Zero-Point Field Induced e+e- Pair Creation for Energy Production

Idea Conceived on 11/24/96

Draft Finished on 6/2/99

Paper Finished on 2/25/13

Javier O. Trevino

Physicist

Innovative Energy Technologies

3402 LeBlanc

San Antonio TX 78247

210-601-0655

jtrevinoaff@gmail.com

Abstract It has recently been demonstrated that strong electric fields induce e+e- pair production.

An isolated monochromatic zero-point field although stochastic in nature, has pockets of

organized photons which may mimic strong transient electric fields. These pockets of

strong “virtual” fields manifest themselves at predictable rates. The resulting pair

production/annihilation rate is used to calculate an energy production rate based on the

pair annihilation energy. Energy production rates on the order of MW/m3 should result

from isolating a zero-point field in the 10-12

to 10-15

m region of the zero-point spectrum.

Purpose The purpose of this paper is to introduce the concept of modifying the zero-point vacuum

field to mimic e+e- pair producing critical electric fields. As the pairs annihilate, they

create real photons that can be used as energy. I calculate the energy production rate as a

function of zero-point spectra and find the critical wavelength which produces useable

rates of energy production.

Background

The idea of extracting energy from empty space may seem like a fantasy, but modern

scientists are exploring that very idea. Current theories include various schemes to use or

capture the zero-point field (e.g. Casimir effect mechanics). While this is a reasonable

theme, I propose a new theme by asking; What about the seething mass of virtual

particle/antiparticle pairs that are a property of the quantum vacuum? If we can extract

e+e- pairs from the quantum vacuum, this would present us with an inexauhstable source

of fuel. The question is, how do we extract these particles?

Schwinger predicted that a sufficiently strong field will cause the vacuum to

spontaneously break down and release particle/antiparticle pairs [4]. When the vacuum is

under the influence of a sufficiently strong field, the vacuum decays to its lowest state by

releasing particles.

It is currently possible to raise these particles out of their virtual existence by applying a

sufficiently strong field (gravitational or electromagnetic). Hawking radiation is one

example of how a strong gravitational field can cause the vacuum to release particles.

There are experiments being conducted with Hi-Z nucleons in which the collision of two

nuclei produces a transient critical electric field on the surface of the nucleus. The field

is strong enough to cause a spontaneous breakdown of the vacuum – thereby releasing

particles. Recently, experiments with a polarized high irradiance Laser succeded in

producing electric fields strong enough to induce a breakdown of the vacuum – releasing

particles from seemingly “out of nowhere.” These examples indicate that strong field

experiments are consistent with Schwingers 1951 prediction that strong fields will

precipitate matter-anti-matter pairs [4].

These methods are a demostration of the reality of strong field pair creation scenarios.

However, the energy and bulk required for the effect is impractical for compact, portable

power applications. Additionally, the production rate is many orders of magnitude

smaller than what is needed to support realistic energy needs. What we need is a way to

mimic the pair producing effects of a strong field in a reasonably compact device. The

resulting pair annihilation energy density rate should approach the MW/m3 range.

Additionally, the device must mimic strong fields with minimal energy input to the

device.

Although this seems to imply an over-unity efficiency device, this is not necessarily the

case. The device should consume very little energy to liberate the matter/antimatter fuel.

Consider a scenario with a large battery. Suppose the battery is equipped with a

normally-open switch. It only takes a few ergs of energy to close the switch, but once

you close it, huge quantities of energy flow through the circuit. However, once the

switch is opened, the energy flow stops. This proposal is very similar. The e+e- pairs are

there – waiting for someone to “open” the vacuum. We already know how to open the

vacuum (Strong Fields). The question is how do we keep the vacuum open?

Specifically, how do we mimic strong electric fields indefinitely?

Strong electric fields are produced by super-dense charge densities, or rapidly changing

magnetic fields. One method utilizes a particle accelerator to slam heavy nuclei together

resulting in densely charged quasi-nuclei that create intense fields at the surface of the

nuclei. Another method uses intense laser pulses slamming into dense electron beams

resulting in backscattered laser radiation which constructively interferes with the

incoming beam to form intense transient electric fields. These methods are innovative,

but they are bulky, inefficient, transient, and produce minute numbers of e+e- pairs.

They are not candidates for portable energy production.

Consider the following: A strong electric field is a dense ensemble of organized virtual

photons. The question is: Where in nature is there already a huge assemblage of virtual

photons? The answer is the zero-point field. For many years, there was debate whether

zero-point photons were real or not. Various phenomenon, like vacuum polarization, and

the Casimir effect indicated that quantum vacuum fluctuations produce measurable

effects. Recent experiments by Lamaroux reveal that these fluctuations are real [1].

Since the zero-point field is an ensemble of randomly oriented virtual photons, why not

explore the idea of “engineering” the zero-point vacuum field to mimic a strong field?

This paper does exactly that. The theory is a semi-classical population based analysis of

the quantum vacuum. We break down the Lorentz invariant zero-point spectral energy

density to monochromatic zero-point fields. From this, and the Schwinger model of field

induced pair production, a calculation is made of the number of monochromatic zero-

point photons necessary to induce a breakdown of the vacuum. A calculation yields the

photon number density and critial field recurrence rate. A calculation is then made to

pinpoint the monochromatic spectrum that has a sufficient energy density to mimic

critical field strengths. The recurrence frequency is determined to deduce the pair

production rate using the Schwinger model. From the pair production rate, a simple final

calculation yields the energy production rate based on the non-relativistic annihilation

energy of the e+e- pairs.

Results of a simplified computer model of the monochromatic zero-point field indicate

that these fields can have pockets of organized photons which mimic locally strong fields.

Introduction The quantum electromagnetic field is an exchange of virtual photons among charged

particles. At any point in time, the quantum and classical field Hamiltonian’s must be

equal. The expression that describes this equivalence is well known and accepted in the

literature [4,5].

)2

1()(

8

1 223

kkV

nBExdH (1)

Quantum Field Theory states that the field operators E, B, and the number operator k

n do

not commute, and there is never a situation where we know exactly how many photons

there are in the field [5]. We only know that an array of them are summed to produce the

Hamiltonian. We use the latter fact to construct a semi-classical population based theory

of virtual field induced pair production. The idea of the electromagnetic vacuum field

interacting with the particle vacuum field is not new. Dyson briefly mentions such an

effect under the guise of “internal” vacuum polarization [6].

Our goal is to mimic strong “real” fields by using the already existing virtual photons

within the zero-point field to create a virtual electric field.

Vacuum Spectra Vacuum zero-point photons are virtual photons that obey a Lorentz invariant spectral

energy density [4,8,9].

32

3

2)(

c

(2)

Integrating over the frequency gives us the monochromatic energy density

32

4

8)()(

cdu

. (3)

The zero-point Hamiltonian is simply the sum of zero-point photon energies and is

expressed as

2

H . (4)

Th energy density of the whole field can be neatly expressed as the summed product of

the whole field number density, and zero-point photon energies

nU

2

(5)

where the number density is the monochromatic energy density divided by the quantized

vacuum photon energy

)(2)(

un . (6)

The expressions above are our tools from which we will compute the parameters

necessary for zero-point field induced pair production.

Computations

Crital Field

The first piece of information we need is to answer the question of how strong of a field

does it take to induce pair creation? We go to Schwingers Field Induced Pair Production

model which describes the probability per unit volume per unit time for pair creation

given by the expression below [4]

j eE

cmj

jc

EeW )][exp(

1

0

32

222

2

0

2

. (7)

E0 is the electrostatic field intensity and j is summed from 1 to infinity. First of all, the

expression can be simplified by observing the fact that the summation converges rapidly.

The function is asymptotic with respect to E and j. When E0 is 1015

V/m, the first term

accounts for almost 100% of the whole sum. When E0 approaches 1020 V/m, the first

term accounts for more than 60% of the whole sum, and W approches the asymptote after

the first 20 terms. In fact, W approches the asymptote after the first 20 terms for all

values exceeding 1020 V/m. See graph below:

Since the first term is within an order of magnitude of the whole sum for all E’s and j’s,

we will just use the first term and perform order of magnitude spreadsheet calculations to

estimate the result. The calculation reveals that the critical field value where the

probability per cubic meter per second approaches 1 is when E0 = 5.5 x 1015 V/m. This

results in an energy density of 2

2

00 E = 1.35 x 1020 J/m3 . Milonni calculates a critical

field of ~ 1018

V/m based on forcing the exp( ) argument to approach 1 [4]. This yields a

probability value of W = 4 x 1056

and an energy density of 4.4 x 1024

J/m3. However,

experiments reveal that critical fields lasting ~ 10-20

seconds in the vicinity of a large

quasi-nucleus create e+e-pairs [2]. So if we let the probability be 1, the time-scale be ~

10-20

seconds, and the volume be based on typical heavy element nuclear volumes (~ 10-

42 m

3), then W = 10

62. This yields a critical field of 3.6 x 10

20 V/m or an energy density

of 2.3 x 1029

J/m3.

The table below outlines the discussion above.

E (V/m) W (prob/m3-sec) Energy Density (J/m

3)

5.5 x 1015

1 1.35 x 1020

1.0 x 1018

1056

4.4 x 1024

3.6 x 1020

1062

2.3 x 1029

0

2E+68

4E+68

6E+68

8E+68

1E+69

1.2E+69

1.4E+69

0 5 10 15 20 25 30

W

j

Rapid Convergence of W respect to j when E = 10^24 V/m

W

Table 1. Properties of the Schwinger Probability Function

From these calculations, it is difficult to say exactly what the threshold for critical field

pair creation is, but, we can safely say that critical energy densities are on the order of

1020

to 1029

J/m3.

Photon Density

Next, we want to know how many monochromatic photons does it take to produce a field

with those energy densities. We will use the monochromatic energy density formula

(Eqn 3) to create a table of results. Since it is easy to calculate other properties other than

number density, the table is expandeded to include quantities such as photon lifetime,

photon range, and individual photon energy density. Properties of monocromatic

bandwidths from wavelengths of 1 meter (Radio Waves) to 1x10-20

m aare shown in the

table below. Keep in mind that the goal is to calculate the number of monochromatic

photons needed to create a critical electric field with energy densities from 1020

to 1029

J/m3.

u n R

(m) (1/s) (1/s) (J) (eV) (J/m3) (#/m

3) (s) (m)

1.00E+00 3.00E+08 1.88E+09 9.93E-26 6.20E-07 6.24E-25 6.28E+00 6.67E-09 2.00E+00

1.00E-01 3.00E+09 1.88E+10 9.93E-25 6.20E-06 6.24E-21 6.28E+03 6.67E-10 2.00E-01

1.00E-02 3.00E+10 1.88E+11 9.93E-24 6.20E-05 6.24E-17 6.28E+06 6.67E-11 2.00E-02

1.00E-03 3.00E+11 1.88E+12 9.93E-23 6.20E-04 6.24E-13 6.28E+09 6.67E-12 2.00E-03

1.00E-04 3.00E+12 1.88E+13 9.93E-22 6.20E-03 6.24E-09 6.28E+12 6.67E-13 2.00E-04

1.00E-05 3.00E+13 1.88E+14 9.93E-21 6.20E-02 6.24E-05 6.28E+15 6.67E-14 2.00E-05

1.00E-06 3.00E+14 1.88E+15 9.93E-20 6.20E-01 6.24E-01 6.28E+18 6.67E-15 2.00E-06

1.00E-07 3.00E+15 1.88E+16 9.93E-19 6.20E+00 6.24E+03 6.28E+21 6.67E-16 2.00E-07

1.00E-08 3.00E+16 1.88E+17 9.93E-18 6.20E+01 6.24E+07 6.28E+24 6.67E-17 2.00E-08

1.00E-09 3.00E+17 1.88E+18 9.93E-17 6.20E+02 6.24E+11 6.28E+27 6.67E-18 2.00E-09

1.00E-10 3.00E+18 1.88E+19 9.93E-16 6.20E+03 6.24E+15 6.28E+30 6.67E-19 2.00E-10

1.00E-11 3.00E+19 1.88E+20 9.93E-15 6.20E+04 6.24E+19 6.28E+33 6.67E-20 2.00E-11

8.25E-12 3.64E+19 2.28E+20 1.20E-14 7.51E+04 1.35E+20 1.12E+34 5.50E-20 1.65E-11

1.00E-12 3.00E+20 1.88E+21 9.93E-14 6.20E+05 6.24E+23 6.28E+36 6.67E-21 2.00E-12

1.00E-13 3.00E+21 1.88E+22 9.93E-13 6.20E+06 6.24E+27 6.28E+39 6.67E-22 2.00E-13

1.00E-14 3.00E+22 1.88E+23 9.93E-12 6.20E+07 6.24E+31 6.28E+42 6.67E-23 2.00E-14

1.00E-15 3.00E+23 1.88E+24 9.93E-11 6.20E+08 6.24E+35 6.28E+45 6.67E-24 2.00E-15

1.00E-16 3.00E+24 1.88E+25 9.93E-10 6.20E+09 6.24E+39 6.28E+48 6.67E-25 2.00E-16

1.00E-17 3.00E+25 1.88E+26 9.93E-09 6.20E+10 6.24E+43 6.28E+51 6.67E-26 2.00E-17

1.00E-18 3.00E+26 1.88E+27 9.93E-08 6.20E+11 6.24E+47 6.28E+54 6.67E-27 2.00E-18

1.00E-19 3.00E+27 1.88E+28 9.93E-07 6.20E+12 6.24E+51 6.28E+57 6.67E-28 2.00E-19

1.00E-20 3.00E+28 1.88E+29 9.93E-06 6.20E+13 6.24E+55 6.28E+60 6.67E-29 2.00E-20

Table 2. Properties of the Monochromatic Spectral Energy Density

Using the number density formula, it is simple to calculate the number of ZPE photons

per cubic wavelength. We simply transform the frequency variable to a wavelength

variable in the number density formula where

c2 . The transformation goes like

2

8

)(2)()(

32

3

c

nN

. (8)

The result is that the number of photons per cubic wavelength is a constant – namely 2.

Referring to the table above, and recalling the energy density requirement for strong field

pair creation is 1020

to 1029

J/m3, the wavelength which can create this energy density is

found.

The result is that it takes approximately six virtual photons with a wavelength no larger

than ~ 8.25 x 10-12

m to induce pair creation.

Structure of The Vacuum Photon Field

Now, let us consider the organization of the vacuum photon field. Consider the

generalized QED electric field [4]

,

*

2/1

])0()0([2

)( eeeaeeaV

itrE rkitirkiti kk

(9)

and compare it to the QED vacuum field [4]

,

*

2/1

0 ])0()0([2

)( eeaeaV

itEtiti kk

(10)

The QED fields are expressed using creation and annhilation operators with a magnitude

represented by the expression under the square root symbol. The only difference between

a “real” electric field and the “virtual” zero-point field, is a dependence on r . The

average of the dot products kr

, gives a measure of the randomness of the photon

propagation vectors. If rkkr

, then the field is directional in the r

direction. . If

0 kr

, then the field is random and has an average value of zero. Notice that when

0 kr

, formula 8 and 9 are identical. From this we can say that a real electric field

is composed of an organized ensemble of virtual photons that adhere to the Hamiltonian

equivalence (eqn 1). On the other hand, the vacuum zero-point photon field is a random

ensemble of virtual photons that follow not only the Hamiltonian equivalence, but also

must adhere to the vacuum spectral energy density (eqn 2).

Random Vector Particle Vacuum Field Expression It is not easy to “see” what is going on in the quantum field expressions (eqn 9,10). In

order to grasp the monochromatic field concept, we are going to construct a random

vacuum field as a sum of randomized monochromatic fields. Our purpose is to discretize

the field so we can simulate it on a computer.

We discard the wave mathematics because the lifetime,

2t and range, tc of

each particle are short (on the order of the inverse frequency and wavelength

respectively). Nothing is “waving.” Each particle appears and dissappears at a time

2.

During its lifetime, it travels a distance in the k

direction. At any time, the number of

photons per unit volume must be 32

3

8 cn

so there are six photons per cubic

wavelength (eqn 8).

Using this scenario, we can express the discretized monochromatic vacuum field as

pixelated fields of cubic wavelength volumes.

]ˆ[2

2/16

032

0

kq

Ej j

ji

(11)

Notice that we use the cube of the photon wavelength as the quantized volume 3q ,

where q is simply the factor that corrects for the effective volume of the photon. Since

each k̂ is random, the vector sum over only six photons is almost never zero.

0 ˆ6

0

j

jk (12)

This implies that on a small scale (wavelength sizes), localized non-zero fields do exist

within a monochromatic random photon field. We will explore more of this later in the

paper in the form of computer screen captures of the pixelated zero-point field.

Since each photon has energy density 32

q

and there are 2 photons per cubic

wavelength, then summing the discretized photon energy densities should yield the

monochromatic energy density )(u (eqn. 3).

32

46

03 82 cq

N

j

j

(14)

Solving for q , we get

Nq . Formally, N = 2, but in discrete mathematics, we let N

= 6. The formal result assures that the spectral energy density is maintained.

So, to recapture the big picture. The monochromatic vacuum field is composed of

pixelated vacuum fields which follow the spectral energy density (eqn 2). Since each

pixelated field adheres to the spectral energy density, then the total monochromatic field

inherently follows the spectral energy density. There are an infinite number of

monochromatic fields. One can easily see that when looking at the total vacuum, at any

fixed position r, there are an infinite number of photons with the jk

’s all adding to zero.

0 )(0 0

j

N

j

k

(15)

Energy Production Rate The question now becomes; If we can isolate a critical wavelength monochromatic zero-

point field, and if periodic localized strong fields occur, then at what rate does pair

production occur? We can then calculate how much energy we can extract from the

resulting gamma radiation as the e+e- pairs annihilate.

Propagation Vector Statistics

First, we divide a cubic volume of space V, into pixels of volume 3 . A pixel contains

six randomly oriented photons. There is a finite probability p, that six photons will point

in the same general direction. This is a strong field condition. If there are N pixels per

cubic meter, then there will be pN strong field pixels. Equivalently, the volume

containing the fields is V. However, only a fraction of the volume contains strong field

conditions, so the effective volume is fV, where f is the fraction. We now need to find f

(or p).

Given n choices, a photon has a 1/n probability of pointing at any one choice. If a sphere

is divided into only two pixels (upper and lower hemisphere), the probability for a photon

to point at either hemishere is ½. Now, what is the probability of six photons pointing to

the same hemisphere. The answer is p = (½)6 = 0.0156. The probability for six photons

to point at a single pixel in a sphere pixelated into n pixels is is p = (1/n)6. Clearly, as n

approaches infinity, the probability approaches zero. In fact, the probability for six

photons to point in the exact same direction is zero. The question is, how close is close

enough? In essence, what is a reasonable value for n such that if six photons point within

the solid angle subtended by 4/n, then the resulting field is strong enough for pair

creation?

We need to explore the sensitivity of the electric field to photon propagation vectors. If

all the vectors point in the same direction, the electric field is maximum. If the vectors

are all pointing away from each other, the electric field is zero. Equation 11 describes the

zero-point electric field as the sum of vectors.

For six monochromatic photons, the field becomes (remember that N = 6/)

654321

2/1

32

0

2/16

032

0

ˆˆˆˆˆˆ

12]ˆ[

12kkkkkkkiE

j

j

j j

j

If r̂ is an arbitrary unit vector, then

)cos()cos()cos()cos()cos()cos(12

ˆ][ 654321

2/1

3

rrE k

So, the field is maximized when the angles are not only the same, but close to zero.

Clearly, we can choose an arbitrary unit vector that minimizes the angles. The part that

is not mathematically obvious is that in order for a strong field condition to occur, we

must minimize the variance in the angles. Equivalently, we must maximize the sum of

one of the components of each unit vector. In the language of vectors we can expand the

unit vectors into components

...ˆˆˆˆˆˆˆˆˆˆˆˆ222111654321 zZyYxXzZyYxXkkkkkk

where the condition 1222 iii ZYX is always true. For six unit vectors, the field is

maximum when the sum of one of the components is nearly six, and the sum of the other

components is near zero. The Maximum Field Condition below summarizes what we

need.

66

1

i

iX and 06

1

i

iY and 06

1

i

iZ

or

06

1

i

iX and 66

1

i

iY and 06

1

i

iZ

or

06

1

i

iX and 06

1

i

iY and 66

1

i

iZ

The question is, how close to six must we be for a strong field condition to be defined?

If we look at the Schwinger function results, we see that W ~ E2 and Ecrit > 10

15 V/m. I

propose that once you exceed the critical field strength, sub-order of magnitude estimates

of E will produce insignificant changes in W. From this, I will simply state that when

exceeding the 1015

V/m threshold, if Eeff > 0.9 Ecrit , then we have sufficient conditions for

pair creation. Obviously, six random photons will have six different angles. To simplify

the analysis, let us assume that the largest random angle is the angle for all six photons.

Now, we solve for the angle and find that = .45 rad (25 deg). This represents a unit

solid angle of .2 steradians. A unit sphere has a surface area of 4 steradians, so this will

divide the sphere into 62 pixels. Thus, n = 62 so p = 1 x 10-11

. Essentially, there is a

probability of 1 x 10-11

that six random photons will point to within 25 degrees of each

other and create a strong field within 90% of the critical field value. This is only valid

for zero-point photons of wavelengths smaller than 10-11

m.

Therefore, the effective volume is Veff = 10-11

V.

Computation Example:

From table 1, we already know that an electric field of 5.5 x 1015

V/m has a Schwinger

number of W = 1 (prob/m3-sec). The wavelength that corresponds to that energy density

in table 2 is approximately 8.25x10-12

m. The volume of pair producing effects only

occupies a fraction of a cubic meter (10-11

V). If we divide a cubic meter into cubic

wavelength pixels, we will have 1033

pixels measuring approximately 10-11

m on each

side (cubic wavelengths). So, the Effective Volume = 10-11

x 1033

pixels ~ 1022

pixels.

Thus, there are only 1022

strong field pixels in a cubic meter. Remember that a pixel

which represents a cubic wavelength contains six zero-point photons. This is because we

calculated that the number of photons per cubic wavelength is constant – namely, there

are 2 photons per cubic wavelength according to equation 8.

Now since we know the Schwinger value W = 1 (prob/m

3-sec), each pixel that has strong

field conditions will produce 1 pair per cubic meter per second. so to calculate the

Effective Probability Rate Pw, we simply multiply W by the effective volume.

Pw = W x Veff which has units of (prob/sec).

Our effective volume is 10-11

m3, thus, we have a pair creation rate of 10

-11 pairs per

second. Assuming that all pairs annihilate into gamma radiation, the resulting energy

production is the non-relativistic gamma radiation energy (10-14

J) multiplied by the

production rate. Therefore the energy production rate is 10-25

Watts per cubic meter.

This result is miniscule and informs us that we need to seek smaller wavelengths for

greater energy production.

The discussion above was primarily aimed at going through the motions of actually

calculating an energy production rate based on the fairly obscure theory described in this

paper.

Now that we’ve done a calculation, I will simply write out results for various

wavelengths in the table below. All entries and results are order of magnitude

calculations.

Wavelength

(m)

Effective

Volume

(m3)

Schwinger

Value (W)

(prob/m3-sec)

No. of e+e-

pairs per

second

Energy Production Rate

(Watts/m3)

10-11

10-11

1 10-11

10-25

10-12

10-11

1060

1049

1035

5x10-15

10-11

1068

1057

1043

Table 3. Energy Production Rate (e+e- annihilation) from “Virtual” Strong Field Induced Pair

Production

These are order of magnitude calculations. Notice that if we reduce the wavelength to

10-12

m, the energy production rate increases very rapidly. One can see that the critical

wavelength where the energy production rate increases dramatically is when the photon

wavelength approches 10-11

m.

Let us now calculate a more precise wavelength that we need to isolate in order to

produce a more reasonable energy production rate – say one megawatt per cubic meter.

Megawatt Calculation

First, we need to generate a table similar to table 1 containing Schwinger values and

energy densities. Notice that the electric field determines the Scwinger number (W), and

the energy density U.

From the Schwinger number (W), we calculate the probability rate since Pw = W x Veff

The energy production rate is (Pw ) x (1 x 10-14

J). As before, we only use the first term in

the Schwinger function because of its rapid convergence.

E W U Prob Rate Energy Production Rate

(V/m) (prob/m^3-sec) (J/m^3) (1/sec) (W/m^3)

1.00E+16 1.63E+24 4.43E+20 1.63E+13 1.63E-01

1.03E+16 8.57E+24 4.65E+20 8.57E+13 8.57E-01

1.05E+16 4.33E+25 4.88E+20 4.33E+14 4.33E+00

1.08E+16 2.11E+26 5.13E+20 2.11E+15 2.11E+01

1.10E+16 9.89E+26 5.39E+20 9.89E+15 9.89E+01

1.13E+16 4.47E+27 5.66E+20 4.47E+16 4.47E+02

1.16E+16 1.95E+28 5.95E+20 1.95E+17 1.95E+03

1.19E+16 8.22E+28 6.25E+20 8.22E+17 8.22E+03

1.22E+16 3.35E+29 6.57E+20 3.35E+18 3.35E+04

1.25E+16 1.32E+30 6.90E+20 1.32E+19 1.32E+05

1.28E+16 5.03E+30 7.25E+20 5.03E+19 5.03E+05

1.31E+16 1.86E+31 7.62E+20 1.86E+20 1.86E+06

1.34E+16 6.67E+31 8.00E+20 6.67E+20 6.67E+06

1.38E+16 2.32E+32 8.41E+20 2.32E+21 2.32E+07

1.41E+16 7.85E+32 8.83E+20 7.85E+21 7.85E+07

1.45E+16 2.58E+33 9.28E+20 2.58E+22 2.58E+08

1.48E+16 8.23E+33 9.75E+20 8.23E+22 8.23E+08

1.52E+16 2.56E+34 1.02E+21 2.56E+23 2.56E+09

1.56E+16 7.75E+34 1.08E+21 7.75E+23 7.75E+09

1.60E+16 2.29E+35 1.13E+21 2.29E+24 2.29E+10

1.64E+16 6.58E+35 1.19E+21 6.58E+24 6.58E+10

1.68E+16 1.85E+36 1.25E+21 1.85E+25 1.85E+11

Table 4. Energy Production Rate (e+e- annihilation) as a function of electric field strength

Then, we generate another table similar to table 2, but with energy densities that contain

the band of energy densities found in the Schwinger table above.

u n R

(m) (1/s) (1/s) (J) (eV) (J/m3) (#/m

3) (s) (m)

6.13E-12 4.90E+19 3.08E+20 1.62E-14 1.01E+05 4.43E+20 2.73E+34 4.08E-20 1.23E-11

6.05E-12 4.96E+19 3.11E+20 1.64E-14 1.02E+05 4.65E+20 2.84E+34 4.03E-20 1.21E-11

5.98E-12 5.02E+19 3.15E+20 1.66E-14 1.04E+05 4.89E+20 2.94E+34 3.98E-20 1.20E-11

5.90E-12 5.08E+19 3.19E+20 1.68E-14 1.05E+05 5.14E+20 3.05E+34 3.94E-20 1.18E-11

5.83E-12 5.15E+19 3.23E+20 1.70E-14 1.06E+05 5.40E+20 3.17E+34 3.89E-20 1.17E-11

5.76E-12 5.21E+19 3.27E+20 1.72E-14 1.08E+05 5.67E+20 3.29E+34 3.84E-20 1.15E-11

5.69E-12 5.27E+19 3.31E+20 1.75E-14 1.09E+05 5.96E+20 3.41E+34 3.79E-20 1.14E-11

5.62E-12 5.34E+19 3.35E+20 1.77E-14 1.10E+05 6.26E+20 3.54E+34 3.75E-20 1.12E-11

5.55E-12 5.41E+19 3.39E+20 1.79E-14 1.12E+05 6.57E+20 3.67E+34 3.70E-20 1.11E-11

5.48E-12 5.47E+19 3.44E+20 1.81E-14 1.13E+05 6.90E+20 3.81E+34 3.65E-20 1.10E-11

5.41E-12 5.54E+19 3.48E+20 1.83E-14 1.14E+05 7.25E+20 3.96E+34 3.61E-20 1.08E-11

5.35E-12 5.61E+19 3.52E+20 1.86E-14 1.16E+05 7.62E+20 4.10E+34 3.57E-20 1.07E-11

5.28E-12 5.68E+19 3.57E+20 1.88E-14 1.17E+05 8.01E+20 4.26E+34 3.52E-20 1.06E-11

5.22E-12 5.75E+19 3.61E+20 1.90E-14 1.19E+05 8.41E+20 4.42E+34 3.48E-20 1.04E-11

5.15E-12 5.82E+19 3.66E+20 1.93E-14 1.20E+05 8.83E+20 4.59E+34 3.44E-20 1.03E-11

5.09E-12 5.89E+19 3.70E+20 1.95E-14 1.22E+05 9.28E+20 4.76E+34 3.39E-20 1.02E-11

5.03E-12 5.97E+19 3.75E+20 1.97E-14 1.23E+05 9.75E+20 4.94E+34 3.35E-20 1.01E-11

4.97E-12 6.04E+19 3.79E+20 2.00E-14 1.25E+05 1.02E+21 5.12E+34 3.31E-20 9.93E-12

4.91E-12 6.11E+19 3.84E+20 2.02E-14 1.26E+05 1.08E+21 5.32E+34 3.27E-20 9.81E-12

4.85E-12 6.19E+19 3.89E+20 2.05E-14 1.28E+05 1.13E+21 5.52E+34 3.23E-20 9.69E-12

4.79E-12 6.27E+19 3.94E+20 2.07E-14 1.29E+05 1.19E+21 5.72E+34 3.19E-20 9.57E-12

4.73E-12 6.34E+19 3.98E+20 2.10E-14 1.31E+05 1.25E+21 5.94E+34 3.15E-20 9.46E-12

Table 5. Zero-point field energy density as a function of Zero-point photon wavelength

From the Scwinger table, we can see that an electric field with an energy density of ~ 7.6

x 1020

J/m3 induces a pair production/annihilation rate of slightly over a megawatt per

cubic meter. The corresponding zero-point field which has the same energy density is

composed of virtual photons with a wavelength of 5.3 x 10-12

m.

In conclusion, if – in a cubic meter of space - we isolate a zero-point field composed

of virtual photons which have a wavelength of 5.3 x 10-12

m, then periodic regions of

organized photons will create pockets of strong “Virtual” fields. These virtual fields

will induce a breakdown of the vacuum – releasing e+e- pairs at a rate of ~ 1020

per

cubic meter per second. As the pairs annihilate, the resulting gamma radiation will

have an energy production rate of ~ 1 Megawatt per cubic meter of device volume.

Now, having said all that, let’s test the validity of the model by programming a computer

to follow the specifications laid out above. In particular, the model will simulate the

statistics of a stochastic monochromatic zero-point field.

Computer Model

It is a relatively simple matter to simulate the monochromatic random photon field

described above, on a computer. Our only criteria are that we follow the spectral energy

density – that is the number of photons per cubic wavelength is constant (2). The

model uses a volume of space with six random photon propagation vectors. The volume

of space is equal to a cubic wavelength. We allow each photon to propagate in a random

direction and expand the vector in a cartesian x,y,z coordinate system. We sum the

photon unit vector elements until the Maximum Field Condition above is met. We count

the number of iterations it takes to meet the condition and associate a probability with

this, P ~ 1/n. Every time the critical condition is met, we form a resultant unit vector and

find the angle it makes with respect to the largest contributing coordinate. This angle is

used to calculate the fraction of surface area subtended onto a unit sphere

f = (arcos())2

The sphere is divided into pixels corresponding to that fractional area and we count the

number of pixels

N = 4/f

This is the same N discussed previously describing the probability of finding six photons

pointing in the same general direction. The probability of finding six photons to point in

the same direction is

P = (1/N)6

The computer model result differs from the theoretical result mainly because the random

number generator is not totally random. Every time I run the model, the random number

seed is the same, so we get similar results every time.

One can see the results of the model in Appendix A. The point to all of this is that there

exists a non-zero probability that six random vectors point in the same general direction

+/- 30 degrees. This represents a strong field condition. The probability P is simply the

fraction of unit volume regions that have strong field conditions. The result is that

isolated monochromatic fields have localized strong fields. That is the whole point of

this paper.

Candidate Devices

The critical wavelength was found to be ~ 5.3 x 10-12

m producing 1 MW per cubic

meter. Casimir geometries with those dimensions may or may not be possible. Current

levels of technology can machine optics to a precision of 100 angstroms. Improved

quantum technologies may allow us to create precision Casimir cavities with subatomic

dimensions. Our goal is to calculate the exclusion bands for various geometries and try

to design a zero-bandwidth cavity centered around 5.3 x 10-12

m. Smaller cavities (10-13

m) will simply vaporize the device due to the extreme energy production rates (>1035

Watts).

Conclusion

The goal was a to develop a theory for a source of e+e- pairs for energy production

applications. I introduced the concept of modifying the zero-point vacuum field to mimic

e+e- pair producing strong electric fields. I developed a semi-classical population based

quantum field theory in which I use zero-point virtual photons to generate a pair creating

critical virtual electric field. I started with the zero-point spectral energy density and

broke it down to monochromatic fields. I then calculated the number of monochromatic

zero-point photons necessary to induce a breakdown of the vacuum. I found that the

photon number density (per cubic wavelength) was a constant. From this, I developed a

simple computer model of a monochromatic zero-point field and found that these fields

have pockets of organized photons which mimic locally strong fields. This resulted in

discovering that the probability of six random photons aligning themselves to within 90%

of the maximum vector potential was 10-11

. This resulted in an interpretation that in a

volume V, about 10-11

V of the volume contains strong field conditions. I then calculated

the pair production rate using the Schwinger model and the computer simulation results.

From the pair production rate, I deduced the energy production rate based on the non-

relativistic annihilation energy. I found that the critical photon wavelength required to

create pairs is ~ 5.3 x 10-12

m producing energy at a rate of 1 MW per cubic meter.

The presentation suggests that IF we can isolate a sufficiently energetic zero-point

electromagnetic field, then the statistics of the spectral energy density function will create

periodic strong localized fields that can mimic the organized virtual photons that produce

real electrostatic fields. If the field has a sufficient energy density (short wavelength

zero-point photons), then we can extract e+e- pairs from the quantum vacuum. The

annihilation of these pairs will result in radiation that can be converted to useable energy.

Acknowledgements Special thanks are due to Harold Puthoff and the small group of Vacuum researchers that

are relentlessly searching for ways to tap the vacuum field energy. The huge volume of

work published gave me the confidence that this work is worth pursuing. Special thanks

are also due to Peter Milonni for publishing his book - The Quantum Vacuum - from

which I learned most of the basics of Quantum Field Theory as applied to the quatum

vacuum.

References

[1] S.K. Lamaroux, Phy Rev. Lett., Vol. 78, No. 1, pp 5 (6 Jan 1997): Demonstration

[2] K.M. Hartmann., Am J. Phy., Vol 53 No. 2, pp, 137, (Feb 1985): Population

Explosion in the Vacuum.

[3] J. Schweppe et al., Phy. Rev. Lett., Vol 51, pp 2261 (1983):

[4] P.W. Milonni, Academic Press 1994, The Quantum Vacuum, Ch 2.

[5] W. Heitler, Dover 0-486-64558-4, The Quantum Theory of Radiation, Ch 2.

[6] F.J. Dyson, Phy. Rev., Vol 75, pp 486 (1949): The Radiation Theories of

Tomonaga, Schwinger,and Feynman.

[7] D.C. Cole and H.E. Puthoff, Phy. Rev. E, Vol 48, pp 1562 (1993): Extracting

Energy and Heat from the Vacuum.

[8] H.E Puthoff, Phy. Rev. A., Vol 40, No. 9, pp 4857 (1989): Source of Vacuum

Electromagnetic Zero-Point Energy

[9] H.E Puthoff, Phy. Rev. D., Vol 35, No. 10, pp 3266 (1987): Ground State of

Hydrogen as a Zero-Point-Determined State.

APPENDIX A

COMPUTER MODEL OF MONOCHROMATIC VACUUM

Zero-Point Field Snapshot #2.

Top chart shows regions of moderately strong fields (E > 0.66 Emax).

Bottom chart shows LOGIC determination of definite strong regions (IF E > 0.8 Emax,

THEN Pixel = Black, ELSE, Pixel = Grey.)

Zero-Point Field Snapshot #3.

Top chart shows regions of moderately strong fields (E > 0.66 Emax).

Bottom chart shows LOGIC determination of definite strong regions (IF E > 0.8 Emax,

THEN Pixel = Black, ELSE, Pixel = Grey.)