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Energy Non-conservation in Quantum Mechanics

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Abstract

We study the conservation of energy, or lack thereof, when measurements are performed in quantum mechanics. The expectation value of the Hamiltonian of a system changes when wave functions collapse in accordance with the standard textbook (Copenhagen) treatment of quantum measurement, but one might imagine that the change in energy is compensated by the measuring apparatus or environment. We show that this is not true; the change in the energy of a state after measurement can be arbitrarily large, independent of the physical measurement process. In Everettian quantum theory, while the expectation value of the Hamiltonian is conserved for the wave function of the universe (including all the branches), it is not constant within individual worlds. It should therefore be possible to experimentally measure violations of conservation of energy, and we suggest an experimental protocol for doing so.

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Notes

  1. The fact that energy is simply “distributed among different branches” depends on our assumption that the Hamiltonian included no off-diagonal matrix elements between different environment states. This is reasonable, but if it were not true, our result that observers can experience violations of energy conservation would be unaltered; we would merely lose the interpretation of the total energy as the sum of the energies of the branches, weighted by the amplitudes-squared.

  2. It would be important to ensure that the spin states of particle 1 did not decohere when it was put into the magnetic field, in effect measuring the spin before the experiment started. If that were the case, the particle would simply be taking different amounts of energy from the magnetic field in two distinct branches of the wave function.

  3. Decoherence and the linearity of the Schrödinger equation ensure that communication between different branches of the wave function is not possible. If it were, we might imagine systematically transferring energy from one branch to another—an “Everett pipeline,” analogous to Polchinski’s “Everett phone” [24]. This might be an argument in favor of the linearity of quantum mechanics, if one wanted to avoid perpetual motion machines. We thank Grant Remmen for this observation.

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Acknowledgements

We would like to thank Anthony Bartolotta, Brad Fillipone, Jason Pollack, Ken Olum, Jonathan Oppenheim, Gil Refael, Grant Remmen, Ashmeet Singh, and Mark Wise for helpful discussions during the course of this project. This research is funded in part by the Walter Burke Institute for Theoretical Physics at Caltech, by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632, and by the Foundational Questions Institute.

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  1. Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, USA

    Sean M. Carroll & Jackie Lodman

  2. Santa Fe Institute, Santa Fe, USA

    Sean M. Carroll

  3. Department of Physics, Harvard University, Cambridge, USA

    Jackie Lodman

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Correspondence to Sean M. Carroll.

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Carroll, S.M., Lodman, J. Energy Non-conservation in Quantum Mechanics. Found Phys 51, 83 (2021). https://doi.org/10.1007/s10701-021-00490-5

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