by Chas Blakemore - Submitted as coursework for PH240, Stanford University, Fall 2017
INTRODUCTION : From the fictional universe of Stargate Atlantis and Marvel Comic's Realm of Kings to NASA's Eagleworks Propulsion laboratory, zero-point energy, also known as vacuum energy, is touted as a potentially limitless and ubiquitous source of energy, if one can only find the means to harness it. [1] Zero-point energy can be formulated in a few different ways, but in its most basic form, it is the minimal yet non-zero energy of a quantum mechanical system. In quantum field theory, zero-point energy can be considered by computing the expected energy of the zero photon mode. [2] In a system with no physical boundaries, the expected energy of the zero photon mode diverges! Yet, if this energy uniformly permeates all of space-time, it is not directly observable.
Conceptual FrameworkFor pedagogical reasons, we will consider the popular formulation of zero-point energy. The most interesting and relevant framework for zero-point energy can be understood from the quantum field theory for photons and electrons: quantum electrodynamics. Glossing over an exceptional amount of mathematical and conceptual background, the energy of a state in quantum field theory is computed as an expectation of a Hamiltonian, , which describes the energy of the state in terms of operators acting on wavefunctions. The final computation usually requires an integral over the allowed momenta of particles in the state.
Following Schwartz, we can compute the energy of a zero-photon vacuum state with infinite boundaries, a system that is in no way physical, realizable or complete in it's description, but pedagogical nonetheless. [2] Again glossing over much of the background information (which we encourage the interested reader to peruse in reference [2]), we find that, Evac = ∝ ∫ k3 dk → ∞. And thus we see that the expected energy of the vacuum state diverges! Of course this isn't truly a physical system as no other particles are present and there are no boundaries, but it illustrates the idea of energy present in empty space...
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OppositionAs was mentioned before, this measurement of thrust generated by pushing against the vacuum of spacetime has significant opposition from the physics community, and rightly so. The most obvious issue is a simple violation of conservation of momentum. NASA has claimed to build a "thruster" which, in theory, can propel itself from rest without ejecting either matter or photons. Trivially, pfinal ≠ pinitial which is a problem! Although unlikely, it is possible that the drive is generating thrust by ejecting some other type of weakly-interacting particle, maybe even through a mechanism not currently encoded by the Standard Model.
All physical arguments aside, the measurement claimed supposedly observes a thrust of 10s of micro-Newtons, with a slow (1/f) drift on the order of 100s of micro-Newtons. [1] Purely from a metrological standpoint, this type of measurement is fraught with error, even using techniques to identify signals by their spectral properties. In time, it's possible that better measurements could be made.
ConclusionsHarnessing energy and thrust from the vacuum of space would be undeniably astounding, and is the main project goal at NASA's Eagleworks labs. Yet, such an idea must contend with physical laws like simple conservation of momentum and would benefit from a well-described mechanism, even if it's only an effective description.
Recently in 2017, vacuum fluctuations have allegedly been demonstrated to provide thrust in a prototype EM drive. This is preposterous measurement that is is physically impossible, both from fundamental arguments and a metrological standpoint, as the brief discussion above demonstrated.
© Chas Blakemore. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
References[1] H. White et al., "Measurement of Impulsive Thrust from a Closed Radio-Frequency Cavity in Vacuum," J. Propul. Power 33, 830 (2017).
[2] M. D. Schwartz, Quantum Field Theory and The Standard Model (Cambridge University Press, 2013).
[3] H. B. G. Casimir, "On the Attraction Between Two Perfectly Conducting Plates," Kon. Ned. Akad. Wetensch. Proc. 51, 793 (1948).
[4] M. J. Sparnaay, "Measurements of the Attractive Forces Between Flat Plates," Physica 24, 751 (1958).
[5] S. K. Lamoreaux, "Demonstration of the Casimir Force in the 0.6 to 6 μm Range," Phys. Rev. Lett. 78, 5 (1997).
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