Relativistic Effects in Atomic Spectra
... In this thesis we investigate relativistic effects in atomic spectra, particularly the spectral gaps in many-electron atoms. We derive explicit perturbative results from first principles of quantum mechanics, the theory of special relativity and Dirac theory. The basis we are working on is the non-r ...
... In this thesis we investigate relativistic effects in atomic spectra, particularly the spectral gaps in many-electron atoms. We derive explicit perturbative results from first principles of quantum mechanics, the theory of special relativity and Dirac theory. The basis we are working on is the non-r ...
Dual Banach algebras
... We can use the structure theorem for weak∗ -continuous ∗-isomorphisms to show that the definition of injectivity does not actually depend on the choice of representation A ⊆ B(H). ...
... We can use the structure theorem for weak∗ -continuous ∗-isomorphisms to show that the definition of injectivity does not actually depend on the choice of representation A ⊆ B(H). ...
Spectral measures in locally convex algebras
... analogue of a self-adjoint (or normal) operator in Hilbert space. A few of the results contained in this paper, specialized to Banach algebras and spaces, have been announced in [21]. We proceed to give a brief survey of the five sections of the paper. The central notion of this paper is t h a t of ...
... analogue of a self-adjoint (or normal) operator in Hilbert space. A few of the results contained in this paper, specialized to Banach algebras and spaces, have been announced in [21]. We proceed to give a brief survey of the five sections of the paper. The central notion of this paper is t h a t of ...
acta physica slovaca vol. 50 No. 1, 1 – 198 February 2000
... Received 10 November 1999, in final form 10 January 2000, accepted 13 January 2000 The work can be considered as an essay on mathematical and conceptual structure of nonrelativistic quantum mechanics (QM) which is related here to some other (more general, but also to more special and “approximative” ...
... Received 10 November 1999, in final form 10 January 2000, accepted 13 January 2000 The work can be considered as an essay on mathematical and conceptual structure of nonrelativistic quantum mechanics (QM) which is related here to some other (more general, but also to more special and “approximative” ...
What is a Dirac operator good for?
... (2) The “Canonical Line Bundle” or “Tautological Line Bundle” or “Hyperplane Bundle” H over CP 1 = S 2 . The canonical line bundle of any complex projective space CP n is the union of the set of all complex lines through the origin in Cn+1 , and the projection to CP n is given by projecting the elem ...
... (2) The “Canonical Line Bundle” or “Tautological Line Bundle” or “Hyperplane Bundle” H over CP 1 = S 2 . The canonical line bundle of any complex projective space CP n is the union of the set of all complex lines through the origin in Cn+1 , and the projection to CP n is given by projecting the elem ...
Quaternion Algebras and Quadratic Forms - UWSpace
... = hd1 , · · · , dn i and V corresponds to f , then we have d(f ) = d1 · · · dn · Ḟ 2 . In this case, d(f ) is called the determinant of V and can be denoted by d(V ). Proposition 1.2.9 Let q = ha, bi, q 0 = hc, di be regular binary quadratic forms. (So that a, b, c, d are all nonzero.) Then q ∼ = q ...
... = hd1 , · · · , dn i and V corresponds to f , then we have d(f ) = d1 · · · dn · Ḟ 2 . In this case, d(f ) is called the determinant of V and can be denoted by d(V ). Proposition 1.2.9 Let q = ha, bi, q 0 = hc, di be regular binary quadratic forms. (So that a, b, c, d are all nonzero.) Then q ∼ = q ...
Equações de Onda Generalizadas e Quantização
... Prof. Dr. Paulo Afonso Faria da Veiga (Institute of Mathematics and Computer Sciences – University of Sao Paulo) Prof. Dr. André Gustavo Scagliusi Landulfo (Centro de Ciências Naturais e Humanas – Federal University of ABC) ...
... Prof. Dr. Paulo Afonso Faria da Veiga (Institute of Mathematics and Computer Sciences – University of Sao Paulo) Prof. Dr. André Gustavo Scagliusi Landulfo (Centro de Ciências Naturais e Humanas – Federal University of ABC) ...
Abstract Vector Spaces, Linear Transformations, and Their
... (4) By Theorem 1.6 span(S ∩ T ), span(S) and span(T ) are subspaces, and by Theorem 1.4 span(S) ∩ span(T ) is a subspace. Now, consider x ∈ span(S ∩ T ). There exist vectors v1 , . . . , vn ∈ S ∩ T and scalars a1 , . . . , an ∈ F such that x = a1 v1 + · · · + an vn . But since v1 , . . . , vn belong ...
... (4) By Theorem 1.6 span(S ∩ T ), span(S) and span(T ) are subspaces, and by Theorem 1.4 span(S) ∩ span(T ) is a subspace. Now, consider x ∈ span(S ∩ T ). There exist vectors v1 , . . . , vn ∈ S ∩ T and scalars a1 , . . . , an ∈ F such that x = a1 v1 + · · · + an vn . But since v1 , . . . , vn belong ...
Rings of functions in Lipschitz topology
... a ring A is the set of all nonzero homomorphisms E: A-R with the weakest topology in which every function *: E*E(x), xCA, on this set is continuous. Let ,S be a set and let A be a subring of Rs which contains constants and is inuerse-closed,i.e.,if f€A with l/l=e forsome e=0, then llf(A. If feA and ...
... a ring A is the set of all nonzero homomorphisms E: A-R with the weakest topology in which every function *: E*E(x), xCA, on this set is continuous. Let ,S be a set and let A be a subring of Rs which contains constants and is inuerse-closed,i.e.,if f€A with l/l=e forsome e=0, then llf(A. If feA and ...
On the use of semi-closed sets and functions in convex analysis
... (b) In [2, Rem. 3.1 3◦ )] one says: ‘Note that for a convex set A of X one has R+ A = X if, and only if, 0 is in the interior of A. So the condition “R+ [dom f − x] = X” is equivalent to “x is the interior of dom f ” (for f convex , which is the case throughout the paper), condition which is much ol ...
... (b) In [2, Rem. 3.1 3◦ )] one says: ‘Note that for a convex set A of X one has R+ A = X if, and only if, 0 is in the interior of A. So the condition “R+ [dom f − x] = X” is equivalent to “x is the interior of dom f ” (for f convex , which is the case throughout the paper), condition which is much ol ...
On the quotient of a b-Algebra by a non closed b
... is called the ”permutation of the variables” isomorphism. In this way, we have an isomorphism q2 (E | F, E | F ; E | F ) −→ q2 (E | F, E | F ; E | F ) We remember also that q2 (E | F, E | F ; E | F ) is defined as q (E | F, q (E | F, E | F )). The multiplication m of our q-algebra is mapped onto an ...
... is called the ”permutation of the variables” isomorphism. In this way, we have an isomorphism q2 (E | F, E | F ; E | F ) −→ q2 (E | F, E | F ; E | F ) We remember also that q2 (E | F, E | F ; E | F ) is defined as q (E | F, q (E | F, E | F )). The multiplication m of our q-algebra is mapped onto an ...
Quantum Error Correction - Quantum Theory Group at CMU
... while there are many possible choices for bases, even orthonormal bases. The choice of basis is a matter of convenience. • Similarly, a quantum code is best thought of not just as a collection of codewords, as in classical codes, but as a subspace P of the Hilbert space Hc of the code carriers, a su ...
... while there are many possible choices for bases, even orthonormal bases. The choice of basis is a matter of convenience. • Similarly, a quantum code is best thought of not just as a collection of codewords, as in classical codes, but as a subspace P of the Hilbert space Hc of the code carriers, a su ...
Quantum Dynamical Systems
... ergodic theory of quantum systems. The basic concepts of the algebraic theory of quantum dynamics – C ∗ - and W ∗ -dynamical systems and their invariant states – are introduced in Subsections 4.1–4.3. In Subsection 4.4, I define a more general notion of quantum dynamical system. The GNS construction ...
... ergodic theory of quantum systems. The basic concepts of the algebraic theory of quantum dynamics – C ∗ - and W ∗ -dynamical systems and their invariant states – are introduced in Subsections 4.1–4.3. In Subsection 4.4, I define a more general notion of quantum dynamical system. The GNS construction ...
Reduced coproducts of compact Hausdorff spaces
... d1-objects, and let Y be a filter of subsets of I. For each J c I, denote the &1-direct product by Hf Ai; and for each pair of subsets J, K c I with J K, let JJK be the canonical projection morphism from HfAi to HfAi. The set Y is directed under reverse inclusion; the resulting direct limit, when it ...
... d1-objects, and let Y be a filter of subsets of I. For each J c I, denote the &1-direct product by Hf Ai; and for each pair of subsets J, K c I with J K, let JJK be the canonical projection morphism from HfAi to HfAi. The set Y is directed under reverse inclusion; the resulting direct limit, when it ...
Why Unsharp Observables? Claudio Carmeli · Teiko Heinonen · Alessandro Toigo
... for all X, Y ∈ B(R). In his recent article [17], Werner showed that the question of joint measurability of position and momentum observables can be reduced to the study of covariant phase space observables. This result leads to the complete characterization of jointly measurable pairs of position an ...
... for all X, Y ∈ B(R). In his recent article [17], Werner showed that the question of joint measurability of position and momentum observables can be reduced to the study of covariant phase space observables. This result leads to the complete characterization of jointly measurable pairs of position an ...
Interval-valued Fuzzy Vector Space
... fuzzy algebra is defined and ingestigated a lot of interesting properties. Each result is illustrated by a suitable exmaple. Keywords: Interval-valued fuzzy sets, interval-valued fuzzy vector space, subspace. AMS Mathematics Subject Classification (2010): 08A72, 15B15 1. Introduction There is a grow ...
... fuzzy algebra is defined and ingestigated a lot of interesting properties. Each result is illustrated by a suitable exmaple. Keywords: Interval-valued fuzzy sets, interval-valued fuzzy vector space, subspace. AMS Mathematics Subject Classification (2010): 08A72, 15B15 1. Introduction There is a grow ...
AI{D RELATED SPACES
... @)+(5): By (4), X has only a finite number of components and they are clopen. Every clopen subset of a topological space is C-embedded and every C*embedded subspace of a strongly O-dimensional space is strongly 0-dimensional. Thus each component of X is connected and strongly O-dimensional, hence ul ...
... @)+(5): By (4), X has only a finite number of components and they are clopen. Every clopen subset of a topological space is C-embedded and every C*embedded subspace of a strongly O-dimensional space is strongly 0-dimensional. Thus each component of X is connected and strongly O-dimensional, hence ul ...
E.6 The Weak and Weak* Topologies on a Normed Linear Space
... of a normed space were introduced in Examples E.7 and E.8. We will study these topologies more closely in this section. They are specific examples of generic “weak topologies” determined by the requirement that a given class of mappings fα : X → Yα be continuous. The “weak topology” corresponding to ...
... of a normed space were introduced in Examples E.7 and E.8. We will study these topologies more closely in this section. They are specific examples of generic “weak topologies” determined by the requirement that a given class of mappings fα : X → Yα be continuous. The “weak topology” corresponding to ...
Extending coherent state transforms to Clifford analysis
... holomorphic functions to Clifford algebra valued functions, satisfying properties generalizing the Cauchy–Riemann conditions. On the other hand, in quantum physics, Clifford algebra or spinor representation valued functions describe some systems with internal degrees of freedom, such as particles wi ...
... holomorphic functions to Clifford algebra valued functions, satisfying properties generalizing the Cauchy–Riemann conditions. On the other hand, in quantum physics, Clifford algebra or spinor representation valued functions describe some systems with internal degrees of freedom, such as particles wi ...
Unitarity as Preservation of Entropy and Entanglement in Quantum
... von Neumann entropy does not require that the evolution is unitary, or anti-unitary.(26) Note also that the assumption of preservation of disorder is not in contradiction with results of statistical mechanics. This point is further discussed in Appendix A. 2.1.2. The Assumption of Probabilistic Line ...
... von Neumann entropy does not require that the evolution is unitary, or anti-unitary.(26) Note also that the assumption of preservation of disorder is not in contradiction with results of statistical mechanics. This point is further discussed in Appendix A. 2.1.2. The Assumption of Probabilistic Line ...
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer)—and ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace (the analog of ""dropping the altitude"" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.