இழைக்கொள்கை
wrote in the Principia, "Absolute space, in its own nature, without relation to anything external,
remains always similar and immovable. Absolute, true, and mathematical time, of itself, and from its
own nature, flows equably without relation to anything external."3
The size of the Planck length can be understood based upon simple reasoning rooted in what
physicists call dimensional analysis. The idea is this. When a theory is formulated as a collection of
equations, the abstract symbols must be tied to physical features of the world if the theory is to make
contact with reality. In particular, we must introduce a system of units so that if a symbol, say, is
meant to refer to a length, we have a scale by which its value can be interpreted. After all, if
equations show that the length in question is 5, we need to know if that means 5 centimeters, 5
kilometers, or 5 light years, etc. In a theory that involves general relativity and quantum mechanics, a
choice of units emerges naturally, in the following way. There are two constants of nature upon which
general relativity depends: the speed of light, c, and Newton's gravitation constant, G. Quantum
mechanics depends on one constant of nature . By examining the units of these constants (e.g., c is a
velocity, so is expressed as distance divided by time, etc.), one can see that the combination G/c3
has the units of a length; in fact, it is 1.616 × 10-33 centimeters. This is the Planck length. Since it
involves gravitational and spacetime inputs (G and c) and has a quantum mechanical dependence ( )
as well, it sets the scale for measurements—the natural unit of length—in any theory that attempts to
merge general relativity and quantum mechanics. When we use the term "Planck length" in the text, it
is often meant in an approximate sense, indicating a length that is within a few orders of magnitude of
10–33 centimeters.
Currently, in addition to string theory, two other approaches for merging general relativity and
quantum mechanics are being pursued vigorously. One approach is led by Roger Penrose of Oxford
University and is known as twistor theory. The other approach—inspired in part by Penrose's work
—is led by Abhay Ashtekar of Pennsylvania State University and is known as the new variables
method. Although these other approaches will not be discussed further in this book, there is growing
speculation that they may have a deep connection to string theory and that possibly, together with
string theory, all three approaches are honing in on the same solution for merging general relativity
and quantum mechanics.
For the mathematically inclined reader, we note that the association between string
vibrational patterns and force charges can be described more precisely as follows. When the motion
of a string is quantized, its possible vibrational states are represented by vectors in a Hilbert space,
much as for any quantum-mechanical system. These vectors can be labeled by their eigenvalues under
a set of commuting hermitian operators. Among these operators are the Hamiltonian, whose
eigenvalues give the energy and hence the mass of the vibrational state, as well as operators
generating various gauge symmetries that the theory respects. The eigenvalues of these latter
operators give the force charges carried by the associated vibrational string state.
8. Based upon insights gleaned from the second superstring revolution (discussed in Chapter
12
Let's briefly summarize the differences between the five string theories. To do so, we note that
vibrational disturbances along a loop of string can travel clockwise or counterclockwise. The Type
IIA and Type IIB strings differ in that in the latter theory, these clockwise/counterclockwise
vibrations are identical, while in the former, they are exactly opposite in form. Opposite has a
precise mathematical meaning in this context, but it's easiest to think about in terms of the spins of the
resulting vibrational patterns in each theory. In the Type IIB theory, it turns out that all particles spin
in the same direction (they have the same chirality), whereas in the Type IIA theory, they spin in both
directions (they have both chiralities). Nevertheless, each theory incorporates supersymmetry. The
two heterotic theories differ in a similar but more dramatic way. Each of their clockwise string
vibrations looks like those of the Type II string (when focusing on just the clockwise vibrations, the
Type IIA and Type IIB theories are the same), but their counterclockwise vibrations are those of the
original bosonic string theory. Although the bosonic string has insurmountable problems when chosen
for both clockwise and counterclockwise string vibrations, in 1985 David Gross, Jeffrey Harvey,
Emil Martinec, and Ryan Rhom (all then at Princeton University and dubbed the "Princeton String
Quartet") showed that a perfectly sensible theory emerges if it is used in combination with the Type II
string. The really odd feature of this union is that it has been known since the work of Claude
Lovelace of Rutgers University in 1971 and the work of Richard Brower of Boston University, Peter
Goddard of Cambridge University, and Charles Thorn of the University of Florida at Gainesville in
1972 that the bosonic string requires a 26-dimensional spacetime, whereas the superstring, as we
have discussed, requires a 10-dimensional one. So the heterotic string constructions are a strange
hybrid—a heterosis—in which counterclockwise vibrational patterns live in 26 dimensions and
clockwise patterns live in 10 dimensions! Before you get caught up in trying to make sense of this
perplexing union, Gross and his collaborators showed that the extra 16 dimensions on the bosonic
side must be curled up into one of two very special higher-dimensional doughnutlike shapes, giving
rise to the Heterotic-O and Heterotic-E theories. Since the extra 16 dimensions on the bosonic side
are rigidly curled up, each of these theories behaves as though it really has 10 dimensions, just as in
the Type II case. Again, both heterotic theories incorporate a version of supersymmetry. Finally, the
Type I theory is a close cousin of the Type IIB string except that, in addition to the closed loops of
string we have discussed in previous chapters, it also has strings with unconnected ends—so-called
open strings.
--------------------------------------------------------------------
With the discovery of M-theory and the recognition of an eleventh dimension, string theorists
have begun studying ways of curling up all seven extra dimensions in a manner that puts them all on
more or less equal footing. The possible choices for such seven-dimensional manifolds are known as
Joyce manifolds, after Domenic Joyce of Oxford University, who is credited with finding the first
techniques for their mathematical construction.
M-theory. Theory emerging from the second superstring revolution that unites the previous five
superstring theories within a single overarching framework. M-theory appears to be a theory
involving eleven spacetime dimensions, although many of its detailed properties have yet to be
understood.
Planck energy. About 1,000 kilowatt hours. The energy necessary to probe to distances as small as
the Planck length. The typical energy of a vibrating string in string theory.
Planck length. About 10-33 centimeters. The scale below which quantum fluctuations in the fabric of
spacetime would become enormous. The size of a typical string in string theory.
Planck mass. About ten billion billion times the mass of a proton; about one-hundredth of a
thousandth of a gram; about the mass of a small grain of dust. The typical mass equivalent of a
vibrating string in string theory.
Planck's constant. Denoted by the symbol h, Planck's constant is a fundamental parameter in
quantum mechanics. It determines the size of the discrete units of energy, mass, spin, etc. into which
the microscopic world is partitioned. Its value is 1.05 × 10-27 grams-cm/sec.
Planck tension. About 1039 tons. The tension on a typical string in string theory.
Planck time. About 10-43 seconds. Time at which the size of the universe was roughly the Planck
length; more precisely, time it takes light to travel the Planck length.
Second superstring revolution. Period in the development of string theory beginning around 1995 in
which some nonperturbative aspects of the theory began to be understood.
Superstring theory. String theory that incorporates supersymmetry.
Type I string theory. One of the five superstring theories; involves both open and closed strings.
Type IIA string theory. One of the five superstring theories; involves closed strings with left-right
symmetric vibrational patterns.
Type IIB string theory. One of the five superstring theories; involves closed strings with left-right
asymmetric vibrational patterns.
Ultramicroscopic. Length scales shorter than the Planck length (and also time scales shorter than the
Planck time).
wrote in the Principia, "Absolute space, in its own nature, without relation to anything external,
remains always similar and immovable. Absolute, true, and mathematical time, of itself, and from its
own nature, flows equably without relation to anything external."3
The size of the Planck length can be understood based upon simple reasoning rooted in what
physicists call dimensional analysis. The idea is this. When a theory is formulated as a collection of
equations, the abstract symbols must be tied to physical features of the world if the theory is to make
contact with reality. In particular, we must introduce a system of units so that if a symbol, say, is
meant to refer to a length, we have a scale by which its value can be interpreted. After all, if
equations show that the length in question is 5, we need to know if that means 5 centimeters, 5
kilometers, or 5 light years, etc. In a theory that involves general relativity and quantum mechanics, a
choice of units emerges naturally, in the following way. There are two constants of nature upon which
general relativity depends: the speed of light, c, and Newton's gravitation constant, G. Quantum
mechanics depends on one constant of nature . By examining the units of these constants (e.g., c is a
velocity, so is expressed as distance divided by time, etc.), one can see that the combination G/c3
has the units of a length; in fact, it is 1.616 × 10-33 centimeters. This is the Planck length. Since it
involves gravitational and spacetime inputs (G and c) and has a quantum mechanical dependence ( )
as well, it sets the scale for measurements—the natural unit of length—in any theory that attempts to
merge general relativity and quantum mechanics. When we use the term "Planck length" in the text, it
is often meant in an approximate sense, indicating a length that is within a few orders of magnitude of
10–33 centimeters.
Currently, in addition to string theory, two other approaches for merging general relativity and
quantum mechanics are being pursued vigorously. One approach is led by Roger Penrose of Oxford
University and is known as twistor theory. The other approach—inspired in part by Penrose's work
—is led by Abhay Ashtekar of Pennsylvania State University and is known as the new variables
method. Although these other approaches will not be discussed further in this book, there is growing
speculation that they may have a deep connection to string theory and that possibly, together with
string theory, all three approaches are honing in on the same solution for merging general relativity
and quantum mechanics.
For the mathematically inclined reader, we note that the association between string
vibrational patterns and force charges can be described more precisely as follows. When the motion
of a string is quantized, its possible vibrational states are represented by vectors in a Hilbert space,
much as for any quantum-mechanical system. These vectors can be labeled by their eigenvalues under
a set of commuting hermitian operators. Among these operators are the Hamiltonian, whose
eigenvalues give the energy and hence the mass of the vibrational state, as well as operators
generating various gauge symmetries that the theory respects. The eigenvalues of these latter
operators give the force charges carried by the associated vibrational string state.
8. Based upon insights gleaned from the second superstring revolution (discussed in Chapter
12
Let's briefly summarize the differences between the five string theories. To do so, we note that
vibrational disturbances along a loop of string can travel clockwise or counterclockwise. The Type
IIA and Type IIB strings differ in that in the latter theory, these clockwise/counterclockwise
vibrations are identical, while in the former, they are exactly opposite in form. Opposite has a
precise mathematical meaning in this context, but it's easiest to think about in terms of the spins of the
resulting vibrational patterns in each theory. In the Type IIB theory, it turns out that all particles spin
in the same direction (they have the same chirality), whereas in the Type IIA theory, they spin in both
directions (they have both chiralities). Nevertheless, each theory incorporates supersymmetry. The
two heterotic theories differ in a similar but more dramatic way. Each of their clockwise string
vibrations looks like those of the Type II string (when focusing on just the clockwise vibrations, the
Type IIA and Type IIB theories are the same), but their counterclockwise vibrations are those of the
original bosonic string theory. Although the bosonic string has insurmountable problems when chosen
for both clockwise and counterclockwise string vibrations, in 1985 David Gross, Jeffrey Harvey,
Emil Martinec, and Ryan Rhom (all then at Princeton University and dubbed the "Princeton String
Quartet") showed that a perfectly sensible theory emerges if it is used in combination with the Type II
string. The really odd feature of this union is that it has been known since the work of Claude
Lovelace of Rutgers University in 1971 and the work of Richard Brower of Boston University, Peter
Goddard of Cambridge University, and Charles Thorn of the University of Florida at Gainesville in
1972 that the bosonic string requires a 26-dimensional spacetime, whereas the superstring, as we
have discussed, requires a 10-dimensional one. So the heterotic string constructions are a strange
hybrid—a heterosis—in which counterclockwise vibrational patterns live in 26 dimensions and
clockwise patterns live in 10 dimensions! Before you get caught up in trying to make sense of this
perplexing union, Gross and his collaborators showed that the extra 16 dimensions on the bosonic
side must be curled up into one of two very special higher-dimensional doughnutlike shapes, giving
rise to the Heterotic-O and Heterotic-E theories. Since the extra 16 dimensions on the bosonic side
are rigidly curled up, each of these theories behaves as though it really has 10 dimensions, just as in
the Type II case. Again, both heterotic theories incorporate a version of supersymmetry. Finally, the
Type I theory is a close cousin of the Type IIB string except that, in addition to the closed loops of
string we have discussed in previous chapters, it also has strings with unconnected ends—so-called
open strings.
--------------------------------------------------------------------
With the discovery of M-theory and the recognition of an eleventh dimension, string theorists
have begun studying ways of curling up all seven extra dimensions in a manner that puts them all on
more or less equal footing. The possible choices for such seven-dimensional manifolds are known as
Joyce manifolds, after Domenic Joyce of Oxford University, who is credited with finding the first
techniques for their mathematical construction.
M-theory. Theory emerging from the second superstring revolution that unites the previous five
superstring theories within a single overarching framework. M-theory appears to be a theory
involving eleven spacetime dimensions, although many of its detailed properties have yet to be
understood.
Planck energy. About 1,000 kilowatt hours. The energy necessary to probe to distances as small as
the Planck length. The typical energy of a vibrating string in string theory.
Planck length. About 10-33 centimeters. The scale below which quantum fluctuations in the fabric of
spacetime would become enormous. The size of a typical string in string theory.
Planck mass. About ten billion billion times the mass of a proton; about one-hundredth of a
thousandth of a gram; about the mass of a small grain of dust. The typical mass equivalent of a
vibrating string in string theory.
Planck's constant. Denoted by the symbol h, Planck's constant is a fundamental parameter in
quantum mechanics. It determines the size of the discrete units of energy, mass, spin, etc. into which
the microscopic world is partitioned. Its value is 1.05 × 10-27 grams-cm/sec.
Planck tension. About 1039 tons. The tension on a typical string in string theory.
Planck time. About 10-43 seconds. Time at which the size of the universe was roughly the Planck
length; more precisely, time it takes light to travel the Planck length.
Second superstring revolution. Period in the development of string theory beginning around 1995 in
which some nonperturbative aspects of the theory began to be understood.
Superstring theory. String theory that incorporates supersymmetry.
Type I string theory. One of the five superstring theories; involves both open and closed strings.
Type IIA string theory. One of the five superstring theories; involves closed strings with left-right
symmetric vibrational patterns.
Type IIB string theory. One of the five superstring theories; involves closed strings with left-right
asymmetric vibrational patterns.
Ultramicroscopic. Length scales shorter than the Planck length (and also time scales shorter than the
Planck time).
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