By Professor Philippe A. Martin, Professor François Rothen (auth.)
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Extra resources for Many-Body Problems and Quantum Field Theory: An Introduction
F3. Note that u" 01 = u 11 + u 22 + u 33 identifies with the divergence V' · u of the displacement. 182) The Lame coefficients >. and f-1 satisfy the inequalities 36 1. 18:~) and which assure stability in the event of an infinitesimal deformation. 184) where, by construction, the trace of the tensor iY"f3 vanishes. 183). 182). 183). 197) The longitudinal deformations have an important characteristic: They are generally associated with a compression and an expansion, which is not the case for the transverse deformations.
The Coulomb gauge is chosen and, for simplicity, it is assumed that each particle has the same charge e and mass m. 119) and the equations that follow, qi = qi(t) and vi = vi(t), for i = 1, 2, ... , n, are the positions and velocities of the n particles at time t, respectively. 120) The term j = i, which is constant (and infinite), corresponds to the selfenergy of a moving charge; it is omitted in this sum. 120) does not explicitly depend on time. Its time dependence comes through that of the particle coordinates.
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.