The Hamiltonian formulation of the Wheeler-Feynman theory of action at a distance in classical electrodynamics as carried out by Kerner, who has derived the nonrelativistic classical Hamiltonian for the interaction of two spinless charged particles interacting by means of half-retarded plus half-advanced potentials, has been generalized to obtain the corresponding relativistic classical Hamiltonian to order ${e}^{4}$ and ${c}^{\ensuremath{-}4}$. Using the procedure of Daughaday and Nigam, the classical Hamiltonian is transformed into the quantum-mechanical Hamiltonian operator.The Hamiltonian formulation of the Wheeler-Feynman theory of action at a distance in classical electrodynamics as carried out by Kerner, who has derived the nonrelativistic classical Hamiltonian for the interaction of two spinless charged particles interacting by means of half-retarded plus half-advanced potentials, has been generalized to obtain the corresponding relativistic classical Hamiltonian to order ${e}^{4}$ and ${c}^{\ensuremath{-}4}$. Using the procedure of Daughaday and Nigam, the classical Hamiltonian is transformed into the quantum-mechanical Hamiltonian operator.

Physical ReviewMay 27, 19661 citation

Nonrelativistic field theoretic setting for gravitational self-interactions

Sergio De Filippo

It is shown that a recently proposed model for the gravitational interaction in non relativistic quantum mechanics is the instantaneous action at a distance limit of a field theoretic model containing a negative energy field. It reduces to the Schroedinger-Newton theory in a suitable mean field approximation. While both the exact model and its approximation lead to estimates for localization lengths, only the former gives rise to an explicit non unitary dynamics accounting for the emergence of the classical behavior of macroscopic bodies.It is shown that a recently proposed model for the gravitational interaction in non relativistic quantum mechanics is the instantaneous action at a distance limit of a field theoretic model containing a negative energy field. It reduces to the Schroedinger-Newton theory in a suitable mean field approximation. While both the exact model and its approximation lead to estimates for localization lengths, only the former gives rise to an explicit non unitary dynamics accounting for the emergence of the classical behavior of macroscopic bodies.

arXiv:gr-qc/0105021May 7, 20010 citations

Lagrangian/Action Formalism As a Flow Scheme

Francesco R. Ruggeri

In a previous note (1), we argued that one may obtain either the Schrodinger or Klein-Gordon equation from the classical free particle Action = Integral dt L = T L(v) by writing v=X/T and using: p= dAction/dx partial and -E = dAction/dt partial. This may be used together with an energy-momentum relationship E=pp/2m (nonrelativistic) or EE=pp + mm (relativistic) to create an eigenfunction equation by using exp(i Action) as a solution. This is equivalent to exp(-iEt+ipx). In (2), we argued that if one generalizes momentum p to Ev where E(v) is energy and uses nonrelativsitic Lagrangian equations: p=dL/dv and H=E= pv-L, one may find the relativistic Lagrangian for a free particle which in turn yields p= d/dx (TL) partial and -E= d/dt (TL) partial i.e. the form of energy and momentum for a free particle in special relativity. In this brief note, we argue that the Action/Lagrangian formalism for a free particle seems to follow directly from a flow(flux) type equation namely: d/dx Action partial = p = Ev (= mo v nonrelativistic limit). If energy is flowing out of a region given by the usual Gaussian flux equation d/dx function (i.e. grad dot vector) represented by d/dx Action partial, then one is losing energy E(v) in this region. Thus d/dt Action partial = -E. As a result, one should analyze the problem in special relativity i.e. use p=E(v) v (but no explicit form for E(v)) and then take the nonrelativistic limit to get the nonrelativistic approximation. As a result, assuming a flow problem with a function Action(x,t) leads to the Lagrangian formalism. From this one may obtain the forms of special relativity, classical equations of motion and quantum mechanics for a free particle (relativistic or nonrelativistic).In a previous note (1), we argued that one may obtain either the Schrodinger or Klein-Gordon equation from the classical free particle Action = Integral dt L = T L(v) by writing v=X/T and using: p= dAction/dx partial and -E = dAction/dt partial. This may be used together with an energy-momentum relationship E=pp/2m (nonrelativistic) or EE=pp + mm (relativistic) to create an eigenfunction equation by using exp(i Action) as a solution. This is equivalent to exp(-iEt+ipx). In (2), we argued that if one generalizes momentum p to Ev where E(v) is energy and uses nonrelativsitic Lagrangian equations: p=dL/dv and H=E= pv-L, one may find the relativistic Lagrangian for a free particle which in turn yields p= d/dx (TL) partial and -E= d/dt (TL) partial i.e. the form of energy and momentum for a free particle in special relativity. In this brief note, we argue that the Action/Lagrangian formalism for a free particle seems to follow directly from a flow(flux) type equation namely: d/dx Action partial = p = Ev (= mo v nonrelativistic limit). If energy is flowing out of a region given by the usual Gaussian flux equation d/dx function (i.e. grad dot vector) represented by d/dx Action partial, then one is losing energy E(v) in this region. Thus d/dt Action partial = -E. As a result, one should analyze the problem in special relativity i.e. use p=E(v) v (but no explicit form for E(v)) and then take the nonrelativistic limit to get the nonrelativistic approximation. As a result, assuming a flow problem with a function Action(x,t) leads to the Lagrangian formalism. From this one may obtain the forms of special relativity, classical equations of motion and quantum mechanics for a free particle (relativistic or nonrelativistic).

Dec 29, 20210 citations

{e}^{4}

and ${c}^{\ensuremath{-}4}$. Using the procedure of Daughaday and Nigam, the classical Hamiltonian is transformed into the quantum-mechanical Hamiltonian operator.