In this thesis, real quantum mechanics (QM) is constructed over the field ${\\cal F}\\sb{P\\sb0\\oplus P\\sb3}$ in the Pauli algebra, where the trivector $i = \\sigma\\sb1\\sigma\\sb2\\sigma\\sb3$ arises naturally as a product of the unit basis vectors in 3-dimensional Euclidean space. In this scheme, i is the volume element of the algebra with $i\\sp2 = -1$ and has geometrical meaning. Components of the state vector in Hilbert space take the form of a real element in the field ${\\cal F}\\sb{P\\sb0\\oplus P\\sb3}$ and the hermitean conjugate of the state vector is just its transpose, so that all the observables, represented by hermitean operators, are symmetric; i itself is an anti-hermitean operator. In this construction, the time reversal operator ${\\cal T}$ is a combination of the transpose "+" and the unitary operator which changes the sign of t. Since ${\\cal T}$ is an anti-automorphism the interpretation of its action must be carefully analyzed in terms of the state vectors and their inner products. Furthermore, the parity transformation is not represented by the spatial inversion $\\tilde\\Pi$ (an antilinear, anti-unitary transformation). Instead, the spatial reversal $\\bar\\Pi$ (an anti-automorphic, linear transformation) is proposed as a parity operator. It is unitary as well as hermitean, and it commutes with the Hamiltonian, H, of a system, thus, it is a constant of motion. This thesis mainly deals with non-relativistic QM for spinless particles.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1991 .W444. Source: Masters Abstracts International, Volume: 31-01, page: 0331. Chair: W. E. Baylis. Thesis (M.Sc.)--University of Windsor (Canada), 1991.In this thesis, real quantum mechanics (QM) is constructed over the field ${\\cal F}\\sb{P\\sb0\\oplus P\\sb3}$ in the Pauli algebra, where the trivector $i = \\sigma\\sb1\\sigma\\sb2\\sigma\\sb3$ arises naturally as a product of the unit basis vectors in 3-dimensional Euclidean space. In this scheme, i is the volume element of the algebra with $i\\sp2 = -1$ and has geometrical meaning. Components of the state vector in Hilbert space take the form of a real element in the field ${\\cal F}\\sb{P\\sb0\\oplus P\\sb3}$ and the hermitean conjugate of the state vector is just its transpose, so that all the observables, represented by hermitean operators, are symmetric; i itself is an anti-hermitean operator. In this construction, the time reversal operator ${\\cal T}$ is a combination of the transpose "+" and the unitary operator which changes the sign of t. Since ${\\cal T}$ is an anti-automorphism the interpretation of its action must be carefully analyzed in terms of the state vectors and their inner products. Furthermore, the parity transformation is not represented by the spatial inversion $\\tilde\\Pi$ (an antilinear, anti-unitary transformation). Instead, the spatial reversal $\\bar\\Pi$ (an anti-automorphic, linear transformation) is proposed as a parity operator. It is unitary as well as hermitean, and it commutes with the Hamiltonian, H, of a system, thus, it is a constant of motion. This thesis mainly deals with non-relativistic QM for spinless particles.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1991 .W444. Source: Masters Abstracts International, Volume: 31-01, page: 0331. Chair: W. E. Baylis. Thesis (M.Sc.)--University of Windsor (Canada), 1991.

Jan 1, 19910 citations

The Geometry of Quantum Physics

James Thomas Wheeler

tx19Jan 1, 19971 citation

The quantum way to diagonalize hermitean matrices

Stefan Weigert

Abstract An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. The method is based on the measurement of quantum mechanical observables which provides the computational resource. In brief, quantum mechanics is able to directly address and output eigenvalues of hermitean matrices. Here, the example of (2×2) matrices is considered in detail. This simple case allows one to illustrate the conceptual features of the general method which is applicable to (N×N) hermitean matrices.Abstract An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. The method is based on the measurement of quantum mechanical observables which provides the computational resource. In brief, quantum mechanics is able to directly address and output eigenvalues of hermitean matrices. Here, the example of (2×2) matrices is considered in detail. This simple case allows one to illustrate the conceptual features of the general method which is applicable to (N×N) hermitean matrices.

arXiv:quant-ph/0305140Fortschritte der PhysikFeb 26, 20032 citations

Geometry of Cyclic Quantum Evolutions

J. Anandan

Springer eBooksJan 1, 19890 citations

THE GEOMETRY OF QUANTUM STATES

Julian Schwinger

World Scientific series in 20th century physicsMay 1, 20003 citations

In this thesis, real quantum mechanics (QM) is constructed over the field

{\\cal F}\\sb{P\\sb0\\oplus P\\sb3}

in the Pauli algebra, where the trivector

i = \\sigma\\sb1\\sigma\\sb2\\sigma\\sb3

arises naturally as a product of the unit basis vectors in 3-dimensional Euclidean space. In this scheme, i is the volume element of the algebra with

i\\sp2 = -1

and has geometrical meaning. Components of the state vector in Hilbert space take the form of a real element in the field

{\\cal F}\\sb{P\\sb0\\oplus P\\sb3}

and the hermitean conjugate of the state vector is just its transpose, so that all the observables, represented by hermitean operators, are symmetric; i itself is an anti-hermitean operator. In this construction, the time reversal operator

{\\cal T}

is a combination of the transpose "+" and the unitary operator which changes the sign of t. Since

{\\cal T}

is an anti-automorphism the interpretation of its action must be carefully analyzed in terms of the state vectors and their inner products. Furthermore, the parity transformation is not represented by the spatial inversion

\\tilde\\Pi

(an antilinear, anti-unitary transformation). Instead, the spatial reversal

\\bar\\Pi

(an anti-automorphic, linear transformation) is proposed as a parity operator. It is unitary as well as hermitean, and it commutes with the Hamiltonian, H, of a system, thus, it is a constant of motion. This thesis mainly deals with non-relativistic QM for spinless particles.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1991 .W444. Source: Masters Abstracts International, Volume: 31-01, page: 0331. Chair: W. E. Baylis. Thesis (M.Sc.)--University of Windsor (Canada), 1991.