## 4 Conservation Laws of the Total Momentum and Energy, Heat Reservoir, Classical Localization  
This early stage dynamics is a purely quantum mechanical one, and hence, it has to be in compliance with the conservation laws of the total momentum and the total energy, exactly. As for the total momentum, being zero from the beginning can be easily seen to be conserved from Equation (8). While, to see the total energy conservation in detail, we have to tacitly assume a direct transfer type interaction between neighboring site-localized phonons with an interaction constant (\(\equiv\Delta\omega_{0},\ 0\ <\ \Delta\omega_{0}\ <\ \omega_{0}\)). Hence, the last term of Equation (1) becomes as,  
\[\sum_{l}\left(-\frac{\partial^{2}}{\partial Q_{l}^{2}}+Q_{l}^{2}\right)\to \omega_{0}\sum_{l}\left(B_{l}^{+}B_{l}+\frac{1}{2}\right)+\Delta\omega_{0} \sum_{<l,\ l^{*}\left(\
eq l\right)>}\left(B_{l}^{+}B_{l}+h.c.\right),\ B_{l} \equiv 2^{-1/2}\left(\frac{\partial}{\partial Q_{l}}+Q_{l}\right), \tag{14}\]  
although it was not written explicitly in the stage of the Section 1.  
By this interaction, the energy difference between the FC state and the STE (shown in Figure 1) is finally radiated as a sound (or heat) wave. It is schematically denoted by a thin brown circle in Figure 3.  
Without this radiation, the symmetry breaking can never be completed. The most important point is that this radiation of phonons occurs from each STE site to infinitely distant numberless lattice sites, simultaneously and equally. If the STE is at \(l^{*}\), infinitely distant numberless lattice sites are such \(l\) s, as shown in Figure 3a, and vice versa, as shown in Figure 3b. That is, the radiation occurs only in the relative (or internal) lattice space, whose central lattice site is occupied by this STE. This is an irreversible process, since this relative space is also infinitely large. While these relative spaces are orthogonal with each other, since  
\[<0|F_{l^{*}}F_{l}^{+}|0>\ =\ 0,\ l^{*}\
eq l. \tag{15}\]  
Consequently, a superposition of states realized in each relative space can be possible, just like Equation (8), even after we have included this irreversible phonon radiation. Since this Equation (14) does not change the total energy, but only makes phonons move, we can now see the total energy is well conserved.  
Figure 2: The schematic nature of the decrease of spatial correlation of an exciton \([C(\Delta),\ \Delta\
eq 0]\), as a function of the time, after the optical excitation. [PERSON] and [PERSON] [15] have calculated the rate of this decreasing in detail, using more realistic models.  
We can now think of the usual master equation method to describe the lattice relaxation [16]. By this method, however, from the beginning, the whole system has to be clearly divided into two; a relevant system on which we focus, and a heat reservoir, which instantaneously absorbs energies released from the relevant system. By tracing out the reservoir variables, we can thus describe the relaxation dynamics of the relevant system. In the electron-phonon coupled systems, the electronic part is often regarded to be the system, while the phonon is regarded to be the reservoir. As we can easily infer from Figure 3, however, such a priori division is impossible in the present problem. The phonons at infinitely distant lattice sites from the STE may be the heat reservoir, but the central SET site, as well as these distant sites, is all in the relative space, being not fixed in the real lattice at all.  
Incidentally, long after this quantum and spontaneous localization, thus, having been completed, an ordinary classical localization may also occur, since the localized exciton can also slowly and diffusively move and will be trapped at dislocation or rare impurity sites, which unavoidably exist in the ubiquitous crystal.