# The post-selection operator current

## Abstract

In this paper, we develop the concept of the post-selection operator current and we use it to extend the methodology of quantum mechanics into related fields of science and engineering. We begin by reviewing some results from standard quantum mechanics. We then introduce the post-selection operator current and note that the concept of the operator current provides a common framework for connecting underlying concepts in classical signal theory and quantum mechanics. Next, we explore the geometry of post-selection making use of the Pancharatnam phase. We illustrate the usefulness of the method through a series of simple examples; at each stage we link the results of quantum mechanics to applications in signal processing. We then simplify some results by introducing a density operator. This simplification allows us to present a number of useful observations about weak values. We conclude by using the operator current explore the relationship between post-selection and gauge invariance. The methods developed in this paper enable us to characterize the time evolution of a post-selected operator. The methods can be applied to other evolutionary/transport equations including the Fokker–Planck equation, and the equation of Brownian motion.

## Keywords

Operator current Ehrenfest theorem Weak measurement Pancharatnam phase Sensor operator current## Notes

### Acknowledgements

Funding for the first author was provided by Office of Naval Research (US) under the auspices of the NSWDD ILIR program. Part of this work was presented at the Perimeter Institute’s conference: *Concepts and Paradoxes in a Quantum Universe* June 20–25, 2016. Thanks to PI for the invitation to participate in this conference. Some of the material in Part III was presented as an invited talk to the ICERM conference on Mathematical and Computational Aspects of Radar Imaging October 16–20, 2017; thanks to ICERM for the invitation to participate in this conference. A.D. Parks and Y. Aharonov introduced the first author to the concept of weak measurement and weak values Their friendship, intellectual and personal, is gratefully acknowledged.

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