scholarly journals Conformal gauges and renormalized equations of motion in massless quantum electrodynamics

1985 ◽  
Vol 97 (1-2) ◽  
pp. 227-255 ◽  
Author(s):  
V. B. Petkova ◽  
G. M. Sotkov ◽  
I. T. Todorov
Keyword(s):  
Quantum Electrodynamics ◽  
Equations Of Motion ◽  
Massless Quantum

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 Quaternionic wave function

2019 ◽  
Vol 34 (02) ◽  
pp. 1950001 ◽  
Author(s):  
Pavel A. Bolokhov
Keyword(s):  
Wave Function ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Variational Calculus ◽  
Nonrelativistic Limit ◽  
Natural Choice ◽  
Complex Wave ◽  
Spinor Formalism ◽  
Function Language

We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.

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.

New anomaly. Nonvanishing emission and scattering of longitudinal photons in massless quantum electrodynamics

1989 ◽  
Vol 227 (3-4) ◽  
pp. 474-478 ◽  
Author(s):  
A.S. Gorsky ◽  
B.L. Ioffe ◽  
A.Yu. Khodjamirian
Keyword(s):  
Quantum Electrodynamics ◽  
Massless Quantum

Operator regularization beyond lowest order

1988 ◽  
Vol 66 (3) ◽  
pp. 268-278 ◽  
Author(s):  
D. G. C. McKeon ◽  
S. S. Samant ◽  
T. N. Sherry
Keyword(s):  
Quantum Electrodynamics ◽  
Loop Order ◽  
Point Function ◽  
Scalar Theory ◽  
Final Model ◽  
Component Field ◽  
Field Formalism ◽  
Massless Quantum ◽  
Limiting Value ◽  
Zumino Model

We pursue operator regularization beyond lowest order. In lowest order, it is the determinants of operators that are regulated; beyond lowest order it is the inverses of operators. As in lowest order, operator regularization to two-loop order and beyond avoids explicit infinities both in the integrals that are evaluated and in the regulating parameter s as it approaches its limiting value of zero. Operator regularization also replaces the Feynman diagrammatic expansion with an expansion due to Schwinger. No explicit symmetry-breaking regulating parameter is inserted into the original Lagrangian. We illustrate our technique by examining the two-point function in [Formula: see text] scalar theory, the effective potential in [Formula: see text] scalar theory, the vacuum polarization in massless quantum electrodynamics, and the two-point function in the Wess–Zumino model using both the superfield and component-field formalism. In all cases we find expressions that are divergence free and remain finite as the regulating parameter approaches its limiting value. In the final model we explicitly show that the supersymmetry Ward identity for the two-point functions is satisfied to two-loop order.

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