Math for Tachyonics
Representing Superluminal Quantities
Calculations involving tachyons are, so far, mere exercises; useful only for stimulating the imagination, since there is no experimental way of detecting tachyons directly, and therefore no actual data on their specific characteristics. Yet, this frees researchers to imagine tachyons of any desired form; the most ready scenario involving superluminal analogs of known particles.
The relativistic mass (m) of a real particle in motion can be related to the particle’s rest-mass (mo) by the equation;
m = mo / [ { 1 - [ (v/c)^2) ] }^(1/2) ] ,
where v is the moving particle’s velocity, and c is lightspeed (~186,000 mi/sec).
The mass (mt) of a moving tachyon, therefore, can be represented by defining it as a negatively-signed imaginary analog of m;
mt = -im ,
where i is the standard imaginary-unit, defined; i = (-1)^(1/2) , and the context or further definitions make clear that all velocities associated with the tachyon are faster-than-light (although this does not distinguish between sub-infinite and infinite superluminal velocity). This sort of scenario is sufficient when no complex calculations are involved, but causes confusion if complex quantities are part of the discussion. In such cases, it is necessary to represent the corresponding tachyonic imaginary-unit in a way that distinguishes it from a standard imaginary-unit. This, of course, means that we require an operator that transforms any bradyonic or luxonic quantity, term, symbol, variable, etc., into an exactly analogous tachyonic quantity.
I have devised such an operator, originally called the “imagination unit”, which I symbolize by using “i” in another font, such as its italicized form,i, and define it as a multiplicative operator that imposes a transformation across the lightspeed barrier, so that it is understood, by convention, to turn a standard quantity into a superluminal analog of itself; meaning, causality is reversed (when compared to the normal causality), speed is faster-than-light (superluminal), and material objects are to be viewed as “actual imaginary” objects, to distinguish them from the “standard imaginary” objects we deal with ordinarily (and for which time is positive, and speeds and/or velocities range only from 0 to c).
That is, we say: If m is the mass of a real particle, then its tachyonic analog, mt, is defined; mt = im , where time is negative, and v > c. We next use i to generate a complete tachyonic number system, in exact analogy to the standard number system. For example, let {…, -2, -1, 0, +1, +2, …} = i{…, -2, -1, 0, +1, +2, …}. This allows calculations for tachyonic quantities to involve complex quantities without confusion between i andi, where all tachyonic numbers (1, 2, ...) are treated algebraically the same as the standard numbers (1, 2, ...), but only in superluminal reference-frames (the product of i and i does not equal -1).
Naturally, the Lorentz transformations can be employed here, so they can be used to study the differences between objects in bradyonic (sublight), luxonic (massless or photonic), and/or tachyonic (superluminal) reference-frames.
One way to get a handle on how the imagination-unit works, as a transformation operator, is to perform a visual inspection of the Velocity Spectrum, denoted;
iV > { iv > ic > [ c > v > (vo = 0) > (ivo = 0abs = ivo) < (vo = 0) < v < c ] < ic < iv } < iV ,
where iV is infinite speed, iv is velocity between tachyonic lightspeed ic and infinite speed (exclusively), c is lightspeed proper,
v is an ordinary velocity (between vo and c, exclusively), vo is a relative zero velocity, ivo is an absolute zero velocity (a standard imaginary), and underlines indicate corresponding quantities for antiparticles.
Here, tachyonic lightspeed, ic, can be defined; ic = 1.00...001c, where the exact number of decimal places to the right is an empirical and statistical unknown.
One-to-one correspondences between bradyonic and tachyonic quantities of any kind, resulting from application of the imagination-unit, can easily be obtained by integration with respect to velocity values on either side of c, exclusive of 0, c, and infinity, so that bradyonic and tachyonic gauges can be related using the Lorentzian or other transformation equations.
Note that the transform indicated by the definition "ic = 1.00...001c" can be used to depict what occurs in superluminal tunneling experiments, in which it appears that photons briefly exhibit superluminal behavior, with fewer decimal places on the right corresponding to greater separation between c and ic.
Formally, the imagination-unit can be defined using a generalized evaluation equation, or as an evaluation function, where the evaluation is taken exclusively between c and infinite-speed, for any time t. For example, write;
[v = infinity]
i = | , t < 0.
[v = c]
Thus, for any function f of an arbitrary variable x, we define tachyonic f(x);
[v = infinity]
if(x) = f(x) | , t < 0.
[v = c]
Observe, however, that this definition is a pure evaluation, and does not specify how bradyonic, luxonic, and tachyonic frames of reference are related. And, while I originally called this operator the “imagination-unit”, it is now known more widely as Richter’s Tachyonics Operator. It is a simple evaluation operator, and does not specify a coordinate system, reference frame, dimensional manifold, or any other such consideration. It thus has universal application, regardless of the coordinate systems being used -- classically, quantum mechanically, relativistically, and so on (depending solely on user preference).
In standard physics, the relativistic energy E of a particle, in terms of the particle's mass (m) and momentum (p), can be given by a formula denoted;
E = (pc)^2 + [mo(c^2)]^2 ,
where p = mv , for any velocity v, and in which the right side can be positive or negative; the positive case applying both to particles and antiparticles alike, leaving the negative case for tachyons and their corresponding antitachyons.
Alternatively, for the representation of a tachyon's energy Et, we can write;
Et = iE = i{(pc)^2 + [mo(c^2)]^2} ,
instead of the usual representation (from Einstein’s theory of Special Relativity). That is, in this scheme, E and Et are dimensionally incompatible values (just as a real number is incompatible with a pure imaginary), because Et must be treated as a tachyonic imaginary. The reason for this representations format is that the standard scheme can cause confusion when complex quantities associated with bradyons and tachyons are considered in the same formulas -- merely because a negatively-signed imaginary-unit does not necessarily imply superluminality in all cases. The difference must always be explained explicitly in the accompanying text. So, in a standard representation of a tachyon, it is unclear, from math symbolism alone, when a bradyonic or tachyonic quantity is specified using the negatively-signed standard imaginary-unit. Employing the new Tachyonics Operator, however, removes this ambiguity, while leaving open the question of reference-system relationships. It does however mandate that natural limits on tachyons (existing at velocities between c and infinite-speed) must correspond to the natural limits of bradyons (between 0 and c). This therefore removes the possibility of infinite solutions cropping up, which renders calculations involving tachyonic quantities meaningless, but which can occur in the standard representation of a tachyon.
Also, the standard interpretation is retained when reference-frames are related by the Lorentz Transformations, in accord with Einstein’s theory of Special Relativity. But the Tachyonics Operator also works in other situations, which modify or omit the Einsteinian paradigm, such as in non-relativistic Quantum Theory, and in purely non-quantum Classical Physics.
A good example that retains the Lorentz formulas, however, while removing the reversed causality of Einsteinian Relativity, is the Lorentzian Relativity proposed by the late Tom Van Flandern, who conducted astronomical experiments that proved gravity is faster-than-light; thereby requiring that Einstein’s Relativity theories are at least reformulated.
One modification of Einsteinian Relativity that can also remove the causality issue stems from the knowledge that the physical constants are not perfectly constant in time. They are now all known to change slowly with time, due to the accelerating expansion of the visible universe. This knowledge removes the requirement of keeping the vacuum constant of lightspeed an invariant for all observers -- consequently implying that some modification of Einstein’s ideas is needed.
I am of the opinion that every scenario is allowed, in practice. The difference comes from what is investigated, and how one goes about it. In other words, I believe there are many kinds of tachyons; some have negative time (according to standard Einsteinian Relativity), some exhibit a “neutral” time (infinite-speed tachyons act instantaneously, so time does not “flow” for them), and some have positive time (as in Traditional Classical and in Classical Lorentzian schemes).
In any case, the Tachyonics Operator can be used to designate superluminal quantities without introducing confusion or conflict with existing paradigms.
Comments Welcome. E-mail: HKurtRichter@yahoo.com.
Representing Superluminal Quantities
Calculations involving tachyons are, so far, mere exercises; useful only for stimulating the imagination, since there is no experimental way of detecting tachyons directly, and therefore no actual data on their specific characteristics. Yet, this frees researchers to imagine tachyons of any desired form; the most ready scenario involving superluminal analogs of known particles.
The relativistic mass (m) of a real particle in motion can be related to the particle’s rest-mass (mo) by the equation;
m = mo / [ { 1 - [ (v/c)^2) ] }^(1/2) ] ,
where v is the moving particle’s velocity, and c is lightspeed (~186,000 mi/sec).
The mass (mt) of a moving tachyon, therefore, can be represented by defining it as a negatively-signed imaginary analog of m;
mt = -im ,
where i is the standard imaginary-unit, defined; i = (-1)^(1/2) , and the context or further definitions make clear that all velocities associated with the tachyon are faster-than-light (although this does not distinguish between sub-infinite and infinite superluminal velocity). This sort of scenario is sufficient when no complex calculations are involved, but causes confusion if complex quantities are part of the discussion. In such cases, it is necessary to represent the corresponding tachyonic imaginary-unit in a way that distinguishes it from a standard imaginary-unit. This, of course, means that we require an operator that transforms any bradyonic or luxonic quantity, term, symbol, variable, etc., into an exactly analogous tachyonic quantity.
I have devised such an operator, originally called the “imagination unit”, which I symbolize by using “i” in another font, such as its italicized form,i, and define it as a multiplicative operator that imposes a transformation across the lightspeed barrier, so that it is understood, by convention, to turn a standard quantity into a superluminal analog of itself; meaning, causality is reversed (when compared to the normal causality), speed is faster-than-light (superluminal), and material objects are to be viewed as “actual imaginary” objects, to distinguish them from the “standard imaginary” objects we deal with ordinarily (and for which time is positive, and speeds and/or velocities range only from 0 to c).
That is, we say: If m is the mass of a real particle, then its tachyonic analog, mt, is defined; mt = im , where time is negative, and v > c. We next use i to generate a complete tachyonic number system, in exact analogy to the standard number system. For example, let {…, -2, -1, 0, +1, +2, …} = i{…, -2, -1, 0, +1, +2, …}. This allows calculations for tachyonic quantities to involve complex quantities without confusion between i andi, where all tachyonic numbers (1, 2, ...) are treated algebraically the same as the standard numbers (1, 2, ...), but only in superluminal reference-frames (the product of i and i does not equal -1).
Naturally, the Lorentz transformations can be employed here, so they can be used to study the differences between objects in bradyonic (sublight), luxonic (massless or photonic), and/or tachyonic (superluminal) reference-frames.
One way to get a handle on how the imagination-unit works, as a transformation operator, is to perform a visual inspection of the Velocity Spectrum, denoted;
iV > { iv > ic > [ c > v > (vo = 0) > (ivo = 0abs = ivo) < (vo = 0) < v < c ] < ic < iv } < iV ,
where iV is infinite speed, iv is velocity between tachyonic lightspeed ic and infinite speed (exclusively), c is lightspeed proper,
v is an ordinary velocity (between vo and c, exclusively), vo is a relative zero velocity, ivo is an absolute zero velocity (a standard imaginary), and underlines indicate corresponding quantities for antiparticles.
Here, tachyonic lightspeed, ic, can be defined; ic = 1.00...001c, where the exact number of decimal places to the right is an empirical and statistical unknown.
One-to-one correspondences between bradyonic and tachyonic quantities of any kind, resulting from application of the imagination-unit, can easily be obtained by integration with respect to velocity values on either side of c, exclusive of 0, c, and infinity, so that bradyonic and tachyonic gauges can be related using the Lorentzian or other transformation equations.
Note that the transform indicated by the definition "ic = 1.00...001c" can be used to depict what occurs in superluminal tunneling experiments, in which it appears that photons briefly exhibit superluminal behavior, with fewer decimal places on the right corresponding to greater separation between c and ic.
Formally, the imagination-unit can be defined using a generalized evaluation equation, or as an evaluation function, where the evaluation is taken exclusively between c and infinite-speed, for any time t. For example, write;
[v = infinity]
i = | , t < 0.
[v = c]
Thus, for any function f of an arbitrary variable x, we define tachyonic f(x);
[v = infinity]
if(x) = f(x) | , t < 0.
[v = c]
Observe, however, that this definition is a pure evaluation, and does not specify how bradyonic, luxonic, and tachyonic frames of reference are related. And, while I originally called this operator the “imagination-unit”, it is now known more widely as Richter’s Tachyonics Operator. It is a simple evaluation operator, and does not specify a coordinate system, reference frame, dimensional manifold, or any other such consideration. It thus has universal application, regardless of the coordinate systems being used -- classically, quantum mechanically, relativistically, and so on (depending solely on user preference).
In standard physics, the relativistic energy E of a particle, in terms of the particle's mass (m) and momentum (p), can be given by a formula denoted;
E = (pc)^2 + [mo(c^2)]^2 ,
where p = mv , for any velocity v, and in which the right side can be positive or negative; the positive case applying both to particles and antiparticles alike, leaving the negative case for tachyons and their corresponding antitachyons.
Alternatively, for the representation of a tachyon's energy Et, we can write;
Et = iE = i{(pc)^2 + [mo(c^2)]^2} ,
instead of the usual representation (from Einstein’s theory of Special Relativity). That is, in this scheme, E and Et are dimensionally incompatible values (just as a real number is incompatible with a pure imaginary), because Et must be treated as a tachyonic imaginary. The reason for this representations format is that the standard scheme can cause confusion when complex quantities associated with bradyons and tachyons are considered in the same formulas -- merely because a negatively-signed imaginary-unit does not necessarily imply superluminality in all cases. The difference must always be explained explicitly in the accompanying text. So, in a standard representation of a tachyon, it is unclear, from math symbolism alone, when a bradyonic or tachyonic quantity is specified using the negatively-signed standard imaginary-unit. Employing the new Tachyonics Operator, however, removes this ambiguity, while leaving open the question of reference-system relationships. It does however mandate that natural limits on tachyons (existing at velocities between c and infinite-speed) must correspond to the natural limits of bradyons (between 0 and c). This therefore removes the possibility of infinite solutions cropping up, which renders calculations involving tachyonic quantities meaningless, but which can occur in the standard representation of a tachyon.
Also, the standard interpretation is retained when reference-frames are related by the Lorentz Transformations, in accord with Einstein’s theory of Special Relativity. But the Tachyonics Operator also works in other situations, which modify or omit the Einsteinian paradigm, such as in non-relativistic Quantum Theory, and in purely non-quantum Classical Physics.
A good example that retains the Lorentz formulas, however, while removing the reversed causality of Einsteinian Relativity, is the Lorentzian Relativity proposed by the late Tom Van Flandern, who conducted astronomical experiments that proved gravity is faster-than-light; thereby requiring that Einstein’s Relativity theories are at least reformulated.
One modification of Einsteinian Relativity that can also remove the causality issue stems from the knowledge that the physical constants are not perfectly constant in time. They are now all known to change slowly with time, due to the accelerating expansion of the visible universe. This knowledge removes the requirement of keeping the vacuum constant of lightspeed an invariant for all observers -- consequently implying that some modification of Einstein’s ideas is needed.
I am of the opinion that every scenario is allowed, in practice. The difference comes from what is investigated, and how one goes about it. In other words, I believe there are many kinds of tachyons; some have negative time (according to standard Einsteinian Relativity), some exhibit a “neutral” time (infinite-speed tachyons act instantaneously, so time does not “flow” for them), and some have positive time (as in Traditional Classical and in Classical Lorentzian schemes).
In any case, the Tachyonics Operator can be used to designate superluminal quantities without introducing confusion or conflict with existing paradigms.
Comments Welcome. E-mail: HKurtRichter@yahoo.com.
For Programmers
Programmers may be interested in the gif code for the defining formulas for the Tachyonics Operator. Below, find the image the code displays shown as the “picture” of the equation, with the line of code that would display it explained beneath it.
The operator here is denoted by i^i, but the superscript is not an exponent.
Integration transformation for some function f of a bradyonic velocity v, changing f(v) to a superluminal analog;
The gif code for displaying this formula is;
i^{i}int_{[0]}^{[c]}f(v)dv = int_{[c]}^{[infinity]}f(v)dv
Integration transformation for function f of bradyonic variable x, with x a function of v, changing f(x) to a superluminal analog;
The gif code for displaying this formula is;
i^{i}int_{[v=0]}^{[v=c]}f(x)dv = int_{[v=c]}^{[v=infinity]}f(x)dv
Here is the defining equation for the Tachyonics Operator;
The gif code for displaying this formula is;
i^{i}f(x) = f(x)|_{[v=c]}^{[v=infinity]}
where f is any function of a generic bradyonic variable x, which need not be an explicit nor implicit function of the velocity v.
If x is not a function of v, then the way x is evaluated depends on which relativity scheme is employed, and how that scheme implies what happens to x evaluated in a superluminal reference-frame. For instance, if x is a function of time (t) in Einsteinian Relativity, then the extra specification that t < 0 (time is negative) would be written after a comma to the right of the above evaluation formula, and/or explained in accompanying text.
Programmers may be interested in the gif code for the defining formulas for the Tachyonics Operator. Below, find the image the code displays shown as the “picture” of the equation, with the line of code that would display it explained beneath it.
The operator here is denoted by i^i, but the superscript is not an exponent.
Integration transformation for some function f of a bradyonic velocity v, changing f(v) to a superluminal analog;
The gif code for displaying this formula is;
i^{i}int_{[0]}^{[c]}f(v)dv = int_{[c]}^{[infinity]}f(v)dv
Integration transformation for function f of bradyonic variable x, with x a function of v, changing f(x) to a superluminal analog;
The gif code for displaying this formula is;
i^{i}int_{[v=0]}^{[v=c]}f(x)dv = int_{[v=c]}^{[v=infinity]}f(x)dv
Here is the defining equation for the Tachyonics Operator;
The gif code for displaying this formula is;
i^{i}f(x) = f(x)|_{[v=c]}^{[v=infinity]}
where f is any function of a generic bradyonic variable x, which need not be an explicit nor implicit function of the velocity v.
If x is not a function of v, then the way x is evaluated depends on which relativity scheme is employed, and how that scheme implies what happens to x evaluated in a superluminal reference-frame. For instance, if x is a function of time (t) in Einsteinian Relativity, then the extra specification that t < 0 (time is negative) would be written after a comma to the right of the above evaluation formula, and/or explained in accompanying text.
More details are available by e-mail from the author: HKurtRichter@yahoo.com
The Superluminal Mind: Modeling Sub-Conscious Thought.
Sci-Fi Speculation by Harrison Kurt Richter
I have been involved in high-level private AI research which has become breakthrough caliber (very proprietary), and brings the Elite closer to building the man/machine hybrid they seek. But other researchers should know that the math I developed, and on which that research is based, is available to anyone for free (open-source). It’s a simple trick, an evaluation operator, but solves a problem for synthetic-brain researchers (modeling the subconscious). This private team models consciousness and the subconscious as superluminal action, describing it as the interaction of gray-matter with a ubiquitous superluminal energy field. Positive initial interest has twice been confirmed to me by them, on their application of my new imaginary-unit, the Tachyonics Operator, in software implementation of modeling both physical and metaphysical aspects of the human brain, in terms amenable to implementation in an AI CPU (or synthetic-brain construct), and which can be used to represent exact analogs of the same processes which occur naturally in the human brain (consciousness, the subconscious, and other metaphysical activity).
The operator allows generation of an alternate-dimensional number-system, with which algorithms can be devised for modeling conscious, subconscious, and “spiritual” characteristics of humans; where the animation and self-awareness of humans is generated by the interaction of gray-matter with a virtual tachyonic energy-field. Viable AI software and hardware systems are therefore designed to mimic the same physiological/metaphysical processes. For example, hardware should involve synapse-like architecture, and software should involve time-dependent learning programs, like Hal, and/or others. Said private applications of the Operator are in the developmental stage, but the Operator itself is open-source information, published for anyone with online access. I especially encourage American researchers to act on this. http://www.tachyonicssociety.WebStarts.com
The software-based models I am working on are revealing transfer functions for implementation at the microprocessor level, inevitably leading to independent self-aware machines, given continued technological progress in this direction. More information and further developments will be relayed as available.
See Also:
http://altscience.wikia.com/wiki/The_Tachyonics_Operator_Explained
http://altscience.wikia.com/wiki/Category:The_GET_Particle_and_Dark_Energy
http://altscience.wikia.com/wiki/Category:Tachyonics_Implies_Unification
http://altscience.wikia.com/wiki/Category:Assertions_on_Tachyonic_Gravity
http://altscience.wikia.com/wiki/Category:Superluminal_Gravitation
This information, and my ideas (if correct), are too important to use for profit (or not worth anything), or to keep under wraps. It should be available to everyone. So, I’ve published my ideas for anyone to read, and to use as they will. All I ask is credit for my original ideas. And I thank you for your interest.
Additional Favorite Sites:
http://tsoaforum.smfforfree2.com/
http://tech.groups.yahoo.com/group/TheTEPG/
http://thedaringsuperluminals.com/fr_home.cfm
Sci-Fi Speculation by Harrison Kurt Richter
I have been involved in high-level private AI research which has become breakthrough caliber (very proprietary), and brings the Elite closer to building the man/machine hybrid they seek. But other researchers should know that the math I developed, and on which that research is based, is available to anyone for free (open-source). It’s a simple trick, an evaluation operator, but solves a problem for synthetic-brain researchers (modeling the subconscious). This private team models consciousness and the subconscious as superluminal action, describing it as the interaction of gray-matter with a ubiquitous superluminal energy field. Positive initial interest has twice been confirmed to me by them, on their application of my new imaginary-unit, the Tachyonics Operator, in software implementation of modeling both physical and metaphysical aspects of the human brain, in terms amenable to implementation in an AI CPU (or synthetic-brain construct), and which can be used to represent exact analogs of the same processes which occur naturally in the human brain (consciousness, the subconscious, and other metaphysical activity).
The operator allows generation of an alternate-dimensional number-system, with which algorithms can be devised for modeling conscious, subconscious, and “spiritual” characteristics of humans; where the animation and self-awareness of humans is generated by the interaction of gray-matter with a virtual tachyonic energy-field. Viable AI software and hardware systems are therefore designed to mimic the same physiological/metaphysical processes. For example, hardware should involve synapse-like architecture, and software should involve time-dependent learning programs, like Hal, and/or others. Said private applications of the Operator are in the developmental stage, but the Operator itself is open-source information, published for anyone with online access. I especially encourage American researchers to act on this. http://www.tachyonicssociety.WebStarts.com
The software-based models I am working on are revealing transfer functions for implementation at the microprocessor level, inevitably leading to independent self-aware machines, given continued technological progress in this direction. More information and further developments will be relayed as available.
See Also:
http://altscience.wikia.com/wiki/The_Tachyonics_Operator_Explained
http://altscience.wikia.com/wiki/Category:The_GET_Particle_and_Dark_Energy
http://altscience.wikia.com/wiki/Category:Tachyonics_Implies_Unification
http://altscience.wikia.com/wiki/Category:Assertions_on_Tachyonic_Gravity
http://altscience.wikia.com/wiki/Category:Superluminal_Gravitation
This information, and my ideas (if correct), are too important to use for profit (or not worth anything), or to keep under wraps. It should be available to everyone. So, I’ve published my ideas for anyone to read, and to use as they will. All I ask is credit for my original ideas. And I thank you for your interest.
Additional Favorite Sites:
http://tsoaforum.smfforfree2.com/
http://tech.groups.yahoo.com/group/TheTEPG/
http://thedaringsuperluminals.com/fr_home.cfm
The Tachyonics Operator