Emeritus physicist, Argonne national laboratoryActive in research on effect of local mathematics and space time dependent number scaling field on physics and geometry
In gauge theories, separate vector spaces, Vx, are assigned to each space time point x with unita... more In gauge theories, separate vector spaces, Vx, are assigned to each space time point x with unitary operators as maps between basis vectors in neighboring Vx. Here gauge theories are extended by replacing the single underlying set of complex scalars, C, with separate sets, Cx, at each x, and including scaling between the Cx. In gauge theory Lagrangians, number scaling shows as a scalar boson field, B, with small coupling to matter fields. Freedom of number scaling is extended to a model with separate number structures assigned to each point x. Separate collections, Ux, of all mathematical systems based on numbers, are assigned to each x. Mathematics available to an observer, Ox, at x is that in Ux. The B field induces scaling between structures in the different Ux. Effects of B scaling on some aspects of physics and geometry are described. The lack of experimentally observed scaling means that B(z) is essentially constant for all points, z, in a region, Z, that can be occupied by us...
Local mathematics consists of a collection of mathematical systems located at each space and time... more Local mathematics consists of a collection of mathematical systems located at each space and time point. The collection is limited to the systems that include numbers in their axiomatic description. A scalar map between systems at different locations is based on the distinction of two conflated concepts, number and number value. The effect that this setup has on theory descriptions of physical and geometric systems is described. This includes a scalar spin 0 field in gauge theories, expectation values in quantum mechanics and path lengths in geometry. The possible relation of the scalar map to consciousness is noted.
This paper extends earlier work on quantum theory representations of natural numbers N, integers ... more This paper extends earlier work on quantum theory representations of natural numbers N, integers I, and rational numbers Ra to describe a space of these representations and transformations on the space. The space is parameterized by 4-tuple points in a parameter set. Each point, (k,m,h,g), labels a specific representation of X = N, I, Ra as a Fock space F^{X}_{k,m,h} of states of finite length strings of qukits q and a string state basis B^{X}_{k,m,h,g}. The pair (m,h) locates the q string in a square integer lattice I \times I, k is the q base, and the function g fixes the gauge or basis states for each q. Maps on the parameter set induce transformations on on the representation space. There are two shifts, a base change operator W_{k',k}, and a basis or gauge transformation function U_{k}. The invariance of the axioms and theorems for N, I, and Ra under any transformation is discussed along with the dependence of the properties of W_{k',k} on the prime factors of k' and k. This suggests that one consider prime number q's, q_{2}, q_{3}, q_{5}, etc. as elementary and the base k q's as composites of the prime number q's.
The usual interpretative rules of quantum mechanics are presented and are shown to be too weak. T... more The usual interpretative rules of quantum mechanics are presented and are shown to be too weak. The reason is that they do not include the intuitive requirement of randomness of the outcome sequence obtained from an infinite repetition of measuring an observable on a state. A strengthening of the rules is proposed which includes, in essence, a precise definition of randomness. The resultant rule is seen to be intuitively more satisfying than the usual rules and to include the expectation value rule and to include essentially all of the spectrum rule of the usual rules. It also suggests that the relationship between the foundations of mathematics and quantum mechanics may be quite deep and complex.
ABSTRACT In this paper a possible framework for a coherent theory of physics and mathematics is p... more ABSTRACT In this paper a possible framework for a coherent theory of physics and mathematics is proposed that is based on the field of reference frames arising from quantum representations of real and complex numbers. The framework is obtained by expanding the frame domains to include space and time lattices. Each frame has finite length qukit strings that also have one qubit. Basis states in a Hilbert space of string states denote values of rational numbers. It is proposed that these strings be considered as hybrid systems in that they are both mathematical and physical systems. As physical systems each type of string has an associated Hamiltonian with a (presently unknown) spectrum of excited states. It follows that each rational number value has an energy associated with it. A generic Schrodinger dynamics for these systems on the lattices is summarized. The iterative structure of the frame field is such that the contents of a stage j frame have images in a stage j-1 (parent) frame. A discussion of parent frame images includes the proposal that images of point locations of child frame lattices are rational states of parent frame hybrid systems. Included is the more speculative proposal that parent frame hybrid systems are images of the points of child frame lattices. The resulting association of energy with images of the lattice points is discussed. Finally there is a discussion of the possible existence of different types of hybrid systems. Besides those associated with images of lattice space points, there may be different types associated with images of lattice time points, wave number lattices, and vectors.
ABSTRACT Ab initio calculations of NO2 and NO2-, using a Dunning [4s3p] basis augmented by 1 comp... more ABSTRACT Ab initio calculations of NO2 and NO2-, using a Dunning [4s3p] basis augmented by 1 component diffuses andp functions were carried out. The SCF energies of NO2 and NO2/- (ground states) as a function of Os, Op, Ns, and Np diffuse function exponents are given and discussed. The curves show some unexpected features which make the optimization of the diffuse function exponents problematic.
The effect of the optical potential on the nucleon momentum distribution in nuclei is discussed i... more The effect of the optical potential on the nucleon momentum distribution in nuclei is discussed in this work. The idea of McCarthy et al., that the refraction and localization effects caused by the real and imaginary parts of the optical potential smears the single-particle momentum distribution, is amplified and applied to nucleon momentum distribution experiments. The assertion is made here that it seems impossible to directly measure in any manner the momentum distribution of nucleons in nuclei. Further it is proposed that much or all of the large discrepancy between the experimental momentum determinations and the shell-model predictions is due to the neglect of these important sources of high-momentum components. The high-energy C12(p, d)C11 pickup reaction data are re-analyzed in the light of these considerations. It is shown that for q2beta2<~8, the 1p-shell harmonic oscillator distribution, (q2beta2)exp(-q2beta2), has sufficient high-momentum components to fit the data. T...
Ab initio calculations of the trans–cis energy difference and the barrier for internal rotation i... more Ab initio calculations of the trans–cis energy difference and the barrier for internal rotation in nitrous acid were made using a double zeta quality basis set and MCSCF+CI methods. The geometry parameters used were the optimized ones obtained by Skaarup and Boggs. The trans–gauche barrier and trans–cis energy difference were calculated here to be 11.4 and 2.5 kcal/mole in good agreement with the experimental values of 11.57±0.2 and 0.4±1.0 kcal/mole, respectively. It was found that exclusion of only one configuration involving excitation of a lone pair on the NOH oxygen from the MCSCF and CI lists reduced the trans–gauche barrier to 6.4 kcal/mole. This shows the importance of lone pair electrons on the oxygen, and not on the nitrogen as previously suggested, to the high barrier.
In gauge theories, separate vector spaces, Vx, are assigned to each space time point x with unita... more In gauge theories, separate vector spaces, Vx, are assigned to each space time point x with unitary operators as maps between basis vectors in neighboring Vx. Here gauge theories are extended by replacing the single underlying set of complex scalars, C, with separate sets, Cx, at each x, and including scaling between the Cx. In gauge theory Lagrangians, number scaling shows as a scalar boson field, B, with small coupling to matter fields. Freedom of number scaling is extended to a model with separate number structures assigned to each point x. Separate collections, Ux, of all mathematical systems based on numbers, are assigned to each x. Mathematics available to an observer, Ox, at x is that in Ux. The B field induces scaling between structures in the different Ux. Effects of B scaling on some aspects of physics and geometry are described. The lack of experimentally observed scaling means that B(z) is essentially constant for all points, z, in a region, Z, that can be occupied by us...
Local mathematics consists of a collection of mathematical systems located at each space and time... more Local mathematics consists of a collection of mathematical systems located at each space and time point. The collection is limited to the systems that include numbers in their axiomatic description. A scalar map between systems at different locations is based on the distinction of two conflated concepts, number and number value. The effect that this setup has on theory descriptions of physical and geometric systems is described. This includes a scalar spin 0 field in gauge theories, expectation values in quantum mechanics and path lengths in geometry. The possible relation of the scalar map to consciousness is noted.
This paper extends earlier work on quantum theory representations of natural numbers N, integers ... more This paper extends earlier work on quantum theory representations of natural numbers N, integers I, and rational numbers Ra to describe a space of these representations and transformations on the space. The space is parameterized by 4-tuple points in a parameter set. Each point, (k,m,h,g), labels a specific representation of X = N, I, Ra as a Fock space F^{X}_{k,m,h} of states of finite length strings of qukits q and a string state basis B^{X}_{k,m,h,g}. The pair (m,h) locates the q string in a square integer lattice I \times I, k is the q base, and the function g fixes the gauge or basis states for each q. Maps on the parameter set induce transformations on on the representation space. There are two shifts, a base change operator W_{k',k}, and a basis or gauge transformation function U_{k}. The invariance of the axioms and theorems for N, I, and Ra under any transformation is discussed along with the dependence of the properties of W_{k',k} on the prime factors of k' and k. This suggests that one consider prime number q's, q_{2}, q_{3}, q_{5}, etc. as elementary and the base k q's as composites of the prime number q's.
The usual interpretative rules of quantum mechanics are presented and are shown to be too weak. T... more The usual interpretative rules of quantum mechanics are presented and are shown to be too weak. The reason is that they do not include the intuitive requirement of randomness of the outcome sequence obtained from an infinite repetition of measuring an observable on a state. A strengthening of the rules is proposed which includes, in essence, a precise definition of randomness. The resultant rule is seen to be intuitively more satisfying than the usual rules and to include the expectation value rule and to include essentially all of the spectrum rule of the usual rules. It also suggests that the relationship between the foundations of mathematics and quantum mechanics may be quite deep and complex.
ABSTRACT In this paper a possible framework for a coherent theory of physics and mathematics is p... more ABSTRACT In this paper a possible framework for a coherent theory of physics and mathematics is proposed that is based on the field of reference frames arising from quantum representations of real and complex numbers. The framework is obtained by expanding the frame domains to include space and time lattices. Each frame has finite length qukit strings that also have one qubit. Basis states in a Hilbert space of string states denote values of rational numbers. It is proposed that these strings be considered as hybrid systems in that they are both mathematical and physical systems. As physical systems each type of string has an associated Hamiltonian with a (presently unknown) spectrum of excited states. It follows that each rational number value has an energy associated with it. A generic Schrodinger dynamics for these systems on the lattices is summarized. The iterative structure of the frame field is such that the contents of a stage j frame have images in a stage j-1 (parent) frame. A discussion of parent frame images includes the proposal that images of point locations of child frame lattices are rational states of parent frame hybrid systems. Included is the more speculative proposal that parent frame hybrid systems are images of the points of child frame lattices. The resulting association of energy with images of the lattice points is discussed. Finally there is a discussion of the possible existence of different types of hybrid systems. Besides those associated with images of lattice space points, there may be different types associated with images of lattice time points, wave number lattices, and vectors.
ABSTRACT Ab initio calculations of NO2 and NO2-, using a Dunning [4s3p] basis augmented by 1 comp... more ABSTRACT Ab initio calculations of NO2 and NO2-, using a Dunning [4s3p] basis augmented by 1 component diffuses andp functions were carried out. The SCF energies of NO2 and NO2/- (ground states) as a function of Os, Op, Ns, and Np diffuse function exponents are given and discussed. The curves show some unexpected features which make the optimization of the diffuse function exponents problematic.
The effect of the optical potential on the nucleon momentum distribution in nuclei is discussed i... more The effect of the optical potential on the nucleon momentum distribution in nuclei is discussed in this work. The idea of McCarthy et al., that the refraction and localization effects caused by the real and imaginary parts of the optical potential smears the single-particle momentum distribution, is amplified and applied to nucleon momentum distribution experiments. The assertion is made here that it seems impossible to directly measure in any manner the momentum distribution of nucleons in nuclei. Further it is proposed that much or all of the large discrepancy between the experimental momentum determinations and the shell-model predictions is due to the neglect of these important sources of high-momentum components. The high-energy C12(p, d)C11 pickup reaction data are re-analyzed in the light of these considerations. It is shown that for q2beta2<~8, the 1p-shell harmonic oscillator distribution, (q2beta2)exp(-q2beta2), has sufficient high-momentum components to fit the data. T...
Ab initio calculations of the trans–cis energy difference and the barrier for internal rotation i... more Ab initio calculations of the trans–cis energy difference and the barrier for internal rotation in nitrous acid were made using a double zeta quality basis set and MCSCF+CI methods. The geometry parameters used were the optimized ones obtained by Skaarup and Boggs. The trans–gauche barrier and trans–cis energy difference were calculated here to be 11.4 and 2.5 kcal/mole in good agreement with the experimental values of 11.57±0.2 and 0.4±1.0 kcal/mole, respectively. It was found that exclusion of only one configuration involving excitation of a lone pair on the NOH oxygen from the MCSCF and CI lists reduced the trans–gauche barrier to 6.4 kcal/mole. This shows the importance of lone pair electrons on the oxygen, and not on the nitrogen as previously suggested, to the high barrier.
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