Review: Hamiltonian Linearization of the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge: A Radiation Gauge for Background-Independent Gravitational Waves in a Post-Minkowskian Einstein Spacetime

In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy , we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of non-harmonic 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) . We define a Hamiltonian linearization of the theory, i.e. gravitational waves, without introducing any background 4-metric, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in . We solve all the constraints of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's , which replace the two polarizations of the TT harmonic gauge, and that linearized Einstein's equations are satisfied. Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.     

The York map as a Shanmugadhasan canonical transformation in tetrad gravity and the role of non-inertial frames in the geometrical view of the gravitational field

A new parametrization of the 3-metric allows to find explicitly a York map by means of a partial Shanmugadhasan canonical transformation in canonical ADM tetrad gravity. This allows to identify the two pairs of physical tidal degrees of freedom (the Dirac observables of the gravitational field have to be built in term of them) and 14 gauge variables. These gauge quantities, whose role in describing generalized inertial effects is clarified, are all configurational except one, the York time, i.e. the trace of the extrinsic curvature of the instantaneous 3-spaces (corresponding to a clock synchronization convention) of a non-inertial frame centered on an arbitrary observer. In the Dirac Hamiltonian is the sum of the weak ADM energy (whose density is coordinate-dependent, containing the inertial potentials) and of the first-class constraints. The main results of the paper, deriving from a coherent use of constraint theory, are: (i) The explicit form of the Hamilton equations for the two tidal degrees of freedom of the gravitational field in an arbitrary gauge: a deterministic evolution can be defined only in a completely fixed gauge, i.e. in a non-inertial frame with its pattern of inertial forces. The simplest such gauge is the 3-orthogonal one, but other gauges are discussed and the Hamiltonian interpretation of the harmonic gauges is given. This frame-dependence derives from the geometrical view of the gravitational field and is lost when the theory is reduced to a linear spin 2 field on a background space-time. (ii) A general solution of the super-momentum constraints, which shows the existence of a generalized Gribov ambiguity associated to the 3-diffeomorphism gauge group. It influences: (a) the explicit form of the solution of the super-momentum constraint and then of the Dirac Hamiltonian; (b) the determination of the shift functions and then of the lapse one. (iii) The dependence of the Hamilton equations for the two pairs of dynamical gravitational degrees of freedom (the generalized tidal effects) and for the matter, written in a completely fixed 3-orthogonal Schwinger time gauge, upon the gauge variable , determining the convention of clock synchronization. The associated relativistic inertial effects, absent in Newtonian gravity and implying inertial forces changing from attractive to repulsive in regions with different sign of , are completely unexplored and may have astrophysical relevance in the interpretation of the dark side of the universe.     

Conservative quantities and their algebra in the self-dual gravity

A general covariant conservation law of energy-momentum in complex general relativity is obtained by way of general displacement transformation in terms of Ashtekar's new variables. The energy is exactly the adm Hamiltonian on the constraint surface on condition that an appropriate time function is chosen. The energy-momentum is gauge covariant and commutes with all the constraints whence they are physical observables. Furthermore, the Poisson brackets of the momentum and the internalSU(2) charges form a 3-Poincaré algebra.     

Equivalent Theories and Changing Hamiltonian Observables in General Relativity

Change and local spatial variation are missing in Hamiltonian general relativity according to the most common definition of observables as having 0 Poisson bracket with all first-class constraints. But other definitions of observables have been proposed. In pursuit of Hamiltonian–Lagrangian equivalence, Pons, Salisbury and Sundermeyer use the Anderson–Bergmann–Castellani gauge generator G, a tuned sum of first-class constraints. Kucha? waived the 0 Poisson bracket condition for the Hamiltonian constraint to achieve changing observables. A systematic combination of the two reforms might use the gauge generator but permit non-zero Lie derivative Poisson brackets for the external gauge symmetry of General Relativity. Fortunately one can test definitions of observables by calculation using two formulations of a theory, one without gauge freedom and one with gauge freedom. The formulations, being empirically equivalent, must have equivalent observables. For de Broglie-Proca non-gauge massive electromagnetism, all constraints are second-class, so everything is observable. Demanding equivalent observables from gauge Stueckelberg–Utiyama electromagnetism, one finds that the usual definition fails while the Pons–Salisbury–Sundermeyer definition with G succeeds. This definition does not readily yield change in GR, however. Should GR’s external gauge freedom of general relativity share with internal gauge symmetries the 0 Poisson bracket (invariance), or is covariance (a transformation rule) sufficient? A graviton mass breaks the gauge symmetry (general covariance), but it can be restored by parametrization with clock fields. By requiring equivalent observables, one can test whether observables should have 0 or the Lie derivative as the Poisson bracket with the gauge generator G. The latter definition is vindicated by calculation. While this conclusion has been reported previously, here the calculation is given in some detail.     

QCD on an Infinite Lattice

We construct a mathematically well–defined framework for the kinematics of Hamiltonian QCD on an infinite lattice in ${\mathbb{R}^3}$ , and it is done in a C*-algebraic context. This is based on the finite lattice model for Hamiltonian QCD developed by Kijowski, Rudolph e.a.. To extend this model to an infinite lattice, we need to take an infinite tensor product of nonunital C*-algebras, which is a nonstandard situation. We use a recent construction for such situations, developed by Grundling and Neeb. Once the field C*-algebra is constructed for the fermions and gauge bosons, we define local and global gauge transformations, and identify the Gauss law constraint. The full field algebra is the crossed product of the previous one with the local gauge transformations. The rest of the paper is concerned with enforcing the Gauss law constraint to obtain the C*-algebra of quantum observables. For this, we use the method of enforcing quantum constraints developed by Grundling and Hurst. In particular, the natural inductive limit structure of the field algebra is a central component of the analysis, and the constraint system defined by the Gauss law constraint is a system of local constraints in the sense of Grundling and Lledo. Using the techniques developed in that area, we solve the full constraint system by first solving the finite (local) systems and then combining the results appropriately. We do not consider dynamics.     

Three-Poincare Algebra in Complex General Relativity

We obtain a general covariant conservation law of energy momentum in complex general relativity by general displacement transformation in terms of Ashtekar new variables. The energy is exactly the ADM Hamiltonian on the constraint surface on condition that an appropriate time function is chosen. The energy momentum is gauge-covariant and commutes with all the constraints whence they are physical observables. Furthermore, the Poisson brackets of the momentum and the internal SU(2) charges form a three-Poincare algebra.     

Dirac Quantization of Noncommutative Abelian Proca field

Dirac formalism of Hamiltonian constraint systems is studied for the noncommutative Abelian Proca field. It is shown that the system of constraints are of second class in agreement with the fact that the Proca field is not gauge invariant. Then, the system of second class constraints is quantized by introducing Dirac brackets in the reduced phase space.     

Hidden Symmetries in the Two-Dimensional Isotropic Antiferromagnet

We discuss the two-dimensional isotropic antiferromagnet in the framework of gauge invariance. Gauge invariance is one of the most subtle useful concepts in theoretical physics, since it allows one to describe the time evolution of complex physical system in arbitrary sequences of reference frames. All theories of the fundamental interactions rely on gauge invariance. In Dirac’s approach, the two-dimensional isotropic antiferromagnet is subject to second-class constraints, which are independent of the Hamiltonian symmetries and can be used to eliminate certain canonical variables from the theory. We have used the symplectic embedding formalism developed by a few of us to make the system under study gauge invariant. After carrying out the embedding and Dirac analysis, we systematically show how second-class constraints can generate hidden symmetries. We obtain the invariant second-order Lagrangian and the gauge-invariant model Hamiltonian. Finally, for a particular choice of factor ordering, we derive the functional Schröodinger equations for the original Hamiltonian and for the first-class Hamiltonian and show them to be identical, which justifies our choice of factor ordering.     

Canonical decomposition of the general relativistic initial value problem

A canonical transformation is employed to implement a conformal transformation of the configuration variables of general relativity. The transformation is so chosen that the spatial constraints become algebraic in the trace of the momentum density. The temporal constraint is then found to have the form of York and O'Murchadha. The role played by the York coordinate condition in decoupling the constraint equations is examined, and a procedure to solve the constraint equations without employing such a coordinate condition is sketched.     

On Dirac's Conjecture

The extended canonical Noether identities and canonical first Noether theorem derived from an extended action in phase space for a system with a singular Lagrangian are formulated. Using these canonical Noether identities, it can be shown that the constraint multipliers connected with the first-class constraints may not be independent, so a query to a conjecture of Dirac is presented. Based on the symmetry properties of the constrained Hamiltonian system in phase space, a counterexample to a conjecture of Dirac is given to show that Dirac's conjecture fails in such a system. We present here a different way rather than Cawley's examples and other's ones in that there is no linearization of constraints in the problem. This example has a feature that neither the primary first-class constraints nor secondary first-class constraints are generators of the gauge transformation.     

Partial and complete observables for Hamiltonian constrained systems

We will pick up the concepts of partial and complete observables introduced by Rovelli in Conceptional Problems in Quantum Gravity, Birkhäuser, Boston (1991); Class Quant Grav, 8:1895 (1991); Phys Rev, D65:124013 (2002); Quantum Gravity, Cambridge University Press, Cambridge (2007) in order to construct Dirac observables in gauge systems. We will generalize these ideas to an arbitrary number of gauge degrees of freedom. Different methods to calculate such Dirac observables are developed. For background independent field theories we will show that partial and complete observables can be related to Kucha?’s Bubble-Time Formalism (J Math Phys, 13:768, 1972). Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions. Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group.     

Scalar Charged Particle in Weyl–Wigner–Moyal Phase Space. Constant Magnetic Field

A relativistic phase-space representation for a class of observables with matrix-valued Weyl symbols proportional to the identity matrix (charge-invariant observables) is proposed. We take into account the nontrivial charge structure of the position and momentum operators. The evolution equation coincides with its analog in relativistic quantum mechanics with nonlocal Hamiltonian under conditions where particle-pair creation does not take place (free particle and constant magnetic field). The differences in the equations are connected with the peculiarities of the constraints on the initial conditions. An effective increase in coherence between eigenstates of the Hamiltonian is found and possibilities of its experimental observation are discussed.     

Gauge invariance and the dirac equation

The gauge invariance of the Dirac equation is reviewed and gauge-invariant operators are defined. The Hamiltonian is shown to be gauge dependent, and an energy operator is defined which is gauge invariant. Gauge-invariant operators corresponding to observables are shown to satisfy generalized Ehrenfest theorems. The time rate of change of the expectation value of the energy operator is equal to the expectation value of the power operator. The virial theorem is proved for a relativistic electron in a time-varying electromagnetic field. The conventional approach to probability amplitudes, using the eigenstates of the unperturbed Hamiltonian, is shown in general to be gauge dependent. A gaugeinvariant procedure for probability amplitudes is given, in which eigenstates of the energy operator are used. The two methods are compared by applying them to an electron in a zero electromagnetic field in an arbitrary gauge. Presented at the Dirac Symposium, Loyola University, New Orleans, May 1981.     

Perturbative Solutions of the Extended Constraint Equations in General Relativity

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface in an asymptotically simple space-time satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on , and are equivalent to the usual constraint equations that satisfies as a space-like hypersurface in a space-time satisfying Einstein’s vacuum equation. This article develops a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the ‘classical’ method of Lichnerowicz and York that is used to solve the usual constraint equations.     

Local Well-Posedness for Membranes in the Light Cone Gauge

In this paper we consider the classical initial value problem for the bosonic membrane in light cone gauge. A Hamiltonian reduction gives a system with one constraint, the area preserving constraint. The Hamiltonian evolution equations corresponding to this system, however, fail to be hyperbolic. Making use of the area preserving constraint, an equivalent system of evolution equations is found, which is hyperbolic and has a well-posed initial value problem. We are thus able to solve the initial value problem for the Hamiltonian evolution equations by means of this equivalent system. We furthermore obtain a blowup criterion for the membrane evolution equations, and show, making use of the constraint, that one may achieve improved regularity estimates.     

Lagrangian and Hamiltonian formalisms for relativistic dynamics of a charged particle with dipole moment

The Lagrangian and Hamiltonian formulations for the relativistic classical dynamics of a charged particle with dipole moment in the presence of an electromagnetic field are given. The differential conservation laws for the energy-momentum and angular momentum tensors of a field and particle are discussed. The Poisson brackets for basic dynamic variables, which form a closed algebra, are found. These Poisson brackets enable us to perform the canonical quantization of the Hamiltonian equations that leads to the Dirac wave equation in the case of spin 1/2. It is also shown that the classical limit of the squared Dirac equation results in equations of motion for a charged particle with dipole moment obtained from the Lagrangian formulation. The inclusion of gravitational field and non-Abelian gauge fields into the proposed formalism is discussed.Received: 4 June 2005, Published online: 27 July 2005     

Constraints and actions in two and three dimensionalN=1 conformal supergravity

We investigate closure of the gauge algebra and constraints inN=1 conformal supergravity in 2 and 3 dimensions. In the 2 dimensional case, contrary to 3 or higher dimensions, some parts of the gauge fields are algebraically unsolvable in the constraint equations on group curvatures. It will be shown that these unsolvable parts are decoupled from the transformation law as well as from the kinetic multiplets. Hence they are absent in the invariant action for matter multiplets coupled to conformal supergravity which is relevant to the old superstrings. Explicit construction of the invariant actions are illustrated for the case of spinning strings and locally supersymmetric σ-models with the Wess-Zumino term.     

Hamilton's principle,covariant Hamiltonian,and canonical formalism in four and five dimensions

A discussion of the 1950s and 1960s on the existence of an explicit covariant canonical formalism is renewed. A new point of view is introduced where Hamilton's principle, based on the existence of a Hamiltonian, is postulated independently from the Lagrange formalism. The Hamiltonian is determined by transformation properties and dimensional considerations. The variation of the action without constraints leads to an explicit covariant canonical formalism and correct equations of motion. The introduction of the charge as a fifth momentum gives rise to a reformulation of classical relativistic point mechanics as a five-dimensionalU(1) gauge theory with a theoretically invisible extra dimension. A generalization to other gauge groups is given. The inversion of the proper time is introduced as a new particle-antiparticle symmetry that allows one to show that in the five-dimensional classical theory all particles have positive energy.     

Of Ghosts, Gauge Volumes, and Gauss's Law

The relationship between the canonical operator and the path integral formulation of quantum electrodynamics is analyzed with a particular focus on the implementation of gauge constraints in the two approaches. The removal of gauge volumes in the path integral is shown to match with the presence of zero-norm ghost states associated with gauge transformations in the canonical operator approach. The path integrals for QED in both the Feynman and the temporal gauges are examined and several ways of implementing the gauge constraint integrations are demonstrated. The upshot is to show that both the Feynman and the temporal gauge path integrals are equivalent to the Coulomb gauge path integral, matching the results developed by Kurt Haller using the canonical formalism. In addition, the Faddeev–Popov form for the Feynman gauge and temporal gauge Lagrangian path integrals are derived from the Hamiltonian form of the path integral.     

1+1维非线性σ模型的BRST变换的等价性和Ward恒等式

在依据Dirac约束规范理论和作推广后的条件下,导出了规范生成元,推导出了1+1维O(3)非线性σ模型的一般条件(β≠0)下的BRST变换,给出了其BRST变换与Dirac规范变换的等价关系,得到了鬼场的新的一般对易关系,且其一般参数β为零时就回到通常的鬼场的对易关系.并由规范生成元导出了BRST荷,进而完成了此模型的一种BRST量子化.还在此基础上进一步导出了此系统的Green函数生成泛函、连通Green函数生成泛函和正规顶角生成泛函,获得了3种不同的Ward恒等式