## 1 Introduction  
The spontaneous symmetry breakage is one of the most important problems of great interest in the solid state physics for these several decades. As already well-known, this problem is closely related, not only to the various mechanisms of crystalline magnets, but also to the BCS mechanism of the superconductivity, and even to the Higgs mechanism of the elementary particle physics [1]. The mechanism for the ferromagnetism of itinerant electrons in a conductive crystal within the mean field approximation [2] is most easy for us to understand the spontaneous symmetry breakage. At first, we start from a hypothetical paramagnetic state of itinerant electrons. It is perfectly symmetric, in the sense that un-spin electrons and down-spin ones equally occupy all of the lattice sites of the crystal, resulting in no macroscopic magnetic (spin) moment, without an externally applied magnetic field. In the next, we hypothetically assume a spatially uniform, but finite unequal, occupation. Under this condition, we estimate the total free energy of the system within the mean field theory. Finally, we determine the real value of this hypothetical finite unequal occupation, so that it will give the lowest free energy. If this lowest energy is even lower than the starting paramagnetic state, without an externally applied magnetic field, we can get a ferromagnetic state that has a spontaneous and macroscopic magnetic (spin) moment. Thus, we can get the symmetry breaking in the space of the electron spin. It should be noted that, during this symmetry breaking transition from the paramagnetic state to the ferromagnetic one, the whole system is assumed to be always in the thermal equilibrium, and hence, the speed of the transition has to be infinitely slow, according to the principle of the thermodynamics. Keeping this point in mind, let us now proceed to the optical region spectroscopy of insulating crystalline solids. In this research field, according to the rapid progress of time resolved laser techniques,real time quantum dynamics of optically created electronic excitations is gradually clarified in detail up to a pico- or femto-second time scale. This advantageous experimental technology has also been intensively applied even to the present spontaneous symmetry breaking problem. As a result, experimental and theoretical studies for this problem have been intensively developed, although it is quite a different way than mentioned above. That is, the real time quantum dynamics of the symmetry breakage. One of its typical results is the spontaneous (self-) localization of an exciton in insulating crystals. The exciton is already well known to be the most elementary optical excitation across the energy gap of insulating crystalline solids [3; 4]. Just after the optical excitation, the exciton is always in a plane-wave state extending all over the crystal. After the lattice relaxation having been completed, however, it is in a localized state, being trapped by the self-induced local lattice distortion around it, provided that the exciton-phonon coupling is short ranged and sufficiently strong. This concept was initiated by [PERSON] [5] and [PERSON] [6] independently, and also developed afterwards rather independently [4; 7]. This localization is intrinsic in the sense that it occurs without extrinsic trapping potentials, say, due to impurities in the crystal [7]. Thus, it is nothing but the spontaneous translational symmetry breakage. Usually, this self-localized exciton still remains within the energy gap of the original insulating crystal, and it is luminescent. Hence, it finally disappears after radiating another photon whose energy is a little smaller than that used for the initial excitation [8]. However, if the exciton-phonon coupling is further strong, it remains frozen as a non-luminescent localized electronic excited state with a large lattice distortion round it [9]. One can now say, it is a tiny photo-induced structural phase transition (PISPT).