Quantum calculations often resemble assembling a complex mechanism: a physicist knows a part will fit, but can't always explain why. One favorite trick is to insert a 'unity' into the middle of a formula — not an ordinary one, but one built from an infinite set of all possible states of the system. This turns a cumbersome expression into a chain of simple integrals. But the mathematical rigor of such a step has long been questionable.

This new work brings order here. The authors construct a general mathematical framework, a theorem on nested integrals, that guarantees that successive insertions of such 'unities' always lead to exact equalities. They rigorously derive conditions under which one can interchange the inner product and the integral, closing a gap often glossed over in physics textbooks. It's like creating a universal blueprint suitable for any quantum 'assembly'.

The power of the approach shows in examples. For simple operators, familiar formulas emerge; for trickier ones (say, overlaps of squeezed states in a nonlinear medium) the theorem yields an exact integral representation that was previously hard to obtain. Another example: matrix elements of a complex two-mode Gaussian network are expressed via double Hermite polynomials. Previously, such a calculation would require clever guesses; now it's a direct consequence of the general procedure.

Thus, the physicist's trick with 'unity' ceases to be wizardry and becomes an honest mathematical tool. This not only strengthens the foundations of quantum theory but also provides a computational method for new problems — from quantum optics to field theory.