Source and implications
of Zero-Point Energy (ZPE)

FEBRUARY 1st, 2025

Table of content

This article is a follow-up to the one on the black body and the ultraviolet catastrophy, which recalls the circumstances in which ZPE was discovered; and, thus, how quantum physics was born, in Max Planck’s time. Here I continue the exploration and transmission of the paper by Nassim Haramein, Cyprien Guermonprez and Olivier Alirol on The origin of mass and the nature of gravity. Where does ZPE come from? How is it experimentally validated? What are the implications of its discovery for physics in general, and for Nassim Haramein’s work in particular? These are the questions that set out the path I invite you to follow with me in this article. Let’s start at the beginning: vacuum contains an energy in its fundamental state, where there are no more particles. This is ZPE, made up of constantly fluctuating electromagnetic waves.

A quantum sea of particles and antiparticles

A quantum fluctuation corresponds to the temporary change in energy level of a particle, at a certain point in space; it is this change that enables the spontaneous creation of a virtual particle/antiparticle pair. We can think of it as the creation of a wave on a surface of water; this wave is actually composed of a wave above sea level, and a wave below, representing the exact opposite of the upper wave. Physicist John Wheeler described this as “quantum foam”. It’s as if we were looking down from space at a seemingly calm ocean, then perceiving the agitation and intensity of its waves as we move closer.

mousse-quantique-wheeler

Illustration of the quantum foam (J. Wheeler, 1955) where the energy level
is directly related to the observation scale at which one measures.
A flat surface at the charge radius rp ∼ 10-15 m can result from intense
and turbulent vacuum fluctuations at the Planck scale ℓ ∼ 10-35 m.
 [1]

Quantum fluctuations occur constantly, at very high frequencies. They involve energy transfers between particles – known as virtual particles – and the quantum vacuum, at very short distances and times (Planck scale [2]). They can have very low amplitudes, as well as high values for certain energies and times. They have all possible wavelengths. Although electric and magnetic fields oscillate with an average value of zero, the energy density – the 0-point energy – is non-zero. [3].

This energy is essential to explain most quantum phenomena. It’s not just a term in an equation: it has been confirmed experimentally. In 1948, the Casimir effect provided tangible proof [4] of the existence of this particle field.

Casimir effect

The device used to demonstrate the Casimir effect is as follows: two plates placed opposite each other create a cavity in the fundamental field. In this way, a certain percentage of the fluctuating modes in the vacuum are eliminated inside the cavity, as the longest wavelengths cannot penetrate. Energy is therefore greater on the outside, creating a pressure gradient that can be likened to a force: the plates are pushed towards each other. Steven Lamoreaux first measured the Casimir effect in 1997.

casimir-effect

The Casimir effect is an extremely weak force, perceptible only
at very small distances (of the order of a few nanometers to micrometers).
To detect it, it is necessary for the plates to be uncharged, otherwise the electrostatic
force generated could be much greater than it is, making it imperceptible.

Nassim Haramein shows that energy density is in fact equivalent to pressure; it is expressed as pressure, physically, mechanically. We will come back to this later. There’s also a dynamic Casimir effect: the plates are made to oscillate very, very rapidly, and photons are observed to emerge directly from the fundamental field. Let’s now look at how these particles are more formally described.

Observables, quantum operators and commutativity relation

Borh's model

A quantum system – a particle in its environment – is described using a wave function. This is a mathematical tool containing the properties of the system under study (energy, position, spin, velocity…) [5], properties that correspond to measurable physical quantities called observables. In quantum physics, observables cannot be manipulated directly; they are represented by quantum operators, which provide information about the system.

« In Bohr’s model, the electron can only exist in certain, discretely separated orbits, where each transition results in spontaneous emission or absorption of a photon. When a photon is absorbed, it is annihilated by jumping into the vacuum state, while an emitted photon is created by jumping out of the vacuum state. Similarly, the electron jumps by being absorbed and emitted from the vacuum at its new orbit. This is mathematically seen through the annihilation and creation operators which are lowering or raising by one quantum the energy level of the system [6]. »

Commutativity describes the relationships between these different operators. Two operators are commutative if the order in which they are applied is irrelevant.

Quantum matrix mechanics

In 1925, German physicist Werner Heisenberg developed quantum matrix mechanics, based on the use of non-commutative algebras to describe physical observables, in this case position x and momentum p. The precise value of these observables is not determined until it is measured; a quantum object therefore has no location until its position is measured. [7]. As the matrices representing x and p are not commutative, depending on whether we first apply the position operator to a wave function and then the momentum operator, or vice versa, we will obtain a different result. This implies a fundamental limitation on the precision with which position and momentum (and hence velocity [8]) can be known simultaneously for the same particle; this limitation is known as the uncertainty principle.

Heisenberg and uncertainty relations

“(…) Can we represent, within the framework of quantum mechanics, a situation where an electron is approximately — i.e. to within a certain imprecision — in a given position, and possesses approximately — i.e. again to within a certain imprecision — a given velocity? And can these imprecisions be made small enough to avoid any contradiction with experience? A brief calculation (…) confirmed that such a situation could be represented mathematically, and that the inaccuracies were linked by what were later called “quantum mechanical uncertainty relations”.”

WERNER HEISENBERG [9]
werner-heisenberg

The momentum and position of a particle are therefore linked by uncertainty relations [10]. This means that the more precisely we try to determine the position of a particle, the less precisely we can know its momentum, and vice versa.

The limit associated with the measurement uncertainty relation is expressed by the following inequality ( remember that momentum is the product of mass and velocity):

ZPE-uncertainty-principle

where (h bar) is Dirac’s constant, m the mass, ∆x the imprecision on the position measurement, ∆p the imprecision on the momentum measurement, ∆v the imprecision on the velocity measurement [11]. then appears as a lower bound for the simultaneous uncertainty of these observables. According to Heisenberg, this uncertainty is not due to a measurement defect, but to the dual wave and corpuscular nature of quantum objects [12].

The uncertainty relation also applies to energy and time [13]. However, energy can still be created without contradicting the principle of energy conservation, since the time interval considered is very short; the existence time of the particles thus produced is therefore infinitesimal, and their total energy zero, due to their subsequent, almost instantaneous annihilation [14].

Let’s take a look at how the behavior of elementary particles like electrons is described.

Dirac's equation

Paul Dirac, a British mathematician and physicist, was looking for an equation describing the behavior of particles when their speed reaches values close to that of light in the vacuum. To this end, he developed the relativistic quantum mechanics of the electron, combining recent discoveries in quantum physics with those of Einstein’s special relativity. His equation, formulated in 1928, admitted results corresponding to the electron, but also results corresponding to particles of negative energy.

Paul_Dirac

“From Dirac’s equation, a relativistic version of the Schrödinger equation, both positive and negative energy states are accessible by the electrons with [a positive solution as true as the negative one]. This observation led Dirac to postulate the existence of the Dirac sea as a pool of disposable electrons with negative energy that can appear out of the vacuum. From this conception Dirac successfully predicted the existence of the anti-electron, or positron, as a hole in the Dirac sea. In 1932, the positron existence was experimentally confirmed by Carl Anderson. This capacity of creation and annihilation of electron-positron pairs out of the electromagnetic vacuum predicted by the Dirac equation and demonstrating the physicality [15] of the quantum vacuum fluctuations [has been confirmed by the Casimir effect].” [16]

Dirac’s equation has led to many important discoveries, such as the prediction of the existence of antimatter. It also introduced the notion of particle spin (intrinsic angular momentum), a fundamental property of subatomic particles.

From Max Planck to Nassim Haramein

Space at the quantum level – at the Planck scale – is made up of nothing but pairs of particles and antiparticles, which rapidly appear and annihilate each other, forming the 0-point fluctuations of the electromagnetic field. Thus, all quantum systems undergo fluctuations even when in their fundamental state; this is due to their wave-like nature, but it is also – according to standard theory – a consequence of the uncertainty principle. The standard explanation is as follows: if an oscillator is to be stationary (i.e. have zero energy), its mass position must be exactly that of equilibrium, and its velocity must be exactly zero, which is not possible. Consequently, there is an energy that is always fluctuating and never zero: the ZPE.

But for Nassim Haramein, it is the motion of particle pairs – and not the uncertainty principle – that generates the ZPE, because if this energy is removed, the equations underlying quantum physics become inconsistent; they become commutative, and Dirac showed that the non-commutativity of observables is closely linked to the equations of quantum mechanics [17]. Put another way, ZPE maintains the non-commutativity of the creation and annihilation operators: it is this energy that leads to the uncertainty principle, not the other way round.

In the next article (soon online), we’ll revisit the concepts of mass and space, and gently begin our descent towards the mechanism leading ZPE to decay to the cosmological scale, allowing mass and forces to emerge.

Key points

  • Although the electric and magnetic fields oscillate with an average value of zero, the energy density – the ZPE – is non-zero. 
  • Dirac’s equation (1928) predicts the creation and annihilation of electron-positron pairs from the electromagnetic vacuum, demonstrating the physicality of quantum fluctuations.
  • The ZPE was confirmed experimentally, notably by the Casimir effect (1948).
  • ZPE is the cause of Heinsenberg’s Uncertainty Principle (1927) – not its consequence – because it maintains the non-commutativity of the creation and annihilation operators of particle/antiparticle pairs.

Notes & references

A quantum sea of particles and antiparticles

[1] Source : The origin of mass and the nature of gravity, p.13
[2] The Planck scale marks a limit below which our current understanding of physics, based on general relativity and quantum mechanics, ceases to apply; and above which the quantum effects of gravity become significant. 
[3] “The electric and magnetic fields oscillate with a zero mean value illustrating the fact that they are not observed at our usual time scale (τ ≫ τ0) since the coherent time τ0 is very small. Historically, this has been described as a field of ’virtual’ particles. However, the energy density calculated as the expectation value of the square of the electric field is non-zero.” Source : The origin of mass and the nature of gravity, op.cit., p.7
[4] Another example is the photoelectric effect, explained by Einstein in 1905 and for which he was awarded the Nobel Prize in 1921. The ZPE plays a crucial role here, as it determines the minimum frequency of light needed to extract an electron from a metal when exposed to light of sufficiently high frequency. See The origin of mass and the nature of gravity p.10 for an exhaustive list.

Observables, quantum operators and commutativity relation

[5] See also the article Quantum physics and reality
[6] The origin of mass and the nature of gravity, op.cit., p.10
[7] Only the statistical distribution of these values is perfectly determined at any given moment; this does not mean, however, that a quantum object is “in more than one place at the same time”.
[8] The momentum is the product of mass and velocity.

Heisenberg and uncertainty relations

[9] HEISENBERG Werner, La Partie et le Tout [The Part and the Whole], Ed. Flammarion, 2010, p.141
[10] Note that in 1927 Heisenberg spoke of relations of uncertainty and not of principle, which goes in the direction of a science of relations put forward on this blog; see the article Quantum physics and reality on this subject.
[11] This uncertainty becomes negligible for macroscopic objects, as their mass is very large, making the ℏ/2m term very small. Consequently, the uncertainty in position and momentum can be neglected, and these objects can be accurately described by Newtonian mechanics.
[12] The position of a particle corresponds to the location of a wave, while its momentum is linked to its wavelength. Knowing the position precisely means compressing the wave into a region of space, which increases the uncertainty about its wavelength, and therefore its momentum. See the article Observers and quantum physics on the subject of wave/particle “duality”.
[13] Although the position/amount of motion pair is the best known, the uncertainty principle actually applies to any pair of conjugate physical quantities (variables linked by non-commutative quantum operators), including energy and time.
[14] “Quasi instantaneous” is to be understood here as meaning a period of time too short to detect the said particles.

Dirac’s equation

[15] Physicality refers to what can be perceived, measured or observed in the material universe.
[16] Source : The origin of mass and the nature of gravity, op.cit., p.10

From Max Planck to Nassim Haramein

[17] « Dirac derived from the non-commutative rule non-commutativite a whole theory of operators mechanics which yields to the definition of the annihilation a and creation a† operators based on the position and
momentum operators. » Source : The origin of mass and the nature of gravity, op.cit., p.7


Photo credit Casimir effect : The origin of mass and the nature of gravity, op.cit.
Photo credit Werner Heisenberg and Paul Dirac : Wikimedia Commons

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