scholarly journals Gauge Explicit Quantum Mechanics and Perturbation Theory

1997 ◽  
Vol 50 (5) ◽  
pp. 869
Author(s):  
A. M. Stewart
Keyword(s):  
Perturbation Theory ◽  
Quantum Mechanics ◽  
Wave Equation ◽  
Statistical Mechanics ◽  
Gauge Invariant

A version of quantum and statistical mechanics, including perturbation theory, is described in which explicit electromagnetic gauge arbitrariness is maintained at every stage. Any gauge may be used for a calculation provided that the wave equation operator is gauge invariant.

scholarly journals Perturbation theory in quantum mechanics

1929 ◽  
Vol 122 (790) ◽  
pp. 589-598 ◽  
Keyword(s):  
Perturbation Theory ◽  
Quantum Mechanics ◽  
Wave Equation ◽  
Existence Theorem ◽  
Infinite Number ◽  
Matrix Method ◽  
Stark Effect ◽  
Electronic Systems ◽  
Linear Differential ◽  
Asymptotic Character

The theory of perturbations of electronic systems has been considered by various writers, and the method is the basis of the solution of most quantum problems. It therefore seems desirable that the theory should be placed on a firm footing, and this necessitates the discussion of the convergence of the series of perturbations, which up to the present has been avoided. This investigation is all the more necessary since the wave equation of the Stark Effect has no solution of the form assumed by Schrödinger in his original discussion of the perturbation theory. Dirac’s theory leads to a system of linear differential equations in an infinite number of variables, and 2 is devoted to the discussion of the existence theorem for such a system by a matrix method. In 3 the application is made to the perturbation theory in the case of perturbations satisfying the conditions assumed in 2. In 4 the theory is extended so as to include more general types of disturbances. It is shown that, though the series of perturbations does not in general converge, yet it usually possesses the same asymptotic character as in the classical theory, and its use can therefore be justified.

scholarly journals The internal conversion of gamma-rays.—Part II

1928 ◽  
Vol 121 (787) ◽  
pp. 447-456
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Wave Equation ◽  
Gamma Rays ◽  
Internal Conversion ◽  
Scalar Potential ◽  
X Ray ◽  
Electro Magnetic Field ◽  
Γ Rays ◽  
Electro Magnetic

In a previous paper the absorption of γ-rays in the K-X-ray levels of the atom in which they are emitted was calculated according to the Quantum Mechanics, supposing the γ-rays to be emitted from a doublet of moment f ( t ) at the centre of the atom. The non-relativity wave equation derived from the relativity wave equation for an electron of charge — ε moving in an electro-magnetic field of vector potential K and scalar potential V is h 2 ∇ 2 ϕ + 2μ ( ih ∂/∂ t + εV + ih ε/μ c (K. grad)) ϕ = 0. (1) Suppose, however, that K involves the space co-ordinates. Then, (K. grad) ϕ ≠ (grad . K) ϕ , and the expression (K . grad) ϕ is not Hermitic. Equation (1) cannot therefore be the correct non-relativity wave equation for a single electron in an electron agnetic field, and we must substitute h 2 ∇ 2 ϕ + 2μ ( ih ∂/∂ t + εV) ϕ + ih ε/ c ((K. grad) ϕ + (grad. K) ϕ ) = 0. (2)

scholarly journals A relativistic basis of the quantum theory.—II

1934 ◽  
Vol 145 (855) ◽  
pp. 645-656 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Wave Equation ◽  
Quantum Theory ◽  
General Expression ◽  
Matrix Equation ◽  
Covariant Derivative ◽  
Riemannian Geometry ◽  
Order Equation ◽  
Second Order ◽  
Space Curvature

The paper is a continuation of the last paper communicated to these 'Proceedings.' In that paper, which we shall refer to as the first paper, a more general expression for space curvature was obtained than that which occurs in Riemannian geometry, by a modification of the Riemannian covariant derivative and by the use of a fifth co-ordinate. By means of a particular substitution (∆ μσ σ = 1/ψ ∂ψ/∂x μ ) it was shown that this curvature takes the form of the second order equation of quantum mechanics. It is not a matrix equation, however but one which has the character of the wave equation as it occurred in the earlier form of the quantum theory. But it contains additional terms, all of which can be readily accounted for in physics, expect on which suggested an identification with energy of the spin.

scholarly journals A MANIFESTLY GAUGE INVARIANT AND UNIVERSAL CALCULUS FORSU(N) YANG–MILLS

2006 ◽  
Vol 21 (23n24) ◽  
pp. 4627-4761 ◽  
Author(s):  
OLIVER J. ROSTEN
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Explicit Dependence ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Group A ◽  
Β Function ◽  
Exact Renormalization Group

Within the framework of the Exact Renormalization Group, a manifestly gauge invariant calculus is constructed for SU (N) Yang–Mills. The methodology is comprehensively illustrated with a proof, to all orders in perturbation theory, that the β function has no explicit dependence on either the seed action or details of the covariantization of the cutoff. The cancellation of these nonuniversal contributions is done in an entirely diagrammatic fashion.

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