Triangle Seminars
Tuesday, 19 Jan 2010
Making sense of non-Hermitian Hamiltonians
Carl Bender
(Washington University, St. Louis)
Abstract:
The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy
eigenvalues are real and that time evolution is unitary.
However, the Hamiltonian H=p2+ix3, which is obviously not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory.
Evidently, the axiom of Dirac Hermiticity is too restrictive. While H=p2+ix3 isnot Dirac Hermitian,
it is PT symmetric, that is, invariant under combined space reflection P and time reversal T. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new
kinds of theories having strange and remarkable properties emerge. Some of these properties have recently been verified in laboratory experiments. If one generalizes classical mechanics into the complex domain, the resulting theories
have equally remarkable properties.
The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy
eigenvalues are real and that time evolution is unitary.
However, the Hamiltonian H=p2+ix3, which is obviously not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory.
Evidently, the axiom of Dirac Hermiticity is too restrictive. While H=p2+ix3 isnot Dirac Hermitian,
it is PT symmetric, that is, invariant under combined space reflection P and time reversal T. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new
kinds of theories having strange and remarkable properties emerge. Some of these properties have recently been verified in laboratory experiments. If one generalizes classical mechanics into the complex domain, the resulting theories
have equally remarkable properties.
Posted by: KCL
Wednesday, 20 Jan 2010
A Three - Generation Calabi-Yau Manifold with Small Hodge Numbers
📍 London
Philip Candelas
(Oxford University)
Abstract:
A complete intersection Calabi-Yau manifold Y, will be presented. This manifold that has Euler number -72 and admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z12 and the non-Abelian dicyclic group Dic3. The quotient manifolds have chi=-6 and Hodge numbers (h11, h21) = (1,4). With the standard embedding of the spin connection in the gauge group, Y gives rise to an E6 gauge theory with 3 chiral generations of particles. The gauge group may be broken further by means of the Hosotani mechanism combined with continuous deformation of the background gauge field.
For the non-Abelian quotient we obtain a model with 3 generations with the gauge group broken to that of the standard model. Moreover there is a limit in which the quotients develop 3 conifold points. These singularities may be resolved simultaneously to give another manifold with (h11, h21) = (2,2) that lies right at the tip of the distribution of Calabi-Yau manifolds. This strongly suggests that there is a heterotic vacuum for this manifold that derives from the 3 generation model on the quotient of Y. The manifold Y may also be realised as a hypersurface in the toric variety. The symmetry group does not act torically, nevertheless we are able to identify the mirror of the quotient manifold by adapting the construction of Batyrev.
A complete intersection Calabi-Yau manifold Y, will be presented. This manifold that has Euler number -72 and admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z12 and the non-Abelian dicyclic group Dic3. The quotient manifolds have chi=-6 and Hodge numbers (h11, h21) = (1,4). With the standard embedding of the spin connection in the gauge group, Y gives rise to an E6 gauge theory with 3 chiral generations of particles. The gauge group may be broken further by means of the Hosotani mechanism combined with continuous deformation of the background gauge field.
For the non-Abelian quotient we obtain a model with 3 generations with the gauge group broken to that of the standard model. Moreover there is a limit in which the quotients develop 3 conifold points. These singularities may be resolved simultaneously to give another manifold with (h11, h21) = (2,2) that lies right at the tip of the distribution of Calabi-Yau manifolds. This strongly suggests that there is a heterotic vacuum for this manifold that derives from the 3 generation model on the quotient of Y. The manifold Y may also be realised as a hypersurface in the toric variety. The symmetry group does not act torically, nevertheless we are able to identify the mirror of the quotient manifold by adapting the construction of Batyrev.
Posted by: IC
The transplanckian S-matrix: recent progress and open problems
📍 London
Gabriele Veneziano
(College de France)
Abstract:
After a quick review of transplanckian-energy string collisions I will focus on a recent S-matrix approach to gravitational scattering and on its surprisingly good consistency with known results in classical gravitational collapse. A number of yet unresolved issues will also be presented.
After a quick review of transplanckian-energy string collisions I will focus on a recent S-matrix approach to gravitational scattering and on its surprisingly good consistency with known results in classical gravitational collapse. A number of yet unresolved issues will also be presented.
Posted by: QMW