Boson, definition

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

The value of a a) is always the same, but the averaging procedure differs in each case. The relations (65)-(67) are a simple consequence of the boson commutation relation [ , = 1 and the definition... 

It is obviously ideally suited to measuring the effect of the electron quantum fluctuations on the phonon frequency. What one immediately learns from Eq. (26) is that the propagator is quasistatic that is, the >m = 0 component dominates for T > co /2tt. This comes from the definition of the Matsubara frequencies for bosons [under Eq. (8)]. As far as the electrons are concerned, the atoms move very slowly (the adiabatic limit). If 2g2 gi> - g3 (see Fig. 5), the electrons are able to screen the slow lattice motion and thus soften the interactions. We are obviously interested in the 2kF phonons, which will be screened most effectively by the dominant 2kF charge response of the one-dimensional electron gas. 

By definition, the Hamiltonian of a system of identical particles is invariant under the interchange of all the coordinates of any two particles. The wave function describing the system must be either symmetric or antisymmetric under this interchange. If the particles have integer spin, the wavefunction is symmetric and the particles are called bosons if they have half-integer spins, the wavefunction is antisymmetric and the particles are fermions. Our discussion will be restricted to electrons, which are fermions. 

Having established that creation and annihilation operators are rank 1 covariant and contravariant tensors, respectively, with respect to the operator ( )L,S, we can define an rath-rank boson operator as consisting of a like number of fermion creation and annihilation operators. Then the normal product of an rath-rank boson operator is a natural definition for the irreducible tensor. 

It is of considerable interest to use the electron bubble as a probe for elementary excitations in finite boson quantum systems—that is, ( He)jy clusters [99, 128, 208, 209, 243-245]. These clusters are definitely liquid down to 0 K [46 9] and, on the basis of quantum path integral simulations [65, 155], were theoretically predicted (see Chapter II) to undergo a rounded-off superfluid phase transition already at surprising small cluster sizes [i.e., Amin = 8-70 (Table VI)], where the threshold size for superfluidity and/or Bose-Einstein condensation can be property-dependent (Section II.D). The size of the ( He)jy clusters employed in the experiments of Toennies and co-workers [242-246] and of Northby and coworkers [208, 209] (i.e., N lO -lO ) are considerably larger than Amin- In this large cluster size domain the X point temperature depression is small [199], that is, (Tx — 2 X 10 — 2 X 10 for V = lO -lO. Thus for the current... 

Here N = N+ + N0 + N- is the total number operator for the -deformed bosons, and the second-order Casimir operator is equal, by definition, to the square of the g-deformed angular momentum operator. These states have the following form... 

On the other hand, the modified boson operators and B are not components of a tensor operator in the sense of the above definitions. However, as shown in [31], one can use them to define a vector operator TjJ, as ... 

With the definition of the effective bath coordinate, the spin-boson Hamiltonian can be rewritten as... 

In the language of intensive boson operators, we simply have to double the definitions (5.6) and (5.7) to account for two families of boson operators and (f and i ), k =, 2. This method, when applied to Eq. (5.24), leads to the following potential surface ... 

Definition The p-product between two states F), Z) e Y is given by the matrix element Y p Z) where p, = diag(l, — 1,1, 1) is called the metric operator. Again, here and in the following, the upper sign refers to the fermionic case and the lower to the bosonic case. For extended states of the form (1) we introduce the following shorthand notation for the /i-product ... 

The atomic theory of matter, which was conjectured on qualitative empirical grounds as early as the sixth century BC, was shown to be consistent with increasing experimental and theoretical developments since the seventeenth century AD, and definitely proven by the quantitative explanation of the Brownian motion by Einstein and Perrin early in the twentieth century [1], It then took no more than a century between the first measurements of the electron properties in 1896 and of the proton properties in 1919 and the explosion of the number of so-called elementary particles - and their antiparticles - observed in modern accelerators to several hundred (most of which are very short lived and some, not even isolated). Today, the standard model assumes all particles to be built from three groups of four basic fermions - some endowed with exotic characteristics - interacting through four basic forces mediated by bosons - usually with zero charge and mass and with integer spin [2],... 

Snatzke G (1991) Helicity of molecules different definitions and application to circular dichroism. In Janos-check R (ed.) Chirality - From Weak Bosons to the oc-Helix, pp. 59-85. Berlin Springer. 

So far, we have discussed quantum Monte Carlo approaches to quantum phase transitions in boson and spin systems. In these systems, the statistical weight in the Monte Carlo procedure is generally positive definite, so there is no sign problem. Note that for spin systems, this is only true if there is no frustration. Frustrated spin systems in general do have a sign problem. 

Each of the improvements which could probably be incorporated into a density functional formalism would act to permit greater localization with reduced hybridization. But would it be enough Localized states are particularly difficult to describe in an ab initio formalism, not being possible even at the level of Hartree-Fock, for example. The issue of how far one can get with the evolutionary improvements discussed above and when one must seek a new approach (such as a slave-boson treatment of the Anderson model using realistic parameters determined by DF-LDA) is a very real one. Despite some progress, there is yet no definitive way from any ab initio calculation to determine whether the f states will behave as an f band metal or as an f core one. Until this can be done, we will not have truly understood the f electron problem. 

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