First Class Constraints Research Articles

On representations of Higher Spin symmetry algebras for mixed-symmetry HS fields on AdS-spaces. Lagrangian formulation

We derive non-linear commutator HS symmetry algebra, which encode unitary irreducible representations of AdS group subject to Young tableaux Y(s1,…, sk) with κ ≥ 2 rows on d-dimensional anti-de-Sitter space. Auxiliary representations for specially deformed non-linear HS symmetry algebra in terms of generalized Verma module in order to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints are found explicitly for the case of HS fields for κ = 2 Young tableaux. The oscillator realization over Heisenberg algebra for obtained Verma module is constructed. The results generalize the method of auxiliary representations construction for symplectic sp(2κ) algebra used for mixed-symmetry HS fields on a flat spaces and can be extended on a case of arbitrary HS fields in AdS-space. Gauge-invariant unconstrained reducible Lagrangian formulation for free bosonic HS fields with generalized spin (s1, s2) is derived.

Remarks on the “Non-canonicity Puzzle”: Lagrangian Symmetries of the Einstein-Hilbert Action

Given the non-canonical relationship between variables used in the Hamiltonian formulations of the Einstein-Hilbert action (due to Pirani, Schild, Skinner (PSS) and Dirac) and the Arnowitt-Deser-Misner (ADM) action, and the consequent difference in the gauge transformations generated by the first-class constraints of these two formulations, the assumption that the Lagrangians from which they were derived are equivalent leads to an apparent contradiction that has been called “the non-canonicity puzzle”. In this work we shall investigate the group properties of two symmetries derived for the Einstein-Hilbert action: diffeomorphism, which follows from the PSS and Dirac formulations, and the one that arises from the ADM formulation. We demonstrate that unlike the diffeomorphism transformations, the ADM transformations (as well as others, which can be constructed for the Einstein-Hilbert Lagrangian using Noether’s identities) do not form a group. This makes diffeomorphism transformations unique (the term “canonical” symmetry might be suggested). If the two Lagrangians are to be called equivalent, canonical symmetry must be preserved. The interplay between general covariance and the canonicity of the variables used is discussed.

Architectural implications of spatial thermal filtering

Process technology scaling, lagging supply voltage scaling, and the resulting exponential increase in power density, have made temperature a first-class design constraint in today's microprocessors. Prior work has shown that the silicon substrate acts as a spatial low-pass filter for temperature. This phenomenon, spatial thermal filtering, has clear implications for thermal management: depending on the size of dissipators, either design-time strategies, such as dividing and distributing functionality spatially, or runtime strategies, such as isolating functionality temporally (duty cycling), may be the most effective way to control peak temperature. To assist designers with such trade-offs, we have performed extensive analysis and simulation to evaluate the extent and effect of spatial filtering on thermal management in a number of microarchitecture design scenarios.We begin our exploration of spatial filtering with an analytical study of the heat conduction problem, followed by a series of studies to validate the effect and extent of spatial filtering under realistic system assumptions. In particular, we investigate the effect of power dissipator size, location, and aspect ratio in the context of high-performance computing. We then extend these experiments with two microarchitectural studies. First, we perform a study of spatial filtering in many-core architectures. Our results show that as cores shrink, the granularity of effective thermal management increases to the point that even turning cores on and off has a limited effect on peak temperature. Second, we investigate spatial filtering in caches. We discover that despite the size and aspect ratio of cache lines, pathological code behavior can heat caches to undesirable levels, accelerating wear-out.

Consistency of the Hamiltonian formulation of the lowest-order effective action of the complete Hořava theory

We perform the Hamiltonian analysis for the lowest-order effective action, up to second order in derivatives, of the complete Ho\ifmmode \check{r}\else \v{r}\fi{}ava theory. The model includes the invariant terms that depend on ${\ensuremath{\partial}}_{i}\mathrm{ln}N$ proposed by Blas, Pujol\`as, and Sibiryakov. We show that the algebra of constraints closes. The Hamiltonian constraint is of second-class behavior and it can be regarded as an elliptic partial differential equation for $N$. The linearized version of this equation is a Poisson equation for $N$ that can be solved consistently. The preservation in time of the Hamiltonian constraint yields an equation that can be consistently solved for a Lagrange multiplier of the theory. The model has six propagating degrees of freedom in the phase space, corresponding to three even physical modes. When compared with the $\ensuremath{\lambda}R$ model studied by us in a previous paper, it lacks two second-class constraints, which leads to the extra even mode.

IMPROVED CANONICAL QUANTIZATION METHOD OF SPINOR ELECTRODYNAMICS WITH SINGULAR LAGRANGIAN

This paper analyzes the general interaction of spinor field and electromagnetic field, i.e. spinor electrodynamics with singular Lagrangian, and the improved canonical quantization method of spinor electrodynamics is given in order to overcome the problems of the second-class constraint linear combination and the first-class constraint number maximization brought by the traditional canonical quantization method. In the improved canonical quantization method, there are no problems of artificial linear combination and first-class constraint number maximization, at the same time, the stability of the system is considered. The improved canonical quantization method is more natural than the traditional way and we show that the results of the canonical quantization of spinor electrodynamics are equal to the results of the traditional way. For a general dynamical system with singular Lagrangian, one can extendingly use the improved canonical quantization method to quantize the system.

Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum, and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum ${p}_{\ensuremath{\theta}}$ breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as ${p}_{i}$ $(i=1,2,3)$ are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between ${p}_{i}$ generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.

The canonical structure of Podolsky's generalized electrodynamics on the null-plane

In this work, we analyze the canonical structure of Podolsky's generalized electrodynamics on the null-plane. We show that the constraint structure presents a set of second-class constraints, which are exclusive of the analysis on the null-plane, and an expected set of first-class ones. An inspection on the field equations leads to the generalized radiation gauge on the null-plane. Dirac Brackets are then introduced by considering the problem of uniqueness under the chosen initial-boundary conditions of the theory.

The kinematical Hilbert space of loop quantum gravity from BF theories

In this work, it is demonstrated how the kinematical Hilbert space of loop quantum gravity (LQG) can be inferred from the configuration space of BF theories via the imposition of the Hamiltonian constraints. In particular, it is outlined how the projection to the representations associated with Ashtekar–Barbero connections provides the correct procedure to implement second-class constraints and the corresponding nontrivial induced symplectic structure. Then, the reduction to SU(2) invariant intertwiners is analyzed and the properties of LQG states under Lorentz transformations are discussed.

A new look at Lorentz-covariant loop quantum gravity

In this work, we study the classical and quantum properties of the unique commutative Lorentz-covariant connection for loop quantum gravity. This connection has been found after solving the second-class constraints inherited from the canonical analysis of the Holst action without the time-gauge. We show that it has the property of lying in the conjugacy class of a pure $\su(2)$ connection, a result which enables one to construct the kinematical Hilbert space of the Lorentz-covariant theory in terms of the usual $\SU(2)$ spin-network states. Furthermore, we show that there is a unique Lorentz-covariant electric field, up to trivial and natural equivalence relations. The Lorentz-covariant electric field transforms under the adjoint action of the Lorentz group, and the associated Casimir operators are shown to be proportional to the area density. This gives a very interesting algebraic interpretation of the area. Finally, we show that the action of the surface operator on the Lorentz-covariant holonomies reproduces exactly the usual discrete $\SU(2)$ spectrum of time-gauge loop quantum gravity. In other words, the use of the time-gauge does not introduce anomalies in the quantum theory.

Generalization of the extended Lagrangian formalism on a field theory and applications

Formalism of extended Lagrangian represent a systematic procedure to look for the local symmetries of a given Lagrangian action. In this work, the formalism is discussed and applied to a field theory. We describe it in detail for a field theory with first-class constraints present in the Hamiltonian formulation. The method is illustrated on examples of electrodynamics, Yang-Mills field and non-linear sigma model.

Remarks on gauge invariance and first-class constraints

Gauge symmetries lead to first-class constraints. This assertion is, of course, true only for non-trivial gauge symmetries, i.e., gauge symmetries that act non-trivially on-shell on the dynamical variables. We illustrate this well-appreciated fact for time reparametrization invariance in the context of modifications of gravity-suggested in a recent proposal by Hořava-in which the Hamiltonian constraint is deformed by arbitrary spatial diffeomorphism invariant terms, where some subtleties are found to arise.

Spinning particle approach to higher spin field theory

We shortly review on the connection between higher-spin gauge field theories and supersymmetric spinning particle models. In such approach the higher spin equations of motion are linked to the first-class constraint algebra associated with the quantization of particle models. Here we consider a class of spinning particle models characterized by local O(N)-extended supersymmetry since these models are known to provide an alternative approach to the geometric formulation of higher spin field theory. We describe the canonical quantization of the models in curved target space and discuss the obstructions that appear in presence of an arbitrarily curved background. We then point out the special role that conformally flat spaces appear to have in such models and present a derivation of the higher-spin curvatures for maximally symmetric spaces.

Flikker

Energy has become a first-class design constraint in computer systems. Memory is a significant contributor to total system power. This paper introduces Flikker, an application-level technique to reduce refresh power in DRAM memories. Flikker enables developers to specify critical and non-critical data in programs and the runtime system allocates this data in separate parts of memory. The portion of memory containing critical data is refreshed at the regular refresh-rate, while the portion containing non-critical data is refreshed at substantially lower rates. This partitioning saves energy at the cost of a modest increase in data corruption in the non-critical data. Flikker thus exposes and leverages an interesting trade-off between energy consumption and hardware correctness. We show that many applications are naturally tolerant to errors in the non-critical data, and in the vast majority of cases, the errors have little or no impact on the application's final outcome. We also find that Flikker can save between 20-25% of the power consumed by the memory sub-system in a mobile device, with negligible impact on application performance. Flikker is implemented almost entirely in software, and requires only modest changes to the hardware.

Flikker

Energy has become a first-class design constraint in computer systems. Memory is a significant contributor to total system power. This paper introduces Flikker, an application-level technique to reduce refresh power in DRAM memories. Flikker enables developers to specify critical and non-critical data in programs and the runtime system allocates this data in separate parts of memory. The portion of memory containing critical data is refreshed at the regular refresh-rate, while the portion containing non-critical data is refreshed at substantially lower rates. This partitioning saves energy at the cost of a modest increase in data corruption in the non-critical data. Flikker thus exposes and leverages an interesting trade-off between energy consumption and hardware correctness. We show that many applications are naturally tolerant to errors in the non-critical data, and in the vast majority of cases, the errors have little or no impact on the application's final outcome. We also find that Flikker can save between 20-25% of the power consumed by the memory sub-system in a mobile device, with negligible impact on application performance. Flikker is implemented almost entirely in software, and requires only modest changes to the hardware.

Quantum charged rigid membrane

The early Dirac proposal to model the electron as a charged membrane is reviewed. A rigidity term, instead of the natural membrane tension, involving linearly the extrinsic curvature of the worldvolume swept out by the membrane is considered in the action modeling the bubble in the presence of an electromagnetic field. We set up this model as a genuine second-order derivative theory by considering a non-trivial boundary term which plays a relevant part in our formulation. The Lagrangian in question is linear in the bubble acceleration and by means of the Ostrogradski–Hamiltonian approach, we observed that the theory comprises the management of both first- and second-class constraints. We thus show that our second-order approach is robust allowing for a proper quantization. We found an effective quantum potential which permits us to compute bounded states for the system. We comment on the possibility of describing brane world universes by invoking this kind of second-order correction terms.

Hamiltonian analysis of nonrelativistic covariant restricted-foliation-preserving diffeomorphism invariant Hořava-Lifshitz gravity

We perform the Hamiltonian analysis of nonrelativistic covariant Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity in the formulation presented recently in arXiv:1009.4885. We argue that the resulting Hamiltonian structure is in agreement with the original construction of nonrelativistic covariant Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity presented in arXiv:1007.2410. Then we extend this construction to the case of restricted-foliation-preserving diffeomorphism invariant Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz theory. We find a well behaved Hamiltonian system with the number of the first- and the second-class constraints that ensure the correct number of physical degrees of freedom of gravity.

Gauge invariance and quantization of Yang-Mills theories in extra dimensions

The gauge structure of the four-dimensional effective theory arising from a pure $S{U}_{5}(N)$ Yang-Mills theory in five dimensions compactified on the orbifold ${S}^{1}/{Z}_{2}$ is reexamined on the basis of Becchi-Rouet-Stora-Tyutin symmetry. In this context, the two scenarios that can arise are analyzed: if the gauge parameters propagate in the bulk, the excited Kaluza-Klein (KK) modes are gauge fields, but they are matter vector fields if these parameters are confined in the 3-brane. In the former case, it is shown that the four-dimensional theory is gauge invariant only if the compactification is carried out by using curvatures instead of gauge fields as fundamental objects. Then, it is shown that the four-dimensional theory is governed by two types of gauge transformations, one determined by the KK zero modes of the gauge parameters, ${\ensuremath{\alpha}}^{(0)a}$, and another by the excited KK modes, ${\ensuremath{\alpha}}^{(n)a}$. The Dirac method and the proper solution of the master equation in the context of the field-antifield formalism are employed to show that the theory is subject to first-class constraints. A gauge-fixing procedure to quantize the KK modes ${A}_{\ensuremath{\mu}}^{(n)a}$ that is covariant under the first type of gauge transformations, which embody the standard gauge transformations of $S{U}_{4}(N)$, is introduced through gauge-fixing functions transforming in the adjoint representation of this group. The ghost sector induced by these gauge-fixing functions is derived on the basis of the Becchi-Rouet-Stora-Tyutin formalism. The effective quantum Lagrangian that links the interactions between light physics (zero modes) and heavy physics (excited KK gauge modes) is presented. Concerning the radiative corrections of the excited KK modes on the light Green's functions, the predictive character of this Lagrangian at the one-loop level is stressed. In the case of the gauge parameters confined to the 3-brane, the known result in the literature is reproduced with some minor variants, although it is emphasized that the exited KK modes are not gauge fields but matter fields that transform under the adjoint representation of $S{U}_{4}(N)$. The Dirac method is employed to show that this theory is subject to both first- and second-class constraints, which arise from the zero and excited KK modes, respectively.

Non-commutative Geometry, Quantum Field Theory and Second-class Constraints

Non-commutative Geometry, Quantum Field Theory and Second-class Constraints

On the relation between operator constraint, master constraint, reduced phase space and path integral quantization

Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space formulation. In this paper we review this rather involved procedure in full generality. Moreover, we demonstrate that the reduced phase space path integral formulation formally agrees with the Dirac's operator constraint quantization and, more specifically, with the master constraint quantization for first-class constraints. For first-class constraints with nontrivial structure functions the equivalence can only be established by passing to Abelian(ized) constraints which is always possible locally in phase space. Generically, the correct configuration space path integral measure deviates from the exponential of the Lagrangian action. The corrections are especially severe if the theory suffers from second-class secondary constraints. In a companion paper we compute these corrections for the Holst and Plebanski formulations of GR on which current spin foam models are based.

On the relation between rigging inner product and master constraint direct integral decomposition

Canonical quantization of constrained systems with first-class constraints via Dirac’s operator constraint method proceeds by the theory of Rigged Hilbert spaces, sometimes also called refined algebraic quantization. This method can work when the constraints form a Lie algebra. When the constraints only close with nontrivial structure functions, the Rigging map can no longer be defined. To overcome this obstacle, the master constraint method has been proposed which replaces the individual constraints by a weighted sum of absolute squares of the constraints. Now the direct integral decomposition (DID) methods, which are closely related to Rigged Hilbert spaces, become available and have been successfully tested in various situations. It is relatively straightforward to relate the rigging inner product to the path integral that one obtains via reduced phase space methods. However, for the master constraint, this is not at all obvious. In this paper we find sufficient conditions under which such a relation can be established. Key to our analysis is the possibility to pass to equivalent, Abelian constraints, at least locally in phase space. Then the master constraint DID for those Abelian constraints can be directly related to the rigging map and therefore has a path integral formulation.