Bose field

This condition applies not only to the vacuum state, but to any arbitrary state which can only mean that aj/zt = 0. To incorporate this condition into the previous set of commutation rules for Bose fields, it is sufficient to change the negative into positive signs, such that... 

It is suggestive that the narrow Kondo resonance states of individual 4f impurities will form heavy quasiparticle bands in a periodic lattice of 4f ions. A satisfactory microscopic theory of heavy-band formation has yet to be developed. The Hamiltonian of eq. (107) can be generalized to the lattice by introducing a Bose field h,- at every lattice site. However, in this model it is no longer practicable to restrict to physical states with = 1 at every site. The most successful approach so far consists in a mean-field approximation for the Bose field (Coleman 1985, 1987, Newns and Read 1987) that is valid for large N and r < It can be applied both for the impurity and the lattice model. It starts from the observation that in the limit with QJN= fixed, the rescaled... 

Bose fields bf = bjVN become classical variables. The square (h,) is the probability of having a 4f° configuration at site i and its availability for hybridization with conductioi electrons. Therefore they can be considered as moving in a classical field a . = (6,) governed by an effective single-particle Hamiltonian ... 

Next, in the continuum limit, the radial gauge is introduced by representing the Bose fields by modulus and phase. 

It may be remarked that since the bosons are taken to be real, their kinetic terms, being proportional to the time derivatives in Eq. 16, drop out due to the periodic boundary conditions imposed on Bose fields ( ()S) = < iot(0)). Strictly speaking it follows from this property that all the Bose fields do no longer have dynamics of their own [25]. 

All these multifarious activities took a lot of Einstein s energies but did not keep him from his physics research. In 1922 he published Ins first paper on unified field theoiy, an attempt at incorporating not only gravitation but also electromagnetism into a new world geometry, a subject that was his main concern until the end of his life. He tried many approaches none of them have worked out. In 1924 he published three papers on quantum statistical mechanics, which include his discoveiy of so-called Bose-Einstein condensation. This was his last contribution to physics that may be called seminal. He did continue to publish all through his later years, however. 

One can see the physical meaning of the operator Y for the case where the field is in a thermal equilibrium state. Indeed, by taking the ensemble average with a Bose-Einstein distribution of field modes at temperature T,... 

The broken-line portion of the v+/(Zeg) curve, which attains a maximum and then falls, was explained by Bose, Paul and Tsai (1981) in terms of the formation of positronium due to positron heating in the electric field, so that the apparent value of Z s) rises as the amount of positronium formation increases. At high electric fields nearly all the positrons form positronium and do not annihilate at the foil. 

Comparable low temperatures have also been obtained by trapping gaseous particles in magnetic fields and lowering their velocity by absorption and reemission of laser energy. Using these methods, a new state of matter, the Bose-Einstein condensate has been created. 

The FrBhlich vibrational model does not address itself directly to ihe problem of the interaction between an external EM-field and the dissipative subsystem, but rather to ihe internal redistribution of photons and phonons upon excitation of the Bose-condensation state. In particular, the frequencies, oo, of the (coherent) vibrations are availalbe only within the framework of a... 

If we now suggest a correspondence between the formal model of FrBhlich and the dissipative subsytem(s) and the equilibrium system discussed in Section 3.1, one may consider the processes which are operative, including interactions with an external EM field, as depicted schematically in Fig. (7). No compelling experimental evidence exists that the equilibrium system made up of the ordinary aqueous dielectrics, Sections 2.3 to 2.7, shows the Bose-condensation which arises in the Frtfhlich theory. In fact, even systems of sufficient complexity to exhibit low-lying vibrational modes and structural subtlety to play a direct role in biochemical reactions at the interface between biochemical and biological processes [a-chymotrypsin (90-91), lysozyme (92 ) and DNA (93)], fail to show features not predicted by the methods of Section 2. Since collisional perturbations, even when non-reactive, will provide a source of the energy inputs, S, this implies the absence of non-linear terms (x=0,A=0), Eq. (14), for such systems. 

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