Attention is given to the interface of mathematics and physics, specifically noting that fundamen... more Attention is given to the interface of mathematics and physics, specifically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to the meta-monogenic (massive) Dirac equation is constructed which form a Hilbert space. A new integral solution is proposed which involves application of a kernel to the right side of the function, instead of to the left as usual. This allows for the introduction of a multivector generalization of the Feynman Path Integral formulation, which shows that particular "geometric groupings" of solutions evolve in the manner to which we ascribe the term "quantum particle". Further, it is shown that the role of usual i is subplanted by the unit time basis vector, applied on the right side of the functions.
The automorphism invariant theory of Crawford[8] has shown great promise, however its application... more The automorphism invariant theory of Crawford[8] has shown great promise, however its application is limited by the paradigm to the domain of spin space. Our conjecture is that there is a broader principle at work which applies even to classical physics. Specifically, the laws of physics should be invariant under polydimensional transformations which reshuffle the geometry (e.g. exchanges vectors for trivectors) but preserves the algebra. To complete the symmetry, it follows that the laws of physics must be themselves polydimensional, having scalar, vector, bivector etc. parts in one multivector equation. Clifford algebra is the natural language in which to formulate this principle, as vectors/tensors were for relativity. This allows for a new treatment of the relativistic spinning particle (the Papapetrou equations) which is problematic in standard theory. In curved space the rank of the geometry will change under parallel transport, yielding a new basis for Weyl's connection and a natural coupling between linear and spinning motion.
A generalized Clifford manifold is proposed in which there are coordinates not only for the basis... more A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted to represent internal structure of matter (e.g. classical or quantum spin). The generalized Dirac operator must now include differentiation with respect to these higher order geometric coordinates. In a Riemann space, where the magnitude and rank of geometric objects are preserved under displacement, these new terms modify the geodesics. One possible physical interpretation is natural coupling of the classical spin to linear motion, providing a new derivation of the Papapetrou equations. A generalized curvature is proposed for the Clifford manifold in which the connection does not preserve the rank of a multivector under parallel transport, e.g. a vector may be "rotated" into a scalar.
A new class of multivector quantum mechanics is defined in which the theoretical gains over stand... more A new class of multivector quantum mechanics is defined in which the theoretical gains over standard formalism are fully illustrated. Multiple generations of particles appear when column spinors are replaced by Clifford multivectors (matrices associated with physical geometry). New gauge fields arise from now-allowable right-side-applied transformations, independent of the usual left-sided ones. The number and group structure of the gauge generators is a function of the dimension and metric of the underlying geometric space, where constraints on a multivector Lagrangian suppress some interactions.
A solution to the 50 year old problem of a spinning particle in curved space has been recently de... more A solution to the 50 year old problem of a spinning particle in curved space has been recently derived using an extension of Clifford calculus in which each geometric element has its own coordinate. This leads us to propose that all the laws of physics should obey new polydimensional metaprinciples, for which Clifford algebra is the natural language of expression, just as tensors were for general relativity. Specifically, phenomena and physical laws should be invariant under local automorphism transformations which reshuffle the physical geometry. This leads to a new generalized unified basis for classical mechanics, which includes string theory, membrane theory and the hypergravity formulation of Crawford[J. Math. Phys., 35, 2701-2718 (1994)]. Most important is that the broad themes presented can be exploited by nearly everyone in the field as a framework to generalize both the Clifford calculus and multivector physics.
The inverse problem, to reconstruct the general multivector wave function from the observable qua... more The inverse problem, to reconstruct the general multivector wave function from the observable quadratic densities, is solved for 3D geometric algebra. It is found that operators which are applied to the right side of the wave function must be considered, and the standard Fierz identities do not necessarily hold except in restricted situations, corresponding to the spin-isospin superselection rule. The Greider idempotent and Hestenes quaterionic spinors are included as extreme cases of a single superselection parameter.
Clifford Algebras in Analysis and Related Topics, 2018
Attention is given to the interface of mathematics and physics, specically noting that fundamenta... more Attention is given to the interface of mathematics and physics, specically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to the meta-monogenic (massive) Dirac equation is constructed which form a Hilbert space. A new integral solution is proposed which i n v olves application of a kernel to the right side of the function, instead of to the left as usual. This allows for the introduction of a multivector generalization of the Feynman Path Integral formulation, which shows that particular \geometric groupings" of solutions evolve in the manner to which w e ascribe the term \quantum particle". Further, it is shown that the role of usual i is subplanted by the unit time basis vector, applied on the right side of the functions.
Spinors, Twistors, Clifford Algebras and Quantum Deformations, 1993
The inverse problem, to reconstruct the general multi vector wave function from the observable qu... more The inverse problem, to reconstruct the general multi vector wave function from the observable quadratic densities, is solved for 3D geometric algebra. It is found that operators which are applied to the right side of the wave function must be considered, and the standard Fierz identities do not necessarily hold except in restricted situations, corresponding to the spin-isospin superselection rule. The Greider idempotent and Hestenes quaterionic spinors are included as extreme cases of a single superselection parameter.
Penelitian ini bertujuan untuk menganalisis kinerja keuangan pemerintah kabupaten dan kota di ind... more Penelitian ini bertujuan untuk menganalisis kinerja keuangan pemerintah kabupaten dan kota di indonesia ditinjau dari pendapatan asli daerah (PAD), dana alokasi umum (DAU), dana alokasi khusus (DAK), dan belanja daerah. Data pada penelitian ini merupakan data sekunder yang diperoleh dari Pusat Informasi dan Komunikasi Badan Pemeriksa Keuangan Republik Indonesia. Jumlah populasi dalam penelitian ini sebanyak 514 kabupaten dan kota, sedangkan sampel sebanyak 439 kabupaten dan kota yang diperoleh melalui metode purposive sampling. Analisis data yang digunakan dalam penelitian ini adalah model estimasi regresi data panel yang dilakukan melalui Uji Chow dan Uji Hausman. Hasil penelitian menunjukkan bahwa PAD dan DAK berpengaruh signifikan terhadap kinerja keuangan pemerintah kabupaten dan kota di Indonesia, Akan tetapi dua variabel lainnya yaitu DAU dan Belanja Daerah tidak terbukti berpengaruh terhadap Kinerja Keuangan Pemerintah.
Systems of equations are invariant under polydimensional transformations which reshuffle the geom... more Systems of equations are invariant under polydimensional transformations which reshuffle the geometry such that what is a line or a plane is dependent upon the frame of reference. This leads us to propose an extension of Clifford calculus in which each geometric element (vector, bivector) has its own coordinate. A new classical action principle is proposed in which particles take paths which minimize the distance traveled plus area swept out by the spin. This leads to a solution of the 50 year old conundrum of 'what is the correct Lagrangian' in which to derive the Papapetrou equations of motion for spinning particles in curved space (including torsion).
A generalized Clifford manifold is proposed in which there are coordinates not only for the basis... more A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted to represent internal structure of matter (e.g. classical or quantum spin). The generalized Dirac operator must now include differentiation with respect to these higher order geometric coordinates. In a Riemann space, where the magnitude and rank of geometric objects are preserved under displacement, these new terms modify the geodesics. One possible physical interpretation is natural coupling of the classical spin to linear motion, providing a new derivation of the Papapetrou equations. A generalized curvature is proposed for the Clifford manifold in which the connection does not preserve the rank of a multivector under parallel transport, e.g. a vector may be "rotated" into a scalar.
An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propaga... more An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propagating electromagnetic signal is presented using geometric algebra. Maxwell's equations can be expressed in a single multivector equation using 3D Clifford algebra (isomorphic to Pauli algebra spinorial formulation of electromagnetism). Subsequently one can more easily solve for the time evolution of both the electric and magnetic field simultaneously in terms of the fields evaluated only on a 3D hypersurface. The form of the special "characteristic" surfaces for which the time derivative of the fields can be singular are quickly deduced with little effort.
Attention is given to the interface of mathematics and physics, specifically noting that fundamen... more Attention is given to the interface of mathematics and physics, specifically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to the meta-monogenic (massive) Dirac equation is constructed which form a Hilbert space. A new integral solution is proposed which involves application of a kernel to the right side of the function, instead of to the left as usual. This allows for the introduction of a multivector generalization of the Feynman Path Integral formulation, which shows that particular "geometric groupings" of solutions evolve in the manner to which we ascribe the term "quantum particle". Further, it is shown that the role of usual i is subplanted by the unit time basis vector, applied on the right side of the functions.
The automorphism invariant theory of Crawford[8] has shown great promise, however its application... more The automorphism invariant theory of Crawford[8] has shown great promise, however its application is limited by the paradigm to the domain of spin space. Our conjecture is that there is a broader principle at work which applies even to classical physics. Specifically, the laws of physics should be invariant under polydimensional transformations which reshuffle the geometry (e.g. exchanges vectors for trivectors) but preserves the algebra. To complete the symmetry, it follows that the laws of physics must be themselves polydimensional, having scalar, vector, bivector etc. parts in one multivector equation. Clifford algebra is the natural language in which to formulate this principle, as vectors/tensors were for relativity. This allows for a new treatment of the relativistic spinning particle (the Papapetrou equations) which is problematic in standard theory. In curved space the rank of the geometry will change under parallel transport, yielding a new basis for Weyl's connection and a natural coupling between linear and spinning motion.
A generalized Clifford manifold is proposed in which there are coordinates not only for the basis... more A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted to represent internal structure of matter (e.g. classical or quantum spin). The generalized Dirac operator must now include differentiation with respect to these higher order geometric coordinates. In a Riemann space, where the magnitude and rank of geometric objects are preserved under displacement, these new terms modify the geodesics. One possible physical interpretation is natural coupling of the classical spin to linear motion, providing a new derivation of the Papapetrou equations. A generalized curvature is proposed for the Clifford manifold in which the connection does not preserve the rank of a multivector under parallel transport, e.g. a vector may be "rotated" into a scalar.
A new class of multivector quantum mechanics is defined in which the theoretical gains over stand... more A new class of multivector quantum mechanics is defined in which the theoretical gains over standard formalism are fully illustrated. Multiple generations of particles appear when column spinors are replaced by Clifford multivectors (matrices associated with physical geometry). New gauge fields arise from now-allowable right-side-applied transformations, independent of the usual left-sided ones. The number and group structure of the gauge generators is a function of the dimension and metric of the underlying geometric space, where constraints on a multivector Lagrangian suppress some interactions.
A solution to the 50 year old problem of a spinning particle in curved space has been recently de... more A solution to the 50 year old problem of a spinning particle in curved space has been recently derived using an extension of Clifford calculus in which each geometric element has its own coordinate. This leads us to propose that all the laws of physics should obey new polydimensional metaprinciples, for which Clifford algebra is the natural language of expression, just as tensors were for general relativity. Specifically, phenomena and physical laws should be invariant under local automorphism transformations which reshuffle the physical geometry. This leads to a new generalized unified basis for classical mechanics, which includes string theory, membrane theory and the hypergravity formulation of Crawford[J. Math. Phys., 35, 2701-2718 (1994)]. Most important is that the broad themes presented can be exploited by nearly everyone in the field as a framework to generalize both the Clifford calculus and multivector physics.
The inverse problem, to reconstruct the general multivector wave function from the observable qua... more The inverse problem, to reconstruct the general multivector wave function from the observable quadratic densities, is solved for 3D geometric algebra. It is found that operators which are applied to the right side of the wave function must be considered, and the standard Fierz identities do not necessarily hold except in restricted situations, corresponding to the spin-isospin superselection rule. The Greider idempotent and Hestenes quaterionic spinors are included as extreme cases of a single superselection parameter.
Clifford Algebras in Analysis and Related Topics, 2018
Attention is given to the interface of mathematics and physics, specically noting that fundamenta... more Attention is given to the interface of mathematics and physics, specically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to the meta-monogenic (massive) Dirac equation is constructed which form a Hilbert space. A new integral solution is proposed which i n v olves application of a kernel to the right side of the function, instead of to the left as usual. This allows for the introduction of a multivector generalization of the Feynman Path Integral formulation, which shows that particular \geometric groupings" of solutions evolve in the manner to which w e ascribe the term \quantum particle". Further, it is shown that the role of usual i is subplanted by the unit time basis vector, applied on the right side of the functions.
Spinors, Twistors, Clifford Algebras and Quantum Deformations, 1993
The inverse problem, to reconstruct the general multi vector wave function from the observable qu... more The inverse problem, to reconstruct the general multi vector wave function from the observable quadratic densities, is solved for 3D geometric algebra. It is found that operators which are applied to the right side of the wave function must be considered, and the standard Fierz identities do not necessarily hold except in restricted situations, corresponding to the spin-isospin superselection rule. The Greider idempotent and Hestenes quaterionic spinors are included as extreme cases of a single superselection parameter.
Penelitian ini bertujuan untuk menganalisis kinerja keuangan pemerintah kabupaten dan kota di ind... more Penelitian ini bertujuan untuk menganalisis kinerja keuangan pemerintah kabupaten dan kota di indonesia ditinjau dari pendapatan asli daerah (PAD), dana alokasi umum (DAU), dana alokasi khusus (DAK), dan belanja daerah. Data pada penelitian ini merupakan data sekunder yang diperoleh dari Pusat Informasi dan Komunikasi Badan Pemeriksa Keuangan Republik Indonesia. Jumlah populasi dalam penelitian ini sebanyak 514 kabupaten dan kota, sedangkan sampel sebanyak 439 kabupaten dan kota yang diperoleh melalui metode purposive sampling. Analisis data yang digunakan dalam penelitian ini adalah model estimasi regresi data panel yang dilakukan melalui Uji Chow dan Uji Hausman. Hasil penelitian menunjukkan bahwa PAD dan DAK berpengaruh signifikan terhadap kinerja keuangan pemerintah kabupaten dan kota di Indonesia, Akan tetapi dua variabel lainnya yaitu DAU dan Belanja Daerah tidak terbukti berpengaruh terhadap Kinerja Keuangan Pemerintah.
Systems of equations are invariant under polydimensional transformations which reshuffle the geom... more Systems of equations are invariant under polydimensional transformations which reshuffle the geometry such that what is a line or a plane is dependent upon the frame of reference. This leads us to propose an extension of Clifford calculus in which each geometric element (vector, bivector) has its own coordinate. A new classical action principle is proposed in which particles take paths which minimize the distance traveled plus area swept out by the spin. This leads to a solution of the 50 year old conundrum of 'what is the correct Lagrangian' in which to derive the Papapetrou equations of motion for spinning particles in curved space (including torsion).
A generalized Clifford manifold is proposed in which there are coordinates not only for the basis... more A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted to represent internal structure of matter (e.g. classical or quantum spin). The generalized Dirac operator must now include differentiation with respect to these higher order geometric coordinates. In a Riemann space, where the magnitude and rank of geometric objects are preserved under displacement, these new terms modify the geodesics. One possible physical interpretation is natural coupling of the classical spin to linear motion, providing a new derivation of the Papapetrou equations. A generalized curvature is proposed for the Clifford manifold in which the connection does not preserve the rank of a multivector under parallel transport, e.g. a vector may be "rotated" into a scalar.
An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propaga... more An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propagating electromagnetic signal is presented using geometric algebra. Maxwell's equations can be expressed in a single multivector equation using 3D Clifford algebra (isomorphic to Pauli algebra spinorial formulation of electromagnetism). Subsequently one can more easily solve for the time evolution of both the electric and magnetic field simultaneously in terms of the fields evaluated only on a 3D hypersurface. The form of the special "characteristic" surfaces for which the time derivative of the fields can be singular are quickly deduced with little effort.
Uploads
Papers by William Pezzaglia