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arXiv:math-ph/0311025

Temparature as Order Parameter of Broken Scale Invariance

Izumi Ojima

Published Nov 16, 2003• 4 citations
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Abstract

In algebraic quantum field theory the (inverse) temperature is shown to be a macroscopic \textit{order parameter} to parametrize mutually disjoint thermal \textit{sectors} arising from the \textit{broken scale invariance} under renormalization-group transformations. This is accomplished in a mathematical formalism for the consistent treatment of \textit{explicitly broken symmetries} such as broken scale invariance, on the basis of a clear-cut criterion for the symmetry breakdown in a unified scheme for sectors proposed recently by the author.

DOI
10.48550/arxiv.math-ph/0311025

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In algebraic quantum field theory the (inverse) temperature is shown to be a macroscopic \textit{order parameter} to parametrize mutually disjoint thermal \textit{sectors} arising from the \textit{broken scale invariance} under renormalization-group transformations. This is accomplished in a mathematical formalism for the consistent treatment of \textit{explicitly broken symmetries} such as broken scale invariance, on the basis of a clear-cut criterion for the symmetry breakdown in a unified scheme for sectors proposed recently by the author.
In algebraic quantum field theory the (inverse) temperature is shown to be a macroscopic order parameter to parametrize mutually dis- joint thermal sectors arising from the broken scale invariance under renormalization-group transformations.This is accomplished in a mathematical formalism for the consistent treatment of explicitly broken symmetries such as broken scale invari- ance, on the basis of a clear-cut criterion for the symmetry breakdown in a unified scheme for sectors proposed recently by the author.
In algebraic quantum field theory the (inverse) temperature is shown to be a macroscopic order parameter to parametrize mutually disjoint thermal sectors arising from the broken scale invariance under renormalization-group transformations. This is accomplished in a mathematical formalism for the consistent treatment of explicitly broken symmetries such as broken scale invariance, on the basis of a clearcut criterion for the symmetry breakdown in a unified scheme for sectors proposed recently by the author.
A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equilibrium and non-equilibrium states to some basic aspects in measurement processes. This is achieved by the help of c → q and q → c channels: the former determines the states to be selected and to be parameterized by the order parameters, and the latter provides the physical interpretations of selected states in terms of order parameters. This formulation extends the traditional range of applicability of the Doplicher-Roberts construction method for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown: in use of the machinery proposed, the physical and mathematical meaning of basic structural ingredients associated with the spontaneously broken symmetry are re-examined, such as the degenerate vacua parameterized by the variables belonging to the relevant homogeneous space, the Goldstone modes and condensates, etc. The geometrical meaning of the space of order parameters is naturally understood in relation with the adjunction as the classifying space of a sector structure. As further examples of applications, some basic notions arising in the mathematical framework of quantum theory are reformulated and examined in connection with control theory.
A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equilibrium and non-equilibrium states to the some basic aspects in measurement processes. This is achieved by the help of \textit{c} →\to→ \textit{q} and \textit{q} →\to→ \textit{c} channels, the former of which determines the states to be selected and to be parametrized by the order parameters defined as the spectrum of the centre constituting the superselection sectors, and the latter of which provides, as classifying maps, the physical interpretations of selected states in terms of order parameters. This formulation extends the traditional range of applicability of the Doplicher-Roberts construction method for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown, as will be reported in a succeeding paper.
A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equi- librium and non-equilibrium states (see [1] for details).This is achieved by the help of carrowqcarrow qcarrowq and qarrowcqarrow cqarrowc channels, the former of which determines the states to be selected and to be parametrized by the order parameters defined as the spectrum of the centre constituting the su- perselection sectors, and the latter of which provides, as classifying maps, the physical interpretations of selected states in terms of order parameters.
The renormalization group should probably be called the renormalization semigroup, but sometimes, contradictory terminology sticks. The renormalization group procedure provides important insights because it shows quantitatively how fine-scale structure is progressively ignored and the physics of critical phenomena depend on these larger-scale, what students might call “structural”, features of the theory. The renormalization group breaks big problems down into small ones. Broken symmetry can be seen as a well-studied paradigm of emergent behaviour in the physical world. By breaking symmetry, these phases forfeit the status of being a “stationary state” of the sort beloved of elementary quantum mechanics treatments. Phase transitions are sharp and there is a clear delineation between the ordered and disordered states. The set of symmetry-breaking phase transitions includes as members those between the ferromagnetic and paramagnetic states and those between the superconducting and normal metal states of certain materials.