L. Fortunato - INFN Padova
... • Some simple models are “naturally” written in terms of creation and annihilation operators. • To them we can always associate an algebra that brings with itself a dynamical symmetry. • By knowing how to deal mathematically with the algebra one can get analytic solutions that can be compared with e ...
... • Some simple models are “naturally” written in terms of creation and annihilation operators. • To them we can always associate an algebra that brings with itself a dynamical symmetry. • By knowing how to deal mathematically with the algebra one can get analytic solutions that can be compared with e ...
QFT on curved spacetimes: axiomatic framework and applications
... U is well defined, since π(A)Ω = 0 if and only if ω(A∗ A) = 0, but then we have also π 0 (A)Ω0 = 0. Furthermore U preserves the scalar product and is invertible and has therefore a unique extension to a unitary operator from H to H0 . This shows that π and π 0 are unitarily equivalent. The represen ...
... U is well defined, since π(A)Ω = 0 if and only if ω(A∗ A) = 0, but then we have also π 0 (A)Ω0 = 0. Furthermore U preserves the scalar product and is invertible and has therefore a unique extension to a unitary operator from H to H0 . This shows that π and π 0 are unitarily equivalent. The represen ...
PARADIGM OF QUASI-LIE AND QUASI-HOM
... collaborators gave rise to a great interest in quantum groups and Hopf algebras. Since then several other versions of (q-) deformed Lie algebras have appeared, especially in physical contexts such as string theory. The main objects for these deformations were infinite-dimensional algebras, primarily ...
... collaborators gave rise to a great interest in quantum groups and Hopf algebras. Since then several other versions of (q-) deformed Lie algebras have appeared, especially in physical contexts such as string theory. The main objects for these deformations were infinite-dimensional algebras, primarily ...
Does Geometric Algebra provide a loophole to Bell`s Theorem?
... real linear subspace generated by the single element M is called the set of trivectors, also known as pseudo-scalars. So we have seen that C`3,0 (R) is a beautiful object, containing within itself the quaternions, the complex numbers, the two-by-two complex matrices, three dimensional real vectors; ...
... real linear subspace generated by the single element M is called the set of trivectors, also known as pseudo-scalars. So we have seen that C`3,0 (R) is a beautiful object, containing within itself the quaternions, the complex numbers, the two-by-two complex matrices, three dimensional real vectors; ...
Classical elliptic current algebras
... 2.0.7. Integration contour. The geometric roots of the difference between these two choices can be explained as follows. These choices of test functions on different coverings Σ of elliptic curve correspond to the homotopically different contours on the elliptic curve. Each test function can be cons ...
... 2.0.7. Integration contour. The geometric roots of the difference between these two choices can be explained as follows. These choices of test functions on different coverings Σ of elliptic curve correspond to the homotopically different contours on the elliptic curve. Each test function can be cons ...
this document - ITP Lecture Archive
... which obeys the Yang–Baxter equation. The R-matrix can be applied to the construction of energy eigenstates. This talk describes how to derive the Bethe equations using the algebra of monodromy matrix elements. Goals: Lax/R-matrix, Yang-Baxter-equation; transfer matrix and conserved charges; monodro ...
... which obeys the Yang–Baxter equation. The R-matrix can be applied to the construction of energy eigenstates. This talk describes how to derive the Bethe equations using the algebra of monodromy matrix elements. Goals: Lax/R-matrix, Yang-Baxter-equation; transfer matrix and conserved charges; monodro ...
Particle Physics on Noncommutative Spaces
... modifications of space-time can modify the high energy behavior of loops. • New ideas to break gauge symmetries: after all lots of ideas come from solid state physics and we have quite a few models in solid state physics that are described by NC gauge theories. This will lead to new phenomenology fo ...
... modifications of space-time can modify the high energy behavior of loops. • New ideas to break gauge symmetries: after all lots of ideas come from solid state physics and we have quite a few models in solid state physics that are described by NC gauge theories. This will lead to new phenomenology fo ...
Lie point symmetries: An alternative approach to wave
... one could equally use Γ3 to construct additional solutions. The only difference is that the creation is by an even number of steps so that one either creates the even series of wave-functions or the odd series. When one is at a higher state, the symmetries Γ2 and Γ4 can both be used as annihilation ...
... one could equally use Γ3 to construct additional solutions. The only difference is that the creation is by an even number of steps so that one either creates the even series of wave-functions or the odd series. When one is at a higher state, the symmetries Γ2 and Γ4 can both be used as annihilation ...
The roots of Arabic algebra
... • Babylonian algebra "did not deal with known and unknown numbers represented by words or symbols. Strictly speaking it did not deal with numbers at all, but with mesurable line segments ... • The operations used to define and solve these problems were not arithmetical but concrete and geometrical . ...
... • Babylonian algebra "did not deal with known and unknown numbers represented by words or symbols. Strictly speaking it did not deal with numbers at all, but with mesurable line segments ... • The operations used to define and solve these problems were not arithmetical but concrete and geometrical . ...
Geometric Quantization - Texas Christian University
... The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical example of this is the cotangent bundle of a manifold. The manifold is the configuration space (ie set of positions), and the tangent bund ...
... The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical example of this is the cotangent bundle of a manifold. The manifold is the configuration space (ie set of positions), and the tangent bund ...
When is a linear functional multiplicative? Krzysztof Jarosz
... then T is multiplicative. Proof. The result immediately follows from the G-K-Ż Theorem. Let a1 , a2 ∈ A and G ∈ M (B) . The functional G ◦ T satisfies the assumptions of the G-K-Ż Theorem, so it is multiplicative and we have G (T (a1 a2 ) − T (a1 ) T (a2 )) = G ◦ T (a1 a2 ) − G ◦ T (a1 ) · G ◦ T (a ...
... then T is multiplicative. Proof. The result immediately follows from the G-K-Ż Theorem. Let a1 , a2 ∈ A and G ∈ M (B) . The functional G ◦ T satisfies the assumptions of the G-K-Ż Theorem, so it is multiplicative and we have G (T (a1 a2 ) − T (a1 ) T (a2 )) = G ◦ T (a1 a2 ) − G ◦ T (a1 ) · G ◦ T (a ...
Shanghai Conference on Representation Theory
... half, we will discuss presentations of these algebras by generators and relations of Hecke type. The presentation problem was raised by G. Williamson in his thesis and its solution in the integral case has potential applications in the (singular) Soergel bimodule categorification. In the second half ...
... half, we will discuss presentations of these algebras by generators and relations of Hecke type. The presentation problem was raised by G. Williamson in his thesis and its solution in the integral case has potential applications in the (singular) Soergel bimodule categorification. In the second half ...
From classical theta functions to topological quantum field theory
... THEOREM. (R. Kirby) Any two surgery descriptions of a 3-manifold are related by handle-slides and addition and deletion of trivial handles. As a corollary of the Stone-von Neumann theorem we obtain: THEOREM. Given a manifold M obtained as surgery on the framed link L, the number Z(M ) = Ω(U+)−b+ Ω( ...
... THEOREM. (R. Kirby) Any two surgery descriptions of a 3-manifold are related by handle-slides and addition and deletion of trivial handles. As a corollary of the Stone-von Neumann theorem we obtain: THEOREM. Given a manifold M obtained as surgery on the framed link L, the number Z(M ) = Ω(U+)−b+ Ω( ...
to be completed. LECTURE NOTES 1
... inconsistency. See [4] for his formulation of quantization in terms of constraint. In this section, I will give a formulation that applies to affine algebraic varieties with a symplectic structure, which seems to be one focus of [4]. Recall that the classical Hamiltonian H(q, p) = ...
... inconsistency. See [4] for his formulation of quantization in terms of constraint. In this section, I will give a formulation that applies to affine algebraic varieties with a symplectic structure, which seems to be one focus of [4]. Recall that the classical Hamiltonian H(q, p) = ...
An Analysis of the Quantum Penny Flip Game using Geometric
... as well as providing a simple and direct derivation of the winning transformation, which we demonstrate can be parametrized by two angles θ, φ. For comparison we derive the same general winning strategy by conventional means using density matrices. KEYWORDS: quantum game, penny flip, geometric algeb ...
... as well as providing a simple and direct derivation of the winning transformation, which we demonstrate can be parametrized by two angles θ, φ. For comparison we derive the same general winning strategy by conventional means using density matrices. KEYWORDS: quantum game, penny flip, geometric algeb ...
Quantum Field Theory on Curved Backgrounds. II
... with algebra g, and let X ∈ g. The map tD → tX(t ∈ R) is a homomorphism of Lie(R) → g, so by the Lemma there is a unique analytic homomorphism ξX : R → G such that d ξX (D) = X. Conversely, if η is an analytic homomorphism of R → G, and if we let X = d η (D), it is obvious that η = ξX . Thus X 7→ ξX ...
... with algebra g, and let X ∈ g. The map tD → tX(t ∈ R) is a homomorphism of Lie(R) → g, so by the Lemma there is a unique analytic homomorphism ξX : R → G such that d ξX (D) = X. Conversely, if η is an analytic homomorphism of R → G, and if we let X = d η (D), it is obvious that η = ξX . Thus X 7→ ξX ...
QUANTUM GROUPS AND DIFFERENTIAL FORMS Contents 1
... dN (S) = 0. The stronger assumption d2 (S) = 0, though unnecessary for Lemma 1, simplifies the analysis done in this paper. Else for xi ∈ S, one needs to introduce higher differential symbols d2 xi , d3 xi , and so on; see the paper of Kerner and Abramov [11] and the references therein for such cons ...
... dN (S) = 0. The stronger assumption d2 (S) = 0, though unnecessary for Lemma 1, simplifies the analysis done in this paper. Else for xi ∈ S, one needs to introduce higher differential symbols d2 xi , d3 xi , and so on; see the paper of Kerner and Abramov [11] and the references therein for such cons ...
Particle Physics on Noncommutative Spaces
... modifications of space-time can modify the high energy behavior of loops. • New ideas to break gauge symmetries: after all lots of ideas come from solid state physics and we have quite a few models in solid state physics that are described by NC gauge theories. This will lead to new phenomenology fo ...
... modifications of space-time can modify the high energy behavior of loops. • New ideas to break gauge symmetries: after all lots of ideas come from solid state physics and we have quite a few models in solid state physics that are described by NC gauge theories. This will lead to new phenomenology fo ...
1 Introduction and Disclaimer
... In this section, we construct the spaces M(r) whose cohomologies are the tensor products of the basic representation of Y . We constructed Hilbn C2 by symplectic reduction: Hilbn C2 = T ∗ (End(Cn ) ⊕ Hom(C1 , Cn ))//θ0 Gl(n). We can similarly define M(r, n) = T ∗ (End(Cn ) ⊕ Hom(Cr , Cn ))//θ0 Gl(n) ...
... In this section, we construct the spaces M(r) whose cohomologies are the tensor products of the basic representation of Y . We constructed Hilbn C2 by symplectic reduction: Hilbn C2 = T ∗ (End(Cn ) ⊕ Hom(C1 , Cn ))//θ0 Gl(n). We can similarly define M(r, n) = T ∗ (End(Cn ) ⊕ Hom(Cr , Cn ))//θ0 Gl(n) ...
DIFFERENTIAL EQUATIONS ON HYPERPLANE COMPLEMENTS II Contents 1 3
... together with the initial condition H(0) = H0 . The case n = 1 is covered by Proposition 3.1. Assume now that n ≥ 2, set U ( j) = U ∩ {z j = · · · = zn = 0} and assume by induction on j = 1, . . . , n − 1 the existence and uniqueness of a holomorphic function H ( j) : U ( j) → GL(F) which satisfies ...
... together with the initial condition H(0) = H0 . The case n = 1 is covered by Proposition 3.1. Assume now that n ≥ 2, set U ( j) = U ∩ {z j = · · · = zn = 0} and assume by induction on j = 1, . . . , n − 1 the existence and uniqueness of a holomorphic function H ( j) : U ( j) → GL(F) which satisfies ...
Whittaker Functions and Quantum Groups
... ) be the states of the top and bottom rows. The partition function then is a row transfer matrix ΘS (δ, ε). The partition function with several rows is the product of the row transfer matrices. Theorem 4. (Baxter) If ∆S = ∆T then ΘS and ΘT commute. Though we will not explain this point, the commuta ...
... ) be the states of the top and bottom rows. The partition function then is a row transfer matrix ΘS (δ, ε). The partition function with several rows is the product of the row transfer matrices. Theorem 4. (Baxter) If ∆S = ∆T then ΘS and ΘT commute. Though we will not explain this point, the commuta ...
The notion of four-momentum in TGD
... 2. Does EP reduce to one aspect of QCC? This would require that classical Noether four-momentum identified as inertial momentum equals to the quantal four-momentum assignable to the states of super-conformal representations and identifiable as gravitational four-momentum. There would be only one ind ...
... 2. Does EP reduce to one aspect of QCC? This would require that classical Noether four-momentum identified as inertial momentum equals to the quantal four-momentum assignable to the states of super-conformal representations and identifiable as gravitational four-momentum. There would be only one ind ...
Light-like -deformations and scalar field theory via Drinfeld twist
... [Mµν , Mλρ ] = −i(ηνλ Mµρ − ηµλ Mνρ − ηνρ Mµλ + ηµρ Mνλ ), [Mµν , Pλ ] = −i(ηνλ Pµ − ηµλ Pν ). ...
... [Mµν , Mλρ ] = −i(ηνλ Mµρ − ηµλ Mνρ − ηνρ Mµλ + ηµρ Mνλ ), [Mµν , Pλ ] = −i(ηνλ Pµ − ηµλ Pν ). ...
Talk Dave Hewitt - Loughborough University
... role of quasi-variables. In H. Chick, K. Stacey, J. Vincent and J. Vincent (Eds), Proceedings of the 12th Study conference of the International Commission on Mathematical Instruction: The Future of the Teaching and Learning of Algebra, (Vol. 1, pp. 258-264). Melbourne, Australia: The University of M ...
... role of quasi-variables. In H. Chick, K. Stacey, J. Vincent and J. Vincent (Eds), Proceedings of the 12th Study conference of the International Commission on Mathematical Instruction: The Future of the Teaching and Learning of Algebra, (Vol. 1, pp. 258-264). Melbourne, Australia: The University of M ...
Schweigert.pdf
... algebras – that arise when studying specific small subsets of these consistency conditions (see e.g. [13, 10, 2]). Moreover, Morita equivalence, combined with orbifold technology, allows for an elegant proof of T-dualities for arbitrary topology of the world sheet. We have already seen that modules o ...
... algebras – that arise when studying specific small subsets of these consistency conditions (see e.g. [13, 10, 2]). Moreover, Morita equivalence, combined with orbifold technology, allows for an elegant proof of T-dualities for arbitrary topology of the world sheet. We have already seen that modules o ...