Covariant Hamiltonian Field Theory: Path Integral Quantization

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative quantum field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field systems are equivalent.     

THE REPRESENTATION THEORY in QUANTUM MECHANICS with NEGATIVE METRIC

In this paper the coherent state for the negative metric boson is established, which, originating from its particle number representation provides a basis for constructing its coordinate and nomenturn representations in the Euclidean space. However,it is demonstrated by us that these two representations do not exist in the Minkowski space. Thus this formulation provides a physical foundation for the path integral quantiration theory of the negative metric boson in the Euclidean space.     

An interpretation within philosophy of the relationship between classical mechanics and quantum mechanics

A mapping of a finite directed graph onto a curve in space-time is considered. The mapping induces the dynamics of a free particle moving along the curve. The distinction between the Lagrangian and the Hamiltonian formulation of particle mechanics is expressed in terms of the distinction between referring to a particle in space and time and referring to the points in space which the particle occupies, respectively. These elements are combined to yield an interpretation of Feynman's path integral formulation of quantum mechanics. Describing a bound state of a system as a particle is discussed.     

The Quantal Canonical Symmetries for a Singular Lagrangian System with Subsidiary Constraints

The quantization for a system containing subsidiary constraints (in configuration space) with a singular Lagrangian is studied, in certain case which can be brought into the theoretical framework of constrained Hamiltonian system. A modified Dirac-Bergmann algorithm for the calculation of all phase-space constraints in those systems is derived. The path integral quantization is formulated by using the Faddeev-Senjanovic scheme. The classical and quantum canonical symmetries (Noether theorem in canonical formalism) are established for such a system. An example is given to illustrate that the connection between the symmetry and conservation law in classical theory are not always validity in the quantum theory.     

Feynman formulas and path integrals for some evolution semigroups related to τ-quantization

This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n-fold iterated integrals as n → ∞) and their connections with phase space Feynman path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamiltonian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of stochastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals.     

A path-integral formulation of the Pauli equation using two real spin coordinates

Feynman's path-integral quantum-mechanical formulation is generalised for particles of spin 1/2. In the one-particle case, the path-integral formulation uses paths in a Euclidean real five-dimensional space, two coordinates (u, v) being reserved for spin. The path integral is proven to correspond exactly to the Pauli equation. A canonical density-matrix formulation is also dealt with. Basic ideas are to start with differential spin operators instead of the Pauli matrices and apply them to functions ψ=ψ ₁(r,t)u +gy ₂(r,t)v whereψ ₁,ψ ₂ are the Pauli wave functions. Then a ‘nilpotent’ spin ‘kinetic-energy’ term is added to the Hamiltonian. This enables us to find a non-matrix spin-dependent Lagrangian which is used as usual in the action of a path integral of the Feynman type. Integral relations are derived from which the path integral can be transformed into components of the Pauli matrix Green's function (propagator) or the canonical density matrix. As an example, a path-integral calculation of the normal Zeeman splitting is carried out.     

Hamiltonian Formulation and Action Principle for the Lorentz-Dirac System

The possibility of constructing a Lagrangian and Hamiltonian formulation is examined for a radiating point-like charge usually described by the classical Lorentz-Dirac equation. It turns out that the latter equation cannot be obtained from the variational principle, and, furthermore, has nonphysical solutions. It is proposed to consider a physically equivalent set of reduced equations which admit a Hamiltonian formulation with non-canonical Poisson brackets. As an example, the effective dynamics of a non-relativistic particle moving in a homogeneous magnetic field is considered. The proposed Hamiltonian formulation may be considered as a first step to a consistent quantization of the Lorentz-Dirac system.     

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.     

A novel method to quantize systems of damped motion

A novel method to quantize systems of damped motion is proposed in the frameworks of canonical quantization and path integral quantization. It can be afforded by considering a Lagrangian multiplied by a time-dependent function, which may represent an effective interaction with “environment.”     

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.     

On Covariant Quantization of the Massive Self-dual 3-forms in 7 Dimensions

Massive self-dual 3-forms in 7 dimensions are analyzed from the point of view of the Hamiltonian path integral quantization. The quantization procedure relies on the quantization of a first-class system equivalent with the original theory. The first-class system is constructed in the framework of gauge unfixing approach and Batalin-Fradkin method. The Hamiltonian path integral of the first-class system takes a manifestly Lorentz-covariant form.     

An alternative approach to canonical quantizationof the radiation damping

Inspired by some works on quantization of dissipative systems, in particular the ones treating the damped harmonic oscillator and by a paper due to Lukierski, we consider the dissipative system of a charge interacting with its own radiation, which is the origin of radiation damping. Using the indirect Lagrangian representation we obtained a Lagrangian formalism with a Chern-Simons-like term. A Hamiltonian analysis is also done in commutative and non-commutative scenarios, which leads to the quantization of the system. Received: 23 August 2005, Published online: 21 October 2005 PACS: 41.60.-m, 04.60.Ds     

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.     

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.     

When does a non-linear point transformation generate an extra O(ℏ2) potential in the effective Lagrangian?

It is known that when a non-linear point transformation is made in the path integral describing a quantum mechanical system, the effective Lagrangian picks up an extra potential which is not obtained by a naive change of integration variables. In field theory on the contrary it is known that such a term is not needed to keep dimensionally regularized complete Green functions invariant. We explain the difference as due to the different regularization procedures used in the two cases. This observation is then used to derive the expression for the extra potential for both the Lagrangian and the Hamiltonian version of the path integral.     

Quantum dynamics of a massless relativistic string (II)

The massless relativistic free string is studied in the gauge x₀ = τ. It is found that the classical solutions include transverse and longitudinal vibrations. The problem is treated both in the Lagrangian and Hamiltonian formalism. Different ways of quantizing the system are investigated. The path integral quantization leads to a Poincaré invariant quantum theory in any number of dimensions.     

Electromagnetic form factor via Bethe-Salpeter amplitude in Minkowski space

For a relativistic system of two scalar particles, we find the Bethe-Salpeter amplitude in Minkowski space and use it to compute the electromagnetic form factor. The comparison with Euclidean space calculation shows that the Wick rotation in the form factor integral induces errors which increase with the momentum transfer Q². At JLab domain (Q ² = 10 GeV^2/c²), they are about 30%. Static approximation results in an additional and more significant error. On the contrary, the form factor calculated in light-front dynamics is almost indistinguishable from the Minkowski space one.     

Maxwell–Chern–Simons场的Casimir效应

利用约束理论对AbelMaxwell-Chem-Simons场进行路径积分量子化. 并利用复变函数论中Plana求和公式,计算(2+1)维空间中两个平行导线型边界的Casimir效应. 不引任何截断参数,而得出有限的解析表达式.