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GENERAL ¨ ARTICLE
Fermi Transport
Joseph Samuel
Joseph Samuel is a
theoretical physicist at the
Raman Research Institute.
He is interested in
geometric and topological
aspects of physics
including general
relativity, the geometric
phase in quantum
mechanics and optics. Of
late he has been pursuing
analogies between the
cosmological constant in
quantum gravity and the
surface
of fluid
K. tension
V. Rashmi
is currently an M. E. student at the Indian Institute of Science. She received her B. Tech.
membranes.
It’s from
hard work
degree
the National Institute of Technology Karnataka, Surathkal. Her current research
and he
likes
to
interestsrelax
are by
in coding theory, error-correction in networks and wireless communication.“
making exquisitely
textured appams for his
friends.
Keywords
Spinning particles, geometric
transport laws.
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Figure 1. It shows the sphere of directions
in green, with three radial directions labeled
1, 2, 3. The polarisation vectors are shown
in red perpendicular to the radius vector
and therefore tangential to the sphere. If
the direction of a light ray changes from 1 to
2 to 3 to 1, the polarisation vector is rotated
as shown. On its return to the 1 direction,
the polarisation vector is as shown in blue,
rotated relative to the initial polarisation shown in red. The angle of rotation
is equal to the area of the geodesic triangle 1-2-3-1, which in this case is an
octant of the sphere 4S/8, i.e., S/2, which is a right angle. This is also equal
to the angle excess – the sum of the angles of the geodesic triangle 1231
minus S.
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GENERAL ¨ ARTICLE
Geometric effects
rotate the plane of
polarisation just as
a sugar solution
does.
Maxwell's
equations tell us
that light is a
transverse wave.
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GENERAL ¨ ARTICLE
Fermi's transport
rule is parallel
transport followed
by projection.
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GENERAL ¨ ARTICLE
The Fermi
transport rule can
be derived from
Maxwell's
equations.
1
G
, Czech. Math. J., Vol.11, p.588, 1961.
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Box 1. Continued...
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GENERAL ¨ ARTICLE
Box 1. Continued...
2
L D Landau and E M Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, 1960.
Spin is a fourvector orthogonal
to the momentum
four-vector.
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1
To identify two points means to
regard them as the same. For
example, identifying the two
ends of your shoe lace would
make it into a loop. In the present
case we identify points in the
two spaces Hp1 and Hp2 by the
rule described in the text.
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GENERAL ¨ ARTICLE
Figure 2. The Pseudosphere: It shows part of a two-dimensional surface
of constant negative curvature embedded in three-dimensional space.
Unlike the sphere, the whole of the pseudosphere cannot be described
as a subset of Euclidean space without self intersections. We show a part
of the pseudosphere. However, the whole of the pseudosphere can be
embedded in Minkowski space. In fact the mass shell of a relativistic
particle is an example of such embedding. The sum of the angles of a
geodesic triangle on a pseudosphere add up to less than S radians.
Triangles have angle deficits rather than angle excesses due to the
negative curvature. The pseudosphere is an example of hyperbolic
geometry, which has fascinated artists like M C Escher, whose work you
can find on the internet.
Geometrically
natural answer
gives the physically
correct answer
for transport.
The earth's axis
precesses ever so
slightly due to
Fermi transport.
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GENERAL ¨ ARTICLE
We all live on a
spinning gyroscope.
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Abstractions help
us see what is
common to diverse
situations.
The geometrical
idea of a
connection
permeates many
branches of
modern physics.
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GENERAL ¨ ARTICLE
Figure 3. Fermi transport and
Kattabomman transport.
1
J Samuel and Rajaram Nityanada, Transport along Null Curves, J. Phys. A., Vol.33, p.2895, 2000.
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GENERAL ¨ ARTICLE
Acknowledgements: It is a pleasure to thank Ayan Guha for his
delightful cartoon (Figure 3) and N Mukunda, Biman Nath and
Supurna Sinha for reading through this article and suggesting
changes, and Mr. Manjunath for TeXing the article.
Suggested Reading
[1]
E Fermi, Atti. Accad. Naz. Lincei. Cl. Sci. Fis. Mat. & Nat., Vol.31, No.184,
[2]
S Weinberg, Gravitation and Cosmology, John Wiley, NY, 1972.
p.306, 1922.
[3]
Address for Correspondence
Joseph Samuel
C Misner, K Thorne and J A Wheeler, Gravitation, W H Freeman & Co., NY,
1970.
[4]
Raman Research Institute
A G Walker, Relative Coordinates, Proc. Royal Soc., Edinburgh, Vol.52,
p.345, 1932.
Bangalore 560 080 India
[5]
Email: [email protected]
[6]
J H White, Am. J. Math., Vol.91, p.693, 1969.
F B Fuller, Proc. Natl. Aca. Sci., USA, Vol.68, p.815, 1971; Vol.75, p.3557,
1978.
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