Declaration For Greater Independence

(The "Correspondence Principle" is a) philosophical guideline for the selection of new theories in physical science, requiring that they explain all the phenomena for which a preceding theory was valid. Formulated in 1923 by the Danish physicist Niels Bohr, this principle is a distillation of the thought that had led him in the development of his atomic theory, an early form of quantum mechanics.

Early in the 20th century, atomic physics was in turmoil. The results of experimentation presented a seemingly irrefutable picture of the atom: tiny electrically charged particles called electrons continuously moving in circles around an oppositely charged and extraordinarily dense nucleus. This picture was, however, impossible in terms of the known laws of classical physics, which predicted that such circulating electrons should radiate energy and spiral into the nucleus. Atoms, however, do not gradually lose energy and collapse. Bohr and others who tried to encompass the paradoxes of atomic phenomena in a new physical theory noted that the old physics had met all challenges until physicists began to examine the atom itself. Bohr reasoned that any new theory had to do more than describe atomic phenomena correctly; it must be applicable to conventional phenomena, too, in such a way that it would reproduce the old physics: this is the correspondence principle.

The correspondence principle applies to other theories besides quantum theory. Thus the mathematical formulations for the behaviour of objects moving at exceedingly high speeds, described by relativity physics, reduce for low values of speed to the correct descriptions of the motions of daily experience.

The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior.

Quantum theory

A heuristic postulate called the correspondence principle was introduced to quantum theory by Niels Bohr: it states that, in effect, some kind of continuity argument should apply to the classical limit of quantum systems as the value of Planck's constant normalized by the action of these systems tends to zero. Often, this is approached through "quasi-classical" techniques.

More rigorously, the mathematical operation involved in classical limits is a group contraction, approximating physical systems where the relevant action is much larger than Planck's constant h, so the "deformation parameter" h/S can be effectively taken to be zero. Thus typically, quantum commutators (equivalently, Moyal brackets) reduce to Poisson brackets, in a group contraction.

In quantum mechanics, due to Heisenberg's uncertainty principle, an electron can never be at rest; it must always have a non-zero kinetic energy, a result not found in classical mechanics. For example, if we consider something very large relative to an electron, like a baseball, the uncertainty principle predicts that it cannot really have zero kinetic energy, but the uncertainty in kinetic energy is so small that the baseball can effectively appear to be at rest, and hence it appears to obey classical mechanics. In general, if large energies and large objects (relative to the size and energy levels of an electron) are considered in quantum mechanics, the result will appear to obey classical mechanics. It is less clear how the classical limit applies to chaotic systems, a field known as quantum chaos.

Quantum mechanics and classical mechanics are usually treated with entirely different formalisms: quantum theory using Hilbert space, and classical mechanics using a representation in phase space. It is possible to bring the two into a common mathematical framework in various ways. In the phase space formulation of quantum mechanics, which is statistical in nature, logical connections between quantum mechanics and classical statistical mechanics are made, enabling natural comparisons between them. Conversely, in the less well-known approach presented in 1932 by Koopman and von Neumann, the dynamics of classical mechanics have been formulated in terms of an operatorial formalism in Hilbert space, a formalism used conventionally for quantum mechanics.

In a crucial paper (1933), Dirac explained how classical mechanics is an emergent phenomenon of quantum mechanics: destructive interference among paths with non-extremal macroscopic actions S » h obliterate amplitude contributions in the path integral he introduced, leaving the extremal action Sclass, thus the classical action path as the dominant contribution, an observation further elaborated by Feynman in his 1942 PhD dissertation.

Relativity and other deformations

Other familiar deformations in physics involve the deformation of classical Newtonian into relativistic mechanics (special relativity), with deformation parameter v/c; the classical limit involves small speeds, so v/c→Ο, and the systems appear to obey Newtonian mechanics.

Similarly for the deformation of Newtonian gravity into General Relativity, with deformation parameter Schwarzschild-radius/characteristic-dimension, we find that objects once again appear to obey classical mechanics (flat space), when the mass of an object times the square of the Planck length is much smaller than its size and the sizes of the problem addressed.

Wave optics might also be regarded as a deformation of ray optics for deformation parameter λ/a. Likewise, thermodynamics deforms to statistical mechanics with deformation parameter 1/N.

It was during the first quarter of the 20th century that the concept of the sacred became dominant in the comparative study of religions. Nathan Söderblom, an eminent Swedish churchman and historian of religions, asserted in 1913 that the central notion of religion was "holiness" and that the distinction between sacred and profane was basic to all "real" religious life. In 1917 Rudolf Otto's Heilige (Eng. trans., The Idea of the Holy, 1923) appeared and exercised a great influence on the study of religion through its description of religious man's experience of the "numinous" (a mysterious, majestic presence inspiring dread and fascination), which Otto, a German theologian and historian of religions, claimed, could not be derived from anything other than an a priori sacred reality. Other scholars who used the notion of sacred as an important interpretive term during this period included the sociologist Émile Durkheim in France, and the psychologist-philosopher Max Scheler in Germany. For Durkheim, sacredness referred to those things in society that were forbidden or set apart; and since these sacred things were set apart by society, the sacred force, he concluded, was society itself. In contrast to this understanding of the nature of the sacred, Scheler argued that the sacred (or infinite) was not limited to the experience of a finite object. While Scheler did not agree with Otto's claim that the holy is experienced through a radically different kind of awareness, he did agree with Otto that the awareness of the sacred is not simply the result of conditioning social and psychological forces. Though he criticized Friedrich Schleiermacher, an early 19th-century Protestant theologian, for being too subjective in his definition of religion as "the consciousness of being absolutely dependent on God," Otto was indebted to him in working out the idea of the holy. Söderblom recorded his dependence on the scholarship of the history of religions (Religionswissenschaft), which had been a growing discipline in European universities for about half a century; Durkheim had access to two decades of scholarship on nonliterate peoples, some of which was an account ofactual fieldwork. Scheler combined the interests of an empirical scientist with a philosophical effort that followed in the tradition of 19th-century attempts to relate human experiences to the concept of a reality (essence) that underlies human thoughts and activities.

Since the first quarter of the 20th century many historians of religions have accepted the notion of the sacred and of sacred events, places, people, and acts as being central in religious life if not indeed the essential reality in religious life. For example, phenomenologists of religion such as Gerardus van der Leeuw and W. Brede Kristensen have considered the sacred (holy) as central and have organized the material in their systematic works around the (transcendent) object and (human) subject of sacred (cultic) activity, together with a consideration of the forms and symbols of the sacred. Such historians of religions as Friedrich Heiler and Gustav Mensching organized their material according to the nature of the sacred, its forms and structural types. Significant contributions to the analysis and elaboration of the sacred have been made by Roger Caillois, a sociologist, and by Mircea Eliade, an eminent historian of religions.

The term sacred has been used from a wide variety of perspectives and given varying descriptive and evaluative connotations by scholars seeking to interpret the materials provided by anthropology and the history of religions. In these different interpretations, however, common characteristics were recognized in the sacred, as it is understood by participant individuals and groups: it is separated from the common (profane) world; it expresses the ultimate total value and meaning of life; and it is the eternal reality, which is recognized to have been before it was known and to be known in a way different from that through which common things are known.

The term sacred comes from Latin sacer ("set off, restricted"). A person or thing was designated as sacred when it was unique or extraordinary. Closely related to sacer is numen ("mysterious power, god"). The term numinous is used at present as a description of the sacred to indicate its power, before which man trembles. Various terms from different traditions have been recognized as correlates of sacer: Greek hagios, Hebrew qadosh, Polynesian tapu, Arabic haram; correlates of numen include the Melanesian mana, the Sioux wakanda, the old German haminja (luck), and Sanskrit Brahman.

Besides the dichotomy of sacred–profane the sacred includes basic dichotomies of pure–unpure and pollutant–"free." In ancient Rome the word sacer could mean that which would pollute someone or something that came into contact with it, as well as that which was restricted for divine use. Similarly, the Polynesian tapu ("tabu") designated something as not "free" for common use. It might be someone or something specially blessed because it was full of power, or it might be something accursed, as a corpse. Whatever was tabu had special restrictions around it, for it was full of extraordinary energy that could destroy anyone unprotected with special power himself. In this case the sacred is whatever is uncommon and may include both generating and polluting forces. On the other hand there is the pure–impure dichotomy, in which the sacred is identified with the pure and the profane is identified with the impure. The pure state is that which produces health, vigour, luck, fortune, and long life. The impure state is that characterized by weakness, illness, misfortune, and death. To acquire purity means to enter the sacred realm, which could be done through purification rituals or through the fasting, continence, and meditation of ascetic life. When a person became pure he entered the realm of the divine and left the profane, impure, decaying world. Such a transition was often marked by a ritual act of rebirth.

Because the sacred contains notions both of a positive, creative power and a danger that requires stringent prohibitions, the common human reaction is both fear and fascination. Otto elaborated his understanding of the holy from this basic ambiguity. Only the sacred can fulfill man's deepest needs and hopes; thus, the reverence that man shows to the sacred is composed both of trust and terror. On the one hand, the sacred is the limit of human effort both in the sense of that which meets human frailty and that which prohibits human activity; on the other hand, it is the unlimited possibility that draws mankind beyond the limiting temporal–spacial structures that are constituents of human existence.

Not only is there an ambivalence in the individual's reaction to the numinous quality of the sacred but the restrictions, the tabus, can be expressive of the creative power of the sacred. Caillois has described at length the social mechanism of nonliterate societies, in which the group is divided into two complementary subgroups (moieties), and has interpreted the tabus and the necessary interrelationship of the moieties as expressions of sacredness. Whatever is sacred and restricted for one group is "free" for the other group. In a number of respects—e.g., in supplying certain goods, food, and wives—each group is dependent on the other for elemental needs. Here the sacred is seen to be manifested in the order of the social–physical universe, in which these tribal members live. To disrupt this order, this natural harmony, would be sacrilege, and the culprit would be severely punished. In this understanding of the sacred, a person is, by nature, one of a pair; he is never complete as a single unit. Reality is experienced as one of prescribed relationships, some of these being vertical, hierarchical relationships and others being horizontal, corresponding relationships.

Another significant ambiguity is that the sacred manifests itself in concrete forms that are also profane. The transcendent mystery is recognized in a specific concrete symbol, act, idea, image, person, or community. The unconditioned reality is manifested in conditioned form. Eliade has elucidated this "dialectic of the sacred," in which the sacred may be seen in virtually any sort of form in religious history: a stone, an animal, or the sea. The ambiguity of the sacred taking on profane forms also means that even though every system of sacred thought and action differentiates between those things it regards as sacred or as profane, not all people find the sacred manifested in the same form; and what is profane for some is sacred for others.

To Democritus the existence of the void was a necessary element in atomistic theory. Without the void the atoms could not be separated from each other and could not move. In the 17th century Descartes rejected the existence of the void, whereas Newton's conception of action at a distance was in perfect harmony with the acceptance of the void and the drawing of a sharp distinction between occupied and nonoccupied space. The success of the Newtonian law of gravitation was one of the reasons that atomic theories came to prevail in the 18th century. Even with respect to the phenomena of light, the corpuscular and hence atomic theory of Newton, which held that light is made of tiny particles, was adopted almost universally, in spite of Huygens's brilliant development of the wave hypothesis.

When, at the beginning of the 19th century, the corpuscular theory of light in its turn was abandoned in favour of the wave theory, the case for the existence of the void had to be reopened, for the proponents of the wave theory did not think in terms of action at a distance; the propagation of waves seemed to presuppose instead a medium not only with geometrical properties but with physical ones as well. At first the physical properties of the medium, the ether, were described in the language of mechanics; later they were described in that of the electromagnetic field theory of J.C. Maxwell. Yet, to a certain extent, the old dichotomy between occupied and nonoccupied space continued to exist. According to the ether theory, the atoms moved without difficulty in the ether, whereas the ether pervaded all physical bodies.

In contemporary science this dichotomy has lost its sharpness, owing to the fact that the distinction between material phenomena, which were supposed to be discontinuous, and the phenomena of light, which were supposed to be continuous, appears to be only a relative one. In conclusion, it can be claimed that, although modern theories still speak of space and even of "empty" space, this "emptiness" is not absolute: space has come to be regarded as the seat of the electromagnetic field, and it certainly is not the void in the sense in which the term was used by Democritus.

(The paradoxes of Zeno refer to) statements made by the Greek philosopher Zeno of Elea, a 5th-century-BC disciple of Parmenides, a fellow Eleatic, designed to show that any assertion opposite to the monistic teaching of Parmenides leads to contradiction and absurdity. Parmenides had argued from reason alone that the assertion that only Being is leads to the conclusions that Being (or all that there is) is (1) one and (2) motionless. The opposite assertions, then, would be that instead of only the One Being, many real entities in fact are, and that they are in motion (or could be). Zeno thus wished to reduce to absurdity the two claims, (1) that the many are and (2) that motion is.

Plato's dialogue, the Parmenides, is the best source for Zeno's general intention, and Plato's account is confirmed by other ancient authors. Plato referred only to the problem of the many, and he did not provide details. Aristotle, on the other hand, gave capsule statements of Zeno's arguments on motion; and these, the famous and controversial paradoxes, generally go by names extracted from Aristotle's account: the Achilles (or Achilles and the tortoise), the dichotomy, the arrow, and the stadium.

The Achilles paradox (q.v.) is designed to prove that the slower mover will never be passed by the swifter in a race. The dichotomy paradox is designed to prove that an object never reaches the end. Any moving object must reach halfway on a course before it reaches the end; and because there are an infinite number of halfway points, a moving object never reaches the end in a finite time. The arrow paradox endeavours to prove that a moving object is actually at rest. The stadium paradox tries to prove that, of two sets of objects traveling at the same velocity, one will travel twice as far as the other in the same time.

If, in each case, the conclusion seems necessary but absurd, it serves to bring the premise (that motion exists or is real) into disrepute, and it suggests that the contradictory premise, that motion does not exist, is true; and indeed, the reality of motion is precisely what Parmenides denied.

In logic, (the Achilles paradox is) an argument attributed to the 5th-century BC Greek philosopher Zeno, and one of his four paradoxes described by Aristotle in the treatise Physics. The paradox concerns a race between the fleet-footed Achilles and a slow-moving tortoise. The two start moving at the same moment, but if the tortoise is initially given a head start and continues to move ahead, Achilles can run at any speed and will never catch up with it. Zeno's argument rests on the presumption that Achilles must first reach the point where the tortoise started, by which time the tortoise will have moved ahead, even if but a small distance, to another point; by the time Achilles traverses the distance to this latter point, the tortoise will have moved ahead to another, and so on.

The Achilles paradox cuts to the root of the problem of the continuum. Aristotle's solution to it involved treating the segments of Achilles' motion as only potential and not actual, since he never actualizes them by stopping. In an anticipation of modern measure theory, Aristotle argued that an infinity of subdivisions of a distance that is finite does not preclude the possibility of traversing that distance, since the subdivisions do not have actual existence unless something is done to them, in this case stopping at them.