Quantum observables as Fréchet sensitivity kernels

Can quantum mechanical probabilities and observables be unified under a broader mathematical framework by interpreting them as sensitivity kernels derived from the variational structure of the Schrödinger equation?

The author introduces a variational-adjoint interpretation of quantum mechanics using Lagrange optimization constrained by the Schrödinger equation. By employing a convolution bilinear form—which naturally incorporates time reversal—the framework defines an adjoint wavefunction whose source is linked to a measurement functional or observable. The interaction between the forward wavefunction and this adjoint wavefunction yields a general Fréchet interaction density that quantifies sensitivity to perturbations.

The analysis demonstrates that the interaction between forward and adjoint wavefunctions defines sensitivity kernels for various quantum observables, including momentum, energy, and spin. When the adjoint wavefunction is chosen to be the complex conjugate of the forward wavefunction, the Fréchet interaction density reduces precisely to the standard, non-negative Born probability density. Consequently, quantum probability is reinterpreted as a specific positive-definite form of sensitivity rather than an isolated postulate.

This work bridges quantum mechanics with time-symmetric interpretations and optimization methods widely used in classical physics and inverse problems. By framing the Born rule as part of a general class of sensitivity kernels, it opens new conceptual pathways for understanding quantum measurements and potential applications in quantum control and metrology.