Mathematical Alchemy in Physics

There is in physics a sentiment and trust that the more perfect logical structure of mathematics can save the day for some of the not so logical conceptual leaps of modern physics. This investigation aims at a better logical balance between the two, wherever this is possible. The present account emerged in part from experiences in crystal physics, which long ago required a closer relation between physics and its mathematical description. For crystals the demands go well beyond standard needs in physics, in fact they end up gibing with those needed in differential geometry.
Physics and mathematics are so close, because the language of modern physics is primarily one of mathematics. As a result physics has developed near-magic confidence in a never ending potential of some contemporary procedures for bringing in new harvests of results. Even after diminishing returns, more magic is attempted, yet oddly, "magic" hand-writings on the wall warning about overreaching goals are not always heeded. As a result man feels as if Nature has been leading him astray. Since no proof exists of Nature taking delight in intentionally misleading its students, it may well be closer to the truth if students of Nature were to admit to misleading one another.
So, the following has become an unearthing of clues about marginal situations between physics and mathematics. Since this is not a pursuit of a specific physics problem with lengthy calculations, no equations are displayed in this paper. They are mentioned by name instead, which is more than adequate, because all of them are well known items in contemporary physics.
Without equations and long extended deductions, more attention can be given to conceptual aspects. Since equations referred to here all have reputations of great effectiveness in modern physics, one would be reluctant to see them change. The hard earned experiences of numerous workers in past and present plead against undue tinkering with the intrinsic structure of well established tools. Yet everything else, specifically what exactly the tools stand for, is open to further probing. Let it be said, though, intrinsic structure may manifest itself better in suitable mathematical garb revealing its virtues. So, while leaving the tools intact, definition domains and realms of applicability are due for major reassessments. Readers drawn to a romantic sentiment in modern physics with its pronounced element of nonclassical mystique may well be in for an anticlimactic experience. The conceptual reassessment, as here delineated, largely does away with the many nonclassical metaphors of contemporary physics. After all, let us face the reality of life, the task of physics always was one of resolving mysteries, not adding to them.


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Last update 08/27/2001
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