Symplectic Geometry and Geometric Quantization
... as follows from (??) and (??). Remark that with the notation (??), one has the equality X̃ = XλX (where the ‘big’ X of XλX does not refer the element X ∈ G). If moreover the correspondence X → λX is a Lie algebra homomorphism (i.e. λkY+Z = kλY + λZ , ∀k ∈ R, λ[Y,Z] = {λY , λZ }), then the action is ...
... as follows from (??) and (??). Remark that with the notation (??), one has the equality X̃ = XλX (where the ‘big’ X of XλX does not refer the element X ∈ G). If moreover the correspondence X → λX is a Lie algebra homomorphism (i.e. λkY+Z = kλY + λZ , ∀k ∈ R, λ[Y,Z] = {λY , λZ }), then the action is ...
Momentum Maps, Dual Pairs and Reduction in
... Definition 2.9 Let (M, ω, ψ, J) be a hamiltonian G-space. A star product ? is strongly invariant if ~i [J(v), f ]? = {J(v), f }, for all v ∈ g and f ∈ C ∞ (M ). In this case, the quantum g-action has a quantum momentum map given P by iJ itself. Recall that given a symplectic connection on M and a se ...
... Definition 2.9 Let (M, ω, ψ, J) be a hamiltonian G-space. A star product ? is strongly invariant if ~i [J(v), f ]? = {J(v), f }, for all v ∈ g and f ∈ C ∞ (M ). In this case, the quantum g-action has a quantum momentum map given P by iJ itself. Recall that given a symplectic connection on M and a se ...
Hydrogen Atom.
... vector and the Laplace-Runge-Lenz vector. When the dynamical symmetry is broken, as in the case of the KleinGordon equation, the classical orbit is a precessing ellipse and the bound states with a given principle quantum number N are slightly split according to their orbital angular momentum values ...
... vector and the Laplace-Runge-Lenz vector. When the dynamical symmetry is broken, as in the case of the KleinGordon equation, the classical orbit is a precessing ellipse and the bound states with a given principle quantum number N are slightly split according to their orbital angular momentum values ...
Course Notes - Mathematics for Computer Graphics トップページ
... the transformations, geometric meanings of addition and scalar product are not trivial. We often want to have geometrically meaningful weighted sum (linear combination) of transformations, which is not an easy task. These kinds of practical demands therefore have inspired graphics researchers to exp ...
... the transformations, geometric meanings of addition and scalar product are not trivial. We often want to have geometrically meaningful weighted sum (linear combination) of transformations, which is not an easy task. These kinds of practical demands therefore have inspired graphics researchers to exp ...
The development of hoops involves some neglected and some new
... numbers created by 2-folding. The shape includes the squared radius and the polar angle; two extra degrees of freedom (conserved linear sizes) are destroyed by the folding process. If a size is negative, A.A will differ from A2 (though they both fold to the correct complex number; folding is many-to ...
... numbers created by 2-folding. The shape includes the squared radius and the polar angle; two extra degrees of freedom (conserved linear sizes) are destroyed by the folding process. If a size is negative, A.A will differ from A2 (though they both fold to the correct complex number; folding is many-to ...
Quantum Probability Theory
... and it follows that P(Eε ) = 0. Since this holds for all ε > 0, we have |f | ≤ 1 almost everywhere w.r.t. P. So f ∈ L∞ (Ω, Σ, P). Finally, since the operators X and Mf are both bounded and coincide on the dense subspace L∞ of H, they are equal. Theorem 1.2 (Gel’fand). Let A be a commutative von Neum ...
... and it follows that P(Eε ) = 0. Since this holds for all ε > 0, we have |f | ≤ 1 almost everywhere w.r.t. P. So f ∈ L∞ (Ω, Σ, P). Finally, since the operators X and Mf are both bounded and coincide on the dense subspace L∞ of H, they are equal. Theorem 1.2 (Gel’fand). Let A be a commutative von Neum ...
10 Time Reversal Symmetry in Quantum Mechanics
... If x → x, p → −p under time reversal, then this commutation relation no longer holds. How to solve this problem? It was realized by Wigner that in quantum mechanics, time reversal has to be defined in a very special way different from all other symmetries. Time reversal operator is anti-unitary: it ...
... If x → x, p → −p under time reversal, then this commutation relation no longer holds. How to solve this problem? It was realized by Wigner that in quantum mechanics, time reversal has to be defined in a very special way different from all other symmetries. Time reversal operator is anti-unitary: it ...
Iterants, Fermions and the Dirac Equation
... Probability and generalizations of classical probability are necessary for doing science. One should keep in mind that the quantum mechanics is based on a model that takes the solution of the Schroedinger equation to be a superposition of all possible observations of a given observer. The solution ...
... Probability and generalizations of classical probability are necessary for doing science. One should keep in mind that the quantum mechanics is based on a model that takes the solution of the Schroedinger equation to be a superposition of all possible observations of a given observer. The solution ...
A categorification of a quantum Frobenius map
... some basic familiarity of the work [EQ16b, EQ16a, KLMS12], but we will recall all necessary facts from the references when needed so as to make the current work readable. In Section 2, we first summarize some basic hopfological constructions of p-differential graded (p-DG) algebras that will be used ...
... some basic familiarity of the work [EQ16b, EQ16a, KLMS12], but we will recall all necessary facts from the references when needed so as to make the current work readable. In Section 2, we first summarize some basic hopfological constructions of p-differential graded (p-DG) algebras that will be used ...
O. Frink and G. Grätzer, The closed subalgebras of a topological
... The topological family .r ('?!) consists of intersections of finite unions of members of'?!. It contains all sets ofthe form E U F and 0 U F, where F is a finite subset of N. It contains the set N U {a} E U 0, but not the set N. We now verify that condition (ii) does not hold, by taking the set A of ...
... The topological family .r ('?!) consists of intersections of finite unions of members of'?!. It contains all sets ofthe form E U F and 0 U F, where F is a finite subset of N. It contains the set N U {a} E U 0, but not the set N. We now verify that condition (ii) does not hold, by taking the set A of ...
RANDOM WORDS, QUANTUM STATISTICS, CENTRAL LIMITS
... for the distribution of the spectrum λ of a GUE matrix [23] as a disguised classical central limit. (Here C is a constant that depends on k but not λ.) The classical argument is rigorous and it establishes a precise estimate. The quantum argument can be read rigorously or non-rigorously, depending o ...
... for the distribution of the spectrum λ of a GUE matrix [23] as a disguised classical central limit. (Here C is a constant that depends on k but not λ.) The classical argument is rigorous and it establishes a precise estimate. The quantum argument can be read rigorously or non-rigorously, depending o ...
cluster algebras in algebraic lie theory
... relations. Although there can be infinitely many generators and relations, they are all obtained from a finite number of them by means of an inductive procedure called mutation. The precise definition of a cluster algebra will be recalled in §2 below. Several examples arising in Lie theory were alre ...
... relations. Although there can be infinitely many generators and relations, they are all obtained from a finite number of them by means of an inductive procedure called mutation. The precise definition of a cluster algebra will be recalled in §2 below. Several examples arising in Lie theory were alre ...
Math 210C. Weyl groups and character lattices 1. Introduction Let G
... Let G be a connected compact Lie group, and S a torus in G (not necessarily maximal). In class we saw that the centralizer ZG (S) of S in G is connected conditional on Weyl’s general conjugacy theorem for maximal tori in compact connected Lie groups. The normalizer NG (S) is a closed subgroup of G t ...
... Let G be a connected compact Lie group, and S a torus in G (not necessarily maximal). In class we saw that the centralizer ZG (S) of S in G is connected conditional on Weyl’s general conjugacy theorem for maximal tori in compact connected Lie groups. The normalizer NG (S) is a closed subgroup of G t ...
Posterior distributions on certain parameter spaces obtained by using group theoretic methods adopted from quantum physics
... all ω ∈ Ω, is the identity element e of G. An effective group G is called transitive on the set Ω, if for any two elements ω1 , ω2 ∈ Ω, there is some g ∈ G, such that ω2 = gω1 . If G is transitive on a set Ω, then Ω is called a homogeneous space for the group G. For example, the rotation group in th ...
... all ω ∈ Ω, is the identity element e of G. An effective group G is called transitive on the set Ω, if for any two elements ω1 , ω2 ∈ Ω, there is some g ∈ G, such that ω2 = gω1 . If G is transitive on a set Ω, then Ω is called a homogeneous space for the group G. For example, the rotation group in th ...
The semantics of the canonical commutation relation
... and is at the same time a well-understood object of model-theoretic studies, a quantum Zariski geometry, see [11]. The quoted paper established the duality (1) between quantum algebras A at roots of unity (rational Weyl algebras in our case) and corresponding quantum Zariski geometries VA , which ex ...
... and is at the same time a well-understood object of model-theoretic studies, a quantum Zariski geometry, see [11]. The quoted paper established the duality (1) between quantum algebras A at roots of unity (rational Weyl algebras in our case) and corresponding quantum Zariski geometries VA , which ex ...
Hypercontractivity for free products
... distortion must be done to make Fn fit in. Theorem A iii) refines this, providing optimal time estimates in the symmetric algebra Ansym . We also obtain optimal time Lp → L2 hypercontractive estimates for linear combinations of words with length less than or equal to one. Apparently, our probabilist ...
... distortion must be done to make Fn fit in. Theorem A iii) refines this, providing optimal time estimates in the symmetric algebra Ansym . We also obtain optimal time Lp → L2 hypercontractive estimates for linear combinations of words with length less than or equal to one. Apparently, our probabilist ...
Representations of Lorentz and Poincaré groups
... so that the adjoint representation of the Lorentz group is six-dimensional. In other words, we could write the Lie algebra as [J˜a , J˜b ] = if ab c J˜c where a, b, c = 1, . . . , 6, by defining (J˜1 , J˜2 , J˜3 ) = J where J i ≡ 12 ijk J jk are the generators of rotations, and (J˜4 , J˜5 , J˜6 ) = ...
... so that the adjoint representation of the Lorentz group is six-dimensional. In other words, we could write the Lie algebra as [J˜a , J˜b ] = if ab c J˜c where a, b, c = 1, . . . , 6, by defining (J˜1 , J˜2 , J˜3 ) = J where J i ≡ 12 ijk J jk are the generators of rotations, and (J˜4 , J˜5 , J˜6 ) = ...
Endomorphism Bialgebras of Diagrams and of Non
... spaces and of bialgebras from diagrams of vector spaces, remote as they may seem, are closely related, in fact that the case of an endomorphism space of a non-commutative space is a special case of a coendomorphism bialgebra of a certain diagram. Some other constructions of endomorphism spaces from ...
... spaces and of bialgebras from diagrams of vector spaces, remote as they may seem, are closely related, in fact that the case of an endomorphism space of a non-commutative space is a special case of a coendomorphism bialgebra of a certain diagram. Some other constructions of endomorphism spaces from ...
Frenkel-Reshetikhin
... The most substantial examples of quantum groups are certain q-deformations of the linear space of regular functions on a simple Lie group G. Its dual algebra Uq(g) is naturally identified with a q-deformation of the universal enveloping algebra U(g) of a simple Lie algebra g corresponding to G. One ...
... The most substantial examples of quantum groups are certain q-deformations of the linear space of regular functions on a simple Lie group G. Its dual algebra Uq(g) is naturally identified with a q-deformation of the universal enveloping algebra U(g) of a simple Lie algebra g corresponding to G. One ...
A quantum random walk model for the (1 + 2) dimensional Dirac
... i ri ai si , where ri , si ∈ A and ai ∈ S for finitely many non-zero i s}. Letting ri = si = 1 in the definition gives S ⊂ I(S). Let J be an ideal containing S, then it is necessarily closed under right and left multiplication of A. Thus by construction I(S) ⊆ J and I(S) is the ideal generated by S. ...
... i ri ai si , where ri , si ∈ A and ai ∈ S for finitely many non-zero i s}. Letting ri = si = 1 in the definition gives S ⊂ I(S). Let J be an ideal containing S, then it is necessarily closed under right and left multiplication of A. Thus by construction I(S) ⊆ J and I(S) is the ideal generated by S. ...
On Classical and Quantum Objectivity - Philsci
... generated by the Hamiltonian vector field vf induced by f . The relation between the observable and the induced Hamiltonian vector field vf is given by the expression ivf ω = df , where ivf ω denotes the contraction of the 2-form ω with the vector vf .1 Therefore, the classical observables f ∈ C ∞ ( ...
... generated by the Hamiltonian vector field vf induced by f . The relation between the observable and the induced Hamiltonian vector field vf is given by the expression ivf ω = df , where ivf ω denotes the contraction of the 2-form ω with the vector vf .1 Therefore, the classical observables f ∈ C ∞ ( ...
QUANTUM LOGIC AND NON-COMMUTATIVE GEOMETRY
... observables make transparent the interpretation of quantum mechanics as a “theory of quantum probability” on a Topological NC phase space • As Streater pointed out: “Though the classical axioms were yet to be written down by Kolmogorov, Heisenberg, with help of the Copenhagen interpretation, invente ...
... observables make transparent the interpretation of quantum mechanics as a “theory of quantum probability” on a Topological NC phase space • As Streater pointed out: “Though the classical axioms were yet to be written down by Kolmogorov, Heisenberg, with help of the Copenhagen interpretation, invente ...
ON THE EQUATIONAL THEORY OF PROJECTION LATTICES OF
... domains of operators φ ∈ R. By (2), H0 is essentially dense. Define, recursively, Hn+1 as the intersection of Hn and all preimages φ−1 (Hn ) where φ T ∈ R. By (3) and (4), Hn+1 is essentially dense. By (4), the intersection Hω = n<ω Hn is essentially dense and, by (1), dense in H. By construction, H ...
... domains of operators φ ∈ R. By (2), H0 is essentially dense. Define, recursively, Hn+1 as the intersection of Hn and all preimages φ−1 (Hn ) where φ T ∈ R. By (3) and (4), Hn+1 is essentially dense. By (4), the intersection Hω = n<ω Hn is essentially dense and, by (1), dense in H. By construction, H ...
Chapter 12: Symmetries in Physics: Isospin and the Eightfold Way
... M eV /c2 ’s which is minuscule compared to the typical energy scale of hadrons (i.e. strongly interacting particles) which is about a GeV /c2 . This is why isospin is such a good symmetry and why isomultiplets have nearly identical masses. As it later turned out, the up and down quarks are not the o ...
... M eV /c2 ’s which is minuscule compared to the typical energy scale of hadrons (i.e. strongly interacting particles) which is about a GeV /c2 . This is why isospin is such a good symmetry and why isomultiplets have nearly identical masses. As it later turned out, the up and down quarks are not the o ...