Regularization of the equations of the quantum theory for a scalar neutral field with selfinteraction in the case of two spatial degrees of freedom

1971 ◽  
Vol 9 (3) ◽  
pp. 1191-1198 ◽  
Author(s):  
L. G. Zastavenko
Keyword(s):  
Quantum Theory ◽  
Degrees Of Freedom ◽  
Neutral Field ◽  
Scalar Neutral Field ◽  
Spatial Degrees Of Freedom

scholarly journals Semi-classical approximations based on Bohmian mechanics

2020 ◽  
Vol 35 (14) ◽  
pp. 2050070 ◽  
Author(s):  
Ward Struyve
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
Quantum Gravity ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Bohmian Mechanics ◽  
Expectation Values ◽  
Usual Approach ◽  
Classical Approximation

Semi-classical theories are approximations to quantum theory that treat some degrees of freedom classically and others quantum mechanically. In the usual approach, the quantum degrees of freedom are described by a wave function which evolves according to some Schrödinger equation with a Hamiltonian that depends on the classical degrees of freedom. The classical degrees of freedom satisfy classical equations that depend on the expectation values of quantum operators. In this paper, we study an alternative approach based on Bohmian mechanics. In Bohmian mechanics the quantum system is not only described by the wave function, but also with additional variables such as particle positions or fields. By letting the classical equations of motion depend on these variables, rather than the quantum expectation values, a semi-classical approximation is obtained that is closer to the exact quantum results than the usual approach. We discuss the Bohmian semi-classical approximation in various contexts, such as nonrelativistic quantum mechanics, quantum electrodynamics and quantum gravity. The main motivation comes from quantum gravity. The quest for a quantum theory for gravity is still going on. Therefore a semi-classical approach where gravity is treated classically may be an approximation that already captures some quantum gravitational aspects. The Bohmian semi-classical theories will be derived from the full Bohmian theories. In the case there are gauge symmetries, like in quantum electrodynamics or quantum gravity, special care is required. In order to derive a consistent semi-classical theory it will be necessary to isolate gauge-independent dependent degrees of freedom from gauge degrees of freedom and consider the approximation where some of the former are considered classical.

scholarly journals The quantum theory of the emission and absorption of radiation

1927 ◽  
Vol 114 (767) ◽  
pp. 243-265 ◽  
Keyword(s):  
Quantum Theory ◽  
Radiation Field ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Hamiltonian Function ◽  
Correct Treatment ◽  
Principle Of Relativity ◽  
Satisfactory Theory ◽  
General Method

The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed sufficiently to form a fairly complete theory of dynamics. One can treat mathematically the problem of any dynamical system composed of a number of particles with instantaneous forces acting between them, provided it is describable by a Hamiltonian function, and one can interpret the mathematics physically by a quite definite general method. On the other hand, hardly anything has been done up to the present on quantum electrodynamics. The questions of the correct treatment of a system in which the forces are propagated with the velocity of light instead of instantaneously, of the production of an electromagnetic field by a moving electron, and of the reaction of this field on the electron have not yet been touched. In addition, there is a serious difficulty in making the theory satisfy all the requirements of the restricted principle of relativity, since a Hamiltonian function can no longer be used. This relativity question is, of course, connected with the previous ones, and it will be impossible to answer any one question completely without at the same time answering them all. However, it appears to be possible to build up a fairly satisfactory theory of the emission of radiation and of the reaction of the radiation field on the emitting system on the basis of a kinematics and dynamics which are not strictly relativistic. This is the main object of the present paper. The theory is noil-relativistic only on account of the time being counted throughout as a c-number, instead of being treated symmetrically with the space co-ordinates. The relativity variation of mass with velocity is taken into account without difficulty. The underlying ideas of the theory are very simple. Consider an atom interacting with a field of radiation, which we may suppose for definiteness to be confined in an enclosure so as to have only a discrete set of degrees of freedom. Resolving the radiation into its Fourier components, we can consider the energy and phase of each of the components to be dynamical variables describing the radiation field. Thus if E r is the energy of a component labelled r and θ r is the corresponding phase (defined as the time since the wave was in a standard phase), we can suppose each E r and θ r to form a pair of canonically conjugate variables. In the absence of any interaction between the field and the atom, the whole system of field plus atom will be describable by the Hamiltonian H ═ Σ r E r + H o equal to the total energy, H o being the Hamiltonian for the atom alone, since the variables E r , θ r obviously satisfy their canonical equations of motion E r ═ — ∂H/∂θ r ═ 0, θ r ═ ∂H/∂E r ═ 1.

The Motion of the Human Center of Mass and its Relationship to Mechanical Impedance

1966 ◽  
Vol 8 (5) ◽  
pp. 399-405 ◽  
Author(s):  
Edmund B. Weis ◽  
Frank P. Primiano
Keyword(s):  
Transfer Function ◽  
Mechanical System ◽  
Degrees Of Freedom ◽  
Center Of Mass ◽  
Mechanical Impedance ◽  
Second Order ◽  
Analysis Tool ◽  
Isotropic Theory ◽  
Spatial Degrees Of Freedom

This report concerns the development of a relationship between the human mechanical impedance and the coupling of the human center of mass to the environment. The mechanical impedance is a common analysis tool in biomechanics while the analysis of the coupling of the center of mass to the environment is technically more difficult, if not impossible. The development is based on linear, passive, isotropic theory and shows that the transfer function which expresses the relation between the motion of the center of mass and the motion of the source is similar to a linear second order mechanical system in each of the translational spatial degrees of freedom.

Dynamics of bell-nonlocality for two atoms interacting with a vacuum multi-mode noise field

2016 ◽  
Vol 14 (03) ◽  
pp. 1650011 ◽  
Author(s):  
Yu-Jie Liu ◽  
Li Zheng ◽  
Dong-Mei Han ◽  
Huan-Lin Lü ◽  
Tai-Yu Zheng
Keyword(s):  
Degrees Of Freedom ◽  
Internal State ◽  
Bell Inequality ◽  
Entanglement Dynamics ◽  
Noise Field ◽  
Chsh Inequality ◽  
Internal States ◽  
Bell Nonlocality ◽  
Multi Mode ◽  
Spatial Degrees Of Freedom

We investigate the internal-state Bell nonlocal entanglement dynamics, as measured by CHSH inequality of two atoms interacting with a vacuum multi-mode noise field by taking into account the spatial degrees of freedom of the two atoms. The dynamics of Bell nonlocality of the atoms with the atomic internal states being initially in a Werner-type state is studied, by deriving the analytical solutions of the Schrödinger equation, and tracing over the degrees of freedom of the field and the external motion of the two atoms. In addition, through comparison with entanglement as measured by concurrence, we find that the survival time of entanglement is much longer than that of the Bell-inequality violation. And the comparison of the quantum correlation time between two Werner-type states is discussed.

scholarly journals The interaction of electric charges

1930 ◽  
Vol 126 (803) ◽  
pp. 696-728 ◽  
Keyword(s):  
Quantum Theory ◽  
Gauge Transformation ◽  
Degrees Of Freedom ◽  
Frame Of Reference ◽  
Degree Of Freedom ◽  
Classical Dynamics ◽  
Proper Distance ◽  
Special Degree ◽  
Odd Degree ◽  
Absolute Quantity

1. The present investigation is a sequel to two earlier papers in these 'Proceedings.' In II a theory of the electron was proposed which led to a prediction of the value of the constant 2π e 2 / hc . The theory involved an appeal to the analogies of classical dynamics, which frequently prove useful though precarious; it has been my purpose to substitute a more satisfactory geometrical basis. In a problem of this kind, concerned with the whole question of the significance of the methods of quantum theory, it is unlikely that finality can have been reached even at the second attempt; but I think that the progress is now sufficient to justify publication. According to II the value of hc /2π e 2 was 136. I remarked that, as it represented the number of degrees of freedom of a system, small mistakes were unlikely; nevertheless I appear to have made such a mistake, and the new prediction is 137 (16). The 136 symmetrical degrees of freedom are a generalisation of rotations and translations in space; but it is characteristic of a pair of electrons that they possess one special degree of freedom unlike the others which has no analogue in the theory of a single electron, viz., an alteration of the proper distance between them; and whereas the 136 rotations are relative to the frame of reference employed, the odd degree of freedom represents alteration of an absolute quantity (the interval). The mistake in the earlier theory was not so much in overlooking this degree of freedom (for it there appeared as the rotation which "interchanges the identity of the two electrons") as in not recognising its distinctness from the others. No one of the 136 relativity transformations can play the part of this non-relativity or gauge transformation.

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