualifications.