Equivalence Between Two Different Field-Dependent BRST Formulations

Finite field-dependent BRST (FFBRST) transformations were constructed by integrating infinitesimal BRST transformation in a closed form. Such a generalized transformations have been extended in various branch of physics and found many applications. Recently BRST transformation has also been generalized with same goal and motivation in slightly different manner. In this work we have shown that the later formulation is conceptually equivalent to the earlier formulation. We justify our claim by producing the same result of later formulation using earlier FFBRST formulation.     

Linking covariant and canonical general relativity via local observers

Hamiltonian gravity, relying on arbitrary choices of ‘space,’ can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between ‘spatial’ and ‘temporal’ variables. The key is viewing dynamical fields from the perspective of a field of observers—a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall into Hamiltonian form, recovering the Ashtekar formulation. We argue this provides a bridge between Ashtekar variables and covariant phase space methods. We also outline a framework where the ‘space of observers’ is fundamental, and spacetime geometry itself may be observer-dependent.     

Non-perturbative to perturbative QCD via the FFBRST

Recently a new type of quadratic gauge was introduced in QCD in which the degrees of freedom are suggestive of a phase of abelian dominance. In its simplest form it is also free of Gribov ambiguity. However this gauge is not suitable for usual perturbation theory. The finite field dependent BRST (FFBRST) transformation is a method established to interrelate generating functionals for different effective versions of gauge fixed field theories. In this paper we propose a FFBRST transformation suitable for transforming the theory in the new quadratic gauge into the standard Lorenz gauge Faddeev–Popov version of the effective lagrangian. The task is made interesting by the fact that the effective lagrangian is invariant under two different BRST transformations which leads to suitable extension of the previous procedures to accomplish the required result. We are thus able to identify a field redefinition to go from a non-perturbative phase of QCD to perturbative QCD.     

Finite BRST Mapping in Higher-Derivative Models

We continue the study of finite field-dependent BRST (FFBRST) symmetry in the quantum theory of gauge fields. An expression for the Jacobian of path integral measure is presented, depending on a finite field-dependent parameter, and the FFBRST symmetry is then applied to a number of well-established quantum gauge theories in a form which incudes higher-derivative terms. Specifically, we examine the corresponding versions of the Maxwell theory, non-Abelian vector field theory, and gravitation theory. We present a systematic mapping between different forms of gauge-fixing, including those with higher-derivative terms, for which these theories have better renormalization properties. In doing so, we also provide the independence of the S-matrix from a particular gauge-fixing with higher derivatives. Following this method, a higher-derivative quantum action can be constructed for any gauge theory in the FFBRST framework.     

Shape dynamics in 2 + 1 dimensions

Shape Dynamics is a formulation of General Relativity where refoliation invariance is traded for local spatial conformal invariance. In this paper we explicitly construct Shape Dynamics for a torus universe in 2 + 1 dimensions through a linking gauge theory that ensures dynamical equivalence with General Relativity. The Hamiltonian we obtain is formally a reduced phase space Hamiltonian. The construction of the Shape Dynamics Hamiltonian on higher genus surfaces is not explicitly possible, but we give an explicit expansion of the Shape Dynamics Hamiltonian for large CMC volume. The fact that all local constraints are linear in momenta allows us to quantize these explicitly under a certain assumption on the kinematic Hilbert space, and the quantization problem for Shape Dynamics turns out to be equivalent to reduced phase space quantization. We consider the large CMC-volume asymptotics of conformal transformations of the wave function. We then discuss the similarity of Shape Dynamics on the 2-torus with the explicitly constructible strong gravity Shape Dynamics Hamiltonian in higher dimensions.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

Superfield formulation of the phase space path integral

We give a superfield formulation of the path integral on an arbitrary curved phase space, with or without first class constraints. Canonical tranformations and BRST transformations enter in a unified manner. The superpartners of the original phase space variables precisely conspire to produce the correct path integral measure, as Pfaffian ghosts. When extended to the case of second-class constraints, the correct path integral measure is again reproduced after integrating over the superpartners. These results suggest that the superfield formulation is of first-principle nature.     

Translational invariance of the Einstein–Cartan action in any dimension

We demonstrate that from the first order formulation of the Einstein– Cartan action it is possible to derive the basic differential identity that leads to translational invariance of the action in the tangent space. The transformations of fields is written explicitly for both the first and second order formulations and the group properties of transformations are studied. This, combined with the preliminary results from the Hamiltonian formulation (Kiriushcheva and Kuzmin in arXiv:0907.1553 [gr-qc]), allows us to conclude that without any modification, the Einstein–Cartan action in any dimension higher than two possesses not only rotational invariance but also a form of translational invariance in the tangent space. We argue that not only a complete Hamiltonian analysis can unambiguously give an answer to the question of what a gauge symmetry is, but also the pure Lagrangian methods allow us to find the same gauge symmetry from the basic differential identities.     

Emergent fermions and anyons in the kitaev model

We study the gapped phase of the Kitaev model on the honeycomb lattice using perturbative continuous unitary transformations. The effective low-energy Hamiltonian is found to be an extended toric code with interacting anyons. High-energy excitations are emerging free fermions which are composed of hard-core bosons with an attached string of spin operators. The excitation spectrum is mapped onto that of a single particle hopping on a square lattice in a magnetic field. We also illustrate how to compute correlation functions in this framework. The present approach yields analytical perturbative results in the thermodynamical limit without using the Majorana or the Jordan-Wigner fermionization initially proposed to solve this problem.     

BRST symmetry for Regge–Teitelboim-based minisuperspace models

The Einstein–Hilbert action in the context of higher derivative theories is considered for finding their BRST symmetries. Being a constraint system, the model is transformed in the minisuperspace language with the FRLW background and the gauge symmetries are explored. Exploiting the first order formalism developed by Banerjee et al. the diffeomorphism symmetry is extracted. From the general form of the gauge transformations of the field, the analogous BRST transformations are calculated. The effective Lagrangian is constructed by considering two gauge-fixing conditions. Further, the BRST (conserved) charge is computed, which plays an important role in defining the physical states from the total Hilbert space of states. The finite field-dependent BRST formulation is also studied in this context where the Jacobian for the functional measure is illustrated specifically.     

Diffeomorphism invariance in the Hamiltonian formulation of General Relativity

It is shown that when the Einstein-Hilbert Lagrangian is considered without any non-covariant modifications or change of variables, its Hamiltonian formulation leads to results consistent with principles of General Relativity. The first-class constraints of such a Hamiltonian formulation, with the metric tensor taken as a canonical variable, allow one to derive the generator of gauge transformations, which directly leads to diffeomorphism invariance. The given Hamiltonian formulation preserves general covariance of the transformations derivable from it. This characteristic should be used as the crucial consistency requirement that must be met by any Hamiltonian formulation of General Relativity.     

Effective Hamiltonian in the extended Krenciglowa-Kuo method

The extended Krenciglowa-Kuo (EKK) method allows us to calculate the effective Hamiltonian in a non-degenerate model space. We show that the EKK method can be implemented numerically in two iterative schemes, which are explained in detail with emphasis on convergence conditions. Using test calculations in a simple model, we clarify how and on what conditions we can calculate the effective Hamiltonian.     

The double of a Jacobian quasi-bialgebra

We show that the construction of the double of a Lie bialgebra can be extended to the case where a vector space is only equipped with the structure of a Jacobian quasi-bialgebra (also called a Lie quasi-bialgebra). In this case, the double is itself a Jacobian quasi-bialgebra and it is quasi-triangular. The more general case of the double of a proto-Lie bialgebra is also discussed. In the first section, the notions of exact, strictly exact, quasi-triangular and triangular Jacobian quasi-bialgebras are defined and their equivalence classes under twisting are studied.     

约束哈密顿系统在相空间中的精确不变量与绝热不变量

研究小干扰力作用下约束哈密顿系统对称性的摄动问题.建立了非保守约束哈密顿系统的正则方程,在增广相空间中研究了系统的对称性与精确不变量.基于力学系统的高阶绝热不变量的概念,给出了系统的各阶绝热不变量的形式及存在条件,并建立了绝热不变量与对称变换之间的对应关系 关键词： 约束哈密顿系统 对称性 摄动 不变量     

Null lifts and projective dynamics

We describe natural Hamiltonian systems using projective geometry. The null lift procedure endows the tangent bundle with a projective structure where the null Hamiltonian is identified with a projective conic and induces a Weyl geometry. Projective transformations generate a set of known and new dualities between Hamiltonian systems, as for example the phenomenon of coupling-constant metamorphosis. We conclude outlining how this construction can be extended to the quantum case for Eisenhart–Duval lifts.     

Canonical and D-transformations in theories with constraints

We describe a class of transformations in a super phase space (we call them D-transformations) which play the role of ordinary canonical transformations in theories with second-class constraints. Namely, in such theories they preserve the form invariance of equations of motion, their quantum analogs are unitary transformations, and the measure of integration in the corresponding Hamiltonian path integral is invariant under these transformations.     

Relativistic rigid particles: Classical tachyons and quantum anomalies

Causal rigid particles whose action includes anarbitrary dependence on the world-line extrinsic curvature are considered. General classes of solutions are constructed, includingcausal tachyonic ones. The Hamiltonian formulation is developed in detail except for one degenerate situation for which only partial results are given and requiring a separate analysis. However, for otherwise generic rigid particles, the precise specification of Hamiltonian gauge symmetries is obtained with in particular the identification of the Teichmüller and modular spaces for these systems. Finally, canonical quantisation of the generic case is performed paying special attention to the phase space restriction due to causal propagation. A mixed Lorenz-gravitational anomaly is found in the commutator of Lorentz boosts with world-line reparametrisations. The subspace of gauge invariant physical states is therefore not invariant under Lorentz transformations. Consequences for rigid strings and membranes are also discussed.