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Canonical transformation
About: Canonical transformation is a research topic. Over the lifetime, 1854 publications have been published within this topic receiving 38019 citations.
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TL;DR: In this article, the first-order Lagrange equations are replaced by subsidiary conditions on the coordinates and velocities of the Hessian matrix of the second-order derivatives of the Lagrangian.
Abstract: A Lagrangian is degenerate when the Hessian matrix whose elements consist of all the second‐order derivatives of the Lagrangian with respect to the generalized velocities has (for simplicity) a constant singular rank everywhere in the space of the arguments of the Lagrangian. This singularity entails a definite number of first‐order Lagrange equations, which act as subsidiary conditions on the coordinates and velocities. Consistency of these subsidiary conditions with the Langrange system requires them to be an invariant system with respect to the second‐order Lagrange equations. An invariant system is analogous to a system of first integrals except that absolute constants appear where arbitrary constants characterize first integrals. The usual definitions of momenta and of Hamiltonian make the Hamiltonian a function of the functionally dependent canonical variables only. Introduction of the momentum variables into the subsidiary conditions on the coordinates and velocities yields under certain circumstances additional subsidiary conditions on the canonical variables only. All the subsidiary conditions on the canonical variables are determined before setting up the multiplier rule for the canonical equations of motion. The multiplier rule is exploited to deduce the invariant system among the subsidiary conditions, the explicit modifications of the canonical equations by the other susidiary conditions, and Dirac's formula for the corresponding modified Poisson brackets. The modifications are caused by the reduction in the number of independent canonically conjugate pairs. A canonical transformation adapted to the subsidiary conditions, which is found with the help of Lie's theory of function groups, transforms the canonical system from the multiplier rule into a canonical system in terms of physical variables. The invariant system is then used to reduce the order of the resulting canonical system by following Levi‐Civita. This reduced canonical system is suitable for integration or quantization.
154 citations
TL;DR: In this article, a canonical transformation that takes the usual Yang-Mills action into one whose Feynman diagram expansion generates the MHV rules was constructed, where the off-shell continuation appears as a natural consequence of using light-front quantisation surfaces.
Abstract: We construct a canonical transformation that takes the usual Yang-Mills action into one whose Feynman diagram expansion generates the MHV rules. The off-shell continuation appears as a natural consequence of using light-front quantisation surfaces. The construction extends to include massless fermions.
145 citations
TL;DR: In this article, the canonical forms into which any pure three-qubit state can be cast are analyzed and the minimal forms, i.e., the ones with the minimal number of product states built from local bases, are also presented and lead to a complete classification of pure threequbit states.
Abstract: In this paper we analyse the canonical forms into which any pure three-qubit state can be cast. The minimal forms, i.e. the ones with the minimal number of product states built from local bases, are also presented and lead to a complete classification of pure three-qubit states. This classification is related to the values of the polynomial invariants under local unitary transformations by a one-to-one correspondence.
135 citations
TL;DR: The canonical transformation theory of Yanai and Chan as mentioned in this paper provides a rigorously size-extensive description of dynamical correlation in multireference problems, and a new formulation of the theory based on the extended normal ordering procedure of Mukherjee and Kutzelnigg is presented.
Abstract: The canonical transformation theory of Yanai and Chan [J. Chem. Phys. 124, 194106 (2006)] provides a rigorously size-extensive description of dynamical correlation in multireference problems. Here we describe a new formulation of the theory based on the extended normal ordering procedure of Mukherjee and Kutzelnigg [J. Chem. Phys. 107, 432 (1997)]. On studies of the water, nitrogen, and iron oxide potential energy curves, the linearized canonical transformation singles and doubles theory is competitive in accuracy with some of the best multireference methods, such as the multireference averaged coupled pair functional, while computational timings (in the case of the iron oxide molecule) are two to three orders of magnitude faster and comparable to those of the complete active space second-order perturbation theory. The results presented here are greatly improved both in accuracy and in cost over our earlier study as the result of a new numerical algorithm for solving the amplitude equations.
135 citations
TL;DR: In this article, the basic properties of the Backlund transformations (BT's) from the Hamiltonian point of view were revised from the perspective of the periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system.
Abstract: Using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system as examples, we revise the basic properties of the Backlund transformations (BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variablefor the parameter λ of the BT Bsuch thatbelongs to the spectrum of the Lax operator L(λ). As a consequence, the generating function of the composition B�1 ◦ . . . ◦ Bn of n BT's gives rise also to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized byis introduced.
134 citations