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Candidate for a 5th dimension October 3rd, 2000
Space and time have been unified by relativity theory. What in our world could be
added as a 5th component to the three space directions and the time axis ?
"As things now stand there is no further indication in our fund of experience
of such an odd world parameter, but we are free to interpret our spacetime-
continuum as a four-dimensional part of a five-dimensional space" wrote Kaluza 1921
(meeting reports of the Prussian Akademy of Sience, phys.-math. class p. 966 - 972).
One indication he had found: The curvature tensor in a five-dimensional space
consists of fifteen components, of which ten as expected are members of the
gravity equation, while four additional components surprisingly fit into
the electromagnetic field equation (for the fifteenth component the matter
is quite different).
This indication, that the electromagnetic potentials might be a manifestation
of a 5th dimension, shall be the reason to take a closer look on these potentials.
Which are the characteristics of the electromagnetic potentials and which point
to a connection with a 5th coordinate ?
1. Of course the electromagnetic potentials have been introduced in classical theory
as auxiliary fields, but in advanced theories they are real, irreplacable fields
with physical substance (Schr”dinger's Equation, QED, Proca's Equation)
2. In the Bohm-Aharonov experiment the bare presence of electromagnetic potential
in regions without electric or magnetic field causes a phase shift in the
quantum mechanical wave funktion (like that of an additional path).
3. The classical Maxwell theory of electric and magnetic fields does not fix
the electromagnetic potentials, but leaves freedom for the choice of an
additional parameter, with which the electromagnetic potentials can be modified
without changing anything in the results of the derived fields (gauge transformation).
4. A successive application of a general coordinate transformation and
a gauge transformation to the electromagnetic potentials (invariance group
of the relativistic Maxwell equation) is structurally similar (isomorph)
to a rotation and streching of a vector in five dimensional space
(linear transformation in a five dimensional manifold).
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