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arXiv · 2608.23191

Quantum observables as Fr\'echet sensitivity kernels

Rafael Abreu
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Abstract

We present a variational--adjoint interpretation of quantum mechanics in which the interaction between forward and adjoint wavefunctions defines a general sensitivity kernel of the system. This Fr\'echet interaction density quantifies how perturbations in the wavefunction (or system parameters) influence a chosen observable. The familiar Born probability density appears as a special case when the adjoint wavefunction is chosen as the complex conjugate of the forward wavefunction, for which the interaction density becomes real and non-negative. Within this framework, probability is a particular positive-definite form of sensitivity described by the Fr\'echet interaction density. In general, the variational--adjoint interpretation also produces Fr\'echet sensitivity kernels associated with other quantum observables, including momentum, energy, and spin. This suggests that the Born probability density belongs to a broader class of Fr\'echet sensitivity kernels associated with quantum observables. The proposed interpretation also provides a connection with time-symmetric interpretations of quantum mechanics and possible future applications in quantum control and quantum metrology.

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Rafael Abreu. 2026-08-24. Quantum observables as Fr\'echet sensitivity kernels. https://arxiv.org/abs/2608.23191

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