The fundamental particles that populate our universe (electrons, photons, quarks …) all possess a characteristic angular momentum or spin.  According to the Standard Model of particle physics, such angular momentum is not produced by an Earth-like physical rotation about a spin axis; rather, it is there because the particle is in a quantum state that has intrinsic spin as a conserved quantum number.  Depending on their spins, fundamental particles come in two contrasting quantum-statistic "personality types": individualistic fermions vs. group-oriented bosons.

In units of the modified Planck constant ħ (1.05457... x 10⁻³⁴ kg m²/sec), fermions have half-integer intrinsic spins (¹/₂, ³/₂, ⁵/₂ …).  They exhibit a "territorial" behavior described by the Pauli Exclusion Principle, allowing only one particle to occupy each quantum state.  Fermions have matter and antimatter versions that have opposite parities.  (Parity: does the wave function change sign under mirror reflection?).  Fermions must be rotated by twice 360 degrees to return to their original state.

All of the basic stable and semi-stable particles found in our universe are fermions, including electrons, muons, and neutrinos, as well as the composite spin-¹/₂ protons and neutrons, which are composed of three spin-¹/₂ quarks.  However, composite nuclei, which are made of neutrons and protons, can be of either fermions or bosons, depending on whether their neutron and proton numbers are odd or even.  For example, the helium-4 nucleus, composed of two neutrons and two protons, is a spin-0 boson, while the lithium-7 nucleus, composed of three protons and four neutrons, is a spin-³/₂ fermion.

All bosons have integer spins (0, 1, 2 …) and tend to congregate together, piling up many particles in the same quantum state.  Bose-Einstein condensates (see AV-77 in the March-1996 issue of Analog) are the most spectacular example of this behavior, with huge numbers of boson particles piling up in the same quantum state with the same wave function.  All of the mediating particles of the fundamental forces are bosons.  These include the photon of electromagnetism, the eight gluons of the strong interaction, the W± and Z⁰ of the weak interaction, and the Higgs particle, which gives other fundamental particles their mass.  These fundamental boson particles have no antimatter twins, and they return to the same state after rotation through only the usual 360 degrees.

Up to now, the physics of wormholes, specifically Morris-Thorne wormholes, has been investigated under the implicit assumption that their matter component has no important quantum effects and is made of boson particles.  This has led to the conclusion that wormholes have an intrinsic tendency to pinch off and close very rapidly, and that exotic (negative mass-energy) matter must be used to stabilize them.  Since our current technology can only produce a very small and localized quantity of negative mass-energy with the Casimir effect, we have no realistic prospects for creating and using such wormholes.  Further, their instability implies that creation of "natural" wormholes in the super-hot chaos that followed the Big Bang would be very unlikely.

The new development that we will discuss in this column is that European theorists Blazquez-Salenco, Knoll, and Radu (B-SKR) have reconsidered the physics of wormholes, but using the assumption they are made with fermion particles.  They employ Einstein-Dirac-Maxwell (EDM) theory, a semiclassical way of uniting aspects of quantum mechanics and general relativity.  Their calculation uses Planck units that set G = c = ħ  = 1, so their results are in units of the Planck length, mass, and time, and the Planck unit charge, which is about 3.3 electron charges.  Their calculations assume that the matter components of the wormhole are two Dirac fermions with half-integer spins arranged to have opposite spin orientations.  The resulting wormhole has mass M and is threaded by lines of electric flux that enter one wormhole mouth (giving the appearance of a negative charge -Q_e) and emerge from the other wormhole mouth (giving the appearance of a positive charge +Q_e).   There is also a requirement that Q_e > M in Planck units.  The threaded-field-line configuration has been described by John Wheeler as "charge without charge," since there is an electric field trapped by the topology of the wormhole, but no actual electric charge is present.  The two fermion particles occupying the wormhole throat have entangled wave functions, and appear to observers at the two wormhole ends as an entangled particle and antiparticle pair.

The EDM formalism allows the fermionic matter to be described by quantum wave functions rather than by quantum fields.  (We note that the EDM approach avoids the ugly self-energy infinities and absurdly large cosmological constant that are major problems for standard quantum field theory and contribute to its incompatibility with general relativity.)   The EDM semi-classical approach leads to a more tractable model.  If there was no electric flux through the wormhole throat, a discontinuity at the throat would require a layer of matter.  However, the inclusion of the wormhole-threading electric field tends to smooth the wormhole geometry, eliminating the discontinuity and the need for matter, exotic or otherwise, at the wormhole throat.

The result of the B-SKR calculation is a stationary spherically-symmetric wormhole that requires no exotic matter for stability.  They model this system analytically and numerically and show the behavior over a range of parameters.  Using their graphs, I conclude that a fermionic wormhole with an electric charge Q_e of about 330 electron charges will have a mass M of around 100 Planck masses (about 1.8 milligrams) and a throat radius of a few hundred Planck lengths.  (Recall that some estimates of the mass of Morris-Thorne wormholes indicated a negative mass for stability of a few Jupiter masses.)  A Planck length is about 1.62 x 10^-35 m, so the wormhole throat aperture would be much too small to pass wavelengths of visible light or even gamma rays.  A fermionic wormhole end would look something like a very massive multiply-charged particle.

B-SKR point out that although the configuration they analyze only includes two fermions, their approach can be extended to include states with an arbitrarily large number of fermions.  Adding fermions would presumably increase the size, mass, and quantum effects.  Increasing the throat-threading electric field would linearly increase the throat area. Therefore, perhaps by making a wormhole with a very large number of fermions and increasing the size of the threading electric field, one could arrive at a throat aperture that might be large enough for light-wave communication or even for the passage of matter.

Thus, the B-SKR calculation raises the possibility of stable particle-like wormholes that are very small but have a mass and electric charge that are manageable in the laboratory.  In my AV column "Shooting Wormholes to the Stars" (AV-162, May-2012 Analog), I described how one mouth of a hypothetical stable electrically charged wormhole with the charge-to-mass ratio around that of a proton could be accelerated to very close to the speed of light in an existing accelerator (e.g. the CERN LHC) and aimed at a distant star.

Because of the wonders of relativistic time dilation, the arrival time at the star is greatly reduced, as viewed through the wormhole's aperture.  The arrival time as viewed in the external world is T=L/c, where L is the distance to the star and c is the speed of light.  For example, the sun-like star Tau Ceti is 11.9 light years from Earth, so as viewed from Earth the arrival time of a near-lightspeed wormhole end would be about T=11.9 years.

The arrival time as viewed through a wormhole is T' = T/g, where g is the Lorentz factor [g= (1-v/c)^-½] and v is the wormhole-end velocity after acceleration.  For reference, the maximum energy protons accelerated in the CERN LHC have a Lorentz factor of 6,930.  Thus, the arrival time at Tau Ceti of an LHC-accelerated wormhole-end would be 15 hours.  In other words, even if T is a few decades, centuries, or millennia, T' can be a few days, weeks, or months.  Effectively, the accelerated wormhole becomes a time machine, connecting the present with an arrival far in the future.

The fermionic wormholes described by B-SKR are almost a fit for this concept.  They are stable, particle-like, and have an electric charge that can be used for acceleration.  However, the estimated wormhole mass is much too large (100 Planck masses is about 10²¹ proton masses) to be accelerated in a synchrotron like the LHC.  A linear accelerator could perhaps perform the needed acceleration, but it would need to be a huge specially-constructed linear accelerator that could handle the tiny charge-to-mass ratio of the accelerated wormhole mouth and give the accelerated wormhole end a near-lightspeed relativistic velocity.

Perhaps producing a fermionic wormhole built from a very large number of fermions and a large threading field would save the day.  It would grow in mass, size, and charge.  If the aperture could be made large enough to pass electrons, one could direct a beam of accelerated electrons through the negatively-charged wormhole mouth and boost the effective charge by threading more electric lines of force through it.   This might result in a wormhole with a large enough charge-to-mass to be accelerated in a synchrotron and with a large enough aperture for viewing, steering, and fast interstellar exploration.  In other words, the B-SKR theory needs to be extended to cover more fermions and larger threading electric fields.

One interesting implication of the B-SKR calculation is the possibility that, because of their simplicity and stability, fermionic wormholes might have been produced naturally in the super-hot era just after the Big Bang.  If that happened, primordial fermionic wormholes should still be around, and it might be sensible to search for them.

They might be a super-heavy component of cosmic rays.  One might search among the particles arriving from space for those that have a charge of many electron charges, but with a very large mass, so that they produce an electric pulse as they go by but with a trajectory that does not bend in a magnetic field

Alternatively, they might be trapped in rocks and minerals, awaiting discovery.  In a mass spectrograph, they could be in principle be pulled out of a vaporized sample by the electric potential, but would be so heavy that they would move in an essentially undeflected straight line in the magnetic field.  Or they might be so heavy that they would have been pulled by gravity to the center of the Earth as it formed.  Even in that case, such wormhole ends might still be found in meteorites that formed in a gravity free environment.

We should go out and look for them.

John G. Cramer's 2016 nonfiction book (Amazon gives it 5 stars) describing his transactional interpretation of quantum mechanics, The Quantum Handshake - Entanglement, Nonlocality, and Transactions, (Springer, January-2016) is available online as a hardcover or eBook at:  http://www.springer.com/gp/book/9783319246406 or https://www.amazon.com/dp/3319246402.

SF Novels by John Cramer:  Printed editions of John's hard SF novels Twistor and Einstein's Bridge are available from Amazon at https://www.amazon.com/Twistor-John-Cramer/dp/048680450X and https://www.amazon.com/EINSTEINS-BRIDGE-H-John-Cramer/dp/0380975106 .  His new novel, Fermi's Question is coming soon from Baen Books.

Alternate View Columns Online: Electronic reprints of 216 or more "The Alternate View" columns by John G. Cramer published in Analog between 1984 and the present are currently available online at: http://www.npl.washington.edu/av .

References:

Fermionic Wormholes:

Jose Luis Blázquez-Salcedo, Christian Knoll, and Eugen Radu, "Transversable wormholes in Einstein-Dirac-Maxwell theory", Physical Review Letters 126, 101102 (2021); ArXiv: 2010.07317v2 [gr-qc].

Bosonic Wormholes:

Michael S. Morris, Kip S. Thorne , and Ulvi Yurtsever, "Wormholes, Time Machines, and the Weak Energy Condition", Physical Review Letters 61, 1446 (1988).

Matt Visser, "Traversable wormholes: Some simple examples", Phys. Rev. D 39, 3182 (1989).

Exit to the website.