Showing papers on "Canonical transformation published in 1982"
TL;DR: Maslov asymptotic ray theory is used to describe body waves in inhomogeneous media, but caustics, shadows, critical points, etc. have to be treated as special cases as discussed by the authors.
Abstract: Asymptotic ray theory is widely used to describe body waves in inhomogeneous media, but caustics, shadows, critical points, etc. have to be treated as special cases. Unfortunately, these singularities are often the points of greatest interest as they are caused by inhomogeneities in the model. Transform methods, e.g., the reflectivity method and WKBJ seismograms, are used to investigate waves at these singular points but are restricted to laterally homogeneous media. Maslov asymptotic theory uses the ideas of asymptotic ray theory and transform methods, combining the advantages—simplicity and generality—of both techniques. In this paper, Maslov asymptotic theory is developed for the computation of body-wave seismograms. The eikonal equation of asymptotic ray theory is equivalent to Hamilton9s canonical equations, and the ray trajectories can be considered in the phase space of position and slowness. Normal asymptotic ray theory gives the wave solution in the spatial domain. However, the asymptotic solution for other generalized coordinates in phase space can also be found. For instance, normal transform methods find the solution in the mixed domain where the horizontal slowness replaces the coordinate. Maslov asymptotic theory extends this idea to inhomogeneous media, and the asymptotic solution in a mixed domain (position and slowness) is obtained by a canonical transformation from the spatial domain. The method is useful as the singularities in the mixed and spatial domains are at different locations, and Maslov theory provides a uniform result, combining the solutions in the different domains. These transforms between the mixed-frequency and spatial-time domains are evaluated exactly using the WKBJ seismogram algorithm. This avoids the oscillatory integrals of asymptotic theory and stabilizes the numerical solution by providing the smoothed, discrete seismograms directly. The result is a rigorous but simple method for computing body-wave seismograms in inhomogeneous media. The theory is developed in outline, and numerical examples are included.
260 citations
TL;DR: In this paper, explicit formulae are given for many-body systems of particles interacting in one dimension, analogous to the Backlund transformation, and transformations analogous to Backlund transform are discussed.
Abstract: Canonical transformations analogous to the Backlund transformation are discussed. Explicit formulae are given for many-body systems of particles interacting in one dimension.
48 citations
01 Aug 1982
TL;DR: In this paper, a proof of the gauge-invariant renormalizability of general gauge theories in arbitrary gauges is given, and it is shown that a canonical change of variables in the initial effective action also generates only a canonical transformation of variable in the renormalized action and in the vertex generating functional.
Abstract: A proof is given of the gauge-invariant renormalizability of general gauge theories in arbitrary gauges. We show that a canonical change of variables in the initial effective action also generates only a canonical change of variables in the renormalized action and in the vertex generating functional. We note that the gauge condition enters into the effective action as a canonical transformation. As a consequence, changing the gauge condition is equivalent to a canonical transformation of the renormalized action and the vertex generating functional. This, in turn, implies the gauge invariance of the renormalized S matrix.
30 citations
TL;DR: In this article, a canonical transformation method was used to derive the Fock-Tani Hamiltonian for electron-hydrogen scattering and the H-bound and resonance channels were exhibited as explicit terms in the second-quantized interaction.
Abstract: : A canonical transformation method previously applied to positron-hydrogen scattering is used to derive the Fock-Tani Hamiltonian for electron-hydrogen scattering. The H- bound and resonance channels are exhibited as explicit terms in the second-quantized interaction. (Author)
18 citations
TL;DR: Inverse monodromy transform (IMT) as discussed by the authors is a canonical transformation that parallels the Inverse Scattering Transform (IST) in the sense that it preserves the Painleve property.
Abstract: Publisher Summary This chapter discusses the inverse monodromy transform, explaining how it is a canonical transformation. The Inverse Monodromy Transform (IMT) parallels the Inverse Scattering Transform (IST). The finite dimensional solution manifold for these flows is not necessarily compact, not a torus, and so the KAM theorem does not directly apply. The potential connection between a possible preservation of the solution manifold and the preservation of the Painleve property is an intriguing one. Now the contours are the same as those used in the integral definitions of Airy functions.
16 citations
TL;DR: In this paper, the many-fermion problem is formulated in terms of boson variables that represent collective and particle-hole excitations of the system and an expansion scheme is presented which in a systematic and physically reasonable way allows the inclusion of the boson degrees of freedom in increasing detail.
Abstract: The many-fermion problem is formulated in terms of boson variables that represent collective and particle-hole excitations of the system. An expansion scheme is presented which in a systematic and physically reasonable way allows the inclusion of the boson degrees of freedom in increasing detail. The RPA canonical transformation can be readily performed among the discretised boson operators. The authors apply the scheme to the homogeneous electron gas and use the exp S method to derive explicit expressions for the static properties for the system.
14 citations
TL;DR: In this paper, it was shown that Haller's interpretation of the unitary transformation from the minimal coupling to the multipolar form of the Hamiltonian in nonrelativistic quantum electrodynamics not only conflicts with the Maxwell-Lorentz equations but also ascribes a special status to the minimal-coupling Hamiltonian which is not warranted by the general method of canonical quantization.
Abstract: It is shown that Haller's interpretation of the unitary transformation from the minimalcoupling to the multipolar form of the Hamiltonian in nonrelativistic quantum electrodynamics not only conflicts with the Maxwell-Lorentz equations but also ascribes a special status to the minimal-coupling Hamiltonian which is not warranted by the general method of canonical quantization. The correct interpretation of the transformation as the quantum analog of a classical canonical transformation is reiterated. It is also shown that the transformation is equivalent to a gauge transformation of the electromagnetic potentials.
11 citations
TL;DR: In this paper, it was shown that the posterior probabilities of discriminant analysis are invariant under canonical transformation, and this result was used to justify the application of canonical variates for classification, thus integrating both approaches to discrimination.
Abstract: The usual computing procedures in discriminant analysis involve both classificatory and separatory functions (Geisser 1977). The first of these concerns classification of samples from a mixture of populations. Statistical assessment of classification procedures is based on rates of correct classification (Glick 1972,1973; Lachenbruch 1975; Michaelis 1973) or on the "loss" due to misclassification (Anderson 1958, Lachenbruch and Goldstein 1979). Separatory methods, on the other hand, deal with the transformation of data so that population differences are highlighted. This is done by means of "canonical variates," which define a subspace of reduced dimensionality wherein data often can be displayed to advantage. The canonical approach was first suggested by the work of Fisher (1936), and is closely associated with the multivariate analysis of variance (Anderson 1958, Rao 1965). Applied researchers often fail to recognize the statistical relationships between classificatory and separatory discrimination, in large part because mathematical forms, system dimensionalities, and even objectives differ between the two approaches. Kshirsagar and Arseven (1975) previously used a sample-based argument to show that full-rank canonical transforms can be used for classification. However, a key feature of the canonical analysis is the reduction of dimensionality. Thus it is important to know whether the same property holds for the reduced set of canonical variates. A simple matrix argument is used below to show that for certain distributions the posterior probabilities of discriminant analysis are invariant under canonical transformation. This result is used to justify the application of canonical variates for classification, thus integrating both approaches to discrimination.
9 citations
TL;DR: In this article, the canonical transformation relating an oscillator s-d boson Hamiltonian with a microscopic collective Hamiltonian derived from a system of A particles interacting through harmonic oscillator forces was derived.
Abstract: In this paper we derive the canonical transformation relating an oscillator s-d boson Hamiltonian with a microscopic collective Hamiltonian derived from a system of A particles interacting through harmonic oscillator forces. As the former has the symmetry group U (6) this will also be the case for the latter.
9 citations
01 Jan 1982
9 citations
TL;DR: In this paper, a third-order intermediary for planetary motion is developed, which contains all contributions that arise from the assumption of circular, coplanar orbits for the disturbing masses.
Abstract: By use of a new canonical transformation procedure, a third-order intermediary for planetary motion is developed. The intermediary contains all contributions that arise from the assumption of circular, coplanar orbits for the disturbing masses. The results are expressible in terms of elliptic integrals of the first, second, and third kinds.
TL;DR: In this article, sufficient conditions are given for the possibility to construct quantum mechanical wavefunctions by the sole knowledge of an appropriate sequence of classical canonical transformations which map a given Hamiltonian onto the new position variable.
Abstract: Sufficient conditions are given for the possibility to construct quantum mechanical wavefunctions by the sole knowledge of an appropriate sequence of classical canonical transformations which map a given Hamiltonian onto the new position variable. The transformation matrix element for each individual step of this sequence is given by the semiclassical limit expression of these matrix elements; it is a function of the generator of this transformation step only. The wavefunction, i.e. the transformation matrix element for the total transformation, is obtained as a multiple integral over the transformation matrix elements of the various intermediate steps. The practicability of this procedure is demonstrated by several examples. The authors consider time-independent systems with one degree of freedom.
TL;DR: In this paper, the free electromagnetic field is canonically quantized in a gauge-invariant way by interpreting the Fourier coefficients of the magnetic induction field B as generalised coordinates, and the coefficient of the electric field E as their conjugate momenta.
Abstract: The free electromagnetic field is canonically quantised in a gauge-invariant way by interpreting the Fourier coefficients of the magnetic induction field B as generalised coordinates, and the coefficients of the electric field E as their conjugate momenta. The usual commutation relations among the components of E and B are obtained. A canonical transformation, corresponding to a rotation in generalised phase space, is made on the Fourier coefficients. This transformation is shown to give a duality transformation on the electric and magnetic fields. The free-field Maxwell equations and the commutation relations are invariant under duality transformations. However, if interactions are introduced, the invariance under duality transformations is broken, and the original canonical theory should be used.
TL;DR: In this article, the results of a previous work concerning a method for performing the canonical formalism for constrained systems are extended when the canonical transformation proposed in that paper is explicitly time dependent.
Abstract: The results of a previous work, concerning a method for performing the canonical formalism for constrained systems, are extended when the canonical transformation proposed in that paper is explicitly time dependent.
TL;DR: In this article, the motion of Bloch electrons in homogeneous magnetic fields is reduced without approximations to, at most, two dimensions in the general three-dimensional case, i.e., for arbitrary crystal potential, arbitrary field-lattice geometry, and all rational fields.
Abstract: The motion of Bloch electrons in homogeneous magnetic fields is reduced without approximations to, at most, two dimensions in the general three-dimensional case, i.e., for arbitrary crystal potential, arbitrary field-lattice geometry, and all rational fields. This is done by fully exploiting a canonical transformation and by constructing with the aid of ray-group projection operators generalized $k\ensuremath{-}q$ functions, which separate off one degree of freedom. Previous ad hoc reductions to one dimension for essentially two-dimensional situations are recovered and explained. The solutions of the resulting lower-dimensional effective Schr\"odinger equations are functions of generalized coordinates. They are converted into the real-space wave functions by means of a contact transformation; their local and global properties are investigated. The results presented allow first-principles calculations of diamagnetic band structures and wave functions to realistic systems.
TL;DR: In this article, the authors considered the Weyl group of relativistic off-mass-shell particles and showed that the condition can be satisfied only by states belonging to a subset of subspaces of the on-mass shell Hilbert space.
Abstract: Working in the context of the Weyl group, which describes off-mass-shell relativistic particles, we impose “gauge-fixing” constraints involvingR0,R+, andD as matrix element conditions to be satisfied by the on-mass-shell states of a massive particle. We evaluate the matrix elements inp-space using five sets of co-ordinates: (p2,p), (p2,p+,pT), (p2,p−,pT), (p2,π), and (p2,π+,πT) where\(\pi ^\mu \equiv p^\mu /(p^2 )^{\tfrac{1}{2}} \). We find that, only in the case ofR0 with (p2,p) coordinates,R+ with (p2,p+,pT) coordinates, andD with (p2, π) or (p2,π+,πT) coordinates, can the condition be satisfied by arbitrary on-mass-shell states. In all other cases, the condition can be satisfied only by states belonging to a subset of subspaces of the on-mass-shell Hilbert space, i.e it forces a violation of the superposition principle. These results constitute thep-space quantum version of Shanmugadhasan's theorem for constrained classical systems which states that there exists, at least locally in phase space, a canonical transformation to a set of variables in which the second-class constraints become canonical pairs equal to zero with the other canonical coordinates independent of the second-class constraints.
01 Jan 1982
TL;DR: Benney's equations were treated as a Vlasov equation in this paper, and the generating series for the conserved quantities satisfies a Hamilton-Jacobi equation associated with a canonical transformation.
TL;DR: In this paper, the blocking effect is introduced through a canonical transformation in the projected BCS-Tamm-Dancoff approximation, and it is suggested that the blocking effects may play an important role in the description of the low-lying states in odd-mass nuclei.
TL;DR: In this paper, the NMR spectra of 23 Na in a single crystal of sodium chloride rotating at the magic angle were calculated by the canonical transformation technique, and the authors showed that the canonical transform can be used to obtain the complete NMR spectrum of the crystal.
01 Feb 1982
TL;DR: In this paper, a brief discussion of the following subjects is presented: (1) introduction, truncated and untruncated Lagrange's and Hamilton's equations; (2) variational principles, the inverse problem and the direct universality of Birkhoff's formalism; (3) BER systems and its transformation theory; (4) Lie-admissible and general first-order system; and (5) transformation theory for Lie-ADR systems.
Abstract: In this paper a brief discussion of the following subjects is presented: (1) introduction, truncated and untruncated Lagrange's and Hamilton's equations; (2) variational principles, the inverse problem and the direct universality of Birkhoff's formalism; (3) Birkhoffian systems and its transformation theory; (4) Lie-admissible and general first-order system; and (5) transformation theory for Lie-admissible and general first-order systems.
TL;DR: In this paper, it was shown that a Gaussian measure induces a Kubo-Martin-Schwinger state for the linear system, and that the preimage of this measure under the canonical transformation implemented by a wave operator is a KMLS state.
Abstract: A smoothed nonlinear Klein-Gordon equation is regarded as the equation of evolution of a classical dynamical system with an infinite-dimensional phase space. It is proved that the wave operators are canonical transformations of this system that linearize it. It is shown that a Gaussian measure induces a Kubo-Martin-Schwinger state for the linear system, and that the preimage of this measure under the canonical transformation implemented by a wave operator is a Kubo-Martin-Schwinger state for the original nonlinear system.Bibliography: 8 titles.
TL;DR: Deenen et al. as mentioned in this paper showed that the general representation problem of non-bijective canonical transformations involves infinite sums of the quotient and tensor product spaces defined by Plebanski and Seligman.
Abstract: A recent letter by J. Deenen on 'non-bijective canonical transformations in quantum mechanics' (see ibid., vol.14, p.L273, 1981) is analysed and found to be incomplete. The general representation problem of non-bijective canonical transformations is shown to involve infinite sums of the quotient and tensor product spaces defined by Plebanski and Seligman (1982).
TL;DR: In this article, it is shown that a continual integral can be reduced to a finite multiple integral by using canonical transformations. But this method is not suitable for the case of a homogeneous force field.
Abstract: Continual integrals are considered for the matrix elements of an evolution operator of quantum-mechanical systems. It is shown that a continual integral can be reduced to a finite multiple integral by using canonical transformations. The method developed is illustrated by an example of a homogeneous force field.
TL;DR: In this paper, the condition that the vacuum expectation value of the Heisenberg scalar field reproduces static soliton solution to the classical equation is satisfied when the LSZ asymptotic quantum field undergoes a canonical transformation.
Abstract: In an quantum sine-Gordon model in 1+1 dimensions, the condition that the vacuum expectation value of the Heisenberg scalar field reproduces static soliton solution to the classical equation is satisfied when the LSZ asymptotic quantum field undergoes a canonical transformation. This can be viewed as a “quantum image” of the Backlund transformation and the soliton is described in terms of condensation of quanta. Their confinement is ascribed to the non-implementability of unitary transformations among unitary inequivalent representations of the canonical commutation relations.