The symmetry of subsystem choice

The example described above raises an intriguing question, pertaining to our understanding of the relation between a part and the whole. 

There are an infinite number of ways to reconstruct the same system from parts. These ways are not equivalent in practical calculations, if for any reason we are unable to compute all the interactions in the system. However, if we have a theory (in our case the multipole method) that is able to compute the interactions, including the long-range forces, then it turns out the final result is virtually independent of the choice of unit cell motif. This arbitrariness of choice of subsystems looks analogous to the arbitrariness of the choice of coordinate stem. The final results do not depend on the coordinate stem used, but still the numerical results (as well as the effort to get the solution) do. 

The separation of the whole stem into subsystems is of key importance to many physical approaches, but we rarely think of the freedom associated with the choice. For example, an atomic nucleus does not in general represent an elementary particle, and yet in quantum mechanical calculations we treat it as a point particle, without an internal structure and we are successful. Further, in the Bo-golyubov transformation, the Hamiltonian is represented by creation and annihilation operators, each being a linear combination of the creation and annihilation 

SYMMlilKY WflH RBSPliCl JU DIVISION IN IO SUBSYSlliMS The mmetry of objects is important for the description of them, and therefore may be viewed as of limited interest. The symmetry of the laws of Nature, i.e. of the theory that describes all objects (whether symmetric or not) is much more important. This has been discussed in detail in Chapter 2 (cf. p. 61), but it seems that we did not list there a fundamental symmetry of any correct theory the symmetry with respect to the choice of subsystems. A correct theory has to describe the total system independently of what we decide to treat as subsystems. 

We will meet this problem once more in intermolecular interactions (Chapter 13). However, in the periodic stem it has been possible to use, in computational practice, the synunetry described above. 

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Several computational aspects of the calculation of interaction-induced electric properties must be considered in theoretical investigations. These concern the choice of basis set and method. In general, the symmetry of the AB supersystem might be basically different form that of the interacting subsystems A and B. In the simple case where A and B are spherical atoms, AB is of Dooh or Cqou symmetry. Basis set dependence and method sensitivity must be carefully examined. The choice of suitable basis sets is known to be a highly non-trivial matter in hyperpolarizability calculations. One expects the degree of difficulty to increase in interaction-induced hyperpolarizability calculations. Another problem of some complexity is the choice... 

Our problem resembles an excerpt from Dreams of a Final Theory by Steven Weinberg pertaining to gauge symmetry. Tire symmetry underlying it has to do with changes in our point of view about the identity of the different types of elementary particle. Tims it is possible to have a particle wave function that is neither definitely an electron nor definitely a neutrino, until we look at it. Here we have freedom in the choice of subsystems as well, and a correct theory has to reconstimte the description of the whole system. 

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