Hamiltonian formulation of f(Riemann) theories of gravity

Hamiltonian structure of the teleparallel formulation of GR

We apply Dirac's Hamiltonian approach to study the canonical structure of the teleparallel form of general relativity without matter fields. It is shown, without any gauge fixing, that the Hamiltonian has the generalized Dirac-ADM form, and constraints satisfy all the consistency requirements. The set of constraints involves some extra first class constraints, which are used to find additional gauge symmetries and clarify the gauge structure of the theory.

Hamiltonian and Lagrangian BRST quantization in Riemann Manifold

The BRST quantization of particle motion on the hypersurface $V_{(N-1)}$ embedded in Euclidean space $R_N$ is carried out both in Hamiltonian and Lagrangian formalism. Using Batalin-Fradkin-Fradkina-Tyutin (BFFT) formalism, the second class constrained obtained using Hamiltonian analysis are converted into first class constraints. Then using BFV analysis the BRST symmetry is constructed. We have given a simple example of these kind of system. In the end we have discussed Batalin-Vilkovisky formalism in the context of this (BFFT modified) system.

Hamiltonian analysis of the BFCG formulation of General Relativity

We perform the complete Hamiltonian analysis of the BFCG action for General Relativity. We determine all the constraints of the theory and classify them into the first-class and the second-class constraints. We also show how the canonical formulation of BFCG General Relativity reduces to the Einstein-Cartan and triad canonical formulations. The reduced phase space analysis also gives a 2-connection which is suitable for the construction of a spin-foam basis which will be a categorical generalization of the spin-network basis from Loop Quantum Gravity.

Solution of the noncanonicity puzzle in General Relativity: a new Hamiltonian formulation

We study the transformation leading from Arnowitt, Deser, Misner (ADM) Hamiltonian formulation of General Relativity (GR) to the $ΓΓ$ metric Hamiltonian formulation derived from the Lagrangian density which was firstly proposed by Einstein. We classify this transformation as gauged canonical - i.e. canonical modulo a gauge transformation. In such a study we introduce a new Hamiltonian formulation written in ADM variables which differs from the usual ADM formulation mainly in a boundary term firstly proposed by Dirac. Performing the canonical quantization procedure we introduce a new functional…

Various Hamiltonian formulations of f(R) gravity and their canonical relationships

Various Hamiltonian formulations of f(R) gravity can be found in the literature. Some authors follow the Ostrogradsky treatment of higher derivative theories and introduce as extra variables first order time derivatives of the metric (typically the extrinsic curvature). Some others take advantage of the conformal equivalence of f(R) theory with Einstein's gravity coupled to a scalar field and introduce as an extra variable the scalar curvature R itself, which includes second time derivatives of the metric. We show that, contrarily to some claims, these formulations are related by canonical tra…

Hamiltonian and Lagrangian BRST quantization in Riemann Manifold II

We have previously developed the BRST quantization on the hypersurface $V_{N-1}$ embedded in N dimensional Euclidean space $R_N$ in both Hamiltonian and Lagrangian formulation. We generalize the formalism in the case of L dimensional manifold $V_L$ embedded in $R_N$ with $1\leq L < N$. The result is essentially the same as the previous one. We have also verified the results obtained here using a simple example of particle motion on a torus knot.

From Lagrangian to Hamiltonian formulations of the Palatini action

We work on the Lagrangian and the Hamiltonian formulations of the Palatini action. In the Lagrangian formulation, we find that we need to assume the metric compatibility and the torsion zero or to assume the tetrad compatibility to describe General Relativity. In the Hamiltonian formulation, we obtain the Einstein's equations only with assuming the tetrad compatibility. The Hamiltonian from assuming the metric compatibility and the torsion zero should be used to quantize General Relativity.

A Krichever--Novikov Formulation of W--Algebras on Riemann Surfaces

It is shown how the theory of classical $W$--algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. An intriguing relation between the theory of $A_1$ embeddings into simple Lie algebras and the holomorphic geometry of Riemann surfaces is exihibited.

New Developments in the Hamiltonian Formulation of the Gravitational Standard-Model Extension

This chapter of the proceedings for the Ninth Meeting on CPT and Lorentz Symmetry is dedicated to the Hamiltonian formulation of the minimal gravitational Standard-Model Extension. Some theoretical questions associated with the latter shall be reviewed. First, we recall the properties of the Hamiltonian, which was computed elsewhere, and discuss how it is linked to the modified Einstein equations. Second, we describe how the covariant and Hamiltonian formulations are shown to be consistent with each other.

Hamiltonian Formulation of Two Body Problem in Wheeler-Feynman electrodynamics

A Hamiltonian formulation for the classical problem of electromagnetic interaction of two charged relativistic particles is found.

Linking the ADM formulation to other Hamiltonian formulations of general relativity

We obtain the Arnowitt-Deser-Misner formulation of general relativity in $n$ dimensions ($n \geq 3$) from its either $SO(n-1,1)$ [$SO(n)$] or $SO(n-1)$ Palatini Hamiltonian formulations and vice versa [we recall that $SO(n-1,1)$ [$SO(n)$] requires no gauge fixing whereas $SO(n-1)$ involves the time gauge]. Similarly, the Hamiltonian formulation of general relativity in terms of Ashtekar-Barbero variables can also be directly obtained from the Arnowitt-Deser-Misner Hamiltonian formulation and vice versa, which is an alternative approach to the way followed by Barbero. We give the relevant maps …

Comment on ``Anyon in an External Eletromagnetic Field: Hamiltonian and Lagrangian Formulations''

We comment on a recent paper by Chaichian et al. (Phys.Rev.Lett. 71(1993)3405).

Lagrangian and Hamiltonian formulations of the Palatini action

We derive the Ashtekar-Barbero variables in Loop Quantum Gravity(LQG) starting from the Palatini action. We find that we need either the torsion-free condition or the compatibility condition for the Palatini action to describe General Relativity.

ADM formulation and Hamiltonian analysis of Coincident General Relativity

We consider a simpler geometrical formulation of General Relativity based on non-metricity, known as Coincident General Relativity. We study the ADM formulation of the theory and perform a detailed Hamiltonian analysis. We explicitly show the propagation of two physical degrees of freedom, as it should, even though the role of boundary terms and gauge conditions is significantly altered. This might represent an alternative promising new route for numerical relativity and canonical quantum gravity. We also give an outlook on the number of propagating degrees of freedom in non-linear extension o…

Hamiltonian Formalism of Bimetric Gravity In Vierbein Formulation

This paper is devoted to the Hamiltonian analysis of bimetric gravity in vierbein formulation. We identify all constraints and determine their nature. We also show an existence of additional constraint so that the scalar mode can be eliminated.

Hamiltonian formulation of nonAbelian noncommutative gauge theories

We implement the Hamiltonian treatment of a nonAbelian noncommutative gauge theory, considering with some detail the algebraic structure of the noncommutative symmetry group. The first class constraints and Hamiltonian are obtained and their algebra derived, as well as the form of the gauge invariance they impose on the first order action.

Finite Nilpotent BRST transformations in Hamiltonian formulation

We consider the finite field dependent BRST (FFBRST) transformations in the context of Hamiltonian formulation using Batalin-Fradkin-Vilkovisky method. The non-trivial Jacobian of such transformations is calculated in extended phase space. The contribution from Jacobian can be written as exponential of some local functional of fields which can be added to the effective Hamiltonian of the system. Thus, FFBRST in Hamiltonian formulation with extended phase space also connects different effective theories. We establish this result with the help of two explicit examples. We also show that the FFBR…

Hamiltonian analysis of fermions coupled to the Holst action

We report three manifestly Lorentz-invariant Hamiltonian formulations of minimally and nonminimally coupled fermion fields to the Holst action. These formulations are achieved by making a suitable parametrization of both the tetrad and the Lorentz connection, which allows us to integrate out some auxiliary fields without spoiling the local Lorentz symmetry. They have the peculiarity that their noncanonical symplectic structures as well as the phase-space variables for the gravitational sector are real. Moreover, two of these Hamiltonian formulations involve half-densitized fermion fields. We a…

Gauge-invariant Hamiltonian formulation of lattice Yang-Mills theory and the Heisenberg double

It it known that to get the usual Hamiltonian formulation of lattice Yang-Mills theory in the temporal gauge $A_{0}=0$ one should place on every link the cotangent bundle of a Lie group. The cotangent bundle may be considered as a limiting case of a so called Heisenberg double of a Lie group which is one of the basic objects in the theory of Lie-Poisson and quantum groups. It is shown in the paper that there is a generalization of the usual Hamiltonian formulation to the case of the Heisenberg double.

Riemann surfaces, separation of variables and classical and quantum integrability

We show that Riemann surfaces, and separated variables immediately provide classical Poisson commuting Hamiltonians. We show that Baxter's equations for separated variables immediately provide quantum commuting Hamiltonians. The construction is simple, general, and does not rely on the Yang--Baxter equation.

Hamiltonian formulation of Noncommutative D3--Brane

Lagrangians of the Abelian Gauge Theory and its dual are related in terms of a shifted action. We show that in d=4 constrained Hamiltonian formulation of the shifted action yields Hamiltonian description of the dual theory, without referring to its Lagrangian. We apply this method, at the first order in the noncommutativity parameter theta, to the noncommutative U(1) gauge theory possessing spatial noncommutativity. Its dual theory is effectively a space--time noncommutative U(1) gauge theory. However, we obtain a Hamiltonian formulation where time is commuting. Space-time noncommutative D3--b…

Hamiltonian formulation of the D-brane action and the light-cone Hamiltonian

We present the Hamiltonian formulation of the bosonic Dirichlet p-brane action. We rewrite the recently proposed quadratic D-brane action in terms of generalized shift vector and lapse function. The first class and the second class constraints are explicitly separated for the bosonic case. We then impose the gauge conditions in such a way that only time-independent gauge transformations are left. In this gauge we obtain the light-cone Hamiltonian which is quadratic in the field momenta of scalar and vector fields. The constraints are explicitly solved to eliminate part of the canonical variabl…

Stable interactions between the extended Chern-Simons theory and a charged scalar field with higher derivatives: Hamiltonian formalism

We consider constrained multi-Hamiltonian formulation for the extended Chern-Simons theory with higher derivatives of arbitrary finite order. The order $n$ extension of the theory admits $(n-1)$-parametric series of conserved tensors. The $00$-component of any representative of the series can be chosen as Hamiltonian. The theory admits a series of Hamiltonian formulations, including the canonical Ostrogradski formulation. The Hamiltonian formulations with different Hamiltonians are not connected by canonical transformations. Also, we demonstrate the inclusion of stable interactions with charge…

弦论哈密顿形式中的对称性

Symmetries in the Hamiltonian formulation of string theory

本文探讨了弦论哈密顿形式中对称性的产生机制，指出差分同构和Weyl对称性可通过Castellani程序自然出现，而非依赖场依赖的重新参数化，从而保持哈密顿与拉格朗日描述的一致性。

On Hamiltonian formulation of the Einstein-Hilbert action in two dimensions

It is shown that the well-known triviality of the Einstein field equations in two dimensions is not a sufficient condition for the Einstein-Hilbert action to be a total divergence, if the general covariance is to be preserved, that is, a coordinate system is not fixed. Consequently, a Hamiltonian formulation is possible without any modification of the two dimensional Einstein-Hilbert action. We find the resulting constraints and the corresponding gauge transfromations of the metric tensor.

New approach to Hamiltonian formulation of Hall magnetohydrodynamics

We present a new Hamiltonian formulation of barotropic Hall magnetohydrodynamics in two complementary approaches based on Dirac's constraint analysis. In one case the Hamiltonian is canonical involving physical variables only but the brackets are noncanonical, while in the other case the phase space is enlarged to retain the canonical structure of brackets.

Hamiltonian Algebroids and deformations of complex structures on Riemann curves

Starting with a Lie algebroid ${\cal A}$ over a space $M$ we lift its action to the canonical transformations on the affine bundle ${\cal R}$ over the cotangent bundle $T^*M$. Such lifts are classified by the first cohomology $H^1({\cal A})$. The resulting object is a Hamiltonian algebroid ${\cal A}^H$ over ${\cal R}$ with the anchor map from $\G({\cal A}^H)$ to Hamiltonians of canonical transformations. Hamiltonian algebroids generalize Lie algebras of canonical transformations. We prove that the BRST operator for ${\cal A}^H$ is cubic in the ghost fields as in the Lie algebra case. The Poiss…

A Hamiltonian Formulation of the Pais-Uhlenbeck Oscillator that Yields a Stable and Unitary Quantum System

We offer a new Hamiltonian formulation of the classical Pais-Uhlenbeck Oscillator and consider its canonical quantization. We show that for the non-degenerate case where the frequencies differ, the quantum Hamiltonian operator is a Hermitian operator with a positive spectrum, i.e., the quantum system is both stable and unitary. A consistent description of the degenerate case based on a Hamiltonian that is quadratic in momenta requires its analytic continuation into a complex Hamiltonian system possessing a generalized PT-symmetry (an involutive antilinear symmetry). We devise a real descriptio…

Hidden Nambu mechanics - A variant formulation of Hamiltonian systems -

We propose a variant formulation of Hamiltonian systems by the use of variables including redundant degrees of freedom. We show that Hamiltonian systems can be described by extended dynamics whose master equation is the Nambu equation or its generalization. Partition functions associated with the extended dynamics in many degrees of freedom systems are given. Our formulation can also be applied to Hamiltonian systems with first class constraints.

An alternative Hamiltonian formulation for the Pais-Uhlenbeck oscillator

Ostrogradsky's method allows one to construct Hamiltonian formulation for a higher derivative system. An application of this approach to the Pais-Uhlenbeck oscillator yields the Hamiltonian which is unbounded from below. This leads to the ghost problem in quantum theory. In order to avoid this nasty feature, the technique previously developed in [Acta Phys. Polon. B 36 (2005) 2115] is used to construct an alternative Hamiltonian formulation for the multidimensional Pais-Uhlenbeck oscillator of arbitrary even order with distinct frequencies of oscillation. This construction is also generalized …

Equivalence of the modified Villain formulation and the dual Hamiltonian method in the duality of the XY-plaquette model

Regarding the duality of the XY-plaquette model in the $2+1d$ system, We compared and discussed the equivalence and difference between the modified Villain formulation, recently introduced by Gorantla et al. to study duality in exotic field theories such as fracton theory, and the standard Villain formulation and dual Hamiltonian method we introduced in our previous paper to study duality between Josephson junctions and quantum phase slips in the $1+1d$ system.

The Hamiltonian Formulation of Higher Order Dynamical Systems

Using Dirac's approach to constrained dynamics, the Hamiltonian formulation of regular higher order Lagrangians is developed. The conventional description of such systems due to Ostrogradsky is recovered. However, unlike the latter, the present analysis yields in a transparent manner the local structure of the associated phase space and its local sympletic geometry, and is of direct application to {\em constrained\/} higher order Lagrangian systems which are beyond the scope of Ostrogradsky's approach.

Hamiltonian formulation of the effective kinetic theory for superfluid Fermi liquids

We present in a local form the time dependent effective description of a superfluid Fermi liquid which includes Landau damping effects at $T\neq 0$. This is achieved by the introduction of an additional variable, the quasiparticle distribution function, which obeys a simple kinetic equation. The transport equation is coupled with first order equations for the Goldstone mode and the particle density. We prove that a main feature of this formulation is its Hamiltonian structure relative to a certain Poisson bracket. We construct the Hamiltonian to quadratic order.

On the Duffin-Kemmer-Petiau Formulation of the Covariant Hamiltonian Dynamics in Field Theory

We show that the De Donder-Weyl (DW) covariant Hamiltonian field equations of any field can be written in Duffin-Kemmer-Petiau (DKP) matrix form. As a consequence, the (modified) DKP beta-matrices (5 X 5 in four space-time dimensions) are of universal significance for all fields admitting the DW Hamiltonian formulation, not only for a scalar field, and can be viewed as field theoretic analogues of the symplectic matrix, leading to the so-called ``k-symplectic'' (k=4) structure. We also briefly discuss what could be viewed as the covariant Poisson bracket given by beta-matrices and the correspo…

变分问题与拉格朗日-达朗贝尔方程的哈密顿表述

Variational problem and Hamiltonian formulation of the Lagrange-d'Alembert equations with nonlinear nonholonomic constraints

本文提出通过特殊变分问题获得常微分方程组，并推导出非完整非线性约束下拉格朗日-达朗贝尔方程的哈密顿表述。

A Riemann-Hilbert formulation for the finite temperature Hubbard model

Inspired by recent results in the context of AdS/CFT integrability, we reconsider the Thermodynamic Bethe Ansatz equations describing the 1D fermionic Hubbard model at finite temperature. We prove that the infinite set of TBA equations are equivalent to a simple nonlinear Riemann-Hilbert problem for a finite number of unknown functions. The latter can be transformed into a set of three coupled nonlinear integral equations defined over a finite support, which can be easily solved numerically. We discuss the emergence of an exact Bethe Ansatz and the link between the TBA approach and the results…