Goldstone theorem

The 4>2 particle is known as a Goldstone boson. The important point is that this phenomenon is general. The Goldstone theorem [48] which states that massless scalars occur whenever a continuous symmetry of a physical system is spontaneously broken (or, more accurately, is not apparent in the ground state) will be accepted without further proof, however, compare [49],... 

Apart from subtle exceptions, an isolated molecule differs from a molecule in a crystal in that the isolated molecule has no shape, whereas in a crystal it acquires shape, but loses its identity as an independent entity. This paradoxical situation is best understood through the famous Goldstone theorem, which for the present purpose is interpreted to state that any phase transition, or symmetry broken, is induced by a special interaction. When a molecule is introduced into an environment of other molecules of its own kind, a phase transition occurs as the molecule changes its ideal (gas) behaviour to suit the non-ideal conditions, created by the van der Waals interaction with its neighbours. An applied electric or magnetic field may induce another type of transformation due to polarization of the molecular charge density, which may cause alignment of the nuclei. When the field is switched off the inverse transformation happens and the structure disappears. The Faraday effect (6.2.3) is one example. 

As far as the continuous symmetry breaking is concerned, the Goldstone theorem states [4] that this will generate hydrodynamic modes, that is, gapless excitations. The order parameter is multicomponent (n > 1) a vector i for magnetism breaking the rotational symmetry, a complex variable i j = ip e/e for charge-density waves and for superconductivity which... 

According to the Goldstone theorem [79] the three real fields Pi x) would introduce three massless Goldstone bosons to the theory. These can, however, be gauged away if one takes advantage of the local SU 2)i, gauge symmetry given by... 

Where the vacuum does not exhibit the same invariance of the Lagrangian, one says that a spontaneous breakdown of the symmetry occurs. General theorems have been derived which elucidate the dynamical consequences on the whole system of the interplay between a symmetric Lagrangian and a nonsymmetric vacuum. The Goldstone theorem is a... 

For nematic liquid crystals, the synunetry is reduced and we need additional variables. The nematic is degenerate in the sense that all equilibrium orientations of the director are equivalent. According to the Goldstone theorem the parameter of degeneracy is also a hydrodynamic variable for a long distance process 0 and the relaxation time should diverge, x—>oo. In nematics, this parameter is the director n(r), the orientational part of the order parameter tensor. For a finite distortion of the director over a large distance (L—>oo), the distortion wavevector 0 and the... 

Spontaneous symmetry breaking the Goldstone theorem and the Higgs phenomenon... 

We may recall that the desirability of ensuring size-extensivity for a closed-shell state was one of the principal motivations behind the formulation of the MBPT for the closed-shells. The linked cluster theorem of Bruckner/25/, Goldstone/26/ and Hubbard/27/, proving that each term in the perturbation series for energy can be represented by a linked (connected) diagram directly reflects the size-extensivity of the theory. Hubbard/27/ and Coester/30/ even pointed out immediately after the inception of MBPT/25,26/, that the size-extensivity is intimately related to a cluster expansion structure of the associated wave-operator that is not just confined only to perturbative theory. The corresponding non-perturbative scheme for the closed-shells was first described by Coester and Kummel/30,31/ in nuclear physics and this was transcribed to quantum chemistry... 

The 1inked-cluster theorem for energy, from the above analysis, is a consequence of the connectivity of T, and the exponential structure for ft. Size-extensivity is thus seen as a consequence of cluster expansion of the wave function. Specfic realizations of the situation are provided by the Bruckner—Goldstone MBPT/25,26/, as indicated by Hubbard/27/, or in the non-perturbative CC theory as indicated by Coester/30,31/, Kummel/317, Cizek/32/, Paldus/33/, Bartlett/21(a)/ and others/30-38/. There are also the earlier approximate many-electron theories like CEPA/47/, Sinanoglu s Many Electron Theory/28/ or the Cl methods with cluster correction /467. 

Goldstone s approach exploited a time-dependent PT in the interaction picture and was based on the Gell-Mann and Low adiabatic theorem [42], as was the work of Hubbard... 

In the Goldstone program below [cf. Section 4], Wick s theorem is fulfilled implicitly by performing a pairwise permutation of the operators until all operators are in normal order. This ensures that all contractions are taken into account easily. Of course, this step-wise procedure gives the same results as a more sophisticated implementation of Wick s theorem which, in particular, is helpful for large operator strings. 

The above are examples of a general theorem due to Goldstone (1961 see also Jona-Lasinio and Nambu, 1961 a, b Goldstone, Salam and Weinberg, 1962) and are not linked specifically to the particular group 0(n) for every broken generator in a spontaneous symmetry breaking there exists a massless scalar boson. 

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