scholarly journals Quantum Electrodynamics in Strong Magnetic Fields. I. Electron States

1983 ◽  
Vol 36 (6) ◽  
pp. 755 ◽  
Author(s):  
DB Melrose ◽  
AJ Parle
Keyword(s):  
Vertex Function ◽  
Quantum Electrodynamics ◽  
Laguerre Polynomials ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Coordinate Space ◽  
Positron States ◽  
Symmetry Properties ◽  
Moment Operator ◽  
Independent Quantity

Dirac's equation in the presence of a static magnetic field is solved in terms of both cartesian and cylindrical coordinates, and solutions are found for three different spin operators. Choosing the spin to correspond to the parallel component Ilz of the magnetic moment operator leads to wavefunctions (a) which are symmetric between electron and positron states and (b) which are eigenfunctions of the Hamiltonian including radiative corrections. A vertex function [y:',~(k)l~ is defined and shown to be proportional to a gauge independent quantity [T::~(k)l". Symmetry properties of [T~:~(k)l~ are derived in the case where the spin corresponds to Ilz. The use of the vertex function is illustrated by deriving the electron propagator in coordinate space from the vacuum expectation value. Properties of functions J~, _n(x) which appear extensively and are related to generalized Laguerre polynomials are derived and summarized in the Appendix.

scholarly journals Quantum Electrodynamics in Strong Magnetic Fields. II. Photon Fields and Interactions

1983 ◽  
Vol 36 (6) ◽  
pp. 775 ◽  
Author(s):  
DB Melrose
Keyword(s):  
Quantum Electrodynamics ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Wave Dispersion ◽  
Wave Interactions ◽  
Strong Magnetic Fields ◽  
Nonlinear Responses ◽  
S Matrix ◽  
Photon Interactions ◽  
Covariant Version

A theory is developed which synthesizes the classical theory of wave-wave interactions in a medium and the QED theory of photon-photon interactions in a vacuum with or without static fields. First, a covariant version of the theory of wave dispersion in an arbitrary medium is outlined. Then the photon propagator is calculated both by solving the wave equation directly and from the vacuum expectation value, and the latter is generalized by introducing a statistically averaged propagator. The hierarchy of nonlinear responses of the medium is introduced and used to define a nonlinear interaction Hamiltonian. The expansion of the S matrix with this interaction Hamiltonian is re-ordered to correspond to a hierarchy of processes ordered according to the number of external photons. The theory is used to treat three-wave and four-wave interactions and both classical and QED applications are discussed. A covariant form of the Onsager relations and calculations of the hierarchy of response tensors using covariant classical kinetic theory and the Heisenberg-Euler Lagrangian are presented in appendices.

Global symmetries and Noether’s theorem

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Global Symmetries ◽  
Symmetry Transformation ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Quantum Field ◽  
Expectation Value ◽  
Minkowski Space Time ◽  
Integral Measure ◽  
Colour Charge ◽  
Internal Symmetries

Noethers theorem shows that continuous global symmetries lead classically to conservation laws. Such symmetries can be divided into spacetime and internal symmetries. The invariance of Minkowski space-time under global Poincaré transformations leads to the conservation of the four-momentum and the total angular momentum. Examples for conserved charges due to internal symmetries are electric and colour charge. The vacuum expectation value of a Noether current is shown to beconserved in a quantum field theory if the symmetry transformation keeps the path-integral measure invariant.

scholarly journals Field Theoretical Approach for Signal Detection in Nearly Continuous Positive Spectra II: Tensorial Data

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 795
Author(s):  
Vincent Lahoche ◽  
Mohamed Ouerfelli ◽  
Dine Ousmane Samary ◽  
Mohamed Tamaazousti
Keyword(s):  
Principal Component Analysis ◽  
Renormalization Group ◽  
Signal Detection ◽  
Detection Threshold ◽  
Principal Component ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Component Analysis ◽  
Slight Generalization ◽  
Nearly Continuous

The tensorial principal component analysis is a generalization of ordinary principal component analysis focusing on data which are suitably described by tensors rather than matrices. This paper aims at giving the nonperturbative renormalization group formalism, based on a slight generalization of the covariance matrix, to investigate signal detection for the difficult issue of nearly continuous spectra. Renormalization group allows constructing an effective description keeping only relevant features in the low “energy” (i.e., large eigenvalues) limit and thus providing universal descriptions allowing to associate the presence of the signal with objectives and computable quantities. Among them, in this paper, we focus on the vacuum expectation value. We exhibit experimental evidence in favor of a connection between symmetry breaking and the existence of an intrinsic detection threshold, in agreement with our conclusions for matrices, providing a new step in the direction of a universal statement.

scholarly journals Vacuum Polarization in a Zero-Width Potential: Self-Adjoint Extension

Universe ◽  
2021 ◽  
Vol 7 (5) ◽  
pp. 127
Author(s):  
Yuri V. Grats ◽  
Pavel Spirin
Keyword(s):  
Scalar Field ◽  
Vacuum Polarization ◽  
Dimensional Regularization ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Field Approximation ◽  
Dimensional Euclidean Space ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Adjoint Extension

The effects of vacuum polarization associated with a massless scalar field near pointlike source with a zero-range potential in three spatial dimensions are analyzed. The “physical” approach consists in the usage of direct delta-potential as a model of pointlike interaction. We use the Perturbation theory in the Fourier space with dimensional regularization of the momentum integrals. In the weak-field approximation, we compute the effects of interest. The “mathematical” approach implies the self-adjoint extension technique. In the Quantum-Field-Theory framework we consider the massless scalar field in a 3-dimensional Euclidean space with an extracted point. With appropriate boundary conditions it is considered an adequate mathematical model for the description of a pointlike source. We compute the renormalized vacuum expectation value ⟨ϕ2(x)⟩ren of the field square and the renormalized vacuum averaged of the scalar-field’s energy-momentum tensor ⟨Tμν(x)⟩ren. For the physical interpretation of the extension parameter we compare these results with those of perturbative computations. In addition, we present some general formulae for vacuum polarization effects at large distances in the presence of an abstract weak potential with finite-sized compact support.

AN IMPROVED ONE-LOOP ANALYSIS OF THE λϕ4 THEORY AT FINITE TEMPERATURE

1987 ◽  
Vol 02 (03) ◽  
pp. 713-728 ◽  
Author(s):  
SWEE-PING CHIA
Keyword(s):  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Zero Temperature ◽  
Coupling Constant ◽  
Loop Analysis ◽  
Temperature Dependent ◽  
Effective Coupling ◽  
Loop Approximation ◽  
The One ◽  
Dependent Mass

The λϕ4 theory with tachyonic mass is analyzed at T ≠ 0 using an improved one-loop approximation in which each of the bare propagators in the one-loop diagram is replaced by a dressed propagator to take into account the higher loop effects. The dressed propagator is characterized by a temperature-dependent mass which is determined by a self-consistent relation. Renomalization is found to be necessarily temperature-dependent. Real effective potential is obtained, giving rise to real effective mass and real coupling constant. For T < Tc, this is achieved by first shifting the ϕ field by its zero-temperature vacuum expectation value. The effective coupling constant is found to exhibit the striking behavior that it approaches a constant nonzero value as T → ∞.

scholarly journals THE TACHYON FIELD BELOW THE MASS BARRIER

1994 ◽  
Vol 09 (20) ◽  
pp. 3497-3502 ◽  
Author(s):  
D.G. BARCI ◽  
C.G. BOLLINI ◽  
M.C. ROCCA
Keyword(s):  
Green Function ◽  
Eigenvalue Problem ◽  
Plane Wave ◽  
Time Evolution ◽  
Mass Parameter ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Tachyon Field ◽  
Expectation Value ◽  
Wave Solutions

We consider a tachyon field whose Fourier components correspond to spatial momenta with modulus smaller than the mass parameter. The plane wave solutions have then a time evolution which is a real exponential. The field is quantized and the solution of the eigenvalue problem for the Hamiltonian leads to the evaluation of the vacuum expectation value of products of field operators. The propagator turns out to be half-advanced and half-retarded. This completes the proof4 that the total propagator is the Wheeler Green function.4,7

scholarly journals Minimal scoto-seesaw mechanism with spontaneous CP violation

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
D. M. Barreiros ◽  
F. R. Joaquim ◽  
R. Srivastava ◽  
J. W. F. Valle
Keyword(s):  
Dark Matter ◽  
Cp Violation ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Discrete Symmetry ◽  
Double Beta Decay ◽  
Neutrino Masses ◽  
Seesaw Mechanism ◽  
Cosmological Observations ◽  
Scalar Singlet

Abstract We propose simple scoto-seesaw models to account for dark matter and neutrino masses with spontaneous CP violation. This is achieved with a single horizontal $$ {\mathcal{Z}}_8 $$ Z 8 discrete symmetry, broken to a residual $$ {\mathcal{Z}}_2 $$ Z 2 subgroup responsible for stabilizing dark matter. CP is broken spontaneously via the complex vacuum expectation value of a scalar singlet, inducing leptonic CP-violating effects. We find that the imposed $$ {\mathcal{Z}}_8 $$ Z 8 symmetry pushes the values of the Dirac CP phase and the lightest neutrino mass to ranges already probed by ongoing experiments, so that normal-ordered neutrino masses can be cornered by cosmological observations and neutrinoless double beta decay experiments.

THE TWISTED EUCLIDEAN GREEN’S FUNCTION IN THE SPACETIME OF A COSMIC STRING

1992 ◽  
Vol 01 (02) ◽  
pp. 371-377 ◽  
Author(s):  
B. LINET
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Minkowski Spacetime ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Massive Scalar ◽  
Elementary Functions ◽  
Integral Expression ◽  
Massive Scalar Field ◽  
Expectation Value

In a conical spacetime, we determine the twisted Euclidean Green’s function for a massive scalar field. In particular, we give a convenient form for studying the vacuum averages. We then derive an integral expression of the vacuum expectation value <Φ2(x)>. In the Minkowski spacetime, we express <Φ2(x)> in terms of elementary functions.

Functional Solution Scheme for Relativistic Strong Coupling Theor

1964 ◽  
Vol 19 (7-8) ◽  
pp. 828-834
Author(s):  
G. Heber ◽  
H. J. Kaiser
Keyword(s):  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Functional Integral ◽  
The Self ◽  
Matrix Elements ◽  
Point Function ◽  
Expectation Value ◽  
Lattice Space ◽  
S Matrix ◽  
Functional Solution

The vacuum expectation value of the S-matrix is represented, following HORI, as a functional integral and separated according to Svac=exp( — i W) ∫ D φ exp( —i ∫ dx Lw). Now, the functional integral involves only the part Lw of the Lagrangian without derivatives and can be easily calculated in lattice space. We propose a graphical scheme which formalizes the action of the operator W = f dx dy δ (x—y) (δ/δ(y))⬜x(δ/δ(x)) . The scheme is worked out in some detail for the calculation of the two-point-function of neutral BOSE fields with the self-interaction λ φM for even M. A method is proposed which under certain convergence assumptions should yield in a finite number of steps the lowest mass eigenvalues and the related matrix elements. The method exhibits characteristic differences between renormalizable and nonrenormalizable theories.

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