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.