Main Problems in Constructing Quantum Theory Based on Finite Mathematics
As shown in our publications, quantum theory based on a finite ring of characteristic <i>p</i> (FQT) is more general than standard quantum theory (SQT) because the latter is a degenerate case of the former in the formal limit <inline-formula><math xmlns="http://www.w3.org/1...
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| description | As shown in our publications, quantum theory based on a finite ring of characteristic <i>p</i> (FQT) is more general than standard quantum theory (SQT) because the latter is a degenerate case of the former in the formal limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. One of the main differences between SQT and FQT is the following. In SQT, elementary objects are described by irreducible representations (IRs) of a symmetry algebra in which energies are either only positive or only negative and there are no IRs where there are states with different signs of energy. In the first case, objects are called particles, and in the second antiparticles. As a consequence, in SQT it is possible to introduce conserved quantum numbers (electric charge, baryon number, etc.) so that particles and antiparticles differ in the signs of these numbers. However, in FQT, all IRs necessarily contain states with both signs of energy. The symmetry in FQT is higher than the symmetry in SQT because one IR in FQT splits into two IRs in SQT with positive and negative energies at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. Consequently, most fundamental quantum theory will not contain the concepts of particle–antiparticle and additive quantum numbers. These concepts are only good approximations at present since at this stage of the universe the value <i>p</i> is very large but it was not so large at earlier stages. The above properties of IRs in SQT and FQT have been discussed in our publications with detailed technical proofs. The purpose of this paper is to consider models where these properties can be derived in a much simpler way. |
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| spelling | doaj-art-222beafde18848a492eaf9cd577efec22025-08-20T02:50:38ZengMDPI AGMathematics2227-73902024-11-011223370710.3390/math12233707Main Problems in Constructing Quantum Theory Based on Finite MathematicsFelix M. Lev0Independent Researcher, San Diego, CA 92101, USAAs shown in our publications, quantum theory based on a finite ring of characteristic <i>p</i> (FQT) is more general than standard quantum theory (SQT) because the latter is a degenerate case of the former in the formal limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. One of the main differences between SQT and FQT is the following. In SQT, elementary objects are described by irreducible representations (IRs) of a symmetry algebra in which energies are either only positive or only negative and there are no IRs where there are states with different signs of energy. In the first case, objects are called particles, and in the second antiparticles. As a consequence, in SQT it is possible to introduce conserved quantum numbers (electric charge, baryon number, etc.) so that particles and antiparticles differ in the signs of these numbers. However, in FQT, all IRs necessarily contain states with both signs of energy. The symmetry in FQT is higher than the symmetry in SQT because one IR in FQT splits into two IRs in SQT with positive and negative energies at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. Consequently, most fundamental quantum theory will not contain the concepts of particle–antiparticle and additive quantum numbers. These concepts are only good approximations at present since at this stage of the universe the value <i>p</i> is very large but it was not so large at earlier stages. The above properties of IRs in SQT and FQT have been discussed in our publications with detailed technical proofs. The purpose of this paper is to consider models where these properties can be derived in a much simpler way.https://www.mdpi.com/2227-7390/12/23/3707finite mathematicsstandard mathematicsfinite quantum theorystandard quantum theory |
| spellingShingle | Felix M. Lev Main Problems in Constructing Quantum Theory Based on Finite Mathematics Mathematics finite mathematics standard mathematics finite quantum theory standard quantum theory |
| title | Main Problems in Constructing Quantum Theory Based on Finite Mathematics |
| title_full | Main Problems in Constructing Quantum Theory Based on Finite Mathematics |
| title_fullStr | Main Problems in Constructing Quantum Theory Based on Finite Mathematics |
| title_full_unstemmed | Main Problems in Constructing Quantum Theory Based on Finite Mathematics |
| title_short | Main Problems in Constructing Quantum Theory Based on Finite Mathematics |
| title_sort | main problems in constructing quantum theory based on finite mathematics |
| topic | finite mathematics standard mathematics finite quantum theory standard quantum theory |
| url | https://www.mdpi.com/2227-7390/12/23/3707 |
| work_keys_str_mv | AT felixmlev mainproblemsinconstructingquantumtheorybasedonfinitemathematics |