Pure spinor integration from the collating formula

We use the technique developed by Becchi and Imbimbo to construct a well-defined BRST-invariant path integral formulation of pure spinor amplitudes. The space of pure spinors can be viewed from the algebraic geometry point of view as a collection of open sets where the constraints can be solved and a set of free and independent variables can be defined. On the intersections of those open sets, the functional measure jumps and one has to add boundary terms to construct a well-defined path integral. The result is the definition of the pure spinor integration measure constructed in terms of differential forms on each single patch.     

WARD IDENTITIES IN PHASE SPACE AND THEIR APPLICATIONS

Starting from the phase space path integral, we have derived the Ward identities in canonical formalism for a system with regular and singular Lagrangian. This formulation differs from the traditional discussion based on path integral in configuration space. It is pointed out that the quantum canonical equations for systems with singular Lagrangians are different from the classical ones obtained from Dirac's conjecture, The preliminary applications of Ward identities in phase space to the Yang-Mills theory are given. Some relations among the proper vertices and propagators are deduced,the PCAC, AVV vertices and generalized PCAC expressions are also obtained. We have also pointed out that some authors in their early work had ignored the treatment of the constraints.     

Canonical Quantization of Systems with Time-Dependent Constraints

The Hamilton-Jacobi method of constrained systems is discussed. The equations of motion for a singular system with time dependent constraints are obtained as total differential equations in many variables. The integrability conditions for the relativistic particle in a plane wave lead us to obtain the canonical phase space coordinates without using any gauge fixing condition. As a result of the quantization, we get the Klein-Gordon theory for a particle in a plane wave. The path integral quantization for this system is obtained using the canonical path integral formulation method.     

Canonical and D-transformations in theories with constraints

We describe a class of transformations in a super phase space (we call them D-transformations) which play the role of ordinary canonical transformations in theories with second-class constraints. Namely, in such theories they preserve the form invariance of equations of motion, their quantum analogs are unitary transformations, and the measure of integration in the corresponding Hamiltonian path integral is invariant under these transformations.     

N = 1 superspace formulation of N = 2 and N = 4 super-Yang-Mills models with central charge

The component models of N = 2 and N = 4 supersymmetric Yang-Mills theories of Sohnius, Stelle and West are reformulated in terms of N = 1 superfields. The non-supersymmetric constraints are supersymmetrized generalizing the linear multiplet in the presence of the non-abelian gauge superfield and (in the N = 4 case) a doublet of chiral superfields. The extended supersymmetry transformations preserving constraints are explicitly given in terms of N = 1 superfields. We are able to introduce the constraints back into the lagrangian using superfield Lagrange multipliers. The on-shell equivalence of this formulation with the formulation of Fayet with one (for N = 2) and three (for N = 4) chiral superfields is shown. The abelian N = 2 model is worked out to show the connection between full superspace treatment and the N = 1 superfield formulation.     

New framework for the Feynman path integral

The well-known Fourier integral solution of the free diffusion equation in an arbitrary Euclidean space is reduced to Feynmannian integrals using the method partly contained in the formulation of the Fresnelian integral. By replacing the standard Hilbert space underlying the present mathematical formulation of the Feynman path integral by a new Hilbert space, the space of classical paths on the tangent bundle to the Euclidean space (and more general to an arbitrary Riemannian manifold) equipped with a natural inner product, we show that our Feynmannian integral is in better agreement with the qualitative features of the original Feynman path integral than the previous formulations of the integral.     

Stochastic quantization and path integral formulation of Fokker-Planck equation

The operator formalism (Fokker-Planck dynamics) associated to a general n-dimensional, non-linear drift, non-constant diffusion Fokker-Planck equation, is derived by a stochastic quantization from the corresponding path integral formulation in phase space.     

On the spectra of noncommutative 2D harmonic oscillator

The spectra and wave functions of the 2-dimensional harmonic oscillator in a noncommutative plane are revised by using the path integral formulation in coordinate space and momentum space, respectively. We perform the path integral formulation in coordinate space first. Then we study this problem in momentum space. The propagator is computed both in coordinate space and in momentum space. The modification due to noncommutativity of eigenvalues and eigenfunctions is studied. Both the small and large noncommutative parameter limits are discussed. PACS 11.10.Ef     

Reduced Order Podolsky Model

We perform the canonical and path integral quantizations of a lower-order derivatives model describing Podolsky’s generalized electrodynamics. The physical content of the model shows an auxiliary massive vector field coupled to the usual electromagnetic field. The equivalence with Podolsky’s original model is studied at classical and quantum levels. Concerning the dynamical time evolution, we obtain a theory with two first-class and two second-class constraints in phase space. We calculate explicitly the corresponding Dirac brackets involving both vector fields. We use the Senjanovic procedure to implement the second-class constraints and the Batalin-Fradkin-Vilkovisky path integral quantization scheme to deal with the symmetries generated by the first-class constraints. The physical interpretation of the results turns out to be simpler due to the reduced derivatives order permeating the equations of motion, Dirac brackets and effective action.     

Path integral and effective Hamiltonian in loop quantum cosmology

We study the path integral formulation of Friedmann universe filled with a massless scalar field in loop quantum cosmology. All the isotropic models of $k=0,+1,-1$ are considered. To construct the path integrals in the timeless framework, a multiple group-averaging approach is proposed. Meanwhile, since the transition amplitude in the deparameterized framework can be expressed in terms of group-averaging, the path integrals can be formulated for both deparameterized and timeless frameworks. Their relation is clarified. It turns out that the effective Hamiltonian derived from the path integral in deparameterized framework is equivalent to the effective Hamiltonian constraint derived from the path integral in timeless framework, since they lead to same equations of motion. Moreover, the effective Hamiltonian constraints of above models derived in canonical theory are confirmed by the path integral formulation.     

Reduction in path integrals on a Riemannian manifold with group action

A new method for the factorization of the path-integral measure in path integrals for a particle motion on a compact Riemannian manifold with a free isometric unimodular group action is proposed. It is shown that path-integral measure is not invariant under the factorization. An integral relation between the path integral given on the total space of the principal fiber bundle and the path integral on the base space of this bundle (the orbit space of the group action) is obtained.     

Tensor analysis on conformal supergeometry and the ghost action of the superstring

The orthogonal decomposition technique is extended to the physical and ghost tensor superfields on the two-dimensional conformal supergeometry. This method makes the meaning of the superfield functional integral clearer. The form of the ghost action of the superstring in the superspace formulation is discussed on this basis.     

On the BFT-BFV quantization of gauge invariant systems with linear second class constraints

We study the Hamiltonian path integral formulation for generic systems with first class and linear second class constraints.ift01001@ufrj     

Topological Chern—Simons vortices in the O (3) σ -model

Based on the phase-space path integral (functional integral) for a system with a regular or singular Lagrangian, the generalized Ward identities for phase space generating functional under the global transformation in phase space are derived respectively. The canonical Noether theorem at the quantum level is also established. It is pointed out that the connection between the symmetries and conservation laws in classical theories, in general,is no longer preserved in quantum theories. The advantage of our formulation is that we do not need to carry out the integration over the canonical momenta as usually performed. Applying the present formulation to Yang-Mills theory, the quantal BRS conserved quantity and Ward-Takahashi identity for BRS tranformation are derived; the Ward identities for gaugeghost proper vertices and new quantal conserved quantity are also found. In comparison of quantal conservation laws with those one deriving from configuration-space path integral using the Faddeev-Popov(F-P) trick is discussed. A precise study of path-integral quantisation for a nonlinear sigma model with Hopf and Chern-Simons (CS) terms is reexamined. It has been shown that the angular momentum at the quantum level is equal to classical (Noether ) one. Applying our formulation to non-Abelian CS theory, the quantal conserved angular momentum of this system is obtained which differs from classical one in that one needs to take into account the contribution of angular momenta of ghost fields.     

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.     

Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- and Three Dimensional Euclidean Space

Path integral formulations for the Smorodinsky-Winternitz potentials in two- and three-dimensional Euclidean space are presented. We mention all coordinate systems which separate the Smorodinsky-Winternitz potentials and state the corresponding path integral formulations. Whereas in many coordinate systems an explicit path integral formulation is not possible, we list in all soluble cases the path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave-functions.     

The action of the supersymmetric N = 2 gauge theory in harmonic superspace

We consider a solution of constraints for the analytic representation of the harmonic superfield connection in the N = 2 supersymmetric gauge theory. A simple geometric expression for the action of this theory is obtained in the form of the integral over the N = 2 superspace.     

Distinguished role of Senjanovic measure in a path integral quantization of constrained systems

It is shown that path integral measure, for any system with constraints, can be cast in the form found by Senjanovic, for systems with the second-class constraints, only. This simplifies, and gives a unifying perspective of quantization of the constrained dynamics. Also, it permits, in the process of elimination, from path integral, the constrained variables, to avoid commutativity requirement on admissible gauges, imposed for rather technical reasons, by Faddeev.