EQUIVALENCE OF THE LAGRANGIAN AND HAMILTONIAN BRST CHARGES FOR THE BOSONIC STRING IN THE HARMONIC AND CONFORMAL GAUGES

1988 ◽  
Vol 03 (05) ◽  
pp. 1081-1101 ◽  
Author(s):  
V. DEL DUCA ◽  
L. MAGNEA ◽  
P. VAN NIEUWENHUIZEN
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Point Of View ◽  
Bosonic String ◽  
Lagrangian Formalism ◽  
Brst Charge ◽  
Gauge Fixing ◽  
Conformal Gauge ◽  
Brst Invariance

We consider the BRST formalism for the bosonic string in arbitrary gauges, both from the Hamiltonian and from the Lagrangian point of view. In the Hamiltonian formulation we construct the BRST charge Q(H) following the Batalin-Fradkin-Fradkina-Vilkovisky (BFFV) formalism in phase space. In the Lagrangian formalism, we use the Noether procedure to construct the BRST charge Q(L) in configuration space. We then discuss how to go from configuration to phase space and demonstrate that the dependence of Q(L) on the gauge fixing disappears and that both charges become equal. We work through two gauges in detail: the conformal gauge and the de Donder (harmonic) gauge. In the conformal gauge one must use equations of motion, and a simple canonical transformation is found which exhibits the equivalence. In the de Donder gauge, nontrivial canonical transformations are needed. Our results overlap with work by Beaulieu, Siegel and Zwiebach on the de Donder gauge, but since we only require BRST invariance and not anti-BRST invariance, we need simpler field redefinitions; moreover, we stay off-shell.

BRST-QUANTIZATION OF BOSONIC STRINGS AND HYPERTWISTORS

1990 ◽  
Vol 05 (25) ◽  
pp. 2057-2062 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Phase Space ◽  
Bosonic Strings ◽  
Fock Space ◽  
Brst Quantization ◽  
Geometric Quantization ◽  
Bosonic String ◽  
Total Space ◽  
Complex Structures ◽  
Brst Invariance

We consider the relation between geometric quantization of bosonic strings and the Becchi-Rouet-Stora-Tyutin (BRST) approach. We introduce the space of hypertwistors as the total space of the bundle of complex structures over the phase space of bosonic string. The conditions of the BRST invariance, determining the Fock space of the bosonic string, have been formulated in terms of the ghost bundle over the hypertwistor space.

Geometric Hamiltonian quantum mechanics and applications

2016 ◽  
Vol 13 (Supp. 1) ◽  
pp. 1630017 ◽  
Author(s):  
Davide Pastorello
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Projective Space ◽  
Star Product ◽  
Hamiltonian Formulation ◽  
Classical Mechanics ◽  
Point Of View ◽  
Quantum Theories ◽  
Geometric Point ◽  
Set Up

Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed.

scholarly journals BRST INVARIANCE AND DE RHAM TYPE COHOMOLOGY OF 't HOOFT–POLYAKOV MONOPOLE

2010 ◽  
Vol 25 (29) ◽  
pp. 2529-2539 ◽  
Author(s):  
SOON-TAE HONG
Keyword(s):  
Cohomology Group ◽  
Topological Charge ◽  
Group Structure ◽  
Quantum Field ◽  
Brst Charge ◽  
Dirac Quantization ◽  
Gauge Fixing ◽  
De Rham ◽  
Brst Invariance ◽  
Popov Ghost

We exploit the 't Hooft–Polyakov monopole to construct closed algebra of the quantum field operators and the BRST charge Q BRST . In the first-class configuration of the Dirac quantization, by including the Q BRST -exact gauge-fixing term and the Faddeev–Popov ghost term, we find the BRST invariant Hamiltonian to investigate the de Rham type cohomology group structure for the monopole system. The Bogomol'nyi bound is also discussed in terms of the first-class topological charge defined on the extended internal two-sphere.

scholarly journals Chiral Schwinger model with Faddeevian anomaly and its BRST quantization

2020 ◽  
Vol 80 (2) ◽  
Author(s):  
Sanjib Ghosal ◽  
Anisur Rahaman
Keyword(s):  
Phase Space ◽  
Brst Quantization ◽  
Hamiltonian Formulation ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Gauge Invariant ◽  
Schwinger Model ◽  
Gauge Fixing

Abstract We consider chiral Schwinger model with Faddeevian anomaly, and carry out the quantization of both the gauge-invariant and non-invariant version of this model has been. Theoretical spectra of this model have been determined both in the Lagrangian and Hamiltonian formulation and a necessary correlation between these two are made. BRST quantization using BFV formalism has been executed which shows spontaneous appearance of Wess–Zumino term during the process of quantization. The gauge invariant version of this model in the extended phase space is found to map onto the physical phase space with the appropriate gauge fixing condition.

Hamiltonian mechanics

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Configuration Space ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Hamiltonian Mechanics ◽  
Lagrange Equations ◽  
First Order ◽  
Point Transformations ◽  
First Order Differential Equations

This chapter gives a brief overview of Hamiltonian mechanics. The complexity of the Newtonian equations of motion for N interacting bodies led to the development in the late 18th and early 19th centuries of a formalism that reduces these equations to first-order differential equations. This formalism is known as Hamiltonian mechanics. This chapter shows how, given a Lagrangian and having constructed the corresponding Hamiltonian, Hamilton’s equations amount to simply a rewriting of the Euler–Lagrange equations. The feature that makes the Hamiltonian formulation superior is that the dimension of the phase space is double that of the configuration space, so that in addition to point transformations, it is possible to perform more general transformations in order to simplify solving the equations of motion.

scholarly journals A geometric Hamiltonian description of composite quantum systems and quantum entanglement

2015 ◽  
Vol 12 (07) ◽  
pp. 1550069 ◽  
Author(s):  
Davide Pastorello
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Quantum Entanglement ◽  
Hamiltonian Formulation ◽  
Classical Mechanics ◽  
Hamiltonian Formalism ◽  
Quantum Systems ◽  
Point Of View ◽  
Entanglement Measure ◽  
Hamiltonian Description

Finite-dimensional Quantum Mechanics can be geometrically formulated as a proper classical-like Hamiltonian theory in a projective Hilbert space. The description of composite quantum systems within the geometric Hamiltonian framework is discussed in this paper. As summarized in the first part of this work, in the Hamiltonian formulation the phase space of a quantum system is the Kähler manifold given by the complex projective space P(H) of the Hilbert space H of the considered quantum theory. However the phase space of a bipartite system must be P(H1 ⊗ H2) and not simply P(H1) × P(H2) as suggested by the analogy with Classical Mechanics. A part of this paper is devoted to manage this problem. In the second part of the work, a definition of quantum entanglement and a proposal of entanglement measure are given in terms of a geometrical point of view (a rather studied topic in recent literature). Finally two known separability criteria are implemented in the Hamiltonian formalism.

Shape Dynamics and the Linking Theory

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Line Element ◽  
Hamiltonian Formulation ◽  
Operational Definition ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Problem Of Time ◽  
Definition Of ◽  
Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

scholarly journals Semi-Classical Localisation Properties of Quantum Oscillators on a Noncommutative Configuration Space

2015 ◽  
Vol 22 (04) ◽  
pp. 1550021 ◽  
Author(s):  
Fabio Benatti ◽  
Laure Gouba
Keyword(s):  
Phase Space ◽  
Configuration Space ◽  
Classical Limit ◽  
Coherent States ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Wigner Functions ◽  
Quantum Mechanical Oscillators ◽  
Mechanical Oscillators ◽  
Space Noncommutativity

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

scholarly journals On the Dirac eigenvalues as observables of the on-shell N = 2D = 4 Euclidean supergravity

Open Physics ◽  
2008 ◽  
Vol 6 (4) ◽  
Author(s):  
Ion Vancea
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Gauge Transformations

AbstractWe generalize previous works on the Dirac eigenvalues as dynamical variables of Euclidean gravity and N =1 D = 4 supergravity to on-shell N = 2 D = 4 Euclidean supergravity. The covariant phase space of the theory is defined as the space of the solutions of the equations of motion modulo the on-shell gauge transformations. In this space we define the Poisson brackets and compute their value for the Dirac eigenvalues.

scholarly journals Finite BRST–anti-BRST transformations in generalized Hamiltonian formalism

2014 ◽  
Vol 29 (27) ◽  
pp. 1450159 ◽  
Author(s):  
Pavel Yu. Moshin ◽  
Alexander A. Reshetnyak
Keyword(s):  
Dynamical Systems ◽  
Path Integral ◽  
Hamiltonian Path ◽  
Hamiltonian Formalism ◽  
Lagrangian Formalism ◽  
Yang Mills ◽  
Gauge Dependence ◽  
Field Dependent ◽  
Gauge Fixing ◽  
Constrained Dynamical Systems

We introduce the notion of finite BRST–anti-BRST transformations for constrained dynamical systems in the generalized Hamiltonian formalism, both global and field-dependent, with a doublet λa, a = 1, 2, of anticommuting Grassmann parameters and find explicit Jacobians corresponding to these changes of variables in the path integral. It turns out that the finite transformations are quadratic in their parameters. Exactly as in the case of finite field-dependent BRST–anti-BRST transformations for the Yang–Mills vacuum functional in the Lagrangian formalism examined in our previous paper [arXiv:1405.0790 [hep-th]], special field-dependent BRST–anti-BRST transformations with functionally-dependent parameters λa= ∫ dt(saΛ), generated by a finite even-valued function Λ(t) and by the anticommuting generators saof BRST–anti-BRST transformations, amount to a precise change of the gauge-fixing function for arbitrary constrained dynamical systems. This proves the independence of the vacuum functional under such transformations. We derive a new form of the Ward identities, depending on the parameters λaand study the problem of gauge dependence. We present the form of transformation parameters which generates a change of the gauge in the Hamiltonian path integral, evaluate it explicitly for connecting two arbitrary Rξ-like gauges in the Yang–Mills theory and establish, after integration over momenta, a coincidence with the Lagrangian path integral [arXiv:1405.0790 [hep-th]], which justifies the unitarity of the S-matrix in the Lagrangian approach.

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