Using this language in \(\S\) 2.1 the definition of the classical limit is given. 2 the algebraic formalism suitable to describe classical and quantum theories is introduced. 1.1 we introduce the concept of phase transitions and spontaneous symmetry breaking, first from a more physical point of view often used in condensed matter physics followed by a mathematical approach common in algebraic quantum theory. This paper is structured in the following way. This principle, what we call Butterfield’s Principle, is taken to be our definition of emergence. in systems described with a finite (respectively, non-zero) parameter, meaning large but finite N in the case of quantum spin systems or tiny but positive values of Planck’s constant \(\hbar\) in the case of classical systems. This can be interpreted in the sense that a robust behavior of the physical phenomenon should occur before the pertinent limit, i.e. And it is this weaker behavior which is physically real.” “There is a weaker, yet still vivid, novel and robust behavior that occurs before we get to the limit, i.e. The solution to this paradox is based on Butterfield’s second claim This does therefore not show that the limit at \(N=\infty\) is “physically real”. On the other hand, as we know from decades of experience and experimental results, emergent phenomena such as SSB or phase transitions are observed in real materials, and therefore in finite systems despite the fact that due to uniqueness principles in Theory, they seem forbidden in such systems. Indeed, theoretically speaking SSB and phase transitions can only show up in classical or infinite quantum theories seen as a classical, respectively thermodynamic limit of an underlying finite quantum theory and these features can be deduced after taking the limit of some parameter, explained in more detail in \(\S\) 2.2 (see also ). This is confirmed by two important examples one encounters in physics, namely spontaneous symmetry breaking (SSB) and phase transitions. \(\hbar \rightarrow 0\)) enables one to deduce novel and robust behavior compared to that of systems described for finite N (resp. On the one hand, we emphasize that this notion of emergence may be viewed as a form of reduction (in the sense of deduction), in that, considering \(N \rightarrow \infty\) (resp. This concept allows studying emergent phenomena inherent in nature in a mathematically rigorous way, namely by the theory of continuous field of \(C^*\)-algebras and in view of the classical limit, by strict deformation quantization. N corresponding to the size of a lattice describing quantum statistical mechanics goes to infinity, or in the spirit of the classical limit, Planck’s constant \(\hbar\) to zero. Footnote 2 In this paper we rather focus on asymptotic emergence in the context of limits, in which emergent behavior occurs in a “higher-level” theory being a limit of a sequence of “lower-level” theories, typically as some parameter, e.g. A particular case for which this mismatch leads to significant ambiguity is the manifestation of emergence, that is, according to Butterfield’s first claim, a theoretical framework that describes phenomena as having behavior that is new and robust with respect to some comparison class. However, describing Nature by means of a suitable theoretical model does not always have to give a correct result consistent with experimental predictions: it can create a serious mismatch between theory and reality. Depending on the purpose of the study such assumptions are usually well founded and therefore they (often) do not affect the description of the physical system of interest. Footnote 1 This for example already happens in the study of the hyperfine structure of the spectrum of the Hydrogen atom, where one often “neglects” the relativistic interactions which are strictly speaking always present. In any event, unless perhaps in very simple cases, one can not avoid making assumptions and thus will consider a certain model that approaches reality. This can be done at various levels of rigor. Examples include spontaneous symmetry breaking, the theory of phase transitions, Bose-Einstein condensation, and so on. Much of modern mathematical physics concerns the search for and formalization of mathematical structures concerning the occurrence of physical phenomena in Nature.
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