The Lattice World, Quantum Foam and the
Universe-WideMetamaterial
David Thomas Crouse1
1The City University of New York,
[email protected] [email protected]
Abstract— The concept of a universe-wide gravity crystal that
combines Heisenbergś LatticeWorld and Wheelerś Quantum Foam is
described. It is assumed that space is a crystal with alattice
constant equal to the Planck length and a basis of a Planck mass.
Inertial anomalies arecalculated that include a parameter that
connects the external gravitational field and gravita-tional flux.
Similar to the electric permittivity and magnetic permeability of
metamaterials, thisparameter can take on positive, zero and
negative values.
In order to provide a solution to the self-energy of the
electron, Werner Heisenberg in 1930introduced the concept of space
being discretized in cells of volume r3o , with ro = ~/cMproton
[1].In correspondence with Niels Bohr, Heisenberg also states that
he believes that the discretenessof space underlies the quantum
mechanical uncertainty relations [2]. However, he rather
quicklysets aside this concept because of its obvious and inherent
problems - one being the breaking ofcontinuous rotational and
translational symmetries of space. Another problem pointed out by
Bohrwas that an absolute “minimum” possible length in one frame of
reference is length-contracted toa smaller value in a different
frame of reference, leading to a contradiction. However the
conceptdid not disappear - being further discussed by Hiesenberg,
Wolfgang Pauli, Gleb Wataghin, andrecently by a whole host of
physicist in the field of quantum gravity [3].
From Heisenberg in the 1930’s, one jumps ahead in time to 1957,
when John Wheeler introducedthe concept of quantum foam as part of
a theory emerging at this time called quantum geometro-dynamics
(QG), or quantum gravity as it is more widely known by today [4].
His objective for thepaper was to introduce the theory of QG and
show that all of classical physics is purely geometricaland based
throughout on the most firmly established principles of
electromagnetism and generalrelativity [4]. Using concepts within
the field of quantum electrodynamics, Wheeler argued thatfor a
possible history to contribute to the transition probability, the
phase of the exponent, whichis dependent on the metric g, should be
small (∼ 1 rad) to avoid destructive interference [4]. Thislimits
fluctuations in g (i.e., ∆g) that can occur over a volume of space
L3 to be on the order of∆g ∼ Lp/L, where Lp is the Planck length
(Lp = 1.61x10−35 m). The fluctuations in the metric∆g remains small
relative to g until L approaches Lp, at which point, ∆g ∼ g and
“the characterof the space undergoes an essential change... and
multiple connectedness develops” [4] that re-sults in an array of
“wormholes” in the topology of space with a lattice constant of
approximatelyLp; he calls this wormhole array a ”quantum foam”.
These wormholes have an electric charge ofqfluct = 12e that
produces an intense electromagnetic field energy E that has
associated with it a
mass of mp = c−2E =
√hcG = 2.17 × 10
−5g.
In this paper, the spatial order of Heisenberg’s lattice world
is imposed upon the quantum foamsuch that it forms a gravity
crystal (GC) throughout all of space. It is assumed that the
“localcompensation” of electromagnetic energy and gravitational
energy discussed by Wheeler either doesnot occur or is incomplete.
It is also assumed that the particles either are stable, and do not
comeinto and out of existence as Wheeler describes, or that they
are present over most of a time periodgiven by the Planck time tp =
5.39x10
−44 s and only fleeting into and out of existence every tp
stepin time so as to satisfy Heisenberg’s Uncertainty Principle.
With these assumptions, the empericalpseudopotential method (EPM)
[5] and tight binding method (TBM)[6] are used to calculate
thedispersion curves of non-relativistic and relativistic particles
within the GC, as well as the particles’effective mass as a
function of momentum and gravitational mass. It is shown that
particles in thiscrystal can appear to violate Einstein’s
Equivalence Principle equating inertial and gravitationalmasses,
but only because the the system includes not only the particle, but
the crystal as well.The inertial mass (mi) can diverge from the
gravitational mass (mg) as the particle’s momentum
Mass (kg x 10-8)
-8
-4
0
4
8
R Γ X M Γ1.51 0.5 0.5 10
Energy (eV x 1029)
Γ X
M
R
b3
b1
b2
0
1
2
3
4
R Γ X M Γ1.51 0.5 0.5 10
Lp
Lp
Real Space
Lp
Crystal Momentum (in units of π/LP)
Figure 1: Left : The gravity crystal as the fine structure of
space. The basis is assumed to be particle of massmp that produces
the spherically symmetric periodic Coulombic potential. The crystal
is cubic with a latticeconstant of Lp. The dispersion curve (Top
Right) and effective mass (Bottom Right) of a non-relativisticpoint
particle of mass mp within the crystal calculated using EPM. The
TBM yields similar results but ismost accurate for heavier
particles. It is seen that the effective inertial mass is not
constant and is dependenton the crystal momentum of the particle.
At low velocities, it is equal to the gravitational mass (blue
dashedline), but at high velocities it can be large and positive,
near zero or negative.
increases and can assume effective values either much larger or
smaller than the gravitational mass,be near zero, or even be
negative. Inertial anomalies of black holes and galaxies are then
discussed.
Once the effective inertial mass is calculated, the concept of
the GC is connected to the fieldof metamaterials. This is done by
introducing a constitutive relation between the gravitationalfield
and gravitational flux (similar to electric and magnetic fluxes in
materials) with a constant ofproportionality (a gravitational
permeability µg) that can unexpectedly take on values other
thanthose normally encountered (unity in the case of gravity and
for optical metamaterials anythingother than double negative
materials (i.e., � < 0 and µ < 0)). Thus, the inclusion of
the crystaĺseffects is done via µg while keeping the inertial mass
and gravitational masses equal to each other.Values for µg can vary
from large and positive, to near-zero, to negative values. However
if thisconcept proves to be true, then this new metamaterial is not
meta at all - it is not beyond nature,but the fabric of nature
itself.
In conclusion, this paper revisits two historical concepts
introduced by Werner Heisenberg andJohn Wheeler concerning the
discretization of space in the form of a universe-wide gravity
crystal.The calculation of the dispersion curves is performed, as
well as the effective masses of particleswithin this crystal. The
inertial anomalies that the crystal produces and its connection to
the fieldof metamaterials are discussed.
REFERENCES
1. W. Heisenberg. The Self-Energy of the Electron. In A. Miller,
editor, Early Quantum Electro-dynamics, pages 121-128. Cambridge
Univeristy Press, 1994/1930.
2. B. Carazza and H. Kragh. Heisenberg’s Lattice World: The 1930
Theory Sketch. AmericanJournal of Physics, 63:597-598, 1995.
3. Hagar, Amit. Discrete or continuous?: the quest for
fundamental length in modern physics.,pages 69-78 Cambridge
University Press, 2014.
4. Wheeler, John A. ”On the nature of quantum geometrodynamics.”
Annals of Physics 2.6(1957): 604-614.
5. Vasileska, Dragica, Stephen M. Goodnick, and Gerhard Klimeck.
Computational Electronics:Semiclassical and Quantum Device Modeling
and Simulation. CRC press, 2010.
6. Ashcroft, N. W. (1976). ND Mermin Solid state physics. pages
176-185, Saunders College,Philadelphia..
