Igor Khavkine

Covariant phase space, constraints, gauge and the Peierls formula

It is well known that both the symplectic structure and the Poisson brackets of classical field theory can be constructed directly from the Lagrangian in a covariant way, without passing through the noncovariant canonical Hamiltonian formalism. This is true even in the presence of constraints and gauge symmetries. These constructions go under the names of the covariant phase space formalism and the Peierls bracket. We review both of them, paying more careful attention, than usual, to the precise mathematical hypotheses that they require, illustrating them in examples. Also an extensive historical overview of the development of these constructions is provided. The novel aspect of our presentation is a significant expansion and generalization of an elegant and quite recent argument by Forger and Romero showing the equivalence between the resulting symplectic and Poisson structures without passing through the canonical Hamiltonian formalism as an intermediary. We generalize it to cover theories with constraints and gauge symmetries and formulate precise sufficient conditions under which the argument holds. These conditions include a local condition on the equations of motion that we call hyperbolizability, and some global conditions of cohomological nature. The details of our presentation may shed some light on subtle questions related to the Poisson structure of gauge theories and their quantization.

Formation of an electronic nematic phase in interacting fermion systems

We study the formation of an electronic nematic phase characterized by a broken point-group symmetry in interacting fermion systems within the weak coupling theory. As a function of interaction strength and chemical potential, the phase transition between the isotropic Fermi liquid and nematic phase is first order at zero temperature and becomes second order at a finite temperature. The transition is present for all typical, including quasi-two-dimensional, electronic dispersions on the square lattice and takes place for arbitrarily small interaction when at van Hove filling, thus suppressing the Lifshitz transition. In connection with the formation of the nematic phase, we discuss the origin of the first-order transition and competition with other broken symmetry states.

Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction

Within this chapter we introduce the overall idea of the algebraic formalism of QFT on a fixed globally hyperbolic spacetime in the framework of unital \(*\)-algebras. We point out some general features of CCR algebras, such as simplicity and the construction of symmetry-induced homomorphisms. For simplicity, we deal only with a real scalar quantum field. We discuss some known general results in curved spacetime like the existence of quasifree states enjoying symmetries induced from the background, pointing out the relevant original references. We introduce, in particular, the notion of a Hadamard quasifree algebraic quantum state, both in the geometric and microlocal formulation, and the associated notion of Wick polynomials.

Dual computations of non-Abelian Yang-Mills theories on the lattice

In the past several decades there have been a number of proposals for computing with dual forms of non-Abelian Yang-Mills theories on the lattice. Motivated by the gauge-invariant, geometric picture offered by dual models and successful applications of duality in the U(1) case, we revisit the question of whether it is practical to perform numerical computation using non-Abelian dual models. Specifically, we consider three-dimensional SU(2) pure Yang-Mills as an accessible yet nontrivial case in which the gauge group is non-Abelian. Using methods developed recently in the context of spin foam quantum gravity, we derive an algorithm for efficiently computing the dual amplitude and describe Metropolis moves for sampling the dual ensemble. We relate our algorithms to prior work in non-Abelian dual computations of Hari Dass and his collaborators, addressing several problems that have been left open. We report results of spin expectation value computations over a range of lattice sizes and couplings that are in agreement with our conventional lattice computations. We conclude with an outlook on further development of dual methods and their application to problems of current interest.

Local and gauge invariant observables in gravity

It is well known that general relativity (GR) does not possess any non-trivial local (in a precise standard sense) and diffeomorphism invariant observables. We propose a generalized notion of local observables, which retain the most important properties that follow from the standard definition of locality, yet is flexible enough to admit a large class of diffeomorphism invariant observables in GR. The generalization comes at a small price, that the domain of definition of a generalized local observable may not cover the entire phase space of GR and two such observables may have distinct domains. However, the subset of metrics on which generalized local observables can be defined is in a sense generic (its open interior is non-empty in the Whitney strong topology). Moreover, generalized local gauge invariant observables are sufficient to separate diffeomorphism orbits on this admissible subset of the phase space. Connecting the construction with the notion of differential invariants, gives a general scheme for defining generalized local gauge invariant observables in arbitrary gauge theories, which happens to agree with well-known results for Maxwell and Yang-Mills theories.

q-deformed spin foam models of quantum gravity

We numerically study Barrett–Crane models of Riemannian quantum gravity. We have extended the existing numerical techniques to handle q-deformed models and arbitrary spacetime triangulations. We present and interpret expectation values of a few selected observables for each model, including a spin–spin correlation function which gives insight into the behaviour of the models. We find the surprising result that, as the deformation parameter q goes to 1 through roots of unity, the limit is discontinuous.

Supercurrent in nodal superconductors

In recent years, a number of nodal superconductors have been identified;

Quantum astrometric observables I: time delay in classical and quantum gravity

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The Calabi complex and Killing sheaf cohomology

-wave superconductors in high

Evaluation of new spin foam vertex amplitudes

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