A note on the Hamiltonian theory of quantization. II

1947 ◽  
Vol 43 (2) ◽  
pp. 196-204 ◽  
Author(s):  
T. S. Chang
Keyword(s):  
Variational Principles ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Field Equations ◽  
Gauge Invariant ◽  
Commutation Relations ◽  
Hamiltonian Theory ◽  
Higher Derivatives ◽  
Missing Momenta ◽  
Derivatives Of

It is pointed out that the equations of motion for any field obtained by varying a Lagrangian subject to auxiliary conditions are exactly equivalent to a certain set of canonical equations and that the commutation relations between the dynamical variables for the latter equations are Lorentz-invariant. By extending the theory to Lagrangians containing higher derivatives of the field quantities, it is shown that any given set of field equations can be put into the canonical form, though it is not derived from variational principles. The question of Lagrangians with missing momenta is also considered. It is shown that if the Lagrangian is ‘gauge-invariant’, some of the p's must be missing and the corresponding Eulerian equations can be replaced by equations containing no q and then can be replaced by initial conditions. The commutation relations between gauge-invariant quantities are Lorentz-invariant. For Lagrangians which are not gauge-invariant but are such as to have missing momenta, the passage to quantum theory will in general give rise to non-Lorentz-invariant commutation relations. In both cases, the equations of motion can be cast in canonical forms.

Field theories with high derivatives

1948 ◽  
Vol 44 (1) ◽  
pp. 76-86 ◽  
Author(s):  
T. S. Chang
Keyword(s):  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Functional Derivative ◽  
Momentum Tensor ◽  
Field Theories ◽  
Field Equations ◽  
Gauge Invariant ◽  
Image Position ◽  
Derivatives Of

The relativistic field theories of elementary particles are extended to cases where the field equations are derived from Lagrangians containing all derivatives of the field quantities. Expressions for the current, the energy-momentum tensor, the angular-momentum tensor, and the symmetrized energy-momentum tensor are given. When the field interacts with an electromagnetic field, we introduce a subtraction procedure, by which all the above expressions are made gauge-invariant. The Hamiltonian formulation of the equations of motion in a gauge-invariant form is also given.After considering the Lagrangian L as a scalar in a general relativity transformation and thus a function of gμν and their derivatives, the functional derivative ofwith respect to gμν (x) at a point where the space time is flat is worked out. It is shown that this differs from the symmetrized energy-momentum tensor given in the above sections by a term which vanishes when certain operators Sij are antisymmetrical or when the Lagrangian contains the first derivatives of the field quantities only and whose divergence to either μ or ν vanishes.

scholarly journals Fab Four: When John and George Play Gravitation and Cosmology

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
J.-P. Bruneton ◽  
M. Rinaldi ◽  
A. Kanfon ◽  
A. Hees ◽  
S. Schlögel ◽  
...  
Keyword(s):  
Scalar Field ◽  
Solar System ◽  
Parameter Space ◽  
Ad Hoc ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Inflationary Model ◽  
Einstein Tensor ◽  
Chameleon Mechanism ◽  
Derivatives Of

Scalar-tensor theories of gravitation attract again a great interest since the discovery of the Chameleon mechanism and of the Galileon models. The former allows reconciling the presence of a scalar field with the constraints from Solar System experiments. The latter leads to inflationary models that do not need ad hoc potentials. Further generalizations lead to a tensor-scalar theory, dubbed the “Fab Four,” with only first and second order derivatives of the fields in the equations of motion that self-tune to a vanishing cosmological constant. This model needs to be confronted with experimental data in order to constrain its large parameter space. We present some results regarding a subset of this theory named “John,” which corresponds to a nonminimal derivative coupling between the scalar field and the Einstein tensor in the action. We show that this coupling gives rise to an inflationary model with very unnatural initial conditions. Thus, we include the term named “George,” namely, a nonminimal, but nonderivative, coupling between the scalar field and Ricci scalar. We find a more natural inflationary model, and, by performing a post-Newtonian analysis, we derive the set of equations that constrain the parameter space with data from experiments in the Solar System.

scholarly journals Cosmology with higher-derivative matter fields

2016 ◽  
Vol 13 (07) ◽  
pp. 1650102 ◽  
Author(s):  
Tiberiu Harko ◽  
Francisco S. N. Lobo ◽  
Emmanuel N. Saridakis
Keyword(s):  
Dark Energy ◽  
Modified Gravity ◽  
State Parameter ◽  
De Sitter ◽  
Field Equations ◽  
Stiff Matter ◽  
Higher Derivative ◽  
Higher Derivatives ◽  
Derivatives Of ◽  
Matter Fields

We investigate the cosmological implications of a new class of modified gravity, where the field equations generically include higher-order derivatives of the matter fields, arising from the introduction of non-dynamical auxiliary fields in the action. Imposing a flat, homogeneous and isotropic geometry, we extract the Friedmann equations, obtaining an effective dark-energy sector containing higher-derivatives of the matter energy density and pressure. For the cases of dust, radiation and stiff matter, we analyze the cosmological behavior, finding accelerating, de Sitter and non-accelerating phases, dominated by matter or dark-energy. Additionally, the effective dark-energy equation-of-state parameter can be quintessence-like, cosmological-constant-like or even phantom-like. The detailed study of these scenarios may provide signatures, that could distinguish them from other candidates of modified gravity.

Dynamics of Flexible Multibody Systems Using Bond Graphs and Lagrange Multipliers

1990 ◽  
Vol 112 (1) ◽  
pp. 30-35 ◽  
Author(s):  
B. Samanta
Keyword(s):  
Rigid Body ◽  
Lagrange Multipliers ◽  
Equations Of Motion ◽  
Constrained Systems ◽  
System Response ◽  
Bond Graphs ◽  
Higher Derivatives ◽  
Model Constraint ◽  
System Equations ◽  
Derivatives Of

A procedure is presented to study the dynamics of interconnected flexible systems using bond graphs. The concept of Lagrange multipliers, which are commonly used in analysis of constrained systems, is introduced in the procedure. The overall motions of each of the component bodies are described in terms of large rigid body motions and small elastic vibrations. Bond graphs are used to represent both rigid body and flexible dynamics of each body in a unified manner. Bond graphs of such sub-systems are coupled to one another satisfying the kinematic constraints at the interfaces to get the overall system model. Constraint reactions are introduced in the form of Lagrange multipliers at the interfaces without disturbing the integral causality in the subsystem models, which leads to easy derivation of system equations. The equations of motion and higher derivatives of the constraint relations are integrated to obtain the constraint reactions and the system response. The procedure is illustrated by an example system and results are in good agreement with those presented earlier.

On the quantization of field theories derived from higher order lagrangians

1948 ◽  
Vol 44 (4) ◽  
pp. 546-559 ◽  
Author(s):  
J. S. de Wet
Keyword(s):  
Energy Momentum Tensor ◽  
Present Theory ◽  
Momentum Tensor ◽  
Higher Order ◽  
Field Theories ◽  
Hamiltonian Form ◽  
Field Equations ◽  
Dirac Quantization ◽  
Higher Derivatives ◽  
Derivatives Of

Heisenberg and Pauli (1) have shown how to quantize field theories derived from a Lagrangian containing first derivatives of the field quantities only. The present paper extends the theory of quantization of fields to the case of higher order Lagrangians, i.e. Lagrangians in which higher derivatives than the first appear. It is shown how such field equations can be put into Hamiltonian form and how the quantization can subsequently be carried out. Both the cases of Einstein-Bose and Fermi-Dirac quantization are discussed. It is established that the quantization is relativistically invariant and consistent with the field equations. An interesting feature of the present theory is that the Hamiltonian proves to be different, in general, from the integral of the 4–4 component of the energy momentum tensor.

scholarly journals New non-perturbative de Sitter vacua in α′-complete cosmology

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Carmen A. Núñez ◽  
Facundo Emanuel Rost
Keyword(s):  
Equations Of Motion ◽  
Generalized Equations ◽  
De Sitter ◽  
Scale Factors ◽  
Spatial Dimensions ◽  
Higher Derivatives ◽  
De Sitter Vacua ◽  
The Stability ◽  
Derivatives Of ◽  
Constant Dilaton

Abstract The α′-complete cosmology developed by Hohm and Zwiebach classifies the O(d, d; ℝ) invariant theories involving metric, b-field and dilaton that only depend on time, to all orders in α′. Some of these theories feature non-perturbative isotropic de Sitter vacua in the string frame, generated by the infinite number of higher-derivatives of O(d, d; ℝ) multiplets. Extending the isotropic ansatz, we construct stable and unstable non-perturbative de Sitter solutions in the string and Einstein frames. The generalized equations of motion admit new solutions, including anisotropic d + 1-dimensional metrics and non-vanishing b-field. In particular, we find dSn+1× Td−n geometries with constant dilaton, and also metrics with bounded scale factors in the spatial dimensions with non-trivial b-field. We discuss the stability and non-perturbative character of the solutions, as well as possible applications.

scholarly journals GRAVITY: A NEW HOLOGRAPHIC PERSPECTIVE

2006 ◽  
Vol 15 (10) ◽  
pp. 1659-1675 ◽  
Author(s):  
T. PADMANABHAN
Keyword(s):  
Order Term ◽  
Equations Of Motion ◽  
Conventional Approach ◽  
Vacuum Energy Density ◽  
General Covariance ◽  
Field Equations ◽  
Metric Tensor ◽  
Semiclassical Gravity ◽  
First Order Correction ◽  
Derivatives Of

A general paradigm for describing classical (and semiclassical) gravity is presented. This approach brings to the center-stage a holographic relationship between the bulk and surface terms in a general class of action functionals and provides a deeper insight into several aspects of classical gravity which have no explanation in the conventional approach. After highlighting a series of unresolved issues in the conventional approach to gravity, we show that (i) principle of equivalence, (ii) general covariance and (iii) a reasonable condition on the variation of the action functional, suggest a generic Lagrangian for semiclassical gravity of the form L = QabcdRabcd with ∇b Qabcd = 0. The expansion of Qabcd in terms of the derivatives of the metric tensor determines the structure of the theory uniquely. The zeroth order term gives the Einstein–Hilbert action and the first order correction is given by the Gauss–Bonnet action. Any such Lagrangian can be decomposed into surface and bulk terms which are related holographically. The equations of motion can be obtained purely from a surface term in the gravity sector. Hence the field equations are invariant under the transformation Tab → Tab + λgab and gravity does not respond to the changes in the bulk vacuum energy density. The cosmological constant arises as an integration constant in this approach. The implications are discussed.

scholarly journals Background independence, gauge invariance and equations of motion for closed string modes

2015 ◽  
Vol 30 (18n19) ◽  
pp. 1550105 ◽  
Author(s):  
B. Sathiapalan
Keyword(s):  
Renormalization Group ◽  
Equations Of Motion ◽  
Closed String ◽  
World Sheet ◽  
Gauge Invariant ◽  
Background Independence ◽  
The World ◽  
Exact Renormalization Group ◽  
Generally Covariant ◽  
Derivatives Of

In an earlier paper, gauge invariant and background covariant equations for closed string modes were obtained from the exact Renormalization Group of the world sheet theory. The background metric (but not the physical metric) had to be flat and hence the method was not manifestly background independent. In this paper, the restrictions on the background metric are relaxed. A simple prescription for the map from loop variables to space–time fields is given whereby for arbitrary backgrounds the equations are generally covariant and gauge invariant. Extra terms involving couplings of the curvature tensor to (derivatives of) the Stueckelberg fields have to be added. The background metric can thus be chosen to be the physical metric without any restrictions. This method thus gives manifestly background independent equations of motion for both open and closed string modes.

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