The Quantum Vacuum and Dark Matter
Since 1933 astrophysicists have known that there is more to this Universe than meets the (telescopeenhanced) eye. Using light from stars and nebulae, astronomers can derive good estimates of the masses of galaxies and galaxy clusters. Astrophysicists can use those masses to calculate the velocities that stars must have in galaxies and that galaxies must have in their clusters in order for those structures to appear as we see them. But the actual velocities inferred from astronomers’ measurements are substantially larger than the velocities that come from those calculations. That fact necessarily implies that galaxies and galaxy clusters contain more mass than astronomers can account through the light that they observe, most of it in the form of dark matter, matter that emits no light. Astrophysicists thus infer that the Universe contains about five times as much dark matter as lightemitting matter.
Further study indicated that the dark matter is not baryonic; that is, it doesn’t consist primarily of protons and neutrons, as does ordinary matter. It must be made of some other kind of particles. The question of what those particles are has flummoxed physicists since the 1970's, when interest in dark matter was revived by Vera Rubin, W. Kent Ford, and their team. All kinds of novelties have been and are being proposed, but efforts to detect those novelties, either coming from space or emerging from particle accelerators, have consistently come up null. Perhaps the dark matter involves something not so novel, some relatively ordinary but unsuspected phenomenon adding mass to the Universe?
I have in mind a phenomenon that is wellknown to physicists, one that they don’t normally think of as adding mass to the Universe. It’s an odd little consequence of Heisenberg’s indeterminacy principle, one which tells us that the vacuum continuously seethes with virtual particles popping into and out of Reality in matterantimatter pairs. Heisenberg’s principle tells us that, when it comes to accounting for conserved quantities, Reality does not acknowledge the existence of actions smaller than Planck’s constant; for relevant example, a small amount of energy δE can come into existence gratuitously if it lasts for only a brief instant δt in accordance with δEδt h = 4.1357x10^{27} Mevsec. Thus, an electronpositron pair (δE = 1.022 Mev) can come into existence gratuitously for as long as 4.047x10^{27} second.
During that brief period the electronpositron pair emits a gravitational field and also responds to any gravitational field existing at the point of manifestation. Thus the particles add mass to the Universe, however temporarily, in a way that reveals its existence. What would it take for this phenomenon to account for the dark matter? The following chart shows the average density of protons and the equivalent density of electronpositron pairs needed to account for the dark matter and dark energy:
% 
Mev/m^{3} 
protons/m^{3} 
e–e+/m^{3} 
kg/m^{3} 

baryonic matter 
4.9 
5,536 
5.90 
5,417 
9.87x10^{27} 
dark matter 
26.8 
30,279 
32.28 
29,627 
53.99x10^{27} 
dark energy 
68.3 
77,765 
82.27 
76,091 
138.66x10^{27} 
total 
100.0 
113,580 
120.45 
111,135 
202.51x10^{27} 
Almost 30,000 electronpositron pairs in a cubic meter on average doesn’t sound like much, but it is sufficient to account for the dark matter if they actually exist. However, those pairs must, in essence, exist continuously and each pair actually exists for a tiny fraction of a second. In order for us to have the equivalent of one pair existing continuously for one second, we must have 2.5x10^{26} pairs pop into and out of existence end to end. Those Heisenberg pops can also happen at random in the course of one second and we will still get the desired average, so to get an average of 29,627 electronpositron pairs in a cubic meter we need to have 7.32x10^{30} manifestations occurring every second in that cube. How does that happen? What makes virtual particles come real, albeit temporarily?
We assume that a virtual particle is a locus where the potential for the existence of various properties (such as electric charge, spin, massenergy, etc.) exists. Those ghosts of particles are independent of each other, interacting only faintly, so they constitute an ideal gas filling the Universe.
Because of the need to obey the conservation laws, each particle will only interact with its own antimatter counterpart in a manifestation into Reality. The two particles acquire equal and opposite electric charges (in accordance with conservation of electric charge), equal and oppositely oriented spins (in accordance with conservation of angular momentum), and so on. They also acquire massenergy sufficient to become real, but only for a brief elapse of time specified by Heisenberg’s indeterminacy principle. One of the other aspects of Heisenberg’s principle, the relation between indeterminacies in linear momentum and position, provides the precipitating factor in realization of the particles.
Indeterminacy in the linear momentum of a particle, δp, correlates with indeterminacy in the particle’s position in the same direction, δx, through Heisenberg’s principle, δpδx h. Physicists incorporate that fact into their description of a particle by asserting that the particle’s properties ride an aleatric wave, which, when the particle is at rest in some frame of observation, consists of a sphericallysymmetric standing wave. We describe that wave through a state function, which generates a complex number at each and every point in phase space: squaring that number yields the probability density of finding the particle properties manifested at the associated sixdimensional point.
When a particle moves, we write its state function as a superposition of the spherical standing wave and a plane wave representing the linear momentum associated with the motion. The δx associated with the spherical wave still projects in all directions, so we infer that the indeterminacy in location projects laterally from the particle as well as forward and backward. We see that inference exemplified in the observation that photons won’t propagate through channels narrower than their wavelengths. But a photon doesn’t have a standingwave component: it is all traveling wave, so the size of the travelingwave component must be what determines the value of äx in Heisenberg’s principle, for particles as well as for photons. So what momentum do we assign to our virtual particles in order to determine their interaction cross sections, the indeterminacies in their positions?
In the brief interval when they exist as real particles, virtual particles can interact with other particles, usually photons of the Universe’s general radiation field (that is, the cosmic background). We thus expect the temperature of the virtual electronpositron gas to have the same value as the temperature of the cosmic background: 2.725 Kelvin. At that temperature the average particle in an ideal gas carries 2.348x10^{4} electron volts of kinetic energy (Boltzmann’s constant = 8.617049x10^{5} electron volts per Kelvin degree per particle). The average virtual electron or positron has a total energy equivalent assigned to it in accordance with
(Eq’n 1)
Solving that equation tells us that the average virtual electron/positron moves 9088 meters per second relative to the rest frame defined by the cosmic background.
That speed gives a particle momentum if the particle has mass. Because it represents an average, we can take that speed as representing the indeterminacy in the momenta of the particles in the virtual gas: we need only multiply that speed by the particle’s mass. Virtual particles have no actual mass; nonetheless, we can calculate a potential indeterminate momentum for them and use it to calculate the indeterminacy in the particle’s position. In this case we have
(Eq’n 2)
which leads us to calculate
(Eq’n 3)
The rest massenergy of an electron is 0.511 Mev and hc^{2}=3.71698x10^{10} Mevm^{2}/sec., so we calculate δx=8x10^{14} meter.
A sphere of that radius has a cross section of 2.01x10^{26} square meter. If we were to cover a sheet measuring one meter square with such spheres, we would need 5x10^{25} of them. If we make those particles positrons, then an electron passing through the sheet will interact with one of them with 100% certainty. Imagine using that sheet to bisect a meterwide cube and imagine moving each particle, in a direction perpendicular to the sheet, by a random distance between +1/2 meter and 1/2 meter. Our electron will still collide with the indeterminacy sphere of one of the positrons with almost 100% certainty, so in order to get the desired rate of manifestations occurring every second in that cubic meter we must have 7.32x10^{30} virtual electrons passing through it in that second.
We expect that the densities of virtual electrons and virtual positrons are equal to each other, yet here we calculate them differing by five orders of magnitude. However, if we increase one density and decrease the other in the same proportion, we still get the number of manifestations that we need every second. Exploiting that fact, we calculate that the density of both the virtual electrons and the virtual positrons must equal 1.913x10^{28} particles per cubic meter on average.
But Virtuality contains more than electrons and positrons. We would also find muons and antimuons, protons and antiprotons, and many other kinds of particles as well. Surely they must add something to the dark matter. Consider the muon, for example.
The muon ponders 105.659 Mev, 206.77 times as much as an electron. That fact means that a virtual muonantimuon pair can come real for an interval 0.0048 times as long as an electronpositron pair does. Like the other virtual particles, the muons and antimuons interact with the Universe when they visit Reality, so they have the same temperature. At a temperature of 2.725 Kelvin the average muon travels 632 meters per second, which means that the average virtual muon has a Heisenberg radius 14.38 times smaller than that of the average virtual electron and a cross section 206.77 times smaller. If we put 206.77 times more antimuons than positrons in our cubic meter, we will still get a collision with almost 100% certainty when one virtual muon passes through that volume. If we have 7.32x10^{30} muonantimuon pairs becoming manifest every second in that cubic meter, we will still get the same contribution to the mass of the Universe as we get from electronpositron pairs: the existence of the muonantimuon pairs lasts for an interval 206.77 times shorter than that of the electronpositron pairs, but they contribute 206.77 times as much mass. For this to happen, the virtual muons and antimuons must each have a density 14.38 times thicker than that of the virtual electrons and positrons.
Virtuality contains particles and antiparticles of all possible kinds, of course, but until we can deduce or measure the densities of those particles, species by species, we can’t say how much each species contributes to the dark matter. Until we can say, it does us no real harm to assume, for the sake of simple calculations, that only electronpositron pairs are involved.
Of course, the sole touchstone of science is observation, either natural or contrived (as in wellcrafted experiments). Astronomers have inferred the existence of dark matter from their observations of the Universe and so far physicists have been unable to determine what constitutes it. Now we have an hypothesis about the constitution of the dark matter and we must test it; otherwise, it’s no good.
In order to prove and verify the hypothesis, we need some experiment that demonstrates the existence of electronpositron pairs in sufficient quantity to account for the dark matter. Such experiments are already being performed, with a device known as a Qthruster.
Qthruster refers to an electromagnetic jet engine that uses the quantum vacuum as propellant. The heart of the device consists of an electric field (E) crossing a magnetic induction field (B) at a right angle. When an electronpositron pair pops into existence, the electric field drives the two particles apart, making them act as a minuscule electric current until they pop back out of existence. That current interacts with the magnetic field, which gives both particles a small change of motion in the direction of the Poynting vector (EXB). That change of motion necessitates that the engine acquire a minuscule change in linear momentum in the opposite direction, in accordance with Newton’s third law of motion. How much thrust could a Qthruster, in theory, obtain?
We look at the electronpositron pair interacting with the electric field. Imagine that the electric field has an intensity of one volt per meter. One Coulomb of electric charge immersed in that field exerts a force of one newton upon anything it touches. Multiplying that force by the time interval over which it acts gives us the change in linear momentum that it confers upon the object that it pushes and dividing the product by the mass of the object gives us the corresponding change in velocity. An electron carries an electric charge of 1.6022x10^{19} Coulomb and ponders 9.11x10^{31} kilograms, so over an interval of 4x10^{27} second it changes its speed by 7.025x10^{16} meter per second. The average change of speed is half of that and it’s the same for the positron, but in the opposite direction.
In a magnetic induction field with an intensity of one Tesla (10^{4} gauss) an electron moving at one meter per second perpendicular to the field’s direction gets pushed in the direction perpendicular to both the field and the electron’s motion with a force of 1.6022x10^{19} newton. During its existence in our one volt per meter electric field, an electronpositron pair acquires 4.5x10^{61} newtonsecond of momentum change and the source of the fields acquires the same amount in the opposite direction. In the course of one second in a cubic meter we have 7.32x10^{30} electronpositron pairs popping into and out of existence, so we have 3.29x10^{30} newtonsecond of momentum or 3.29x10^{30} newton of force exerted by the quantum vacuum in response to the fields.
If we increase the electric field intensity to one million volts per meter (1000 volts across a gap one millimeter wide), we increase that force one millionfold. If the density of virtual particles increases (and inside a galaxy the density will be greater than the Universal average that I used above), then the force increases according to the square of the proportional change in density: we get X more targets and X more projectiles, so we get X^{2} more hits.
Experiments have shown that a Qthruster with much less than a cubic meter of active volume can generate one to four millinewtons of thrust. We expect the virtual particles to have a greater density inside a galaxy than they do in the Universe as a whole, but the experimental result implies that inside the Milky Way and, perhaps, in the presence of stars and planets the density is billions, even trillions, of times thicker than the Universewide average. However, real matter displays similar differences in density, so we should not be surprised if virtual matter does the same.
One problem continues to vex Qthruster theory. What happens to the momentum and energy that the electronpositron pairs acquire during their sojourn in Reality when the particles pop back into Virtuality? The conservation laws tell us that those properties cannot simply cease to exist, so whither do they go?
Suppose that we have observed an electronpositron pair popping into existence and then vanishing back into Virtuality. In the brief time that they exist, the particles don’t move, so there’s no linear momentum or kinetic energy that we must take into account in describing them. But if another team of observers passes us on a fast train, they see the particles moving and, therefore, carrying linear momentum and kinetic energy. Those properties appear when the particles emerge from Virtuality and disappear when the particles lose their reality.
Those properties are not themselves manifestations of Heisenbergian indeterminacy. They exist by virtue of the particles possessing mass, however temporarily, and motion. We have assumed, quite reasonably, that the particles move at some velocity before they become manifest and that they maintain that motion, in accordance with Newton’s first law of motion, after coming real. The conservation laws tell us that the particles can’t simply obtain linear momentum and kinetic energy when they emerge from Virtuality; they must have had those properties before they became real. As the conservation laws require, the momentum and energy are neither created nor destroyed, but are merely hidden and made unavailable for interactions with real particles when the particles carrying them exist in Virtuality. So any momentum and energy that the particles acquire while they are real gets taken with those particles when they return to Virtuality. That analysis applies to the exhaust of the Qthruster.
Thus we have a reasonable hypothesis concerning the nature of the dark matter. Now all we need is one or more experiments that will test the hypothesis in greater detail and either verify or falsify it.
Appendix: Detecting the Quantum Vacuum
We still have the question of how we can detect the quantum vacuum experimentally and even measure the density of its virtual particles. Whatever method we use must exploit the virtual particles in those brief intervals when they come real. Our instruments must be able to detect some sign of an interaction between the Heisenberg pairs and some phenomenon that we can control. The Qthruster provides the basis for one such experiment, but there is another, more direct experiment that physicists can perform.
If an electronpositron pair comes temporarily real and collides with another particle carrying at least 1.022 million electronvolts of kinetic energy, the particles achieve permanent reality. The positron then undergoes mutual annihilation with an electron, releasing a pair of 511kev gamma photons. Thus, if we send energetic particles through an evacuated region of space, we should expect to see a faint flux of gamma photons coming from that region.
Thirty thousand electronpositron pairs manifested at any given instant in one cubic meter doesn’t look like a promising target for any experiment, but that number is just the Universal average. We expect the density to be higher inside a galaxy and close to gravitating bodies, so an experiment on Earth doesn’t look too daunting.
Physicists can do the experiment with existing particle accelerators. We need to use a straight stretch of the vacuum tube in which the particle beam travels (to avoid Bremsstrahlung) and we need to shield it from all external sources of radiation, especially 511kev gamma radiation. That done, the experimenters must send a beam of particles that all carry slight more than 1.022 million electron volts of kinetic energy apiece and put as many particles as possible into the beam. If the above hypothesis gives us a correct picture of the quantum vacuum, then the experiment will produce a faint flux of 511kev gamma photons. Now we just need someone to take a look and see.
eabf