I am very glad to announce that my paper “Non-Gaussian Photon Probability Distribution” has been accepted by the AIP conference Space, Propulsion & Energy Sciences, International Forum (SPESIF) 2010, to be held at John Hopkins in late February 2010.
For those of you who have been following my work for sometime, will recognize that my work is heavily based on experimental data. Therefore, I do expect this paper to substantially increase the rate of change of other scientific discoveries and the development of the technologies of tomorrow.
More details to follow, and abstract below:
This paper investigates the axiom that the photon’s probability distribution is a Gaussian distribution. The Airy disc empirical evidence shows that the best fit, if not exact, distribution is a modified Gamma mΓ distribution (whose parameters are α = r, β = r/√u) in the plane orthogonal to the motion of the photon. This modified Gamma distribution is then used to reconstruct the probability distributions along the hypotenuse from the pinhole, arc from the pinhole, and a line parallel to photon motion. This reconstruction shows that the photon’s probability distribution is not a Gaussian function. However, under certain conditions, the distribution can appear to be Normal, thereby accounting for the success of quantum mechanics. This modified Gamma distribution changes with the shape of objects around it and thus explains how the observer alters the observation. This property therefore places additional constraints to quantum entanglement experiments. This paper shows that photon interaction is a multi-phenomena effect consisting of the probability to interact Pi, the probabilistic function and the ability to interact Ai, the electromagnetic function. Splitting the probability function Pi from the electromagnetic function Ai enables the investigation of the photon behavior from a purely probabilistic Pi perspective. The Probabilistic Interaction Hypothesis is proposed as a consistent method for handling the two different phenomena, the probability function Pi and the ability to interact Ai, thus redefining radiation shielding, stealth or cloaking, and invisibility as different effects of a single phenomenon Pi of the photon probability distribution. Sub wavelength photon behavior is successfully modeled as a multi-phenomena behavior. The Probabilistic Interaction Hypothesis provides a good fit to Otoshi’s (1972) microwave shielding, Schurig et al. (2006) microwave cloaking, and Oulton et al. (2008) sub wavelength confinement; thereby providing a strong case that the photon probability distribution is a modified Gamma mΓ distribution and not a Gaussian distribution.
I will post a copy of this paper to my website iSETI some time before the conference as soon as everything is finalized.
A friend of mine sent me an email with this comment,
“Single photon physics provides secure communication. If I do coding with photons, eavesdropping is not possible. A person who tries to eavesdrop, destroys the photon in the process” Dr. Steve Harris, Stanford University, 2008.
Got me thinking. In a photonics world based on quantum theory this is a correct statement, I would be the last person on Earth to disagree with an esteemed professor such as Steve Harris.
However, given that the photon probability distribution is a modified Gamma distribution and not a Gaussian distribution I was able to construct a different model of the photon that agrees reasonably well with Oulton et al, 2008 results. Caution: this model does need futher testing. This new model suggest that it is not possible to have secure communications with single photon physics.
I have partially completed my work on the photon probability distribution being a modified Gamma distribution and not a Gaussian distribution. To this end I have been testing microwave, light & subwavelength models using this modified Gamma distribution.
The subwavelength modified Gamma model provides a reasonably close fit with R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile and X. Zhang paper A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation. The fit is close but note that there are areas where the models disagree.
Source: Nature Photonics
Figure 1: GaAs nano wire in SiO2 on a metallic plane (100nm<d<500nm, 2nm<h<100nm)
The Oulton et al paper shows the energy distribution for d=200nm & h=100nm, Figure 2.
Source: Nature Photonics
Figure 2: Energy distribution (d=200nm, h=100nm)
My model shows a similar result, Figure 3.
Figure 3: Modified Gamma model (d=200nm, h=100nm)
The Oulton et al paper shows the energy distribution for d=200nm & h=100nm, Figure 4.
Source: Nature Photonics
Figure 4: Energy distribution (d=200nm, h=2nm)
My model shows a similar result, Figure 5.
Figure 5: Modified Gamma model (d=200nm, h=2nm)
The energy in the gap area for Oulton et al model is 15% for h=2nm & up to 20% for h-100nm. The modified Gamma model’s energy in the gap area is about 10% and 15% respectively, about 5% consistently less than Outlon et al. However, the modified Gamma model is sensitive to the dimensions of the SiO2 layer, and I have assumed this to be 800nm across and 400 nm high.
I think this is a great fit for subwavelength, coming out of nowhere, while at the same time giving good assurances at the microwave and optical frequencies.
If you know what the internal dimensions of the WR 430 microwave waveguide are, do contact me and let me know.
I get this question a lot and I am beginning to understand that there is a vast difference in most people’s understanding of analytical methods and numerical methods. This is a question I will have to address in my next revision of the book. Allow me to explain by way of examples. I found that if I jump straight into regression some people don’t get it.
The standard distance traveled formula is
s = ut + (1/2) at^2 (1)
It shows that all the individual terms on the RHS must have the same units as the LHS. If this were a regression equation one would write it in the form:
s = p.(u.t) + q.(a.t^2) + r (2)
where p, q & r are some coefficients of constant value. Noting that r would also have the same units as s. This is a non-linear relationship in t, and to adapt this equation for multiple-linear regression, one would combine the first & second term variables into a ‘meta’ (if that is the right word) variables say x1 (=u.t) & x2 (=a.t^2), giving
s = p.(x1) + q.(x2) + r (3)
Regressing for p, q and r using the known values of u, t and a, in the form of x1 and x2, would give the following solution
p = 1 p has the numerical value of 1, and p does not have any units
q = 1/2 q has the numerical value of 1/2 and q does not have any units
r = 0 r has the numerical value of zero and r has the units m (meters)
When we compare equation (3) with equation (1), we note 2 points,
(a) That you don’t write p is 1 in equation (1) even though the coefficient p still exists, and it is dimensionless.
(b) That you don’t write the r term because r is zero, and even if its units is m.
If we were to insists on writing everything out then equation (1) would always be written as,
s m = 1.u.t m + (1/2).a.t^2 m + 0 m (4)
this is technically correct, but not helpful.
Let’s take another example. In the RMBS* & CMBS* sub-industries, the regression loan default model would take the form,
where t is the loan age in months and P(d) the probability of default is dimensionless. Again the units of each term on the RHS must be the same as the units on the LHS which in this case must be dimensionless. Therefore, the units for each constant, a, b, c, d, e & f would take on the following units,
a would be dimensionless
b would have the units month^-1
c would have the units month^-2
d would have the units month^-3
e would have the units month^-4
f would have the units month^-5
In equation (1) the coefficients were dimensionless. Here is an example where the constant term is not dimensionless.
g = GM/r^2 (6)
A regression version of this equation would take the form,
g = a.M/r^2 +b (7)
Since the every term in the RHS must have the same units as on the LHS, this tells us that,
i. a has the numerical value of 6.67428 x 10^-11 and the units of a would be m^3 kg^-1 s^-2
ii. b has the numerical value of zero and the units of b would be m/s^2
since b=0 equation (7) is simply written as.
g = a.M/r^2 (8)
That is we observe 3 points,
That the coefficients have compensating units.
That when the coefficients are unity, it is not written into the formula even though these coefficients are not dimensionless; they are ‘silent’ and do not appear in the formula.
The units of the coefficients do not have to be the same as those of the other coefficients or the LHS.
Returning to g=τc^2. This equation was derived using multiple-linear regression, so the unsolved regression equation took the form,
g = a.(t1-t2)/(r1-r2) + b (9)
where a and b are regression coefficients. Since, the units of each term on the RHS must be the same as on the LHS,
a .(t1-t2)/(r1-r2) would have the units m/s^2
b would also have the units m/s^2
Since the time dilations t1 and t2 have the units of seconds, s and contracted distances r1 and r2 have the units of meters, m, this informs us that
a would have the units of m^2/s^3
b would have the units m/s^2
to solve for a and b using multiple-linear regression, equation (9) would take the form
g = a.(x1) + b (10)
where x1 = (t1-t2)/(r1-r2). The regression solution shows that
a has the numerical value of c^2, the square of the velocity of light
b has the numerical value of zero
and the regression takes the form
g m/s^2 = c^2.(x1) m/s^2 + 0 m/s^2 (11)
since b is zero and can be ignored as with (3) and (1), equation (11) can now be rewritten as
Well, as I said before, how time flies. The SPESIF 2010 Call For Papers is upon us.
I’m particularly involved in this session as I am honored to be the Session Co-Chairman for the ‘A03.1. Theories, Models and Concepts’ at SPESIF 2010, with Dr Martin Tajmar.
Deadlines:
Initial Abstract Submission: July 15, 2009
Draft Manuscript: August 15, 2009
Final Camera Ready Manuscript: November 15, 2009
For those not familiar with SPESIF, this conference has been described as “out of the box world represented by the excellent SPESIF Conference and before it, STAIF”. Papers presented will be published as American Institute of Physics Conference Proceedings.
The book, “An Introduction to Gravity Modification: A Guide to Using Laithwaite’s and Podkletnov’s Experiments and the Physics of Forces for Empirical Results”, on which this paper is based on can be purchased at:
I have been invited to present ”An Approach to Gravity Modification as a Propulsion Technology” at the Space, Propulsion & Energy Sciences International Forum (SPESIF-2009) at Huntsville, AL, USA, February 24-27, 2009.
This is a peer-reviewed conference and papers will be published as AIP Conference Proceedings. AIP stands for the prestigious American Institute of Physics.
If you would like to attend this conference I have included informational links below.
Abstract.Gravity modification as a portable non-mass effect is feasible. Contemporary experiments such as HFGW and LIGO require mass to model gravitational acceleration and gravitational waves. A different approach to gravitational acceleration, and thus space propulsion technologies is presented here. This paper proposes that gravitational acceleration on any particle is the effect of the deformation of the shape and mass of the particle due to non-inertia transformations present in that local region of the gravitational field. The analytical formulation and numerical integration has led to the discovery of a new formula for gravitational acceleration, g = τc^2, that is neither a function of the mass of the gravitational source nor a function of gravitational waves; where τ is a function of the time dilation present in the local gravitational field. This formula has been tested and verified to be correct in the gravitational fields of the nine planetary bodies in our Solar System, and the Sun; mechanical acceleration, and electromagnetic fields. Thus leading to the inference that g = τc2 is the generic formula for all non-nuclear force fields. The true power of this definition of gravitational acceleration lies in the fact that it now lends itself to a portable technology, as mass is no longer required to derive acceleration. This new relationship for acceleration, describes how an electron moving in a magnetic field causes a force on the electron, and explains why the electron velocity, magnetic field and resulting force relationship is orthogonal. This electron model would be the basis for future propulsion technologies.
Yes, my research has led to the discovery of a new definition for gravitational acceleration. Gravitational acceleration, g, is determined solely by the transformations present in the local spacetime, and without reference to the gravitational field’s mass source.
dt/dr or tau, is the change in time dilation over the change in distance in that local region of spacetime.
This formula is the correct description of accleration, i.e. force, for all non-nuclear forces. At the present time I do not included nuclear forces as I have not tested this formula for nuclear forces.
This formula is thus the basis for gravity modification and force field technologies.
It should not take you more than 5 minutes to verify this formula if you have an Excel Add In like XNumbers.
Will post/talk more at a later date.
Just remembered that I have posted typo errors found in the book An Introduction to Gravity Modification and their corrections, here.
Best,
Ben
Benjamin T Solomon
iSETI LLC
PO Box 831
Evergreen, CO 80437, USA
For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.
Keeping it simple, ‘theory’ refers to axioms and theorems that are taken as true because they are implied by the axioms; ‘provable in the theory’ means derivable from the axioms, and is ‘consistent’ if it never proves a contradiction; ‘can be constructed’ means that some mechanical procedure exists which can construct the statement; ‘elementary arithmetic’ consists merely of addition and multiplication over the natural numbers. That is, one can construct true but unprovable statements. Caution: distilling a theorem into a seven-word statement can cause a theorem to lose some of its richness.
Gödel’s Incompleteness Theorem (GIT) is an amazing piece of work. It is amazing in that there are two apparently opposite interpretations. John D Barrow in his book New Theories of Everything expounds the problems of mathematics in understanding the laws of physics of the Universe. As Barrow states,
Superficially, it appears that all human investigation of the Universe must be limited. Science is based on mathematics; mathematics cannot discover all truths. This is an argument that is often heard.
I believe that this pessimistic interpretation was originally given by Hermann Weyl who was noted to have stated that GIT “has been a constant drain on the enthusiasm”.
I see that there are essentially two related problems with the use of mathematics in physics. Don’t get me wrong, mathematics is absolutely essential to physics. However, mathematics is a tool to describe the physical world, while the physical world is reality. It is the reality that makes mathematics invaluable.
The two problems are ‘what are the implications of Gödel’s Incompleteness Theorem’, and ‘what does it mean when a mathematical description of the physical Universe cannot be solved?’
With respect to the first problem, Barrow states that our interpretation of Gödel’s Incompleteness Theorem is a matter of perspective. If one is a pessimist, then GIT takes on Herman Weyl and Stanley Jaki’s interpretation that (to quote Barrow) GIT ‘prevents us from gaining an understanding of the cosmos as a necessary truth, …, and so is a fundamental barrier to human understanding of the Universe.’
If one is an optimist then GIT takes on Freeman Dyson’s interpretation that GIT ‘proves that the world of pure mathematics is inexhaustible’. My own view is derived from Roger Penrose in his book The Road to Reality. Dyson and Penrose both have an optimistic view of GIT which I share.
I read Jaki’s paper The Jaki-Godel Theorem. This paper is a ‘must read’ for those who are interested in GIT. Actually, if you are not a Theory of Everything (TOE) guy than Jaki’s interpretation is not pessimistic. It is realistic. If you are a TOE guy than GIT is pessimistic, in that to quote Jaki,
Herein lies the ultimate bearing of Gödel’s theorem on physics. It does not mean at all the end of physics. It means only the death knell on endeavours that aim at a final theory according to which the physical world is what it is and cannot be anything else. Gödel’s theorem does not mean that physicists cannot come up with a theory of everything … such a theory cannot be taken for something which is necessarily true. Apart from Gödel’s theorem, such a theory cannot be a guarantee that in the future nothing essentially new would be discovered in the physical universe which would then demand another final theory and so on. Regress to infinity is no answer to a question that keeps generating itself with each answer.
In my book An Introduction to Gravity Modification I had, without prior knowledge of Jaki’s work, come up with a similar interpretation, and state it as the Preferred Theory Axiom, as that theory, Ti, that explains Nature’s method of operation, MN, for all and any appropriate factors, fk. In reality, however, we are not at the point where we can say for sure that one of these theories, Ti, is the one correct method, MN, that Nature operates by and therefore, we do not know what MN really is.
That is, to formally recognize Gödel’s Incompleteness Theorem in my work I have separated the mathematical description of the theory, Ti, from the physical reality of Nature’s method of operation, MN.
I think the issue is not whether one is a pessimist or an optimist as Barrow suggests, but where one’s focus is. Take a look at the seven words again, ‘one can construct true but unprovable statements’. If one’s focus is on ‘one can construct true statements’ than we get the Dyson-Penrose perspective that what we can construct is inexhaustible. If on the other hand, one’s focus is on ‘unprovable statements’ than we get the Weyl-Jaki perspective that at least some of our constructs will be unprovable.
But this raises the second problem, put more succinctly it is, ‘can the Universe be describe in purely provable mathematical terms?’ Barrow’s tentative conclusion is that GIT will limit the sorts of questions we can answer. This is also in part a challenge to GIT as an area of physics, in that if there is a part of the Universe that cannot be described in arithmetic terms, then GIT no longer holds.
Earlier I had presented the second problem as ‘what does it mean when a mathematical description of the physical Universe cannot be solved’. This question places a wedge that separates mathematics from physics.
The Russian mathematician Vladimir Arnold had posed a challenge whether it was possible to have a general mathematical criterion to decide whether any equilibrium was stable. Francisco Doria and Newton da Costa showed that this was not possible, and also showed that a similar question, ‘will the orbit of a particle become chaotic?’ is also undecidable. By undecidable one means if neither that formula nor its negation can be proved within the theory, or keeping it very simple, an outcome can be found to be part of a theory in a finite number of evaluations.
If we look at the Universe, all its processes are ‘solved’ and ‘known’ outcomes. Each ‘iteration’ of the Universe’s physical processes leads to a known or unknown ‘solution’. One can infer that the Universe consists of solved, decidable processes. ‘Decidable’ in that the outcome is known in the next iteration or maybe prior to the next iteration. There is no question that a physical reality cannot be ‘solved’, as in the manner a mathematical equation cannot be solved. The Universe does not stop in mid-process because it cannot ‘solve’ the completion of its processes. The Universe, as we know it, does not have a region or kenos, where the laws of physics are indeterminate. The Indeterminate Kenos (see An Introduction to Gravity Modification for definition of kenos).
Note, I use the term ‘indeterminate’ and not ‘unknown’. ‘Indeterminate’ means that the Universe cannot know or figure out or solve for a solution. ‘Unknown’ means that the Universe knows ‘what is next’ but we as humans do not know.
One could argue that in quantum theory outcomes are not known because they are probabilistic, that the outcomes are indeterminate. This is indeterminate as in ‘not fixed’. It is not indeterminate as in ‘undefined’. Arnold, Doria and da Costa’s work showed that mathematical descriptions of the physical Universe have outcomes that could be undefined because they are undecidable. However, the Universe has all the outcomes clearly defined and also knows which outcome to select, and when. In quantum theory we don’t know which outcome will be selected and therefore term this type of behavior ‘probabilistic’.
So what does this mean? It means that all descriptions and laws of physical reality aka Universe, have both known and unknown solutions. This is where mathematics diverges from the laws of physics. In the physical universe indeterminate or undecidable outcomes are not allowed, but are allowed in mathematics.
That implies a mathematical description of the physical Universe that produces indeterminate, undecidable or unsolvable outcomes is either representationally insufficient because it cannot forecast how a physical process will unfold or incorrect because it cannot facilitate further decidable enhancements.
This also implies that for substantial theory driven technological progress we require physical theories that meet four criteria consistent, solvable, decidable, and verifiable. My work in, An Introduction to Gravity Modification meet all four criteria.
With this in mind let us take at look at Michio Kaku’s classes of impossibilities. Michio Kaku in his book Physics of the Impossible describes three classes of impossibilities. Class I is impossible today but do not violate known laws of physics. Examples are teleportation, and antimatter engines. Class II impossibilities are at the very edge of our understanding of the physical world. Examples are time machines, hyperspace travel, and wormhole travel. And Class III impossibilities violate the known laws of physics. We are in Class 0 impossibility.
It is clearer if we work backwards. Within the limits of our knowledge base we can therefore infer that Class III impossibilities are, inconsistent, unsolvable, undecideable and unverifiable. That is Class III impossibilities do not meet all 4 requirements for a physical theory.
Class II impossibilities are consistent, unsolvable, undecidable and unverifiable. That is they meet only the first of the 4 requirements.
Class I impossibilities are consistent, unsolvable, undecidable but verifiable, that is they meet only 2 of the 4 requirements for a physical theory.
And therefore, we can say that Class 0 impossibilities are consistent, solvable, decidable and verifiable. Note, however, that if a hypothesis is consistent, solvable and/or decidable but unverifiable, it is only of a theoretical interest with no practical or physical value.
Source: Nature Photonics
Source: Nature Photonics
Source: Nature Photonics
