# Zero-point energy

(Redirected from Zero point energy)
Zero-point radiation continually imparts random impulses on an electron, so that it never comes to a complete stop. Zero-point radiation gives the oscillator an average energy equal to the frequency of oscillation multiplied by one-half of Planck's constant

Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have i.e. it is the energy of the system's ground state. Zero-point energy can have several different types of context e.g. it may be the energy associated with the ground state of an atom, a subatomic particle or even the quantum vacuum itself.

In classical mechanics all particles can be thought of as having some energy made up of their potential energy and kinetic energy. Temperature arises from the intensity of random particle motion caused by kinetic energy (brownian motion). As temperature is reduced to absolute zero, it might be thought that all motion ceases and particles come completely to rest. In fact, however, kinetic energy is retained by particles even at the lowest possible temperature. The random motion corresponding to this zero-point energy never vanishes as a consequence of the uncertainty principle of quantum mechanics.[1]

The uncertainty principle states that no object can ever have precise values of position and velocity simultaneously. The total energy of a quantum mechanical object (potential and kinetic) is described by its Hamiltonian which also describes the system as a wave function that oscillates between various energy states (see wave-particle duality). All quantum mechanical systems undergo fluctuations even in their ground state a consequence of their wave-like nature. The uncertainty principle requires every quantum mechanical system to have a fluctating zero-point energy greater than the minimum of its classical potential well. This results in motion even at absolute zero. For example, liquid helium does not freeze under atmospheric pressure at any temperature because of its zero-point energy.

Given the equivalence of mass and energy expressed by Einstein’s E = mc2, any point in space that contains energy must be able to create particles. Virtual particles spontaneously flash into existence at every point in space due to the energy of quantum fluctuations caused by the uncertainty principle. Quantum field theory treats every point of space as a quantum harmonic oscillator. Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, and that all properties of matter are merely vacuum fluctuations arrising from interactions with the zero-point field. [2]

Physics currently lacks a full understanding of how zero-point radiation works, in particular the discrepancy between theorized and observed vacuum energy is a source of major contention.[3][4] Physicists John Wheeler and Richard Feynman calculated that the zero-point radiation of the vacuum to be an order of magnitude greater then nuclear energy, with one teacup containing enough to boil all the world's oceans[5] while experimental evidence from both the expansion of the universe and the Casimir effect show the any such force to be exceptionally weak. This discrepancy is known as the vacuum catastrophe.

Despite these issues, the topic is central to many important areas of physics; active areas of research include the effects of virtual particles,[6] quantum entanglement,[7] the difference (if any) between inertial and gravitational mass,[8][9] a reason for the observed value of the cosmological constant[10] and the nature of dark energy.[11][12]

The concept of zero-point energy was developed by Max Planck in Germany in 1911 as a corrective term added to a zero-grounded formula developed in his original quantum theory in 1900.[13] The term zero-point energy is a translation from the German Nullpunktsenergie.[14]:275ff

## History

Zero-point energy evolved from the historical development of ideas about the vacuum. In the 17th century, it was thought that a totally empty volume of space could be created by simply removing all gases. This was the first generally accepted concept of the vacuum.[15]

Late in the 19th century, however, it became apparent that the evacuated region still contained thermal radiation. The existence of the æther as a substitute for a true void was taken for granted. According to the successful electromagnetic æther theory based upon Maxwellian electrodynamics, the this all-encompassing æther was endowed with energy and hence very different from nothingness. Maxwell himself noted that “To those who maintained the existence of a plenum as a philosophical principle, nature’s abhorrence of a vacuum was a sufficient reason for imagining an all-surrounding æther.” The fact that electromagnetic and gravitational phenomena were easily transmitted in empty space indicated that their associated æthers were part of the fabric of space itself. Maxwell continued: “Æthers were invented for the planets to swim in, to constitute electric atmospheres and magnetic effluvia, to convey sensations from one part of our bodies to another, and so on, till a space had been filled three or four times with æthers.”[16] To some scientists of the period, it seemed that radiation in space might be eliminated by cooling. Thus evolved the second concept of achieving a real vacuum: cool it down to zero temperature after evacuation. Absolute zero temperature was technically impossible to achieve in the 19th century, so it the debate remained unsolved.

In 1900, Max Planck derived the average energy of a single energy radiator, e.g., a vibrating atomic unit, as a function of absolute temperature:[17]

${\displaystyle \epsilon ={\frac {h\nu }{e^{h\nu /kT}-1}}~,}$

where h is Planck's constant, ν is the frequency, k is Boltzmann's constant, and T is the absolute temperature.

In 1912, Max Planck published the first journal article to describe the discontinuous emission of radiation, based on the discrete quanta of energy. In this paper, Planck’s now-famous “blackbody” radiation equation contains the residual energy factor, one half of hf, as an additional term (½hf), dependent on the frequency f, which is always greater than zero (where h = Planck’s constant). It is therefore widely agreed that “Planck’s equation marked the birth of the concept of zero-point energy."[18] In a series of works from 1911 to 1913, Planck proposed his second quantum theory, in which he introduced the zero-point energy. Only the emitted radiation was attributed to discrete energy quanta, while the absorbed radiation could be continuous in energy. From these ideas, he found that the average energy of an oscillator is[13]:sec 2[19]:235ff

${\displaystyle \epsilon ={\frac {h\nu }{2}}+{\frac {h\nu }{e^{h\nu /kT}-1}}~.}$

Soon, the idea of zero-point energy attracted the attention of Albert Einstein and his assistant Otto Stern. They attempted to prove the existence of zero-point energy by calculating the specific heat of hydrogen gas and compared it with the experimental data. However, after assuming they had succeeded and after publishing the findings, they retracted the support of the idea because they found Planck's second theory may not apply to their example.[14]:270ff

In 1916 Walther Nernst proposed that empty space was filled with zero-point electromagnetic radiation.[13]:sec 4 Then in 1925, the existence of zero-point energy was shown to be “required by quantum mechanics, as a direct consequence of Heisenberg's uncertainty principle” in Werner Heisenberg's famous article "Quantum theoretical re-interpretation of kinematic and mechanical relations".[20]:162

## Varieties

The concept of zero-point energy occurs in a number of situations. The idea of a quantum harmonic oscillator and it's associated energy, can apply to either a particle or to the fabric of space itself.

In ordinary quantum mechanics, the zero-point energy is the energy associated with the ground state of the system. The professional physics literature tends to measure frequency, as denoted by ν above, using angular frequency, denoted with ω and defined by ω=2πν. This leads to a convention of writing Planck's constant h with a bar through its top (ħ) to denote the quantity h/2π. In these terms, the most famous such example of zero-point energy is the above E=ħω/2 associated with the ground state of the quantum harmonic oscillator. In quantum mechanical terms, the zero-point energy is the expectation value of the Hamiltonian of the system in the ground state.

The zero-point energy E=ħω/2 causes the ground-state of an harmonic oscillator to advance its phase (color). This has measurable effects when several eigenstates are superimposed.

In quantum field theory, the fabric of space is visualized as consisting of fields, with the field at every point in space and time being a quantum harmonic oscillator, with neighboring oscillators interacting with each other. In this case, one has a contribution of E=ħω/2 from every point in space, resulting in a calculation of infinite zero-point energy in any finite volume; this is one reason renormalization is needed to make sense of quantum field theories. The zero-point energy is again the expectation value of the Hamiltonian; here, however, the phrase vacuum expectation value is more commonly used, and the energy is called the vacuum energy.

In quantum perturbation theory, it is sometimes said that the contribution of one-loop and multi-loop Feynman diagrams to elementary particle propagators are the contribution of vacuum fluctuations, or the zero-point energy to the particle masses.

## Relation to the uncertainty principle

Zero-point energy is fundamentally related to the Heisenberg uncertainty principle.[21] Roughly speaking, the uncertainty principle states that complementary variables (such as a particle's position and momentum, or a field's value and derivative at a point in space) cannot simultaneously be specified precisely by any given quantum state. In particular, there cannot exist a state in which the system simply sits motionless at the bottom of its potential well: for, then, its position and momentum would both be completely determined to arbitrarily great precision. Therefore, instead, the lowest-energy state (the ground state) of the system must have a distribution in position and momentum that satisfies the uncertainty principle−−which implies its energy must be greater than the minimum of the potential well.

Near the bottom of a potential well, the Hamiltonian of a general system (the quantum-mechanical operator giving its energy) can be approximated as a quantum harmonic oscillator,

${\displaystyle {\hat {H}}=V_{0}+{\frac {1}{2}}k\left({\hat {x}}-x_{0}\right)^{2}+{\frac {1}{2m}}{\hat {p}}^{2}~,}$

where V0 is the minimum of the classical potential well.

The uncertainty principle tells us that

${\displaystyle {\sqrt {\left\langle \left({\hat {x}}-x_{0}\right)^{2}\right\rangle }}{\sqrt {\left\langle {\hat {p}}^{2}\right\rangle }}\geq {\frac {\hbar }{2}}~,}$

making the expectation values of the kinetic and potential terms above satisfy

${\displaystyle \left\langle {\frac {1}{2}}k\left({\hat {x}}-x_{0}\right)^{2}\right\rangle \left\langle {\frac {1}{2m}}{\hat {p}}^{2}\right\rangle \geq \left({\frac {\hbar }{4}}\right)^{2}{\frac {k}{m}}~.}$

The expectation value of the energy must therefore be at least

${\displaystyle \left\langle {\hat {H}}\right\rangle \geq V_{0}+{\frac {\hbar }{2}}{\sqrt {\frac {k}{m}}}=V_{0}+{\frac {\hbar \omega }{2}}}$

where ${\displaystyle \omega ={\sqrt {k/m}}}$ is the angular frequency at which the system oscillates.

A more thorough treatment, showing that the energy of the ground state actually saturates this bound and is exactly E0=V0+ħω/2, requires solving for the ground state of the system.

## Vacuum Energy

Main article: Vacuum energy

Vacuum energy, also called the quantum vacuum zero-point energy, is the zero-point energy that relates to the quantum vacuum.[22] According to traditional quantum mechanics particles can be treated as quantum harmonic oscillators. In Quantum field theory (QED) every point is space is thought of as a harmonic oscillator and as a result the vacuum state can be thought of as not being truly empty but instead contains fleeting electromagnetic waves and particles that pop into and out of existence, the vacuum energy can be thought of as the kinetic energy that arrises due to the uncertainty principle applying to these virtual particles.[23] The vacuum energy contains of all the fields in space, which in the Standard Model includes the electromagnetic field, other gauge fields, fermionic fields, and the Higgs field. It is the energy of the vacuum, which in quantum field theory is defined not as empty space but as the ground state of the fields. In cosmology, the vacuum energy is one possible explanation for the cosmological constant[24] and the source of dark energy.[25][26] A related term is zero-point field, which is the lowest energy state of a particular field.[27]

Scientists are not in agreement about how much energy is contained in the vacuum. Quantum mechanics requires the energy to be large as Paul Dirac claimed it is, like a sea of energy. Other scientists specializing in General Relativity require the energy to be small enough for curvature of space to agree with observed astronomy. The Heisenberg uncertainty principle allows the energy to be as large as needed to promote quantum actions for a brief moment of time, even if the average energy is small enough to satisfy relativity and flat space. To cope with disagreements, the vacuum energy is described as a virtual energy potential of positive and negative energy.[28]

## Experimental observations

A phenomenon that is commonly presented as evidence for the existence of zero-point energy in vacuum is the Casimir effect, proposed in 1948 by Dutch physicist Hendrik B. G. Casimir (Philips Research), who considered the quantized electromagnetic field between a pair of grounded, neutral metal plates. The vacuum energy contains contributions from all wavelengths, except those excluded by the spacing between plates. As the plates draw together, more wavelengths are excluded and the vacuum energy decreases. The decrease in energy means there must be a force doing work on the plates as they move. This force has been measured and found to be in good agreement with the theory. However, there is still some debate on whether vacuum energy is necessary to explain the Casimir effect. Robert Jaffe of MIT argues that the Casimir force should not be considered evidence for vacuum energy, since it can be derived in QED without reference to vacuum energy by considering charge-current interactions (the radiation-reaction picture).[29]

The experimentally measured Lamb shift has been argued to be, in part, a zero-point energy effect.[30]

## Gravitation and cosmology

 Unsolved problem in physics: Why does the zero-point energy density of the vacuum not change with changes in the volume of the universe? And related to that, why does the large constant zero-point energy density of the vacuum not cause a large cosmological constant? What cancels it out? [31][32][33] (more unsolved problems in physics)

In cosmology, the zero-point energy offers an intriguing possibility for explaining the speculative positive values of the proposed cosmological constant. [34] In brief, if the energy is "really there", then it should exert a gravitational force.[35] In general relativity, mass and energy are equivalent; both produce a gravitational field. One obvious difficulty with this association is that the zero-point energy of the vacuum is absurdly large. Naively, it is infinite, because it includes the energy of waves with arbitrarily short wavelengths. But since only differences in energy are physically measurable, the infinity can be removed by renormalization. In all practical calculations, this is how the infinity is handled.

## Utilization

Throughout space there is energy. Is this energy static or kinetic? If static our hopes are in vain; if kinetic – and we know it is, for certain – then it is a mere question of time when men will succeed in attaching their machinery to the very wheel work of Nature. Many generations may pass, but in time our machinery will be driven by a power obtainable at any point in the Universe.

Nikola Tesla (1889)[36]

Nikola Tesla was the first to propose that the vacuum energy, or æther, might be harnessed for useful work;[36] ever since then many people have claimed to exploit zero-point energy with a large amount of pseudoscientific literature causing ridicule around the subject.[37][38]

Despite controversy, harnessing zero-point energy is an ongoing area of worldwide research, particularly in China, Germany, Russia and Brazil.[37] The Casimir force between two plates, which is caused by zero-point energy, was first predicted in 1948 by Dutch physicist Hendrik Casimir.[39] Steve K. Lamoreaux initially measured the tiny force in 1997.[40] It had long been assumed that the Casimir force had little practical use; it was assumed the only way to actually gain energy from the two plates is to allow them to come together (getting them apart again would then require more energy), and therefore it is a one-use-only force in nature.[37]

In 1999 however, Fabrizio Pinto, a former scientist at NASA's Jet Propulsion Laboratory at Caltech in Pasadena, published in Physical Review his Gedankenexperiment for a "Casimir engine". The paper showed that continuous positive net exchange of energy from the Casimir effect was possible, even stating in the abstract "In the event of no other alternative explanations, one should conclude that major technological advances in the area of endless, by-product free-energy production could be achieved." [41] Despite this and several similar peer reviewed papers, there is not a consensus as to whether such devices will actually work in practice. Garret Moddel at University of Colorado has highlighted that he believes such a device hinges on the assumption that the Casmimir force is a nonconservative force, he argues that there is sufficient evidence to say that it is a conservative force and therefore even though such an engine can exploit the casamir force for useful work it cannot produce more output energy then has been input into the system.[42]

There have been several promising breakthroughs in the field of thermodynamics; a paper by Armen Allahverdyan and Theo Nieuwenhuizen in 2000[43] and then by Marlan Scully et al. in 2003 published in Science[44] showed the feasibility of extracting zero-point energy for useful work from a single bath, without contradicting the laws of thermodynamics, by exploiting certain quantum mechanical properties.

In 2014 NASA's Eagleworks Laboratories[45] announced in they had successfully validaded the use of a Quantum Vacuum Plasma Thruster which makes use of the Casimir effect for propulsion.[46][47]

The calculation that underlies the Casimir experiment, a calculation based on the formula predicting infinite vacuum energy, shows the zero-point energy of a system consisting of a vacuum between two plates will decrease at a finite rate as the two plates are drawn together. The vacuum energies are predicted to be infinite, but the changes are predicted to be finite. Casimir combined the projected rate of change in zero-point energy with the principle of conservation of energy to predict a force on the plates. The predicted force, which is very small and was experimentally measured to be within 5% of its predicted value, is finite.[48] Even though the zero-point energy is theoretically infinite, there is no evidence to suggest that infinite amounts of zero-point energy are available for use with present technology.

## In popular culture

"Zero point energy" has been invoked in science fiction movies and video games, often as an explanation for "impossible" technology that provides free energy or otherwise contradicts known laws of physics.

Science skeptic and writer Martin Gardner has called claims of such zero-point-energy-based systems "as hopeless as past efforts to build perpetual motion machines".[49] A perpetual motion machine is a device that can operate indefinitely, with optional output of excess energy, without any source of fuel. Such a device would violate the laws of thermodynamics. Despite the science, numerous articles and books have been published addressing and discussing the potential of tapping zero-point-energy from the quantum vacuum or elsewhere. Examples of such are the work of the following authors: Claus Wilhelm Turtur,[50] Jeane Manning, Joel Garbon,[51] John Bedini,[52] Tom Bearden,[53][54][55] Thomas Valone,[56][57][58] Moray B King,[59][60][61] Christopher Toussaint, Bill Jenkins,[62] Nick Cook[63] and William James.[64]

The 2004 video game Half-Life 2 features a weapon called the "zero-point energy field manipulator" also known as the "Gravity gun".

In Disney/Pixar's animated film The Incredibles, the main villain Syndrome refers to his weapons as using zero-point energy.[65][66] The fan fiction community devoted to the character is named "Zero Point" because of this.[67]

In the TV show Stargate Atlantis, Zero Point Modules are advanced power sources built by the Ancients to power their cities and outposts. Weighing only a few kilograms,[68] a single ZPM can power the entire city of Atlantis for thousands of years. ZPMs supposedly extract vacuum energy from a small artificially-created region of subspace,[69] based on the concept of zero-point energy.[70] ZPMs are depicted as more powerful and efficient than fictional Naquadah generators or any conventional energy source on present day Earth.[69]

## Notes

1. ^ Encyclopedia Britannica. 2016. zero-point energy | physics | Britannica.com. [ONLINE] Available at: https://www.britannica.com/science/zero-point-energy. [Accessed 27 September 2016].
2. ^ Battersby, S. 2008. It's confirmed: Matter is merely vacuum fluctuations. New Scientist . [ONLINE] Available at: https://www.newscientist.com/article/dn16095-its-confirmed-matter-is-merely-vacuum-fluctuations/.
3. ^ Shiga, D. 2005. Vacuum energy: something for nothing? New Scientist. [ONLINE] Available at: https://www.newscientist.com/article/mg18825191-800-vacuum-energy-something-for-nothing/.
4. ^ Siegel, E. 2016. What Is The Physics Of Nothing? Forbes [ONLINE] Available at: http://www.forbes.com/sites/startswithabang/2016/09/22/what-is-the-physics-of-nothing/#2766138a1451
5. ^ Pilkington, P. 2003. Zero point energy. The Guardian. [ONLINE] Available at: https://www.theguardian.com/education/2003/jul/17/research.highereducation.
6. ^ Yam, P. 1997. Exploiting Zero-Point Energy. Scientific American. Vol 277. p82-85 [ONLINE] Available at: http://www.forbes.com/sites/startswithabang/2016/09/22/what-is-the-physics-of-nothing/#2766138a1451
7. ^ Choi, C.Q. 2013. Something from Nothing? A Vacuum Can Yield Flashes of Light. Scientific American. [ONLINE] Available at: http://www.scientificamerican.com/article/something-from-nothing-vacuum-can-yield-flashes-of-light/.
8. ^ Haisch, B. Rueda, A. and Puthoff, H. E. 1994. Inertia as a zero-point-field Lorentz force. Physical Review A. Issue 2, Vol 49, p678 [ONLINE] Available at: http://journals.aps.org/pra/abstract/10.1103/PhysRevA.49.678
9. ^ Haisch, B. Rueda, A. and Puthoff, H. E. 1997. Physics of the Zero-Point Field: Implications for Inertia, Gravitation and Mass. Speculations in Science and Technology. Vol 20. p. 99-114 {ONLINE} Available at: https://issuu.com/zerofieldenergy/docs/bernard-haisch--rueda--puthoff---physics-of-the-ze
10. ^ Zinkernage, H. and Rugh, S.E. 2002. The quantum vacuum and the cosmological constant problem. Studies in History and Philosophy of Modern Physics, Vol. 33., pp.663-705 "The cosmological constant problem arises at the intersection between general relativity and quantum field theory, and is regarded as a fundamental problem in modern physics"..."zero-point fluctuations of the quantum fields, as well as other ‘vacuum phenomena’ of quantum field theory, give rise to an enormous vacuum energy density ρ_vac. As we shall see, this vacuum energy density is believed to act as a contribution to the cosmological constant Λ...[but] observations shows that Λ is very small" [ONLINE] https://arxiv.org/pdf/hep-th/0012253v1.pdf
11. ^ Science Daily. 2007. Dark Energy May Be Vacuum. [ONLINE] Available at: https://www.sciencedaily.com/releases/2007/01/070116130456.htm. [Accessed 27 September 2016].
12. ^ Space.com. 2014. Does Dark Energy Spring From the 'Quantum Vacuum?'. [ONLINE] Available at: http://www.space.com/25238-dark-energy-quantum-vacuum-theory.html.
13. ^ a b c Kragh, Helge (2012). "Preludes to dark energy: zero-point energy and vacuum speculations". Archive for History of Exact Sciences. Springer-Verlag. 66 (3): 199–240. arXiv:1111.4623. doi:10.1007/s00407-011-0092-3.
14. ^ a b Albert Einstein (1995). Martin J. Klein; et al., eds. The Collected Papers of Albert Einstein, Volume 4: The Swiss Years: Writings, 1912-1914. Princeton University Press. ISBN 9780691037059.
15. ^ *Conlon, Thomas E. (27 September 2011). Thinking About Nothing: Otto von Guericke and the Magdeburg Experiments on the Vacuum. The Saint Austin Press. ISBN 978-14478-3916-3. Retrieved 6 November 2012.
16. ^ Kragh, H.S. Overduin, J. 2014. The Weight of the Vacuum: A Scientific History of Dark Energy. Springer. 2014 edition. p7. ISBN 978-3-642-55090-4
17. ^ Planck, M (1900). "Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum". Verhandlungen der Deutschen Physikalischen Gesellschaft. 2: 237–245.
18. ^ Bulsara, A.R. et al. “Tuning in to Noise” Physics Today, March, 1996, p. 39
19. ^ Thomas S. Kuhn. Black-Body Theory and the Quantum Discontinuity, 1894-1912. University of Chicago Press. ISBN 978-0-226-45800-7.
20. ^ Kragh, Helge (2002). Quantum Generations: A History of Physics in the Twentieth Century (Reprint ed.). Princeton University Press. ISBN 978-0691095523.
21. ^ W. Heisenberg (1927). "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". Zeitschrift für Physik (in German). 43 (3): 172–198. Bibcode:1927ZPhy...43..172H. doi:10.1007/BF01397280.
22. ^ Scientific American. 1997. FOLLOW-UP: What is the 'zero-point energy' (or 'vacuum energy') in quantum physics? Is it really possible that we could harness this energy? - Scientific American. [ONLINE] Available at: http://www.scientificamerican.com/article/follow-up-what-is-the-zer/. [Accessed 27 September 2016].
23. ^ Encyclopedia Britannica. 2016. principles of physical science - Conservation laws and extremal principles | Britannica.com. [ONLINE] Available at: https://www.britannica.com/science/principles-of-physical-science/Conservation-laws-and-extremal-principles#ref366439. [Accessed 27 September 2016].
24. ^ Rugh, S. E.; Zinkernagel, H. (2002). "The Quantum Vacuum and the Cosmological Constant Problem". Studies in History and Philosophy of Modern Physics, vol. 33 (4): 663–705. arXiv:hep-th/0012253. doi:10.1016/S1355-2198(02)00033-3.
25. ^ Science Daily. 2007. Dark Energy May Be Vacuum. [ONLINE] Available at: https://www.sciencedaily.com/releases/2007/01/070116130456.htm. [Accessed 27 September 2016].
26. ^ Space.com. 2014. Does Dark Energy Spring From the 'Quantum Vacuum?'. [ONLINE] Available at: http://www.space.com/25238-dark-energy-quantum-vacuum-theory.html. [Accessed 27 September 2016].
27. ^ Gribbin, J. (1998). Q is for Quantum: An Encyclopedia of Particle Physics. Touchstone Books. ISBN 0-684-86315-4.
28. ^ Peskin, M. E.; Schroeder, D. V. (1995). An introduction to quantum field theory. Addison-Wesley. pp. 786–791. ISBN 978-0-201-50397-5.
29. ^ Jaffe, R. L. (2005). "Casimir effect and the quantum vacuum". Physical Review D. 72 (2): 021301. arXiv:hep-th/0503158. Bibcode:2005PhRvD..72b1301J. doi:10.1103/PhysRevD.72.021301.
30. ^ Hawton, M. (1993). "Self-consistent frequencies of the electron-photon system". Physical Review A. 48 (3): 1824. Bibcode:1993PhRvA..48.1824H. doi:10.1103/PhysRevA.48.1824.
31. ^ Abbott, L. 1988. The Mystery of the Cosmological Constant. Scientific American. Vol 258. p106-113. "According to theory, the constant, which measures the energy of the vacuum, should be much greater than it is. An understanding of the disagreement could revolutionize fundamental physics" [ONLINE] http://pages.erau.edu/~reynodb2/blog/Abbott_CosmologicalConstant_SciAm.pdf
32. ^ Battersby, S. 2016. Dark energy: Staring into darkness. Nature. Vol 537. p201–204. "It's hard to make the numbers stack up. The vacuum energy needed to produce the observed cosmic acceleration is about 1 joule per cubic kilometre of space; the simplest version of quantum-field theory adds up the energy of those virtual particles to give a value about 120 orders of magnitude higher than that" "Dark energy is key to opening a window on “a completely unexplored region of fundamental physics,” says Mark Trodden, a theoretical cosmologist and director of the Penn Center for Particle Cosmology in Pennsylvania, Philadelphia. Finding the answer would not only change the view of nature, but also foretell the fate of the Universe" [ONLINE] Available at: http://www.nature.com/nature/journal/v537/n7622_supp/full/537S201a.html
33. ^ Zinkernage, H. and Rugh, S.E. 2002. The quantum vacuum and the cosmological constant problem. Studies in History and Philosophy of Modern Physics, Vol. 33., pp.663-705 "The cosmological constant problem arises at the intersection between general relativity and quantum field theory, and is regarded as a fundamental problem in modern physics"..."zero-point fluctuations of the quantum fields, as well as other ‘vacuum phenomena’ of quantum field theory, give rise to an enormous vacuum energy density ρ_vac. As we shall see, this vacuum energy density is believed to act as a contribution to the cosmological constant Λ...[but] observations shows that Λ is very small" [ONLINE] https://arxiv.org/pdf/hep-th/0012253v1.pdf
34. ^ Tarkowski, W. (2004). "A Toy Model of the Five-Dimensional Universe with the Cosmological Constant". International Journal of Modern Physics A. 19 (29): 5051. arXiv:gr-qc/0407024. Bibcode:2004IJMPA..19.5051T. doi:10.1142/S0217751X04019366.
35. ^ Zee, A. (2008). "Gravity and Its Mysteries: Some Thoughts and Speculations" (PDF). AAPPS Bulletin. 18 (4): 32.
36. ^ a b Petersen, I. “Peeking inside an electron’s screen.” Science News. Vol. 151, Feb. 8, 1997, p. 89
37. ^ a b c Amber M. Aiken, Ph.D. "Zero-Point Energy: Can We Get Something From Nothing?" (PDF). U.S. Army National Ground Intelligence Center. Forays into "free energy" inventions and perpetual-motion machines using ZPE are considered by the broader scientific community to be pseudoscience.
38. ^
39. ^ H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proceedings of the Royal Netherlands Academy of Arts and Sciences, vol. 51, pp. 793–795, 1984.[ONLINE] Available at: http://www.mit.edu/~kardar/research/seminars/Casimir/Casimir1948.pdf
40. ^ Lamoreaux, S. K. 1997. Demonstration of the Casimir Force in the 0.6 to 6μm Range. Physical Review Letters, Iss. 1/Vol. 78, , 5475.[ONLINE] Available at: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.78.5
41. ^ Pinto, F, 1999. Engine cycle of an optically controlled vacuum energy transducer. Physical Review B, 21/60, 4457. [ONLINE] Available at: http://journals.aps.org/prb/abstract/10.1103/PhysRevB.60.14740
42. ^ YouTube. 2013. 2013 GlobalBEM Day 3 Tent 1 Livestream - Garret Moddel, Mitchell Rabin, Foster Gamble - 2/4 - YouTube. [ONLINE] Available at: https://www.youtube.com/watch?v=aa1NIf47LNY&feature=youtu.be&t=39m19s.
43. ^ A.E. Allahverdyan, Th.M. Nieuwenhuizen. 2000. Extraction of Work from a Single Thermal Bath in the Quantum Regime. Physical Review Letters, 9/85, 1799 [ONLINE] Available at: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.85.1799
44. ^ Marlan O. Scully, M. Suhail Zubairy, Girish S. Agarwal, Herbert Walther. 2003. Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence. Science, 5608/299, 862-864. [ONLINE] Available at: http://science.sciencemag.org/content/299/5608/862.abstract
45. ^ NASA Technical Reports Server (NTRS) - Eagleworks Laboratories: Advanced Propulsion Physics Research. 2016. NASA Technical Reports Server (NTRS) - Eagleworks Laboratories: Advanced Propulsion Physics Research. [ONLINE] Available at: http://ntrs.nasa.gov/search.jsp?R=20110023492&hterms=eagleworks&qs=Ntx%3Dmode%2520matchallpartial%2520%26Ntk%3DAll%26N%3D0%26Ntt%3Deagleworks.
46. ^ engineering.com . 2012. Propulsion on an Interstellar Scale – the Quantum Vacuum Plasma Thruster. [ONLINE] Available at: http://www.engineering.com/DesignerEdge/DesignerEdgeArticles/ArticleID/5058/Propulsion-on-an-Interstellar-Scale-the-Quantum-Vacuum-Plasma-Thruster.aspx.
47. ^ WIRED UK. 2014. Nasa validates 'impossible' space drive | WIRED UK. [ONLINE] Available at: http://www.wired.co.uk/article/nasa-validates-impossible-space-drive.
48. ^
49. ^ Martin Gardner, "'Dr' Bearden's Vacuum Energy", Skeptical Inquirer, January/February 2007
50. ^ Claus Wilhelm Turtur. Conversion of the zero-point energy of the quantum vacuum into classical mechanical energy (PDF). Bremen Europ. Hochsch.-Verl. Archived from the original on 2010.
51. ^ Jeane Manning, Joel Garbon. Breakthrough power : how quantum-leap new energy inventions can transform our world. Amber Bridge Books.
52. ^ John Bedini, Tom Bearden. Free energy generation : circuits & schematics. Cheniere Press.
53. ^ Tom Bearden. Energy from the vacuum : concepts & principles. Cheniere Press.
54. ^ Tom Bearden. Clash of the geniuses : inventing the impossible. New Science Ideas.
55. ^ Tom Bearden. Virtual State Engineering and Its Implications. Ft. Belvoir Defense Technical Information Center.
56. ^ Thomas Valone. Practical conversion of zero-point energy : feasibility study of the extraction of zero-point energy from the quantum vacuum for the performance of useful work. Integrity Research Institute.
57. ^ Thomas Valone. Zero point energy : the fuel of the future. Integrity Research Institute.
58. ^ Thomas Valone. Future energy : proceedings of the First International Conference on Future Energy. Integrity Research Institute.
59. ^ Moray B King. Tapping the zero-point energy : how free energy and anti-gravity might be possible with today's physics. Adventures Unlimited.
60. ^ Moray B King. Quest for zero point energy : engineering principles for 'free energy' inventions. Adventures Unlimited.
61. ^ Moray B King. The energy machine of T. Henry Moray : zero-point energy & pulsed plasma physics. Adventures Unlimited.
62. ^ Christopher Toussaint, Bill Jenkins. Free energy : the race to zero point. Lightworks Audio & Video.
63. ^ Nick Cook. The hunt for zero point : inside the classified world of antigravity technology. Broadway Books.
64. ^ William James. Zero point : power of the gods. Bloomington.
65. ^ "Buddy Pine character profile". Retrieved 5 September 2012.
66. ^ "Zero Point Energy". Retrieved 5 September 2012.
67. ^ "Zero Point at FanFiction.net". Retrieved 5 September 2012.
68. ^ Stargate SG-1: The DVD Collection 52
69. ^ a b "The Rising" (Stargate Atlantis)
70. ^ "Zero Point Energy goes Hollywood!". Retrieved 2008-03-14.