An Essay on Discrete Foundations for Physics - Revision history

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	==Abstract==		==Abstract==

	<em>We<sup><span style="font-size: x-small;"> </span></sup>base our theory of physics and cosmology on the five<sup><span style="font-size: x-small;"> </span></sup>principles of finiteness, discreteness, finite computability, absolute nonuniqueness, and strict<sup><span style="font-size: x-small;"> </span></sup>construction. Our modeling methodology starts from the current practice of<sup><span style="font-size: x-small;"> </span></sup>physics, constructs a self-consistent representation based on the</em> ordering operator<sup><span style="font-size: x-small;"> </span></sup>calculus ''and provides rules of correspondence that allow us to<sup><span style="font-size: x-small;"> </span></sup>test the theory by experiment. We use <span style="font-size: smaller;"><span style="font-size: x-small;">program universe</span></span> to<sup><span style="font-size: x-small;"> </span></sup>construct a growing collection of bit strings whose initial portions<sup><span style="font-size: x-small;"> </span></sup>(labels) provide the quantum numbers that are conserved in the<sup><span style="font-size: x-small;"> </span></sup>events defined by the construction. The labels are followed by''<sup><span style="font-size: x-small;"> </span></sup>content strings, ''which are used to construct event-based finite aned<sup><span style="font-size: x-small;"> </span></sup>discrete coordinates. On general grounds such a theory has a<sup><span style="font-size: x-small;"> </span></sup>limiting velocity, and positions and velocities do not commute. We<sup><span style="font-size: x-small;"> </span></sup>therefore reconcile quantum mechanics with relativity at an appropriately fundamental<sup><span style="font-size: x-small;"> </span></sup>stage in the construction. We show that 1) events in<sup><span style="font-size: x-small;"> </span></sup>different coordinate systems are connected by the appropriate finite and<sup><span style="font-size: x-small;"> </span></sup>discrete version of the Lorentz transformation, 2) three-momentum is conserved<sup><span style="font-size: x-small;"> </span></sup>in events, and 3) this conservation law is the same<sup><span style="font-size: x-small;"> </span></sup>as the requirement that different paths can ?interfere? only when<sup><span style="font-size: x-small;"> </span></sup>they differ by an integral number of de Broglie wavelengths.<sup><span style="font-size: x-small;"> </span></sup>The labels are organized into the four levels of the''<sup><span style="font-size: x-small;"> </span></sup>combinatorial hierarchy ''characterized by the cumulative cardinals 3, 10, 137, 2<sup><span style="font-size: x-small;">127</span></sup> + 136 <img border="0" align="bottom" alt="~=" src="http://physicsessays.aip.org/stockgif3/sime.gif" /> 1.7 ? 10<sup><span style="font-size: x-small;">38</span></sup>. We justify<sup><span style="font-size: x-small;"> </span></sup>the identification of the last two cardinals as a first<sup><span style="font-size: x-small;"> </span></sup>approximation to'' <img border="0" align="bottom" alt="[barred aitch]" src="http://physicsessays.aip.org/stockgif3/barh.gif" />c/e<sup><span style="font-size: x-small;">2</span></sup> ''and'' <img border="0" align="bottom" alt="[barred aitch]" src="http://physicsessays.aip.org/stockgif3/barh.gif" />c/Gm<img border="0" align="middle" alt="<sub>p</sub><sup>2</sup>" src="http://physicsessays.aip.org/servlet/GetImg?key=PHESEM000002000001000076000001%3A0%3A0%3A28&t=a&d=a" /> = (M<sub><span style="font-size: x-small;">planck</span></sub>/m<sub><span style="font-size: x-small;">p</span></sub>)<sup><span style="font-size: x-small;">2</span></sup> ''respectively. We show that the<sup><span style="font-size: x-small;"> </span></sup>quantum numbers associated with the first three levels can be<sup><span style="font-size: x-small;"> </span></sup>rigorously identified with the quantum numbers of the first generation<sup><span style="font-size: x-small;"> </span></sup>of the standard model of quarks and leptons, with color<sup><span style="font-size: x-small;"> </span></sup>confinement and a first approximation to weak-electromagnetic unification. Our cosmology<sup><span style="font-size: x-small;"> </span></sup>provides an event horizon, a zero-velocity frame for the background<sup><span style="font-size: x-small;"> </span></sup>radiation, a fireball time of about 3.5 ? 10<sup><span style="font-size: x-small;">6</span></sup> years,<sup><span style="font-size: x-small;"> </span></sup>about the right amount of visible matter, and 12.7 times<sup><span style="font-size: x-small;"> </span></sup>as much dark matter. A preliminary calculation of the fine<sup><span style="font-size: x-small;"> </span></sup>structure spectrum of hydrogen gives the Sommerfeld formula and a<sup><span style="font-size: x-small;"> </span></sup>correction to our first approximation for the fine structure constant,<sup><span style="font-size: x-small;"> </span></sup>which leads to'' 1/<img border="0" align="bottom" alt="alpha" src="http://physicsessays.aip.org/stockgif3/agr.gif" /> = 137.035 967 4 ?. ''We can now justify the earlier<sup><span style="font-size: x-small;"> </span></sup>results'' m<sub><span style="font-size: x-small;">p</span></sub>/m<sub><span style="font-size: x-small;">e</span></sub> = 1836.151 497 ? ''and'' m<sub><img border="0" align="bottom" alt="pi" src="http://physicsessays.aip.org/stockgif3/pgr-script.gif" /></sub>/m<sub><span style="font-size: x-small;">e</span></sub> <img border="0" align="bottom" alt="<~" src="http://physicsessays.aip.org/stockgif3/lsim.gif" /> = 274. ''Our estimate of the weak angle<sup><span style="font-size: x-small;"> </span></sup>is'' ''s''''i''''n''<sup><span style="font-size: x-small;">2</span></sup> <img border="0" align="bottom" alt="theta" src="http://physicsessays.aip.org/stockgif3/thgr.gif" /><sub><span style="font-size: x-small;">Weak</span></sub> = 1/4 ''and of the fermi constant'' G<sub><span style="font-size: x-small;">F</span></sub> ? m<img border="0" align="middle" alt="<sub>p</sub><sup>2</sup>" src="http://physicsessays.aip.org/servlet/GetImg?key=PHESEM000002000001000076000001%3A0%3A0%3A28&t=a&d=a" /> = 1 / <img border="0" align="middle" alt="sqrt(2)" src="http://physicsessays.aip.org/servlet/GetImg?key=PHESEM000002000001000076000001%3A0%3A1%3A28&t=a&d=a" />(256)<sup><span style="font-size: x-small;">2</span></sup>. ''Our finite<sup><span style="font-size: x-small;"> </span></sup>particle number rela- tivistic scattering theory should allow us to<sup><span style="font-size: x-small;"> </span></sup>systematically extend these results''. Eteris paribus, caveat lector[[Category:Scientific Paper]]		<em>We<sup><span style="font-size: x-small;"> </span></sup>base our theory of physics and cosmology on the five<sup><span style="font-size: x-small;"> </span></sup>principles of finiteness, discreteness, finite computability, absolute nonuniqueness, and strict<sup><span style="font-size: x-small;"> </span></sup>construction. Our modeling methodology starts from the current practice of<sup><span style="font-size: x-small;"> </span></sup>physics, constructs a self-consistent representation based on the</em> ordering operator<sup><span style="font-size: x-small;"> </span></sup>calculus ''and provides rules of correspondence that allow us to<sup><span style="font-size: x-small;"> </span></sup>test the theory by experiment. We use <span style="font-size: smaller;"><span style="font-size: x-small;">program universe</span></span> to<sup><span style="font-size: x-small;"> </span></sup>construct a growing collection of bit strings whose initial portions<sup><span style="font-size: x-small;"> </span></sup>(labels) provide the quantum numbers that are conserved in the<sup><span style="font-size: x-small;"> </span></sup>events defined by the construction. The labels are followed by''<sup><span style="font-size: x-small;"> </span></sup>content strings, ''which are used to construct event-based finite aned<sup><span style="font-size: x-small;"> </span></sup>discrete coordinates. On general grounds such a theory has a<sup><span style="font-size: x-small;"> </span></sup>limiting velocity, and positions and velocities do not commute. We<sup><span style="font-size: x-small;"> </span></sup>therefore reconcile quantum mechanics with relativity at an appropriately fundamental<sup><span style="font-size: x-small;"> </span></sup>stage in the construction. We show that 1) events in<sup><span style="font-size: x-small;"> </span></sup>different coordinate systems are connected by the appropriate finite and<sup><span style="font-size: x-small;"> </span></sup>discrete version of the Lorentz transformation, 2) three-momentum is conserved<sup><span style="font-size: x-small;"> </span></sup>in events, and 3) this conservation law is the same<sup><span style="font-size: x-small;"> </span></sup>as the requirement that different paths can ?interfere? only when<sup><span style="font-size: x-small;"> </span></sup>they differ by an integral number of de Broglie wavelengths.<sup><span style="font-size: x-small;"> </span></sup>The labels are organized into the four levels of the''<sup><span style="font-size: x-small;"> </span></sup>combinatorial hierarchy ''characterized by the cumulative cardinals 3, 10, 137, 2<sup><span style="font-size: x-small;">127</span></sup> + 136 <img border="0" align="bottom" alt="~=" src="http://physicsessays.aip.org/stockgif3/sime.gif" /> 1.7 ? 10<sup><span style="font-size: x-small;">38</span></sup>. We justify<sup><span style="font-size: x-small;"> </span></sup>the identification of the last two cardinals as a first<sup><span style="font-size: x-small;"> </span></sup>approximation to'' <img border="0" align="bottom" alt="[barred aitch]" src="http://physicsessays.aip.org/stockgif3/barh.gif" />c/e<sup><span style="font-size: x-small;">2</span></sup> ''and'' <img border="0" align="bottom" alt="[barred aitch]" src="http://physicsessays.aip.org/stockgif3/barh.gif" />c/Gm<img border="0" align="middle" alt="<sub>p</sub><sup>2</sup>" src="http://physicsessays.aip.org/servlet/GetImg?key=PHESEM000002000001000076000001%3A0%3A0%3A28&t=a&d=a" /> = (M<sub><span style="font-size: x-small;">planck</span></sub>/m<sub><span style="font-size: x-small;">p</span></sub>)<sup><span style="font-size: x-small;">2</span></sup> ''respectively. We show that the<sup><span style="font-size: x-small;"> </span></sup>quantum numbers associated with the first three levels can be<sup><span style="font-size: x-small;"> </span></sup>rigorously identified with the quantum numbers of the first generation<sup><span style="font-size: x-small;"> </span></sup>of the standard model of quarks and leptons, with color<sup><span style="font-size: x-small;"> </span></sup>confinement and a first approximation to weak-electromagnetic unification. Our cosmology<sup><span style="font-size: x-small;"> </span></sup>provides an event horizon, a zero-velocity frame for the background<sup><span style="font-size: x-small;"> </span></sup>radiation, a fireball time of about 3.5 ? 10<sup><span style="font-size: x-small;">6</span></sup> years,<sup><span style="font-size: x-small;"> </span></sup>about the right amount of visible matter, and 12.7 times<sup><span style="font-size: x-small;"> </span></sup>as much dark matter. A preliminary calculation of the fine<sup><span style="font-size: x-small;"> </span></sup>structure spectrum of hydrogen gives the Sommerfeld formula and a<sup><span style="font-size: x-small;"> </span></sup>correction to our first approximation for the fine structure constant,<sup><span style="font-size: x-small;"> </span></sup>which leads to'' 1/<img border="0" align="bottom" alt="alpha" src="http://physicsessays.aip.org/stockgif3/agr.gif" /> = 137.035 967 4 ?. ''We can now justify the earlier<sup><span style="font-size: x-small;"> </span></sup>results'' m<sub><span style="font-size: x-small;">p</span></sub>/m<sub><span style="font-size: x-small;">e</span></sub> = 1836.151 497 ? ''and'' m<sub><img border="0" align="bottom" alt="pi" src="http://physicsessays.aip.org/stockgif3/pgr-script.gif" /></sub>/m<sub><span style="font-size: x-small;">e</span></sub> <img border="0" align="bottom" alt="<~" src="http://physicsessays.aip.org/stockgif3/lsim.gif" /> = 274. ''Our estimate of the weak angle<sup><span style="font-size: x-small;"> </span></sup>is'' ''s''''i''''n''<sup><span style="font-size: x-small;">2</span></sup> <img border="0" align="bottom" alt="theta" src="http://physicsessays.aip.org/stockgif3/thgr.gif" /><sub><span style="font-size: x-small;">Weak</span></sub> = 1/4 ''and of the fermi constant'' G<sub><span style="font-size: x-small;">F</span></sub> ? m<img border="0" align="middle" alt="<sub>p</sub><sup>2</sup>" src="http://physicsessays.aip.org/servlet/GetImg?key=PHESEM000002000001000076000001%3A0%3A0%3A28&t=a&d=a" /> = 1 / <img border="0" align="middle" alt="sqrt(2)" src="http://physicsessays.aip.org/servlet/GetImg?key=PHESEM000002000001000076000001%3A0%3A1%3A28&t=a&d=a" />(256)<sup><span style="font-size: x-small;">2</span></sup>. ''Our finite<sup><span style="font-size: x-small;"> </span></sup>particle number rela- tivistic scattering theory should allow us to<sup><span style="font-size: x-small;"> </span></sup>systematically extend these results''. Eteris paribus, caveat lector

			[[Category:Scientific Paper\|essay discrete foundations physics]]

	[[Category:Relativity]]		[[Category:Relativity]]

Maintenance script: Imported from text file

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{{Infobox paper
| title = An Essay on Discrete Foundations for Physics
| author = [[H Pierre Noyes]]
| keywords = [[finite]], [[discrete foundations: relativity]], [[elementary quantum particles]], [[cosmology]], [[dark matter]]
| published = 1989
| journal = [[Physics Essays]]
| volume = [[2]]
| number = [[1]]
| num_pages = 25
| pages = 76-99
}}

==Abstract==

We base our theory of physics and cosmology on the five principles of finiteness, discreteness, finite computability, absolute nonuniqueness, and strict construction. Our modeling methodology starts from the current practice of physics, constructs a self-consistent representation based on the ordering operator calculus ''and provides rules of correspondence that allow us to test the theory by experiment. We use program universe to construct a growing collection of bit strings whose initial portions (labels) provide the quantum numbers that are conserved in the events defined by the construction. The labels are followed by'' content strings, ''which are used to construct event-based finite aned discrete coordinates. On general grounds such a theory has a limiting velocity, and positions and velocities do not commute. We therefore reconcile quantum mechanics with relativity at an appropriately fundamental stage in the construction. We show that 1) events in different coordinate systems are connected by the appropriate finite and discrete version of the Lorentz transformation, 2) three-momentum is conserved in events, and 3) this conservation law is the same as the requirement that different paths can ?interfere? only when they differ by an integral number of de Broglie wavelengths. The labels are organized into the four levels of the'' combinatorial hierarchy ''characterized by the cumulative cardinals 3, 10, 137, 2127 + 136 <img border="0" align="bottom" alt="~=" src="http://physicsessays.aip.org/stockgif3/sime.gif" /> 1.7 ? 1038. We justify the identification of the last two cardinals as a first approximation to'' <img border="0" align="bottom" alt="[barred aitch]" src="http://physicsessays.aip.org/stockgif3/barh.gif" />c/e2 ''and'' <img border="0" align="bottom" alt="[barred aitch]" src="http://physicsessays.aip.org/stockgif3/barh.gif" />c/Gm<img border="0" align="middle" alt="p2" src="http://physicsessays.aip.org/servlet/GetImg?key=PHESEM000002000001000076000001%3A0%3A0%3A28&t=a&d=a" /> = (Mplanck/mp)2 ''respectively. We show that the quantum numbers associated with the first three levels can be rigorously identified with the quantum numbers of the first generation of the standard model of quarks and leptons, with color confinement and a first approximation to weak-electromagnetic unification. Our cosmology provides an event horizon, a zero-velocity frame for the background radiation, a fireball time of about 3.5 ? 106 years, about the right amount of visible matter, and 12.7 times as much dark matter. A preliminary calculation of the fine structure spectrum of hydrogen gives the Sommerfeld formula and a correction to our first approximation for the fine structure constant, which leads to'' 1/<img border="0" align="bottom" alt="alpha" src="http://physicsessays.aip.org/stockgif3/agr.gif" /> = 137.035 967 4 ?. ''We can now justify the earlier results'' mp/me = 1836.151 497 ? ''and'' m<img border="0" align="bottom" alt="pi" src="http://physicsessays.aip.org/stockgif3/pgr-script.gif" />/me <img border="0" align="bottom" alt="<~" src="http://physicsessays.aip.org/stockgif3/lsim.gif" /> = 274. ''Our estimate of the weak angle is'' ''s''''i''''n''2 <img border="0" align="bottom" alt="theta" src="http://physicsessays.aip.org/stockgif3/thgr.gif" />Weak = 1/4 ''and of the fermi constant'' GF ? m<img border="0" align="middle" alt="p2" src="http://physicsessays.aip.org/servlet/GetImg?key=PHESEM000002000001000076000001%3A0%3A0%3A28&t=a&d=a" /> = 1 / <img border="0" align="middle" alt="sqrt(2)" src="http://physicsessays.aip.org/servlet/GetImg?key=PHESEM000002000001000076000001%3A0%3A1%3A28&t=a&d=a" />(256)2. ''Our finite particle number rela- tivistic scattering theory should allow us to systematically extend these results''. Eteris paribus, caveat lector[[Category:Scientific Paper]]

[[Category:Relativity]]