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E. Rabinovici
topology, singularity structure, the property of being commutative or not being
commutative. Let us somewhat elaborate on the magic.
• Distance—The simplest example is that of a universe in which one of the
dimensions is extended along identified with a circle of radius R. It turns out
that an experimentalist using strings as his probes will not be able to determine
if the universe is indeed best described but one of it dimensions being a circle of
radius R (in the appropriate string scale) or by being a circle of radius 1/R. For
point particles moving a circle of radius R, the energy spectrum has large gaps,
when R is small, and very small gaps, when the value of R is large. For strings,
which are extended objects, the spectrum consists of an additional part, that of
the strings wrapping around the circle. A point particle can’t wrap around the
circle. This part of the spectrum is narrowly gapped for small values of R and
widely gapped for large values of R, this is because the energy required to wrap a
string, which has its tension, around the circle, is proportional to the length of the
wrapped string. For an experimentalist, who can use point particles as probes, the
distinction is clear, not so for one using string probes. This is but the tip of the
iceberg of an infinite set of ambiguities. Some geometries, judged to be different
by point probes, are thus identified by extended probes.
• Dimensions—One could at least expect that the value describing the number of
spacetime dimensions to be unique. It turns out not to be the case. Ambiguities in
calling the “correct” number of space-time dimensions come in several varieties.
In string theory there are examples of a string moving in certain ten dimensional
space-times which measures exactly the same physics as that measured by a
certain point particle gauge theory in four dimensions. Each system is made out
of totally different elementary constituents, one system is defined to exist in ten
dimensions, the other in four and yet both reproduce the same observables, the
same physics. Another example consists of a strongly coupled string theory in ten
dimensions whose low energy physics is reproduced by an eleven dimensional
supergravity theory. This is true for large systems. For small, string scale,
systems, more magic is manifested. A string in some cases can’t distinguish if it
is moving on a three dimensional sphere or a one dimensional circle. So much
for non-ambiguous dimensions.
• Topology—Objects that can be deformed to each other without using excessive
violence (such as tearing them) are considered to have the same topology. The
surface of a perfect sphere is the same, topologically, as that of the surface of
a squashed sphere, but different, topologically, than that of a bagel/torus. Well,
once again that is always true only as far as point particles probes are concerned.
In string theory there are examples in which the string probes can describe the
same set of possible observations as reflecting motions on objects with different
topologies. In addition in string theory there are cases were one may smoothly
connect two objects of different topology in defiance of the definition given above
for different topologies. Topology can thus be ambiguous.
• Singularities—A crucial problem of General Relativity is the existence of singularities. Bohr has shown how such singularities are resolved in electrodynamics
E. Rabinovici
topology, singularity structure, the property of being commutative or not being
commutative. Let us somewhat elaborate on the magic.
• Distance—The simplest example is that of a universe in which one of the
dimensions is extended along identified with a circle of radius R. It turns out
that an experimentalist using strings as his probes will not be able to determine
if the universe is indeed best described but one of it dimensions being a circle of
radius R (in the appropriate string scale) or by being a circle of radius 1/R. For
point particles moving a circle of radius R, the energy spectrum has large gaps,
when R is small, and very small gaps, when the value of R is large. For strings,
which are extended objects, the spectrum consists of an additional part, that of
the strings wrapping around the circle. A point particle can’t wrap around the
circle. This part of the spectrum is narrowly gapped for small values of R and
widely gapped for large values of R, this is because the energy required to wrap a
string, which has its tension, around the circle, is proportional to the length of the
wrapped string. For an experimentalist, who can use point particles as probes, the
distinction is clear, not so for one using string probes. This is but the tip of the
iceberg of an infinite set of ambiguities. Some geometries, judged to be different
by point probes, are thus identified by extended probes.
• Dimensions—One could at least expect that the value describing the number of
spacetime dimensions to be unique. It turns out not to be the case. Ambiguities in
calling the “correct” number of space-time dimensions come in several varieties.
In string theory there are examples of a string moving in certain ten dimensional
space-times which measures exactly the same physics as that measured by a
certain point particle gauge theory in four dimensions. Each system is made out
of totally different elementary constituents, one system is defined to exist in ten
dimensions, the other in four and yet both reproduce the same observables, the
same physics. Another example consists of a strongly coupled string theory in ten
dimensions whose low energy physics is reproduced by an eleven dimensional
supergravity theory. This is true for large systems. For small, string scale,
systems, more magic is manifested. A string in some cases can’t distinguish if it
is moving on a three dimensional sphere or a one dimensional circle. So much
for non-ambiguous dimensions.
• Topology—Objects that can be deformed to each other without using excessive
violence (such as tearing them) are considered to have the same topology. The
surface of a perfect sphere is the same, topologically, as that of the surface of
a squashed sphere, but different, topologically, than that of a bagel/torus. Well,
once again that is always true only as far as point particles probes are concerned.
In string theory there are examples in which the string probes can describe the
same set of possible observations as reflecting motions on objects with different
topologies. In addition in string theory there are cases were one may smoothly
connect two objects of different topology in defiance of the definition given above
for different topologies. Topology can thus be ambiguous.
• Singularities—A crucial problem of General Relativity is the existence of singularities. Bohr has shown how such singularities are resolved in electrodynamics