Quantum observables as Fréchet sensitivity kernels
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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BibTeX
@misc{abreu2026quantum,
title = {{Quantum observables as Fréchet sensitivity kernels}},
author = {Rafael Abreu},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.23191},
eprint = {2608.23191},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05802219v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - Quantum observables as Fréchet sensitivity kernels
AU - Rafael Abreu
PY - 2026
DA - 2026/10/16
PB - HAL
DO - 10.48550/arxiv.2608.23191
UR - https://hal.science/hal-05802219v1
AB - 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.
ER -