Lie algebra decompositions with applications to quantum dynamics

... subalgebras of gl(n, F) are called the linear Lie algebras, which will be reviewed next. Let us consider the subspace sl(n, F) of trace zero matrices in gl(n, F), that is, sl(n, F) = span{A ∈ gl(n, F) : tr(A) = 0} . Let A, B ∈ gl(n, F). Then, it can easily be seen that tr([A, B]) = 0, i.e., sl(n, F) ...

... subalgebras of gl(n, F) are called the linear Lie algebras, which will be reviewed next. Let us consider the subspace sl(n, F) of trace zero matrices in gl(n, F), that is, sl(n, F) = span{A ∈ gl(n, F) : tr(A) = 0} . Let A, B ∈ gl(n, F). Then, it can easily be seen that tr([A, B]) = 0, i.e., sl(n, F) ...

A Numerical Approach to Virasoro Blocks and the Information

... same property [5]. But in the semiclassical approximation, the blocks develop additional ‘forbidden singularities’ [7] that represent a violation of unitarity. These singularities are a signature of semiclassical black hole physics in AdS3 . They arise because thermal correlators exhibit a Euclidean ...

... same property [5]. But in the semiclassical approximation, the blocks develop additional ‘forbidden singularities’ [7] that represent a violation of unitarity. These singularities are a signature of semiclassical black hole physics in AdS3 . They arise because thermal correlators exhibit a Euclidean ...

Dirac Operators on Noncommutative Spacetimes ?

... Definition 2.2. Let (M, F) be a 4-dimensional twisted manifold. A noncommutative Cartan geometry on (M, F) is a pair of Clifford algebra valued one-forms (V, Ω), satisfying the expansion (2.5), the reality conditions V † = γ0 V γ0 , Ω† = −γ0 Ωγ0 and the limit Ṽ a |λ=0 = ω|λ=0 = ω̃|λ=0 = 0. Let us n ...

... Definition 2.2. Let (M, F) be a 4-dimensional twisted manifold. A noncommutative Cartan geometry on (M, F) is a pair of Clifford algebra valued one-forms (V, Ω), satisfying the expansion (2.5), the reality conditions V † = γ0 V γ0 , Ω† = −γ0 Ωγ0 and the limit Ṽ a |λ=0 = ω|λ=0 = ω̃|λ=0 = 0. Let us n ...

Lectures on Conformal Field Theory arXiv:1511.04074v2 [hep

... Abstract: These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more ge ...

... Abstract: These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more ge ...

Quantum Theory, Groups and Representations: An Introduction (under construction) Peter Woit

... 15.1 An example: the Euclidean group . . . . . . . . . . 15.2 Semi-direct product groups . . . . . . . . . . . . . 15.3 Representations of N o K, N commutative . . . . 15.4 Semi-direct product Lie algebras . . . . . . . . . . 15.5 For further reading . . . . . . . . . . . . . . . . . . ...

... 15.1 An example: the Euclidean group . . . . . . . . . . 15.2 Semi-direct product groups . . . . . . . . . . . . . 15.3 Representations of N o K, N commutative . . . . 15.4 Semi-direct product Lie algebras . . . . . . . . . . 15.5 For further reading . . . . . . . . . . . . . . . . . . ...

11 Harmonic oscillator and angular momentum — via operator algebra

... From these equations it appears that J+ |j, mi is an eigenstate of Jz with eigenvalue h̄(m+1), and that J− |j, mi is an eigenstate of Jz with eigenvalue h̄(m − 1). In the same manner, J+2 |j, mi will be an eigenvector with eigenvalue h̄(m + 2), etc. Starting with the given existence of the eigenstat ...

... From these equations it appears that J+ |j, mi is an eigenstate of Jz with eigenvalue h̄(m+1), and that J− |j, mi is an eigenstate of Jz with eigenvalue h̄(m − 1). In the same manner, J+2 |j, mi will be an eigenvector with eigenvalue h̄(m + 2), etc. Starting with the given existence of the eigenstat ...

Factorization Algebras in Quantum Field Theory Volume 1 (8 May

... This quantization theorem applies to many examples of physical interest, including pure Yang-Mills theory and σ-models. For pure Yang-Mills theory, it is shown in Costello (2011b) that the relevant obstruction groups vanish and that the deformation group is one-dimensional; thus there exists a one-p ...

... This quantization theorem applies to many examples of physical interest, including pure Yang-Mills theory and σ-models. For pure Yang-Mills theory, it is shown in Costello (2011b) that the relevant obstruction groups vanish and that the deformation group is one-dimensional; thus there exists a one-p ...

Symmetry and symmetry breaking, algebraic approach to - IME-USP

... and still undergoes a steady process of evolution, proceeding in stages from simpler forms to levels of organization of ever increasing complexity. Roughly speaking, these stages may be characterized as physical evolution, chemical evolution and biological evolution. Let us begin with a few comments ...

... and still undergoes a steady process of evolution, proceeding in stages from simpler forms to levels of organization of ever increasing complexity. Roughly speaking, these stages may be characterized as physical evolution, chemical evolution and biological evolution. Let us begin with a few comments ...

pdf

... The standard definition of a von Neumann algebra involves reference to a topology, and it is then shown (by von Neumann’s double commutant theorem) that this topological condition coincides with an algebraic condition (condition 2 in the Definition 1.2). But for present purposes, it will suffice to ...

... The standard definition of a von Neumann algebra involves reference to a topology, and it is then shown (by von Neumann’s double commutant theorem) that this topological condition coincides with an algebraic condition (condition 2 in the Definition 1.2). But for present purposes, it will suffice to ...

An Introduction to the Theory of Quantum Groups

... A large part of physics involves studying physical systems and how they evolve over time. Toward this end, one considers the phase space of a physical system which is a manifold consisting of points representing all possible states of a particular system. Each state/point P is described by a set of ...

... A large part of physics involves studying physical systems and how they evolve over time. Toward this end, one considers the phase space of a physical system which is a manifold consisting of points representing all possible states of a particular system. Each state/point P is described by a set of ...

Lecture Note - U.I.U.C. Math

... Quantum groups is a new exciting area of mathematics, which originated from mathematical physics (field theory, statistical mechanics), and developed greatly over the last 15 years. It is connected with many other, old and new, parts of mathematics, and remains an area of active, fruitful research t ...

... Quantum groups is a new exciting area of mathematics, which originated from mathematical physics (field theory, statistical mechanics), and developed greatly over the last 15 years. It is connected with many other, old and new, parts of mathematics, and remains an area of active, fruitful research t ...

On skew Heyting algebras - ars mathematica contemporanea

... A skew Heyting lattice is an algebra S = (S; ∧, ∨, 1) of type (2, 2, 0) such that: • (S; ∧, ∨, 1) is a co-strongly distributive skew lattice with top 1. Each upset u sup is thus a bounded distributive lattice. • for any u ∈ S an operation →u can be defined on u↑ such that (u↑; ∧, ∨, →u , 1, u) is a ...

... A skew Heyting lattice is an algebra S = (S; ∧, ∨, 1) of type (2, 2, 0) such that: • (S; ∧, ∨, 1) is a co-strongly distributive skew lattice with top 1. Each upset u sup is thus a bounded distributive lattice. • for any u ∈ S an operation →u can be defined on u↑ such that (u↑; ∧, ∨, →u , 1, u) is a ...

Thèse de doctorat - IMJ-PRG

... Abstract The main part of this thesis is devoted to study some constructions and structures around quantum shuffle algebras: differential algebras and Kashiwara operators; defining ideals and specialization problem; coHochschild homology and an analogue of Borel-Weil-Bott theorem. In the last chapte ...

... Abstract The main part of this thesis is devoted to study some constructions and structures around quantum shuffle algebras: differential algebras and Kashiwara operators; defining ideals and specialization problem; coHochschild homology and an analogue of Borel-Weil-Bott theorem. In the last chapte ...

Supersymmetry (SUSY)

... Supersymmetry is a deep and rich subject, I have been studying it for about 7 years, but I am still learning. We have only a few lectures and cannot teach it all. We have not established that SUSY is realised in nature. SUSY is currently searched for at the Large Hadron Collider at CERN. If su ...

... Supersymmetry is a deep and rich subject, I have been studying it for about 7 years, but I am still learning. We have only a few lectures and cannot teach it all. We have not established that SUSY is realised in nature. SUSY is currently searched for at the Large Hadron Collider at CERN. If su ...

Introduction to Representations of the Canonical Commutation and

... in the description of the CCR and CAR. This is easily done in the case of the CAR, where there exists an obvious candidate for the C ∗ -algebra of the CAR over a given Euclidean space [17]. If this Euclidean space is of countably inﬁnite dimension, the C ∗ -algebra of the CAR is isomorphic to the so ...

... in the description of the CCR and CAR. This is easily done in the case of the CAR, where there exists an obvious candidate for the C ∗ -algebra of the CAR over a given Euclidean space [17]. If this Euclidean space is of countably inﬁnite dimension, the C ∗ -algebra of the CAR is isomorphic to the so ...

Quantum Groups: A Path to Current Algebra

... of the duality required, I should mention here that an ordinary group gives rise to a quantum group by taking the vector space with the group as basis. When pushed to provide formal proofs of our claims, mathematicians generally resort to set theory. We build our structures on sets and feel satisﬁed ...

... of the duality required, I should mention here that an ordinary group gives rise to a quantum group by taking the vector space with the group as basis. When pushed to provide formal proofs of our claims, mathematicians generally resort to set theory. We build our structures on sets and feel satisﬁed ...

Toward an Understanding of Parochial Observables

... 1997, p. 278, Thm. 4.5.2), asserts that for each state ω on A, there exists a representation (πω , Hω ) of A, known as the GNS representation for ω, and a (cyclic) vector Ωω ∈ Hω such that for all A ∈ A, ω(A) = hΩω , πω (A)Ωω i One may find representations of A on different Hilbert spaces, and in th ...

... 1997, p. 278, Thm. 4.5.2), asserts that for each state ω on A, there exists a representation (πω , Hω ) of A, known as the GNS representation for ω, and a (cyclic) vector Ωω ∈ Hω such that for all A ∈ A, ω(A) = hΩω , πω (A)Ωω i One may find representations of A on different Hilbert spaces, and in th ...

No. 18 - Department of Mathematics

... for the long-range spin-chain, see also section 3.1 and references therein. 8 Because of quantum conformal invariance, one-, two- and three-point functions contain all the information one needs. 9 Form factors are matrix elements of field operators. They satisfy algebraic relations, called formfacto ...

... for the long-range spin-chain, see also section 3.1 and references therein. 8 Because of quantum conformal invariance, one-, two- and three-point functions contain all the information one needs. 9 Form factors are matrix elements of field operators. They satisfy algebraic relations, called formfacto ...

Vladimirov A.A., Diakonov D. Diffeomorphism

... space by (d + 1)-cells or simplices, although the number of edges entering one vertex may not be the same for all vertices. Alternatively, the number of edges coming from all vertices is the same but then the edges lengths may vary, if one attempts to embed the lattice into at space. Since only the ...

... space by (d + 1)-cells or simplices, although the number of edges entering one vertex may not be the same for all vertices. Alternatively, the number of edges coming from all vertices is the same but then the edges lengths may vary, if one attempts to embed the lattice into at space. Since only the ...

Algebraic aspects of topological quantum field theories

... special case n = 2. This is basically due to the fact that the category 2Cob can be described explicitly by a set of generators and relations. In turn this is only possible because a complete classification of 2-manifolds/surfaces exists. Eventually the ultimate goal will be to use this result to d ...

... special case n = 2. This is basically due to the fact that the category 2Cob can be described explicitly by a set of generators and relations. In turn this is only possible because a complete classification of 2-manifolds/surfaces exists. Eventually the ultimate goal will be to use this result to d ...

For screen - Mathematical Sciences Publishers

... In the second section we adduce the main notions and consider some examples. These examples, in particular, show that the Drinfeld–Jimbo enveloping algebra as well as its modifications are quantum enveloping algebras in our sense. In the third section with the help of the Heyneman–Radford theorem we ...

... In the second section we adduce the main notions and consider some examples. These examples, in particular, show that the Drinfeld–Jimbo enveloping algebra as well as its modifications are quantum enveloping algebras in our sense. In the third section with the help of the Heyneman–Radford theorem we ...