DOW-UAP-D138 — AAWSAP DIRD, Warp Drive, Dark Energy, and the Manipulation of Extra Dimensions, April 2010

File: DOW-UAP-D138
Agency: Department of War
Incident date: 4/2/10
Location: Las Vegas, Nevada
Released in: Release 06 (2026-09-18)
File type: PDF

This document is a Defense Intelligence Reference Document (DIRD), a technical reference format used by the Defense Intelligence Agency (DIA) to capture baseline knowledge on a specific topic for later analytic use. DIRDs are best understood as reference and synthesis products rather than as original research. It is one of 38 DIRDs produced under the Advanced Aerospace Weapon System Applications Program (AAWSAP) between 2009 and 2011. Because AAWSAP’s scope permitted a broad range of supporting topics, not every DIRD in the series directly concerns aerospace systems or future threat assessment. The following summary reflects the DIRD’s scope and framing at the time of writing and should not be read as implying current validation of the concepts discussed.

This DIRD examines whether a warp-propulsion concept can be grounded in known theoretical physics by linking general-relativistic warp metrics with dark energy, Casimir effects, and higher-dimensional models from string theory and brane cosmology. The paper hypothesizes that if dark energy arises from vacuum effects and originates in extra dimensions, then a future technology capable of manipulating those dimensions might be capable of altering local spacetime expansion to generate a warp bubble. While framed as a method to mitigate the astronomical energy demands of more traditional warp models, the report acknowledges that this concept relies entirely on unverified assumptions; namely, the physical reality, stability, and macroscopic controllability of extra dimensions. Consequently, while the paper draws on mainstream theoretical physics concepts, the speculative chain linking them lacks empirical support and offers no viable engineering pathway toward a functioning propulsion system.

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TL;DR

April 2010: when warp‑drive proposals were still just Friday‑themed brainstorming.

DIRD PDF by DoW, dated April 2010, incident 4/2/10, Las Vegas, Nevada.
Examines speculative warp‑propulsion linking dark energy, Casimir effects, extra dimensions.
No official conclusion in the file

AI summary

This document is a classified report titled "Warp Drive, Dark Energy, and the Manipulation of Extra Dimensions," prepared by the Defense Intelligence Agency's Acquisition Support Division in April 2010. It was authored by AAP Person 74 and AAP Person 58. The report discusses theoretical concepts related to warp drives, dark energy, and extra dimensions, focusing on how these could enable faster-than-light travel. It references previous research, including a 1994 paper by Miguel Alcubierre, and explores the potential for manipulating spacetime using advanced physics. The document is part of a series of advanced technology reports produced under the AAWSA Program in 2009.

Full text

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Defense
Intelligence
Reference
Document
Acquisition Threat Support
2 April 2010
ICOD : 1 December 2009
DIA-08- 1004-001

Warp Drive, Dark Energy, and
the Manipulation of Extra
Dimensions

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Warp Drive, Dark Energy, and the Manipulation of Extra

Dimensions

Prepared by:
Acquisition Support Division (DW0-3)
Defense Warning Office
Directorate for Analysis
Defense Intelligence Agency
Authors:

AAP Person 74, AAP Person 58

Administrative Note
COPYRIGHT WARN ING: Further dissemination of the photographs in this publication is not authorized.

This product is one in a series of advanced technology reports produced in FY 2009
under the Defense Intelligence Agency, Defense Warning Office's Advanced Aerospace
Weapon System Applications (AAWSA) Program. Comments or questions pertaining to
AAWSA Program
this document should be addressed to !AAP Person 1
Manager, Defense Intelligence Agency, ATTN: CLAR/DWO-3, Bldg 6000, Washington,
DC 20340-5100.

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Contents
Introduction ............................................................................................................v
2. General Relativistic Warp Drives ........................................................................ 1
2.1 Warp Drive Requirements ............................................................................ 2
3. The Cosmological Constant ................................................................................ 4
3.1 Einstein's Equation and the Introduction of/\ .............................................. 4
4. Casimir Energy and the Quantum Vacuum .......................................................... 5
4.1 The Casimir Effect ........................................................................................ 6
5. Extra Space Dimensions ..................................................................................... 8
5.1 Kaluza-Klein Theory ..................................................................................... 8
5.2 Large Extra Dimensions.............................................................................. 10
5.3 Randall Sundrum Brane Models .................................................................. 11
5.4 Extra Dimension Summary ......................................................................... 12
6. Dark Energy as a Higher Dimensional Artifact .................................................. 12
7. Warp Drive and Higher Dimensional Manipulation ........................................... 15
7.1 Adjusting Higher Dimensions for Propulsions ............................................ 16
7.2 The Geometry of Extra Dimensions ............................................................ 17
7.3 Higher Dimensions and Stabilization .......................................................... 17
7 .4 Elementary Warp Drive Calculations .......................................................... 20
7.5 Future Experiments .................................................................................... 22
7 .6 The Development of the Technology .......................................................... 23
8. Summary .......................................................................................................... 24

Figures
Figure 1. York Extrinsic Time (9-) Plot..................................................................... 1
Figure 2. The Interior Region of Parallel Conducting Plates ................................... 7
Figure 3. Internal Structure of a Seemingly One-Dimensional Object .................... 9
Figure 4. Manipulated Extra Dimension ................................................................ 15

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Figure 5. Artist's Conception of a Futuristic Warp Drive Spacecraft ..................... 16
Figure 6. A Toroidal Higher Dimension ................................................................. 17
Figure 7. A Combination of Phenomenologically Viable Fields .............................. 19
Figure 8. False Vacuum Minima ............................................................................ 19
Figure 9. Thick and Thin Shell Warp Bubble ......................................................... 21

Tables
Table 1. Transit Times to Various Exotic Destinations at 100 Times the Speed of
Light .........................................................................................................vi
Table 2. Negative Energy Required for Warp Bubble (Larger Negative Energy) ..... 3
Table 3. Negative Energy Required for Warp Bubble ............................................ 22

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Warp Drive, Dark Energy, and the Manipulation of Extra
Dimensions
Introduction
If one is to realistically entertain the notion of interstellar exploration in
timeframes of a human lifespan, a dramatic shift in the traditional approach to
spacecraft propulsion is necessary. It has been known and well tested since
the time of Einstein that all matter is restricted to motion at sublight velocities
(<< 3 x 108 m/s, the speed of light, or c), and that as matter approaches the
speed of light, its mass asymptotically approaches infinity. This mass increase
ensures that an infinite amount of energy would be necessary to travel at the
speed of light, and, thus, this speed is impossible to reach and represents an
absolute speed limit to all matter traveling through spacetime.
Even if an engine were designed that could propel a spacecraft to an
appreciable fraction of light speed, travel to even the closest stars would take
many decades in the frame of reference of an observer on Earth. Although
these lengthy transit times would not make interstellar exploration impossible,
they would certainly dampen the enthusiasm of governments or private
individuals funding these missions. After all, a mission whose success is
perhaps a century away would be difficult to justify. In recent years, however,
physicists have discovered two loopholes to Einstein's ultimate speed limit:
the Einstein-Rosen bridge ( commonly referred to as a "wormhole") and the
warp drive. Fundamentally, both ideas involve manipulation of spacetime itself
in some exotic way that allows for faster-than-light (FTL) travel.
Essentially, the wormhole involves connecting two potentially distant regions
of space by a topological shortcut. Theoretically, one would enter the
wormhole and instantaneously be transported to the exit located in a distant
region of space. Although no observational evidence of wormholes exists,
theoretically they can exist as a valid solution to general relativity.
The warp drive-the main focus this paper-involves local manipulation of the
fabric of space in the immediate vicinity of a spacecraft. The basic idea is to
create an asymmetric bubble of space that is contracting in front of the
spacecraft while expanding behind it. Using this form of locomotion, the
spacecraft remains stationary inside this "warp bubble," and the movement of
space itself facilitates the relative motion of the spacecraft. The most
attractive feature of the warp drive is that the theory of relativity places no
known restrictions on the motion of space itself, thus allowing for a
convenient circumvention of the speed of light barrier.
An advanced aerospace platform incorporating warp drive technology would
profoundly alter the capacity to explore-and potentially to colonize-the
universe. Because a warp drive is not limited by the speed of light, one can
only guess the top speeds such a technology might be capable of achieving.
For the sake of argument, let's consider the duration of trips taken by a

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spacecraft capable of 100c1 for an array of exotic destinations of possible
interest. As Table 1 shows, trips to the planets within our own solar system
would take hours rather than years, and journeys to local star system would
be measured in weeks rather than hundreds of thousands of years.
Table 1. Transit Times to Various Exotic Destinations
at 100 Times the Speed of Light
Destination
Mars
Jupiter
Neptune
Alpha Centauri
Epsilon Eridani
The Orion Nebula

Transit Time
193 seconds
36 minutes
4 hours
15 days
38 days
1.3 years

Until recently, the warp drive was a concept reserved for science fiction.
However, a 1994 paper by Miguel Alcubierre placed the idea on a more solid
theoretical footing. Alcubierre (Reference 1) demonstrated that a specific
Lorentzian manifold could be chosen that exhibited bubble-like features
reminiscent of the warp drive from the popular Star Trek television series. The
bubble allowed for the surrounding spacetime to move at FTL speeds, and the
inhabitants of the bubble would feel no acceleration effects because spacetime
itself would be in motion instead of the spacecraft and its inhabitants.
A number of papers have emerged in recent years that build on this original
idea. However, these papers do not typically address how one might actually
create the necessary spacetime bubble. Our own research directly addresses
this question from a new and unique perspective and introduces a novel
paradigm shift in the field of warp drive study (Reference 2). More formally,
our work approaches the physics of warp drive from the perspective of
quantum field theory; this diverges from the more traditional approach to
warp drives, which utilizes the physics of general relativity. One of the
improvements the model introduces is a dramatic reduction in the overall
energy required to create such a phenomenon.
The roadmap to this new idea was the observation that spacetime is currently
known to be in a state of accelerated expansion, as demonstrated by the
redshifting of galaxies, and the belief that if the mechanism for this expansion
could be understood, then it might ultimately be controlled. A popular term
used by cosmologists today is "dark energy," an exotic and ubiquitous form of
energy that is believed to constitute over 70 percent of the matter-energy
content of the universe (Reference 3-6). One salient feature of dark energy is
its intrinsic ability to generate negative pressure, causing the fabric of space
to expand in the way that is currently observed (Reference 7).

1 This speed, while somewhat arbitrary, highlights the fact that our galaxy wo uld become far more accessible if or
when one discovers how to surpass the speed of light barrier.

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Although we know what dark energy does, we do not yet fully understand its
nature. We do not understand why it exists or how it is created; we simply
know it provides an ever-present force on spacetime, causing the universe to
expand. Indeed, recent high-precision experimental observations indicate dark
energy may be a cosmological vacuum energy (Reference 8-10). These
observations are based on the magnitudes of high-redshift supernova and
have been a source of high research activity of late owing to the unexpected
discovery that the rate of expansion of the universe is increasing (commonly
referred to as accelerated expansion).
One tantalizing aspect of dark energy is that if it were fully understood, and if
a technology were developed that could generate and harness the exotic
effects of dark energy on the fabric of space, then a warp drive would be one
step closer to technological reality. While a full understanding of the true
nature of dark energy may be many years away, it is entirely feasible that
experimental breakthroughs at the Large Hadron Collider or developments in
the field of M-theory could lead to a quantum leap in our understanding of this
unusual form of energy and perhaps help to direct technological innovations.
Our own research focuses on gaining an understanding of the physical origin
of dark energy. By exploring novel ideas at the forefront of theoretical physics,
one is able to propose a physically viable model incorporating some of the
cutting-edge ideas emerging from string theory and quantum field theory. This
leads to a deeper understanding of the possible origin of dark energy and
allows consideration of a mechanism that would allow a sufficiently advanced
technology to control the dark energy density in any region of space, and thus
the expansion of space. This work has clear implications for the advancement
of warp drive research.
This paper is structured as follows: Section 2 reviews the more traditional
general relativistic warp drives, the energy required to create them, and the
physics required to understand them. Section 3 discusses the cosmological
constant, a term featured in Einstein's equation that regulates the contraction
and expansion of the spacetime. Section 4 introduces the Casimir energy,
which, under certain conditions, may be the phenomenon that physically
generates the cosmological constant. Section 5 discusses higher dimensions in
physics and their importance in the context of Casimir energy calculations.
Section 6 introduces the formulas that demonstrate that the Casimir energies
in higher dimensions may in fact be the dark energy that is responsible for the
accelerated expansion of the universe. Section 7 relates all the previous
concepts together and introduces the novel warp drive paradigm. Section 8
performs original calculations of the energy required to create a superluminal
warp drive. Finally, the paper speculates about the technological progress that
would be necessary to turn this model into a reality.

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2 . General Relativistic Warp Drives
Alcubierre (Reference 1) derived a spacetime metric motivated by cosmological inflation
that would allow arbitrarily short travel times between two distant points in space. The
"warp drive" metric uses coordinates (t, x, y, z) and curve (or worldline) x = Xsh (t), y =
0, z = 0, lying in the t-x plane passing through the origin. Note that Xsh is the x-axis
coordinate position of the moving spacesh ip (or warp bubble) frame. The metric 1
specifying this particular spacetime geometry is (Reference 1):
(2.1)
where c is the speed of light, Vsh(t) is the speed associated with the curve (or warp
bubble speed), and Tsh(t) is the Euclidean distance from the curve. The warp bubble
shape function f (rsh) is any smooth positive function that satisfies f (0) = 1 and
decreases away from the origin to vanish when Tsh > R for some distance R. The
geometry of each spatial slice is flat, and spacetime is flat where f (rsh ) vanishes but is
curved where it does not vanish.
The driving mechanism of Equation (2.1) is the York extrinsic time, 9. This quantity is
defined as ( Reference 1) :

S = v," x," df .
C

The 9 behavior of the warp drive bubble
provides for the simultaneous expansion
of space behind the spacecraft and a
corresponding contraction of space in
front of the spacecraft. Figure 1 illustrates
the 9 behavior of the warp drive bubble
geometry. Thus the spacecraft is
enveloped within a warp bubble and can
be made to exhibit an arbitrarily large
faster-than-light {FTL) speed (Vsh >> c)
as viewed by external coordinate
observers. Even though the worldlines
inside the warp bubble reg ion are
spacelike for all external observers, the
moving spaceship (warp bubble) frame
itself never travels outside of its local
comoving lig ht cone and thus does not
violate special relativity.

d,;"

(2.2)

Figure 1. York Extrinsic Time (S.} Plot

1 A spacetime metric (ds 2), or line element, is a Lorentz-invariant distance fu nction between any two points in

spacetime that is defined by ds2 = 9a,-d><" dx'', where g,"' is the metric tensor which is a 4 x4 matrix that encodes the
geometry of spacetime and dX" is the infinitesimal coord inate separation between two points. The Greek indices ( µ,
v = 0... 3) denote spacetime coordinates, x 0...x 3, such that x 1...x 3 = space coordinates and xD = time coordinate.

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2.1 WARP DRIVE REQUIREMENTS
Implementation of FTL interstellar travel via warp drives requires engineering of
spacetime into very specialized local geometries as shown by Equation (2.1). The
analysis of these via the general relativistic field equation plus the resultant source
matter equations of state demonstrates that such geometries require the use of
"exotic" matter in order to produce the requis ite FTL spacetime modification. Exotic
matter is generally defined by general relativity (GR) physics to be matter that
possesses (renormalized) negative energy density and/or negative stress-tension ( =
positive outward pressure, aka gravitational repulsion). The term is widely
misunderstood and misapplied by the non-GR community. Also, it has been claimed
that FTL spacetimes are not plausible because exotic matter violates the general
relativistic energy conditions. 2 However, this has been shown to be a spurious issue
(Reference 11).
The energy density for the Alcubierre (Reference 1) warp drive that is derived from the
general relativistic field equation is complex, so we instead use a more simple formu la
to express the net energy required, E.varp , to build a warp bubble around a spaceship
(Reference 12):

=_ v2warp c4 R2 cr
Ewarp

(2.3)

=-( l.21 x 1044 ) v:,,up R2 cr,
where G is Newton's un iversal gravitation constant (6.673 x 10- 11 N·m 2 /kg 2 ), Vwarp is the
dimensionless speed of the warp bubble, R (> 0) is the radius of the warp bubble, and cr
(> 0) is proportional to the inverse of the warp bubble wall th ickness L'. (i.e., cr ~ 1/L'. ).
Equation (2.3) characterizes the amount of negative energy that one needs to localize
in the walls of the warp bubble . Table 2 presents a tabulation of the required negative
energy as a function of the "warp factor," Vwarp , One can compare the values of E.varp in
the table with the (positive) rest-energy contained in the Sun (1.79 x 10 47 J). The
consequence of Equation (2.3) and Table 2 is that if one wants to travel at hyperlight
speeds, then the warp bubble energy requirement will be an enormous negative
number. And this remains true even if one eng ineers an arbitrarily low sublight speed
warp bubble. Engineering a warp drive bubble is quite daunting given these results.
2 The condition for ordinary, classical (non-exotic) forms of matter t hat we are fami liar with in nature is that PE > p
and/or PE ;:: 0, where PE is t he energy density and p is t he pressure/stress-tension of some source of matter. These
conditions represent two examples of what are variously called the "standa rd" energy conditions : Weak Energy
Condition (WEC: PE.?: 0, PE + p .?: 0), Null Energy Cond it ion (NEC : PE + p ~ 0), Dominant Energy Condition (DEC),
and Strong Energy Condition (SEC). These energy conditions forbid negative energy density between material
objects to occur in nature, but they are mere hypotheses. The energy conditions were developed to establish a
series of mathematical hypotheses governing the behavior of collapsed - matter singularities in the study of
cosmology and black holes .

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Table 2. Negative Energy Required for Warp Bubble
(Larger Negative Energy)

Warp Factor, Vwarp

Ewarp (J)

10- 5 (= 3 km/s)
10-4 (= 30 km/s)
0.01 (= 3,000 km/s)
0.5 (= 150,000 km/s)
1 (= light speed)
2 (= 600,000 km/s)

- 3.03 X 1040
- 3.03 X 1042
- 3.03 X 1046
-7.59 X 1049
- 3.03 X 1050
- 1.21 X 1051
- 3.03 X 1052
- 3.03 X 1054

10 (= 3.0 X 106 km/s)
100 (= 3.0 x 107 km/s)

Assume: R = 50 m cr = 103 m- 1

Lobo and Visser (Reference 12) constructed an improved model of the warp drive
spacetime by applying linearized gravity to the weak-field warp drive case and testing
the energy conditions to first and second orders of Vwarp , The fundamental basis of their
model is that it specifically includes a finite mass spaceship that interacts with the warp
bubble. Their results verified that all warp drive spacetimes violate the energy
cond itions and will continue to do so for arb itrarily low warp bubble speed. They also
found that the energy condition violations in this class of spaceti mes is generic to the
form of the geometry under consideration and is not a side effect of t he superlum inal
properties. Based on these facts plus Equation (2.3) and Table 2, it appears that for all
conceivable laboratory experiments in which negative energy ca n be created in minute
amounts, the warp bubble speed will be absurdly low.
Coup ling of the finite spaceship mass with t he warp bubble leads to the (quite
reasonable) condition that the net total energy stored in the warp bubble be less than
the total rest-energy of the spacesh ip itself, which places a strong constra int upon the
(dimension less) speed of the warp bubb le (Reference 3) :

< [2__2 ( M .hip R , hip2 ti
v,.,.rpC

R.hip

J]i
(2.4)

where M ship and R ship are the mass and size of the spaceship, respectively, and R is the
rad ius of the warp bubble. Equation (2.4) ind icates that for any reasonable values of
the enginee ring parameters inside the brackets, Vwarp will be absurdly low. This result is
due to the intrinsic nonlinearity of the general relativistic field equation. To illustrate
this poi nt, the example starship parameters from Table 2 (R = 50 m, ~ 1/cr = 10-3 m)
are inserted into Equation (2.4) and assume M ship = 10 6 kg to find that Vwarp ~ 1.72 x
10- 14 (or 5. 16 x 10-6 m/s). Garden snails can crawl faster than th is. And if Rand M ship

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are kept constant, then t... = 3.37 x 10 24 m (or 3.57 x 108 light-years) in order for Vwarp :::;
1, wh ich is an unrealistic requirement on the warp bubble design.
Because this energy requirement is so phenomenally high one finds it of paramount
importance to explore new ideas in the field of warp drive technology . What now follows
is a pedagogically rich review of the novel warp drive concept that we have been
developing since 2005.

3 . The Cosmological Constant
Einstein is famous for a multitude of achievements in the field of physics. Arguably his
most notable contribution is the General Theory of Relativity, a geometric description of
gravitation whose fundamental idea relates the matter and the energy content of the
universe to the geometry of spacetime. Simply put, the presence of matter and energy
causes spacetime to curve, and this curvature controls how matter and energy move
through spacetime. General relativity has been the prevailing theory of gravitation since
1915 and thus far has unambiguously passed observational and experimental tests. It
remains an active area of research and technology is still being developed to test
certain features of the theory . Gravitationa l waves, for example, are one prediction
from GR; however, technology is only now reaching the stage of maturity to allow for
the detection of these waves .

3 .1 EINSTEIN 'S EQUATION AND THE INTRODUCTION OF/\
Upon completion of GR, Einstein applied his theory to the entire universe. He firmly
believed in Mach's principle, and the only way to satisfy this was to assume that space
is globally closed and that the metric tensor should be determined uniquely from the
energy-momentum tensor (Reference 13). He also assumed that the universe was
static, which was a reasonable assumption at the time because observational
astronomy had not advanced to a level that contradicted this paradigm. In 1917, when
a static solution to his equations could not be found, he introduced the cosmologica l
constant A (Reference 14): 3

8nG

RI"' - 2Rg pv = 7

~, v+ Agµv ·

(3.1)

In this equation Rµv is the Ricci curvature tensor, R is the Ricci curvature scalar, Tµv is
the stress-energy-momentum tensor, 4 and gJI" is the spacetime metric. The left-hand
side of Equation (3.1) encodes the curvature in the geometry of spacetime, and the
right-hand side encodes the source of matter-energy that curves spacetime.
The addition of A can be understood as a term in the equation which allows one to
adjust theory to match observation. In Einstein's case, he chose to add A to ensure that
the universe was static and unchanging. In later years, he often referred to this
amendment to his equations as his "biggest blunder." Several years after GR had been
formulated, the astronomer Edwin Hubble discovered the phenomenon of galactic
redshifting, which strongly indicated that the universe was indeed expanding. This
3 Pronounced "lambda ."

4 Tµ,· encodes the density and flux of a matter source's energy and momentum.

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theoretical prediction from GR was ignored by Einstein because of his belief in a static
universe.
Even though Einstein retracted the addition of A into his equations, it is now known that
it does indeed play a role and is typically included in GR equations. Data from precise
astronomical observations strongly suggest that an extremely small, yet non-zero A is a
necessary feature of GR and is responsible for the expansion of the universe that is
observed.
From a physical perspective, A represents an inherent energy density associated with
empty space. One way to envision this is to take a perfectly insulating box into deep
space, and then to remove all matter and all energy from this box so that it encloses a
perfect void. Even in this emptiness, a residua l energy fie ld would remain. According to
GR, the effect of th is energy would be to cause the region of space to expand, albeit at
an extremely small rate. To summarize, A is a ubiquitous, ever present feature of
space, and its presence causes space to expand.
In the late 1990s it emerged that not only is the universe expanding, but the rate of
expansion is, in fact, increasing. Since then, it has become more popular to refer to A
as dark energy, and the remainder of this paper will follow this convention.
Although the role of dark energy is extremely well understood mathematically, and in
the context of its effects on spacetime, its physical nature is still a mystery. One knows
that it is homogeneous, not particularly dense, and that it does not interact with any of
the fundamental forces of nature. One also knows that it exerts negative pressure on
spacetime, which expla ins the observed accelerated expansion (Reference 15, 16). As
there is yet to be a reasonable explanation for the fundamental origin of dark energy,
the problem is considered serious and has been tackled by a large number of eminent
and respected physicists, including previous Nobel prize winners (Reference 17).
Because dark energy is intimately related to the expansion of space, and because this
expansion is exactly the feature that would allow for a warp drive to function, an
understanding of this mysterious energy is of paramount importance in the
development of this novel propulsion technology.

4. Casimir Energy and the Quantum Vacuum
A central theme in th is paper is the notion of the quantum vacuum . To a particle
physicist, the term "vacuum" means the ground state of a quantum field in some
quantum theory for matter. In general, this ground state must obey Lorentz invariance,
at least with regards to three spatial dimensions, meaning that the vacuum must look
identical to all observers.
At all energies probed by experiments to date, the universe is accurately described as a
set of quantum fields. To a non-physicist a quantum field may, at first, be a strange
concept to grasp. This is because one generally likes to visualize the things one thinks
about; for example, an electron and even a photon provides something one can, on
some level, picture in one's minds. Simply put, a quantum field is an intangible
mathematical object whose properties are ideal in explain ing nature. Theories have
reached such an advanced level that the familia r physical images that one appreciates

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must be abandoned for more erudite mathematical constructions which are better
suited at describing the building blocks of nature (Reference 18-20).
If one takes the Fourier transform of a free quantum field,5 each mode of a fixed
wavelength behaves like a simple harmonic oscillator. A quantum mechanical property
of a simple harmonic oscillator is that the ground state exhibits zero-point fluctuations
as a consequence of the Heisenberg Uncertainty Principle. One way to understand these
zero-point fluctuations is to imagine releasing a pendulum and watching as dissipative
forces slowly try to bring the pendulum to a stop. The uncertainty principle would
ensure that the pendulum was never able to come to a complete rest, but instead
would exhibit microscopic oscillations around the equilibrium position indefinitely. Of
course, for a real macroscopic pendulum, these fluctuations would be miniscule and all
but impossible to detect; however, the analogy with a quantum harmonic oscillator
holds well. The expectation value of the energy associated with the ground state energy
of a quantum oscillator is:

(4.1)

In this formula c and n are the speed of light and Planck's reduced constant (1.055 x
10- 34 J.s), respectively, and k is the wave-vector related to the momentum of the
quantum field. One of the features of this ground state energy is that the wave vector
has an infinite degree of freedom. Clearly this sum is divergent; however, this is a
common feature of quantum field theory, and an array of mathematical techniques
known as renormalization exists to deal with the infinities that arise.

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Related media: war.gov pairing

Same release: Release 06

Same agency: DoW

Discussion

The Pentagon paid $21.9 million for warp drives, wormholes and invisibility cloaks: the receipts