Conformal invariance in two-dimensional percolation
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The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique to be described in Part Two). It was rather to describe as concretely as possible, although in hypothetical form, the geometric aspects of universality, especially conformal invariance, in the context of percolation, and to present the numerical results that support the hypotheses. On the other hand, one ulterior purpose is to draw the attention of mathematicians to the mathematical problems posed by the physical notions. Some precise basic definitions are necessary simply to orient the reader. Moreover a brief description of scaling and universality on the one hand and of renormalization on the other is also essential in order to establish their physical importance and to clarify their mathematical content.Cite
We consider two concepts often discussed as significant features of general relativity (particularly when contrasted with the other forces of the Standard Model): background independence and diffeomorphism invariance. We remind the reader of the role of backgrounds both as calculational tools and as part of the formulation of theories. Examining familiar gauge theory constructions, we are able to pinpoint when in the formulation of these theories they become background independent. We then discuss extending the gauge formulation to gravity. In doing so we are able to identify what makes general relativity a background independent theory. We also discuss/dispel suggestions that "active" diffeomorphism invariance is a feature unique to general relativity and we go on to argue against the claim that this symmetry is the origin of background independence of the theory.
Independence
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In this article, we aim at showing that principles of invariance are essential in order to characterize laws of nature in physics. To this end, we will focus on Weyl ’s and Wigner ’s epistemological reflections devoted to symme-tries. We will analyze the concept of invariance, which generalizes Kant ’s first analogy (the principle of the permanence of the substance). Moreover, we will analyze Weyl s’assumption following which the principles of invariance can be viewed as a kind of a priori knowledge in a relativized sense. Finally, we will show that, according to Wigner, symmetries must be considered as conditions which allow us to structure our understanding of the empirical reality.
Invariance principle
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We discuss the issue of motivating the analysis of higher order gravity theories and their cosmologies and introduce a rule which states that these theories may be considered as a vehicle for testing whether certain properties may be of relevance to quantum theory. We discuss the physicality issue arising as a consequence of the conformal transformation theorem, the question of formulating a consistent first order formalism of such theories and also the isotropization problem for a class of generalized cosmologies. We point out that this field may have an important role to play in clarifying issues arising also in general relativity.
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There are two kinds of definitions of perturbation of physical quantities in the framework of generalrelativity:one is direct,the other is geometrical.Correspondingly,there are two types of gauge transformation relatedwith these two definitions.The passive approach is based on the property of general covariance,and the active one isthrough the action of Lie-derivative.Although under a proper coordinate choice,the two approaches seem to agree witheach other,they are different in nature.The geometrical definition of relativistic perturbation and the active approach forgauge transformation are more rigorous in mathematics and less confusing in physical explanation.The direct definition,however,seems to be plagued with difficulties in physical meaning,and the passive approach is more awkward to use,especially for high-order gauge transformations.
General Covariance
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Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra A of observables and of a representation of A on a measure space over the space of generalized connections. This representation is singled out by its elegance and diffeomorphism covariance. Recently, in the context of the quest for semiclassical states, states of the theory in which the quantum gravitational field is close to some classical geometry, it was realized that it might also be worthwhile to study different representations of the algebra A of observables. The content of the present note is the observation that under some mild assumptions, the mathematical structure of representations of A can be analyzed rather effortlessly, to a certain extent: Each representation can be labeled by sets of functions and measures on the space of (generalized) connections that fulfill certain conditions. These considerations are however mostly of mathematical nature. Their physical content remains to be clarified, and physically interesting examples are yet to be constructed.
Diffeomorphism
Representation
Semiclassical physics
General Covariance
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In this paper we study three aspects of generalized classical and operator theories, herein generically called deformations, which do not appear to have propagated in the rather vast literature in the field: (1) the first known studies on classical and operator deformations; (2) their rather serious physical and mathematical shortcomings due to lack of invariance when conventionally formulated; and (3) the ongoing efforts for the achievement of invariant formulations preserving the axiomatic consistency of the original theories. We begin by recalling the mathematical beauty, axiomatic consistency and experimental verifications of the special relativity at both classical and quantum levels, and its main axiomatic properties: universal invariance of the fundamental units of space and time; preservation of hermiticity-observability at all times; uniqueness and invariance of numerical predictions; and other known properties. We then review the first known, generally ignored, classical and operator deformations. We then study the generally ignored problematic aspects of classical and operator deformations in their current formulation which include: lack of invariance of the fundamental units of space and times with consequential inapplicability to real measurements; loss of observability in time; lack of uniqueness and invariance of numerical predictions; violation of causality and probability laws; and, above all, violation of Einstein's special relativity. We finally outline the generally ignored ongoing efforts for the resolutions of the above shortcomings, and show that they require the necessary use of new mathematics specifically constructed for the task. We finally present a systematic study for the identical reformulation of existing classical and operator deformations in an invariant form.
Operator (biology)
Observability
Axiomatic system
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A new attempt is demonstrated that QFTs can be UV finite if they are viewed as the low energy effective theories of a fundamental underlying theory (that is complete and well-defined in all respects) according to the modern standard point of view. This approach works for any interaction model and space-time dimension. It is much simpler in principle and in technology comparing to any known renormalization program.Unlike the known renormalization methods, the importance of the procedure for defining the ambiguities (corresponding to the choice of the renormalization conditions in the conventional program) is fully appreciated in the new approach. It is shown that the high energy theory(s) or the underlying theory(s) in fact 'stipulates (stipulate)' the low energy and effective ones through these definitions within our approach while all the conventional methods miss this important point. Some simple but important nonperturbative examples are discussed to show the power and plausibility of the new approach. Other related issues (especially the IR problem and the implication of our new approach for the canonical quantization procedure) are briefly touched.
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Section (typography)
Intuition
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The context of an operational description is given by the distinction between what we consider as relevant and what as irrelevant for a particular experiment or observation. A rigorous description of a context in terms of a mathematically formulated context-independent fundamental theory is possible by the restriction of the domain of the basic theory and the introduction of a new coarser topology. Such a new topology is never given by first principles, but depends in a crucial way on the abstractions made by the cognitive apparatus or the pattern recognition devices used by the experimentalist. A consistent mathematical formulation of a higher-level theory requires the closure of the restriction of the basic theory in the new contextual topology. The validity domain of the so constructed higher-level theory intersects nontrivially with the validity domain of the basic theory: neither domain is contained in the other. Therefore, higher-level theories cannot be totally ordered and theory reduction is not transitive. The emergence of qualitatively new properties is a necessary consequence of such a formulation of theory reduction (which does not correspond to the traditional one). Emergent properties are not manifest on the level of the basic theory, but they can be derived rigorously by imposing new, contextually selected topologies upon context-independent first principles. Most intertheoretical relations are mathematically describable as singular asymptotic expansions which do not converge in the topology of the primary theory, or by choosing one of the infinitely many possible, physically inequivalent representations of the primary theory (Gelfand Naimark Segal-construction of algebraic quantum mechanics). As examples we discuss the emergence of shadows, inductors, capacitors and resistors from Maxwell's electrodynamics, the emergence of order parameters in statistical mechanics, the emergence of mass as a classical observable in Galilei-relativistic theories, the emergence of the shape of molecules in quantum mechanics, the emergence of temperature and other classical observables in algebraic quantum mechanics.
Domain theory
Closure (psychology)
Mathematical Theory
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