Quantum observables as Fréchet sensitivity kernels

• URL: http://arxiv.org/abs/2608.23191v1
• Date: Mon, 24 Aug 2026 12:42:31 GMT
• Title: Quantum observables as Fréchet sensitivity kernels
• Abstract summary: 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.<n>Within this framework, probability is a positive-definite form of sensitivity described by the Fréchet interaction density.<n>This suggests that the Born probability density belongs to a broader class of Fréchet sensitivity kernels associated with quantum observables.
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• License: http://creativecommons.org/licenses/by/4.0/
• 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 Fréchet 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 Fréchet interaction density. In general, the variational--adjoint interpretation also produces Fréchet 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 Fréchet 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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