What Is Constructed and What Is Formal?

A constructed path-integral representation identifies a mathematical object and proves that a stated limiting procedure produces it. A formal expression can correctly suggest that representation while leaving a measure, operator domain, or limiting argument unspecified. The distinction becomes concrete when a regular potential shrinks to a point interaction: its pointwise limit and its Hamiltonian limit describe different things.

Required background. Quadratic Form Methods constructs one-dimensional point interactions. Delta-Function Potential as Self-Adjoint Extension identifies their domains and resolvents. Trotter Product Formula states the evolution limits used after constructing a Hamiltonian.

Construction identifies an object and a mode of approximation

A regulated integral may be well defined for each cutoff while failing to have a limit as the cutoff is removed. A convergent limit may in turn represent an operator different from the intended Hamiltonian. These are separate mathematical questions.

For an evolution problem, a precise construction can proceed by defining self-adjoint $H_\varepsilon$, proving resolvent convergence to a specified $H$, and then invoking the corresponding spectral convergence theorem for unitary groups or semigroups. For a probability representation, it also identifies the reference process and proves integrability of the inserted weights. An asymptotic expansion supplies coefficients and a remainder in a specified regime; it does not become a convergent construction merely because its coefficients are computable.

The following example makes the identification step visible without requiring a general existence theorem for real-time path measures.

A shrinking potential has a nontrivial operator limit

Choose $v\in C_c^\infty(\mathbb R)$, $v\geq0$, with $\int v(u)\,du=1$. For $g\in\mathbb R$ and $\varepsilon>0$, set

$$V_\varepsilon(x)=\frac g\varepsilon v(x/\varepsilon), \qquad H_\varepsilon=-\alpha\frac{d^2}{dx^2}+V_\varepsilon, \qquad\alpha>0.$$

Each potential is bounded and real, so each Hamiltonian is self-adjoint on $H^2(\mathbb R)$. At every fixed $x\ne0$, however, $V_\varepsilon(x)$ eventually vanishes. Taking this almost-everywhere pointwise limit inside the Hamiltonian formula would suggest the free operator.

The energy forms tell a different story. On the common domain $Q=H^1(\mathbb R)$,

$$q_\varepsilon[\psi]=\alpha\|\psi'\|^2+ g\int v(u)|\psi(\varepsilon u)|^2\,du \longrightarrow q_g[\psi]=\alpha\|\psi'\|^2+g|\psi(0)|^2.$$

The continuous Sobolev representative makes the limit meaningful. The limiting form constructs the delta interaction, with derivative jump $\psi'(0+)-\psi'(0-)=g\psi(0)/\alpha$. Its domain and spectral consequences belong to the model page linked above.

A uniform form estimate upgrades the pointwise calculation

For $\psi\in H^1$, absolute continuity and Cauchy–Schwarz give

$$|\psi(\varepsilon u)-\psi(0)| \leq\sqrt{\varepsilon|u|}\,\|\psi'\|.$$

The Sobolev bound $\|\psi\|_\infty\leq C\|\psi\|_{H^1}$ therefore yields

$$|q_\varepsilon[\psi]-q_g[\psi]| \leq C_{g,v}\sqrt\varepsilon\,\|\psi\|_{H^1}^2.$$

The constant is finite because $v$ has compact support. Polarization gives the corresponding sesquilinear estimate. The trace inequality, uniformly under spatial translation and averaging against $v$, also gives a lower bound and coercive shifted form norm independent of small $\varepsilon$.

Choose a common sufficiently large positive shift $c$. Regard the shifted forms as invertible maps from $Q$ to its dual $Q^*$. Their difference tends to zero in that operator norm, and their inverses remain uniformly bounded. The inverse identity then implies

$$\|(H_\varepsilon+c)^{-1}-(H_g+c)^{-1}\| \leq C'\sqrt\varepsilon.$$

The Hilbert-space resolvents are obtained by composing those inverse form maps with the continuous embeddings $\mathcal H\to Q^*$ and $Q\to\mathcal H$. This is norm-resolvent convergence, with a concrete estimate. It supplies the missing operator identification that pointwise convergence of $V_\varepsilon$ could not.

The associated path expression still needs its order of limits

For each fixed $\varepsilon$, bounded-potential Trotter and Feynman–Kac theorems apply. Their hypotheses are satisfied, but their potential bounds need not be uniform as $\varepsilon\downarrow0$. One may first construct each semigroup and then take the proved Hamiltonian limit.

Replacing $V_\varepsilon$ directly by $g\delta_0$ inside a Brownian time integral is another assertion. A delta distribution cannot be evaluated as an ordinary function along a path. The appropriate probabilistic expression uses local time with a diffusion-dependent normalization. That representation requires its own theorem; the operator construction above does not justify a naive substitution.

Similarly, simultaneous removal of a time mesh and a spatial regulator requires estimates uniform in both parameters. The existence of each iterated limit does not alone prove that the limits commute.

Exercise: a harmless-looking slice normalization survives the limit

Suppose a sequence of $n$-slice operators $U_n(t)$ converges strongly to $U(t)$. Multiply each slice by $1+a/n$, with fixed real $a$. What happens to the limit?

Solution

The modified product is $(1+a/n)^nU_n(t)$ and converges strongly to $e^aU(t)$. Each individual factor tends to one, but the accumulated normalization remains finite and changes the limiting operator. If $a\ne0$ and $U(t)$ is unitary, the modified limit is not unitary. A regulator construction must control the accumulated normalization, not only its per-slice limit.

References

• Albeverio, Sergio, Friedrich Gesztesy, Raphael Høegh-Krohn, and Helge Holden, Solvable Models in Quantum Mechanics, 2nd ed., AMS Chelsea Publishing, 2005, publisher edition — point interactions and zero-range approximations.
• Simon, Barry, Functional Integration and Quantum Physics, 2nd ed., AMS Chelsea Publishing, 2005, doi:10.1090/chel/351 — semigroup, probability, and limiting constructions.