an introduction to ringand modules

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104) invariant. 105) are called restricted gauge transformations. 104) are manifestly covariant under the Lorentz transformations. This gauge is often taken as a starting point for the formulation of relativistic quantum electrodynamics. 1. 106) 'V·A=O It is always possible to satisfy this condition. 106). 108) = -t:o-1 Pe v 1 8\i'V . 108) which, under the name of Poisson's equation, describes the electrostatic scalar potential. 108) is V(x, t) = _1_/ 4nt:o . llO) Over all space, V varies instantaneously when Pe varies.

In condensed matter physics, it is normal to describe to first approximation the excitations of the system (such as those of the phonons) as that of a collection of independent quasi-particles. Finally, in the mean-field method, the wavefunction for n particles is approximated by a product of appropriate single-particle states. The one-particle states are the "building blocks" of the construction of the second quantification formalism of Chap. 3. The collection of states of a quantum particle form a Hilbert space H called the single-particle space.

This symmetry plays a much more fundamental role in quantum physics than in classical physics. In classical dynamics, two identical particles can nevertheless be distinguished over the course of time if we follow their trajectories. These trajectories are continuous and uniquely specified by the initial conditions: the particles, although identical, are distinguishable. This is no longer the case in quantum mechanics, where identification of the particles by their initial conditions and trajectories is not possible, even in principle: the particles become indistinguishable.

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