We discuss the fate of the correspondence principle beyond quantum mechanics, specifically in quantum field theory and quantum gravity, in connection with the intrinsic limitations of the human ability to observe the external world. We conclude that the best correspondence principle is made of unitarity, locality, proper renormalizability (a refinement of strict renormalizability), combined with fundamental local symmetries and the requirement of having a finite number of fields. Quantum gravity is identified in an essentially unique way. (...) The gauge interactions are uniquely identified in form. Instead, the matter sector remains basically unrestricted. The major prediction is the violation of causality at small distances. (shrink)

This book presents a history of the correspondence principle from a new perspective. The author provides a unique exploration of the relation between the practice of theory and conceptual development in physics. In the process, he argues for a new understanding of the history of the old quantum theory and the emergence of quantum mechanics. The analysis looks at how the correspondence principle was disseminated and how the principle was applied as a research tool during (...) the 1920s. It provides new insights into the interaction between theoretical tools and scientific problems and shows that the use of this theoretical tool changed the tool itself in a process of transformation through implementation. This process, the author claims, was responsible for the conceptual development of the correspondence principle. This monograph connects to the vast literature in the history of science, which analyzed theoretical practices as based on tacit knowledge, skills, and calculation techniques. It contributes to the historical understanding of quantum physics and the emergence of quantum mechanics. Studying how physicists used a set of tools to solve problems, the author spells out the ‟skillful guessing” that went into the making of quantum theoretical arguments and argues that the integration and implementation of technical resources was a central driving force for the conceptual and theoretical transformation in the old quantum theory. (shrink)

The correspondence principle made of unitarity, locality and renormalizability has been very successful in quantum field theory. Among the other things, it helped us build the standard model. However, it also showed important limitations. For example, it failed to restrict the gauge group and the matter sector in a powerful way. After discussing its effectiveness, we upgrade it to make room for quantum gravity. The unitarity assumption is better understood, since it allows for the presence of physical particles (...) as well as fake particles (fakeons). The locality assumption is applied to an interim classical action, since the true classical action is nonlocal and emerges from the quantization and a later process of classicization. The renormalizability assumption is refined to single out the special role of the gauge couplings. We show that the upgraded principle leads to an essentially unique theory of quantum gravity. In particular, in four dimensions, a fakeon of spin 2, together with a scalar field, is able to make the theory renormalizable while preserving unitarity. We offer an overview of quantum field theories of particles and fakeons in various dimensions, with and without gravity. (shrink)

Although Bohr’s Correspondence Principle (CP) played a central role in the first days of quantum mechanics, its original version seems to have no present-day relevance. The purpose of this article is to show that the CP, with no need of being interpreted in terms of the quantum-to-classical limit, still plays a relevant role in the understanding of the relationships between the classical and the quantum domains. In particular, it will be argued that a generic version of the CP (...) is very helpful in elucidating the physical meaning of the phenomenon of quantum decoherence. (shrink)

One finds, even in texts by distinguished physicists, diverse enunciations of the correspondence principle. Typical is that quantum mechanics should agree with classical mechanics in some appropriate limit. Most commonly, the limit specified is that of high quantum numbers, or of large masses and orbits of large dimensions. But sometimes it is specified as mean behavior when large numbers quanta are involved, or sometimes even as just the average of quantum mechanical variables. Sometimes, the principle is even (...) taken as a prescription for replacing the classical dynamical observables with an appropriate mathematical operator. In 1918, however, Bohr proposed what he would later call the correspondence principle as a way of deriving amplitudes and polarizations of emitted and absorbed spectral lines. I will begin with Bohr's principle and trace the evolution of correspondence considerations through the 1920's, with a view as to whether in each case it is supposed to play the role of a theorem, an adequacy constraint, an inductive hypothesis or a heuristic. (shrink)

The present paper is intended to take up the basic issue in Marxist epistemology: does the development of human cognition include mechanisms that ensure the attainment of, or approximation to, the absolute truth?But to take up this issue we have to define the concept of absolute truth. This is why the paper begins with comment on the assumptions we adopt. This is followed by explanations of the concept of the absolute truth and that of relative truth and some of derivative (...) concepts, too. Section 3 is concerned with the problem of whether the absolute truth is attainable, and with the role played in the process by the correspondence principle, which is reconstructed in a specified way. (shrink)

In a recent revision (chapter 4 of Nowakowa and Nowak 2000) of an older article Leszek Nowak (1992) has attempted to rebut Niiniluoto’s 1990 critical suggestion that proponents of the Poznań idealizational approach to the sciences have committed a rather elementary logical error in the formal machinery that they advocate for use in the analysis of scientific methodology. In this paper I criticize Nowak’s responses to Niiniluoto’s suggestion, and, subsequently, work out some of the consequences of that criticism for understanding (...) the role that idealization plays in scientific methodology. (shrink)

In an article by Margenau and Cohen various correspondence principles were described in connection with Weyl, Born-Jordan, and symmetrized ordering of quantum mechanical operators. In this article we make an interesting comparison between the aforementioned ordering process and our previous prescriptions.

A vast amount of ink has been spilled in both the physics and the philosophy literature on the measurement problem in quantum mechanics. Important as it is, this problem is but one aspect of the more general issue of how, if at all, classical properties can emerge from the quantum descriptions of physical systems. In this paper we will study another aspect of the more general issue-the emergence of classical chaos-which has been receiving increasing attention from physicists but which has (...) largely been neglected by philosophers of science. (shrink)

In October 1924, The Physical Review, a relatively minor journal at the time, published a remarkable two-part paper by John H. Van Vleck, working in virtual isolation at the University of Minnesota. Van Vleck used Bohr's correspondence principle and Einstein's quantum theory of radiation to find quantum formulae for the emission, absorption, and dispersion of radiation. The paper is similar but in many ways superior to the well-known paper by Kramers and Heisenberg published the following year that is (...) widely credited to have led directly to Heisenberg's Umdeutung paper. As such, it clearly shows how strongly the discovery of matrix mechanics depended on earlier work on the application of the correspondence principle to the interaction of matter and radiation. (shrink)

In October 1924, The Physical Review, a relatively minor journal at the time, published a remarkable two-part paper by John H. Van Vleck, working in virtual isolation at the University of Minnesota. Using Bohr’s correspondence principle and Einstein’s quantum theory of radiation along with advanced techniques from classical mechanics, Van Vleck showed that quantum formulae for emission, absorption, and dispersion of radiation merge with their classical counterparts in the limit of high quantum numbers. For modern readers Van Vleck’s (...) paper is much easier to follow than the famous paper by Kramers and Heisenberg on dispersion theory, which covers similar terrain and is widely credited to have led directly to Heisenberg’s Umdeutung paper. This makes Van Vleck’s paper extremely valuable for the reconstruction of the genesis of matrix mechanics. It also makes it tempting to ask why Van Vleck did not take the next step and develop matrix mechanics himself. (shrink)

Several philosophers of science have claimed that the correspondence principle can be generalized from quantum physics to all of (particularly physical) science and that in fact it constitutes one of the major heuristical rules for the construction of new theories. In order to evaluate these claims, first the use of the correspondence principle in (the genesis of) quantum mechanics will be examined in detail. It is concluded from this and from other examples in the history of (...) science that the principle should be qualified with respect to its nature and relativized with respect to its scope of application. At the same time this conclusion implies a qualification and a relativization of the heuristic power of the principle. Generally speaking, intertheoretical correspondence is primarily of a formal-mathematical and empirical but not of a conceptual nature. Moreover, it only applies to certain parts of the theories involved. Finally, a number of philosophical justifications of the principle are discussed and some conclusions are drawn concerning the debates on theory reduction and on the discovery-justification distinction. (shrink)

This is the second installment of a two-part paper on developments in quantum dispersion theory leading up to Heisenberg’s Umdeutung paper. In telling this story, we have taken a 1924 paper by John H. Van Vleck in The Physical Review as our main guide. In this second part we present the detailed derivations on which our narrative in the first part rests. The central result that we derive is the Kramers dispersion formula, which played a key role in the thinking (...) that led to Heisenberg’s Umdeutung paper. We derive classical formulae for the dispersion, emission, and absorption of radiation and use Bohr’s correspondence principle to construct their quantum counterparts both for the special case of a charged harmonic oscillator (Sect. 5) and for arbitrary non-degenerate multiply-periodic systems (Sect. 6). We then rederive these results in modern quantum mechanics (Sect. 7). (shrink)

This book is devoted to the problems of the growth of science. These prob lems, neglected for a long time by the philosophers of science, have become in the 60's and 70's a subject of vivid discussion. There are philosophers who stress only the dependence of science upon various sociological, psycho logical and other factors and deny any internal laws of the development of knowledge, like approaching the truth. The majority rejects such nihilism and searches for the laws of the (...) growth of science. However, they often overlook the role of the Correspondence Principle which connects the suc cessive scientific theories. On the other hand, some authors, while stressing the role of this principle, overlook logical difficulties connected with it, e. g. the problem of the incompatibility of successive theories, of the falsity of some of their assumptions, etc. I believe the Correspondence Principle to be a basic principle of the pro gress of contemporary physics and, probably, of every advanced science. How ever, this principle must be properly interpreted and the above-mentioned logical difficulties must be solved. Their solution requires, as it seems, revealing the idealizational nature of the basic laws of science, in any case of the quantitative laws of advanced sciences. This point has been recently emphasized by some Polish philosophers, especially in Poznan. (shrink)

This article examines the problem of the origins of the correspondence principle formulated by Bohr in 1920 and intends to test the correctness of the argument that the essential elements of that principle were already present in the 1913 “trilogy”. In contrast to this point of view, moreover widely shared in the literature, this article argues that it is possible to find a connection between the formulation of the correspondence principle and the assessment that led (...) Bohr to abandon the search for a Planck-type theory. In fact, a thorough examination of Bohr’s works shows that the birth of this principle coincided with the depletion of a research program whose origins may date back to Bohr’s stay at the Rutherford’s laboratory (summer 1912). Finally, this article argues that original program of research was abandoned when it became clear that Planck’s quantum hypothesis for the harmonic oscillator was not an adequate support for the theoretical architecture of atomic physics; namely, there was evidence enough to justify a most drastic conclusion, according to Bohr: “I do not think that a theory of the Planck type can be made logical consistent”. (shrink)

This paper presents a more clear formulation of the correspondence principle and explores its justification.

A comprehensive look at the kinds of correspondence principle in physics.

This paper investigates Brentano's criticism of the correspondence theory of truth within the context of a discussion of his ontological assumptions. Brentano's interpretation of the formula Veritas est adaequatio rei et intellectus and of the principle ens et verum convertuntur is shown to fit into the history of these principles and into modern interpretations like that of Tarski.

This paper investigates Brentano's criticism of the correspondence theory of truth within the context of a discussion of his ontological assumptions. Brentano's interpretation of the formula Veritas est adaequatio rei et intellectus and of the principle ens et verum convertuntur is shown to fit into the history of these principles and into modern interpretations like that of Tarski.

No, but the paper argues that Bohr understood his correspondence principle, or at least an aspect of that principle expressed by the notion of rational generalization, as grounded in Hankel’s principle of permanence.

The correspondence principle states that the quantum system will approach the classical system in high quantum numbers. Indeed, the average of the quantum probability density distribution reflects a classical-like distribution. However, the probability of finding a particle at the node of the wave function is zero. This condition is recognized as the nodal issue. In this paper, we propose a solution for this issue by means of complex quantum random trajectories, which are obtained by solving the stochastic differential (...) equation derived from the optimal guidance law. It turns out that point set A, which is formed by the intersections of complex random trajectories with the real axis, can represent the quantum mechanical compatible distribution of the quantum harmonic oscillator system. Meanwhile, the projections of complex quantum random trajectories on the real axis form point set B that gives a spatial distribution without the appearance of nodes, and approaches the classical compatible distribution in high quantum numbers. Furthermore, the statistical distribution of point set B is verified by the solution of the Fokker–Planck equation. (shrink)

Periodic systems are considered whose increments in quantum energy grow with quantum number. In the limit of large quantum number, systems are found to give correspondence in form between classical and quantum frequency-energy dependences. Solely passing to large quantum numbers, however, does not guarantee the classical spectrum. For the examples cited, successive quantum frequencies remain separated by the incrementhI −1, whereI is independent of quantum number. Frequency correspondence follows in Planck's limit,h → 0. The first example is that (...) of a particle in a cubical box with impenetrable walls. The quantum emission spectrum is found to be uniformly discrete over the whole frequency range. This quality holds in the limitn → ∞. The discrete spectrum due to transitions in the high-quantum-number bound states of a particle in a box with penetrable walls is shown to grow uniformly discrete in the limit that the well becomes infinitely deep. For the infinitely deep spherical well, on the other hand, correspondence is found to be obeyed both in emission and configuration. In all cases studied the classical ensemble gives a continuum of frequencies. (shrink)

In the present paper I aim to discuss the philosophical foundations of the early correspondence principle, by comparing the conceptual structure underlying the first correspondence principle with the procedure of analogy that Immanuel Kant introduced in the Critique of Judgment from 1790. On such a comparison, I will seek to demonstrate the consistency of the conceptual ratio according to which the correspondence principle is to the classical "concepts" of space and time, as these a (...) priori forms of intuition, in Kant, are related to the separate faculty of pure intuition. As a result, it will turn out that the conceptual structure of the correspondence principle suits to the Kantian doctrine of a separate faculty of pure intuition, which is divided from the faculty of understanding. The aim is to shed new light on the line of reasoning underlying Niels Bohr's analogical thinking in quantum physics. (shrink)

Kuhn argued against both the correspondence theory of truth and convergent realism. Although he likely misunderstood the nature of the correspondence theory, which it seems he wrongly believed to be an epistemic theory, Kuhn had an important epistemic point to make. He maintained that any assessment of correspondence between beliefs and reality is not possible, and therefore, the acceptance of beliefs and the presumption of their truthfulness has to be decided on the basis of other criteria. I (...) will show that via Kuhn’s suggested epistemic values, specifically via problem-solving, his philosophy can be incorporated into a coherentist epistemology. Further, coherentism is, in principle, compatible with convergent realism. However, an argument for increasing likeness to truth requires appropriate historical continuity. Kuhn maintained that the history of science is full of discontinuity, and therefore, the historical condition of convergent realism is not satisfied.Keywords: Thomas Kuhn; The correspondence theory of truth; The coherence theory of justification; Convergent realism; Rationalism. (shrink)

The aim of this paper is to shed light on Leibniz’s justification of the Principle of Sufficient Reason. It approaches this issue through a close textual analysis of the correspondence with Samuel Clarke and a more abstruse and lesser-known writing, ‘Leibniz’s Philosophical Dream’.

The basic dynamical quantities of classical mechanics, such as position, linear momentum, angular momentum and energy, obtain their fundamental epistomological content by means of their intimate relationship to the symmetries of the space-time manifold which is the arena of physics. The program of canonical quantization can be understood as a two stage process. The first stage is Bohr's Correspondence Principle, whereby the basic dynamical quantities of the quantum theory are required to retain precisely the same relationship to the (...) symmetries of the space-time manifold as do their classical counterparts, thereby preserving their epistemological, as well as measurement-theoretic, significance. Having so identified the basic dynamical variables, functions of these may now be used to identify the subtler symmetries of the proper canonical group. The second and determining stage of the quantization program requires the establishment of a correspondence between some of these subtler symmetries of the classical theory and related symmetries of the quantum theory, the relationship being determined by a common algebraic form for their defining functions. (shrink)

The privacy, real or illusory, afforded by the personal letter allows each participant the philosophical freedom to explore a range of possible opinions, to experiment with different ideas, to hesitate, and to change his or her mind in ways that published articles and books discourage. The private letter also allows the use of language and style of writing to be altered to suit the particular recipient. This is especially evident in Leibniz's correspondence with Des Bosses. Sometimes, however, the intended (...) recipient is not the addressee, as when Leibniz engaged with Locke through Thomas Burnett of Kemney. This situation was not an isolated occurrence in Leibniz's dealings with Locke. In this chapter, we shall see how Leibniz attempted to engage with Locke through a second correspondence and how he adapted the style and presentation of his views, not for the main correspondent, Lady Masham, but for the other intended recipient. We shall also see that the views Leibniz presents confirm his loyalty during this period to an ontology of embodied, perceiving substances. -/- The correspondence with Damaris Masham began shortly after Lady Masham sent Leibniz a copy of her father's The True Intellectual System of the Universe (TIS) at the end of 1703. Learning that the book was on its way, Leibniz's first letter was intended to thank her in advance. (shrink)

This paper aims to discuss a well-known concept from argumentation theory, namely the principle of charity. It will show that this principle, especially in its contemporary version as formulated by Donald Davidson, meets with some serious problems. Since we need the principle of charity in any kind of critical discussion, we propose the way of modifying it according to the presupponendum—the rule written in the sixteenth century by Ignatius Loyola. While also corresponding with pragma-dialectical rules, it also (...) provides additional content. This will be termed the dialectical principle of charity, and it offers a few steps to be performed during an argument in order to make sure that the participants understand each other well and are not deceived by any cognitive bias. The meaning of these results could be of great significance for argumentation theory, pragma-dialectics and the practice of public discourse as it enhances the principle of charity and makes it easier to apply in argumentation. (shrink)

This article seeks to explain the correspondence between human intelligibility and that of the physical world by synthesizing the contributions of Jean Ladrière. Ladrière shows that the objectification function of formal symbolism in mathematics as an artificial language has operative power acquired through algorithm to represent physical reality. In physical theories, mathematics relates to observations through theoretic and empirical languages mutually interacting in a methodological circle, and nonmathematical anticipatory intention guides mathematical algorithmic exploration. Ladrière reasons that mathematics can make (...) the physical world comprehensible because of the presence of a rational principle in both kinds of intelligibility. (shrink)

Niels Bohr’s “correspondence principle” is typically believed to be the requirement that in the limit of large quantum numbers (n→∞) there is a statistical agreement between the quantum and classical frequencies. A closer reading of Bohr’s writings on the correspondence principle, however, reveals that this interpretation is mistaken. Specifically, Bohr makes the following three puzzling claims: First, he claims that the correspondence principle applies to small quantum numbers as well as large (while the statistical (...) agreement of frequencies is only for large n); second, he claims that the correspondence principle is a law of quantum theory; and third, Bohr argues that formal apparatus of matrix mechanics (the new quantum theory) can be thought of as a precise formulation of the correspondence principle. With further textual evidence, I offer an alternative interpretation of the correspondence principle in terms of what I call Bohr’s selection rule. I conclude by showing how this new interpretation of the correspondence principle readily makes sense of Bohr’s three puzzling claims. (shrink)

This paper investigates three morphosyntactic alternations in Estonian – those between the exterior locative cases allative, adessive and ablative and the corresponding postpositionspeale‘onto’,peal‘on’ andpealt‘off’. Based on the Complexity Principle (e.g., Rohdenburg, Günter. 2002. Processing complexity and the variable use of prepositions in English. In Hubert Cuyckens & Günter Radden (eds.),Perspectives on prepositions, 79–100. Tübingen: Niemeyer), we expect cognitively more complex constructions to use more explicit (i.e., morphologically more substantial) marking by means of a postposition. Further, we expect variation to (...) be conditioned similarly in all three semantic categories. For each of the three alternations, a random sample of the two outcomes (case vs. postposition) from the Estonian National Corpus is used, resulting in 3,000 data points. Using properties of the Landmark phrase as independent variables in Bayesian mixed-effects logistic regression models, we predict the choice of postpositions over case-marked realisations. Of the patterns found, only the frequency-related one supports our complexity-related hypothesis. We conclude that the Complexity Principle, in its general form, has little explanatory power for the Estonian constructions at hand and, in particular, that the derived principle of ‘analytic support’ is not generally applicable. We show, however, that the grammatical knowledge of Estonian exterior locative cases and the corresponding postpositions is regulated by our three factors in a relatively uniform way. (shrink)

Our concept of the universe and the material world is foundational for our thinking and our moral lives. In an earlier contribution to the URAM project I presented what I called 'the ultimate organizational principle' of the universe. In that article (Grandpierre 2000, pp. 12-35) I took as an adversary the wide-spread system of thinking which I called 'materialism'. According to those who espouse this way of thinking, the universe consists of inanimate units or sets of material such as (...) atoms or elementary particles. Against this point of view on reality, I argued that it is 'logic', which exists in our inner world as a function of our mind, that is the universal organizing power of the universe. The present contribution builds upon this insight. Then I focussed on rationality; now I am interested in the responsibility that is the driving force behind our effort to find coherence and ultimate perspectives in our cosmos. It is shown that biology fundamentally differs from physics. Biology has its own fundamental principle, which is formulated for the first time in history in a scientific manner by Ervin Bauer. This fundamental principle is the cosmic life principle. I show that if one considers the physical laws as corresponding to reality, as in scientific realism, than physicalism becomes fundamentally spiritual because the physical laws are not material. I point out that the physical laws originate from the fundamental principle of physics which is the least action principle. I show that the fundamental principle of physics can be considered as the "instinct of atoms". Our research has found deep and meaningful connections between the basic principle of physics and the ultimate principles of the universe: matter, life and reason. Therefore, the principle of least action is not necessarily an expression of sterile inanimateness. On the contrary, the principle of physics is related to the life principle of the universe, to the world of instincts behind the atomic world, in which the principles of physics, biology, and psychology arise from the same ultimate principle. Our research sheds new light to the sciences of physics, biology, and psychology in close relation to the basic principles. These ultimate principles have a primary importance in our understanding of the nature of Man and the Universe, together with the relations between Man and Nature, Man and the Universe. The results offer new foundations for our understanding our own role in the Earth, in the Nature and in the Universe. Even the apparently inanimate world of physics shows itself to be animate on long timescales and having a kind of pre- human consciousness in its basic organisation. This hypothesis offers a way to understand when and how the biological laws may direct physical laws, and, moreover, offers a new perspective to study and understand under which conditions can self-consciousness govern the laws of biology and physics. This point of view offers living beings and humans the possibility of strengthening our natural identity, and recognising the wide perspective arising from having access to the deepest ranges of our own human resources and realising the task for which human and individual life has been created. (shrink)

There is no distinct correspondence theory of truth, truth is correspondence to fact. If facts are taken to be true propositions, we wind up with just another version of the correspondence theory's ostensible competitor, the redundancy theory of truth. If instead facts are taken to be Armstrong's states of affairs, or Tractarian facts, or Mellor's _facta<D>, we get a _truthmaker<D> principle, that for every truth there is a truthmaker; something whose existence implies the proposition in question. (...) Truthmaker principles are interesting and useful, but go far beyond truth as correspondence with fact, and so are not really theories of truth at all. (shrink)

If ‘F’ is a predicate, then ‘Fer than’ or ‘more F than’ is a corresponding comparative relational predicate. Concerning such comparative relations, John Broome’s Collapsing Principle states that, for any x and y, if it is false that y is Fer than x and not false that x is Fer than y, then it is true that x is Fer than y. Luke Elson has recently put forward two alleged counter-examples to this principle, allegedly showing that it yields (...) contradictions if there are borderline cases. In this paper, I argue that the Collapsing Principle does not rule out borderline cases, but I also argue that it is implausible. (shrink)

The purpose of the present paper is to develop kinematics of the special relativity with an anisotropy of the one-way speed of light. As distinct from a common approach, when the issue of anisotropy of the light propagation is placed into the context of conventionality of distant simultaneity, it is supposed that an anisotropy of the one-way speed of light is due to a real space anisotropy. In that situation, some assumptions used in developing the standard special relativity kinematics are (...) not valid so that the “anisotropic special relativity” kinematics should be developed based on the first principles, without refereeing to the relations of the standard relativity theory. In particular, using condition of invariance of the interval between two events becomes unfounded in the presence of anisotropy of space since the standard proofs drawing the interval invariance from the invariance of equation of light propagation are not valid in that situation. Instead, the invariance of the equation of light propagation, which is a physical law, should be taken as a first principle. A number of other physical requirements, associativity, reciprocity and so on are satisfied by the requirement that the transformations between the frames form a group. Finally, the correspondence principle is to be satisfied which implies that the coordinate transformations should turn into the Galilean transformations in the limit of small velocities. The above formulation based on the invariance and group property suggests applying the Lie group theory apparatus which includes the following steps: constructing determining equations for the infinitesimal group generators using the invariance condition; solving the determining equations; specifying the solutions using the correspondence principle; defining the finite transformations by solving the Lie equations; relating the group parameter to physical parameters. The transformations derived in such a way, as distinct from the transformations derived in the context of conventionality of distant simultaneity, cannot be converted into the standard Lorentz transformations by a coordinate change. The anisotropic nature of the presented transformations manifests itself in that they do not leave the interval invariant but only provide the conformal invariance of the interval. The relations that represent measurable effects include the conformal factor which depends on the relative velocity of the frames and the anisotropy degree. It is important to note the use of the correspondence principle as a heuristic principle which allows to relate the conformal factor to the anisotropy degree and thus completely specify the transformations and observable quantities. (shrink)