In this article we explore String Theory in a
non-mathematical way, though we will explain the meaning
of many of its mathematical results and give pointers to how
these are derived for the more mathematically inclined. A
good mathematical account (aimed at advanced
undergraduate physics or mathematics students) can be
found in: Barton Zwiebach, 2004, A First Course in String
Theory, Cambridge University Press.
The best current working model of particle physics is the
Standard Model. We have explored aspects of this theory
in other articles (QED, QCD, EWT, symmetry and
fundamental forces). Essentially it describes the force of
electromagnetic, the weak force and the strong forces, in
terms of the exchange of virtual bosons, called gauge
bosons, between the interacting particles. For example,
two electrons repel each other due to their like negative
electric charges, and this repulsion is brought about by the
exchange of virtual photons in the standard model. In this
model, elementary particles, like electrons and photons are
described as point particles, that is they are
zero-dimensional entities with no measurable size.
The standard model is not complete. It fails to describe the
fourth fundamental force of gravity, though it speculates
that gravitational attraction is due to the exchange of virtual
gravitons (a graviton is a boson of spin 2). It also makes
some predictions that are yet to be verified by experiment.
With the incompleteness of the Standard Model, it is natural
to explore possible alternative theories. One such theory is
String Theory. This theory is also incomplete and there is
little empirical evidence to support it, though there is some
experimental evidence for it, in particular String Theory
naturally predicts the existence of gravity and the graviton
(or more specifically it predicts a mathematical construct
that behaves who we expect the graviton to behave).
The best working theory of gravity is currently Einstein's
theory of General Relativity (to be discussed in a
There is currently no experimental evidence demonstrating
the existence of the graviton or of gravity waves. However,
there is good reason to expect gravity waves to exist and
as we have discussed in other articles (atoms, waves) when
a wave is confined to a certain region of pace (as all waves
essentially are as can be demonstrated) it's energy levels
are quantised, that is only certain discrete values of
energy are possible. The classical analogy is a vibrating
guitar string, which can only maintain vibration at certain
discrete energies or frequencies (the fundamental and the
harmonics). It is not clear to me why the graviton should
exist as a particle like the photon, rather than as a
pseudo-particle like the phonon (the quantisation of
vibrations within crystals). However, let us assume for now
that the graviton does exist.
String Theory postulates the existence of strings rather
than point particles. A mathematical string is a
one-dimensional construct with length but no thickness,
however, our strings will be vibrating in many dimensions,
indeed more than the usual 4D (3 of space and one of
time). These strings are minute, perhaps of the order of the
Planck Length, about 10^-33 cm as measured in 4D
space-time. This is considered to be the fundamental
length-scale of the Universe, the quantum of space, so to
speak, with no smaller length possible (or at least no
smaller length that has a definite meaning) - it gives us the
fundamental graininess of space which is not, therefore,
considered truly continuous on this tiny scale.
Picture a vibrating string. We could have a closed string,
whose ends meet, or an open string with both ends free or
attached to supports, like a string between two walls or a
string attached by (frictionless) hoops to two vertical poles,
one on each end. When the string is given kinetic energy it
will vibrate at a certain frequency. The equation of motion is
the wave equation, which describes all the possible
vibrations of our (ideal) string. To solve this equation we
need both an initial condition (such as how far we stretch
the string before we release it) and boundary conditions
that describe how the ends of the string are constrained or
free to move. Two principle types of boundary condition
are: Dirichlet boundary conditions which fix the ends of
the string so that they can not move relative to the string,
e.g. by pinning each end to a wall and stretching the string
between the two walls. Neumann boundary conditions
allow the string ends to move in one or more dimensions,
e.g. by attaching each end of the string to a frictionless
vertical pole allowing each end to slide up and down.
Neumann boundary conditions do not pin the ends in place,
but do constrain the curvature (gradient or differential) of
the string at its end-points. In String Theory we can
generalise these boundary conditions and say that in the
first Dirichlet case, each string end is fixed to a point, a
zero-dimensional object or 'membrane, called a D0-brane.
D-brane is short for 'Dirichlet membrane' but D can
conveniently be thought of as 'dimension'. In the second,
Neumann case, the end-point is attached to a vertical line,
a one-dimensional object, called a D1-brane. [Complexities
arise in higher dimensions of vibration when some of the
string coordinates will have Neumann boundary conditions,
others Dirichlet]. Similarly we can construct D-branes of any
dimensionality, such as a D6-brane.
Although a mathematical construct on one hand, D-branes
are actual physical objects. In general they represent either
the whole of space-time or some region of it. They can
have any number of dimensions up to the maximum number
of actual dimensions, which as we shall see below is 26 in
Relativity - How Many Dimensions?
Our tiny strings are best thought of as energy vibrations, it
makes no sense (in current theories) to ask what our
strings are made of, since like our point particles they are
elementary - indivisible units. Our strings are extraordinary
in two ways, first they vibrate at the speed of light, that is
they have relativistic (or ultrarelativistic) energies. This
means that the classical wave equation has to be modified
to make it consistent with Special Relativity.
In particular, Relativity states that the laws of Nature must
be the same in different reference frames. Someone
sitting on a train is in a different reference frame to
someone standing on the platform. Each frame is moving
relative to the other and relative to the passenger who may
consider themselves stationary, a ball thrown across the
carriage has a different relative velocity than it does
compared to the person standing on the platform. However,
the ball follows a trajectory due to the same physical laws
from either viewpoint. We would not expect the laws to be
different just because one person is moving relative to the
other, for indeed both are moving as the Earth flies through
space, and it is impossible to say if anyone is ever
stationary! We also would not expect the laws of physics to
change in different regions of space. They should be the
same for a spaceship in outer space between the Sun and
Alpha Centauri as in a spaceship in the void between
Betelgeuse and its nearest neighbour. The conditions may
be different, one ship may be exposed to more radiation
than the other, for example, but the laws governing that
radiation should be the same. Similarly, simply rotating the
spaceship through space should not change physical laws,
neither should the laws change from one day to the next!
This is the Principle of Relativity. We use the Lorentz
transform equations to change space-time coordinates
from one reference frame to another and our physical laws
should be Lorentz invariants - the form of the equations
should not change! (Similarly, translating or rotating a
vector does not change its length, which is invariant!).
Above: a closed string representing a tachyon. (A
tachyon is a hypothetical particle that travels
faster than light).
Above and below: the graviton emerges from
String theory as a closed string vibrating at the
speed of light.
An open string stretched taught between two
D-branes and vibrating at the lowest possible
energy/frequency. Such a string would also be a
In a closed string there are two traveling waves
traveling in opposite directions. If these waves
are of the same phase that a standing wave
occurs as shown in these pictures.
An open string vibrating at the next higher
frequency or harmonic, representing a photon!
There is only one wave in such strings, but
reflection off the walls causing interference and
the establishment of standing waves of
Applying Lorentz invariance to the equation of motion for our relativistic string, we find that for such invariance to
hold we need 25 spatial dimensions instead of the usual 3! Thus we have 26 dimensions, 25 of space and one
of time. (Theories with more time dimensions are hard to develop and may be impossible). In other words our
need to conserve momentum (due to the similarity or homogeneity of space everywhere) and energy (due to the
homogeneity of time) we discover that 26 dimensions are needed! [However, see superstrings below).
For our string to describe particles, it must reproduce the quantum mechanics that describes atomic and
sub-atomic particles, in other words the frequencies and energies of our string must be quantised. In a classical
sense they already are - a string with specified conditions at its end-points can only vibrate at certain
frequencies (see waves) since other frequencies cancel out, For an open string with fixed end-points, a
travelling wave would reflect off the D-brane at each end and then we have two traveling waves in opposite
directions. These two travelling waves add together by the Principle of Superposition and the wave interferes
with the reflected wave. Waves can interfere destructively or constructively, according to their wave-lengths and
phases. In the end the resultant wave must have a whole-number of half-wavelengths between the two
D-branes, no other frequencies survive interference, and the resultant wave is a stationary or standing wave (as
shown in the animations). This is why the string of a musical instrument only vibrates with a certain range of
frequencies, the lowest frequency fundamental and the harmonics, each subsequent harmonic having an extra
half-wavelength than the one below it. By the Principle of Superposition, any sum of these elementary harmonics
or modes of vibration is also possible, and a mixture of harmonics gives a musical note.
Similarly, for closed strings we have two interfering waves traveling in each direction around the string and each
wave must end exactly where it began in order to keep going smoothly around the string. Again only certain
Left: the lowest mode of vibration of both closed
strings, shown here, and open strings represent
Below: open strings are under tension as they are
stretched between two D-branes (lines, planes or
hyperplanes of arbitrary dimensionality, not
exceeding the maximum set by the model under
consideration) or with both ends attached to the
same D-brane. The two D-branes may be separate
or they may overlap and so coincide (though are
still drawn separate for clarity!)
This is not the whole story, however. To move from a classical string to a quantum one, we have to replace
observable parameters or variables (observables), such as position and momentum, with quantum
mechanical operators. In Quantum mechanics, a wave function describes a system (of one or more
particles) and we interrogate this wave function with mathematical operators which extract information that
can be observed about the system from the wave function. For example, a momentum operator will give us all
the possible momenta that the system may possess. This uncertainty is another feature of quantum
mechanics, a particle or other system is probabilistic, we can only predict the possible states and the
probability of each one being found, we can not predict what a single given measurement will actually find
with certainty. (Einstein objected to this when he said (apparently atheistically), 'God does not play dice!'
However, this probabilistic model is currently supported by all available evidence and so is the correct to use
in the absence of evidence to the contrary)).
The whole mathematical process so far can be summarised in mathematical terms by the following signposts
(for the mathematically curious, the rest of you can skip this list):
- Obtain the equation of motion for the relativistic string using Hamilton-Lagrange mechanics and the
Euler-Lagrange equation, obtaining the Lagrangian, then the string action (Nambu-Goto action) and
hence the equation of motion. [This requires knowledge of wave mechanics, relativity, including 4-
vectors, and the calculus of variations.]
- Apply the boundary conditions to the equation of motion for open and closed strings to obtain the
appropriate wave equation (solution of the general equation of motion).
- Apply Lagrangian and Lorentz symmetries / invariance to obtain the conserved charges and currents
that live on the world-sheet, e.g. momentum current with its conserved charge, momentum.
- Apply mode expansions (the string is a quantum mechanical harmonic oscillator in 26 dimensions) by
expanding the wave equation coefficients as Fourier series and obtain creation and annihilation
operators from the expansion coefficients. These operators respectively increase or decrease the
energy/frequency of the string. Apply the creation operator to the ground state of the momentum to
find the various possible string states that make up the state space. Obtain the various quantum
operators and their commutation relations (in part by analogy to the quantum mechanics of point
particles). [This requires knowledge of quantum mechanics.]
- Solve the equation of motion to obtain the mass of the string and the mass corresponding to the
various oscillation modes.
- Study the form of the various modes of oscillations obtained and see what particles they may describe.
The apparent form of the wave equation solution to the equation of motion obtained depends on the choice
of coordinates used to describe the string, but will be Lorentz invariant. Using coordinates called the light-
cone coordinates (the light-cone gauge, coordinates that simplify the description of movement of light
beams in space-time) it happens that the result is a wave function satisfying Schrodinger's wave equation,
the same equation used to describe the atom in standard quantum mechanics! (Or at least we get an
equation of identical form). This is very surprising, since Schrodinger's equation does not incorporate the
effects of relativity! Normally we would use the klein-Gordon equation (for bosons) or the Dirac equation (for
fermions). This surprising result is down to a convenient choice of coordinates (Schrodinger's equation is
easier to use than either the Klein-Gordon or Dirac equations).
State Space - Tachyons, photons and gravitons
The state space is a mathematical space containing all the possible excitation states of our strings in 25
spatial dimensions. Quantum mechanical operators are never at rest, they always vibrate with a minimum
energy when in the ground state (unlike a classical pendulum) and a higher energy in an excited (more
For both the closed and open strings, there is one ground state corresponding to a tachyon, a hypothetical
particle that travels faster than light! The first excited state of the open string corresponds to the photon, that
of the closed string to the graviton. Particles like quarks, electrons and neutrinos, which are fermions, do not
appear in our model so far, which is a model for bosons only, a bosonic string theory.
Thus, we can say that there is some empirical evidence for string Theory - it predicts the graviton (as well as
the photon)! General Relativity describes gravity, but it is not a quantum mechanical theory and so is
expected to break down at very small scales (like the Planck length) and for very strong gravitational fields.
String Theory, although incomplete, is a promising candidate to describe quantum gravity.
Tachyons possibly pose a problem. In our model we have 25 spatial dimensions filling the whole of space as
a space-filling D25-brane. It is possible that tachyons exist although they may not interact with ordinary matter
and so possibly could not be used to send signals and violate causality (see Special Relativity). However,
they do suggest that the 25 D-brane (i.e. all of space) is unstable! They predict that the D25-brane will
collapse into closed strings and all open strings will disappear. Further, lower dimensional Dp-branes (with p
< 25) which are sub-regions of space-time, become coherent excited states of tachyons. The conclusion
would seem to be that D-branes are made of tachyons! However, not all the subtleties of tachyon instability
are understood and they may yet play a central role, perhaps existing early on in the Universe and playing
some cosmological role.
Superstrings - fermions and tachyons?
Our String Theory, with its 26 dimensions, does not incorporate fermions! This is a major drawback! However,
more advanced theories that incorporate supersymmetry, called Superstring Theories, do! Interestingly, in
these theories the number of spatial dimensions reduces from 25 to 9, so we have 10 space-time dimensions.
(A related theory, called M theory, which is strictly not a string theory has 10 spatial dimensions or 11 space-
time dimensions. It appears that all these variant theories, string theories, various superstring theories and M
theory may in fact be different aspects of the same unified theory!).
Supersymmetry is hypothetical and postulates that for every type of particle whose spin (quantised intrinsic
angular momentum due to particle rotation about an 'axis') is a whole number (bosons, e,g, the graviton of
spin 0 and the photon of spin 1) there is a corresponding particle with half-integer spin (a fermion, e.g.
electron, neutrino, quark) with the same mass and other (internal) quantum numbers the same. There is no
evidence for supersymmetry and if it exists then it is probably approximate and not exact.
Superstring theory not only incorporates bosons and fermions, but it is often said to do away with the
troublesome tachyon. However, there are situations in Superstring theory in which tachyons can still arise.
For example, a superstring connecting to D-branes can contain a tachyon. Some D-branes carry charge (e.g.
electric or color charge, see below) and are stable to decay because of charge conservation. However, two
similar but oppositely charged D-branes can annihilate whilst conserving net charge as superstrings
connecting them contain tachyons. This is called pair instability.
Where are the extra dimensions?
If space does have more than three dimensions, then
where are the extra ones? One possibility is that they
are compacted. We might have three extended large
dimensions, those we are familiar with, and the extra
six (3 + 6 = 9) may be compacted.
To help visualise this, consider the torus (doughnut
shape) shown on the right. A torus is essentially a
cylinder curved around with its ends joined together.
There are two dimensions on the surface of this torus.
A being living on its surface would have a system
similar to our latitude and longitude 9for the surface of
a sphere) only different - there is one dimension
indicated by the vertical hoops which curves around
the small radius of the torus (running across the
original cylinder, perpendicular to its axis) and a
second dimension running around the large radius of
the torus, horizontally around the donut (along the
surface of the original cylinder parallel to its long axis).
Now imagine that this torus is minute, perhaps of the
order of the Planck length. To us in our large 3 spatial
dimensions, the torus appears to be a point - its two
surface dimensions are hidden! These are called
compactified dimensions. Although small, these
dimensions exist everywhere, we could imagine the
whole of space to be packed with tori like this (perhaps
accounting for the graininess of space on the Planck
Of course this is an aid to visualise, like the strings
above and is not to be taken too literally. In fact, one
superstring model, called the type IIa theory,
postulates that the 6 extra space dimensions are
compactified around a 6-dimensional torus (or tori).
This 6D torus can be imagined as made up of 3 '2D
tori' like the ones we have drawn here, arranged at
right-angles to one-another (most likely in a non-literal
sense, we can not truly visualise higher dimensions).
These compact extra dimensions are so small that they
could only be detected at very high energies and so
may have escaped detection so far.
String Theory, however, also allows for hidden large
dimensions. Supposing we had a relatively large
extra dimension, say about the size of a cell or about 1
micrometre (one millionth of a metre).
Above: an open string with both end-points
attached to the same D-brane.
Note: closed strings can exist independently of
Could such a large extra dimension have escaped detection? Measuring gravitational forces at microscopic
scales is notoriously difficult due to the weakness of gravity on such small scales, with such small masses.
Measurements of gravity, assuming 3 spatial dimensions (General Relativity) hold down to the smallest so far
measured, about 0.1 mm. This suggests that large extra dimensions should be smaller than 0.1 mm.
However, electromagnetic forces have been measured accurately down to 10^-11 cm without deviation from
expected values for 3 spatial dimensions; suggesting that no large extra dimensions exist. However, if we
consider Superstring Theory in which open string ends are attached to a space-filling D3-brane,
corresponding to our familiar space, then since these strings represent the fermions and electromagnetic,
weak and strong forces, then these particles and forces would not experience the extra dimensions, and so
measurements of electromagnetic forces on any scale might not reveal them. Closed strings, representing
gravitons, however, are not dependent on D-branes and gravity will be effected by the extra dimensions.
Thus, more accurate gravity measurements are still needed to test the existence of extra large dimensions.
String Theory allows for the possibility of extra large dimensions. Compacted extra dimensions can be shown
to have little effect on point particles accept at high energies (e.g. by solving Schrodinger's equation for a
point particle in a potential well with extra dimensions). However, in String Theory, extra compact dimensions
have significant effects even at low energies!
Interestingly, calculations in General relativity that assume 5 dimensions (4 spatial and one time) yield the
(Maxwell) equations describing electromagnetism! The importance of this result is not understood. There is
also a curious role for a fourth pseudo-dimension in General relativity, in that according to this theory, gravity
is due to the curving or warping of space-time by the presence of energy (any energy, not just energy in the
form of mass!). In a sense this 4D space-time curves into a hidden 5th dimension, though the theory does
not postulate that this is any kind of visible 5th dimension. It is impossible for us to say with certainty, whether
theories that require extra dimensions to work imply that these dimensions really exist or whether these
dimensions are simply mathematical constructs that allow the model to work. However, the possibilities are
Strings, Superstrings and Particles
We have already seen how a bosonic string theory gives rise to particles like gravitons (closed strings) and
photons (open strings) but what about quarks, leptons and the weak force gauge bosons (the Zo, W- and w+
bosons) and gluons? Where do these particles enter the theory?
Point particles carry charges, such as electric charge (electromagnetism, QED) or color charge (strong force,
QCD). Strings also carry charge. The theory also predicts that D-branes to which open strings attach carry
Maxwell fields, that is they carry fields such as the electromagnetic field. To conserve charge the charge
carried by a string has to spread out along the attached D-branes, which it does by field lines that exist on
the D-brane. Furthermore, each end of the string has opposite charge. In this model, the ends of open
strings behave like charged point particles. Perhaps we can think of the string representing a
particle/anti-particle pair, such as an electron/positron pair, with one end of the string the electron, the other
the positron. Likewise we could have a string representing a quark/anti-quark pair. When particles
spontaneously appear in vacuum fluctuations (or certain other processes) they are formed in pairs
(conserving charge) which move apart (conserving momentum). Interestingly, no matter how far apart the two
become, they remain mysteriously linked by what has been called, 'ghostly action at a distance'. When some
measurement changes the state of one particle, the partner also changes instantly in a complimentary
manner. (For example, this process occurs with pairs of photons with respects to measurements of their
polarisation). Perhaps string theory explains this, if the two particles are opposite ends of the same string?
Things get more complicated, however. To introduce all the particles of the Standard Model it is necessary to
construct elaborate arrangements of D-branes and strings. No model has yet recreated all aspects of the
Standard Model (check updates for this), however, good progress has been made.
We can set up subspaces of Dp-branes (in the model we will discuss, a type IIa (2a) superstring theory, p =
6, so we have D6-branes). In this theory the 6 extra spatial dimensions are compacted around a
6-dimensional torus (a T6). Wrapped around this T6 are D-branes and their open strings. This entity
represents a particle. The exact particle types depends which branes the string connects.
Gluons can be modelled as open strings connecting three parallel color branes: one brane for the red color
charge, one for the blue and one for the green (see QCD). (Actually the three branes coexist, but they are
drawn as three separate parallel lines for convenience). An open string connecting, for example, the red and
blue branes would be a rb (red-blue) gluon, since gluons carry mixed color-charges. This model gives us the
8 types if gluon as required by the Standard Model. Gluons are unusual as gauge bosons, in that not only do
they mediate the strong force, but they also carry strong force charge and so gluons can interact by
Similarly two parallel coincident branes give us the electroweak force, with the massless photon (the
conveyor of the electromagnetic force) and three bosons for the three conveyors of the weak force (the Z0,
W+ and W- bosons). However, two parallel branes result in only 2 of the 3 weak force bosons having mass,
whereas in fact all three do. It happens that if the two branes intersect at right-angles, instead of being
parallel, that the three weak force bosons all acquire mass, whilst the photon remains massless as required.
A neutrino could be depicted as a string connecting these two branes (neutrinos only respond to the weak
force and gravity, they do not 'see' the strong force or electromagnetism).
Quarks are more tricky, since they experience all three forces (electromagentic, weak (collectively the
electroweak force) and strong force) plus gravity. They can be represented by making the two electroweak
branes intersect the three color branes. A quark (or quark/antiquark pair?) is then represented as an open
string connecting one of the electroweak branes with one of the color branes. However, complications arise
because fermions possess helicity. Since they have a spin (quantum mechanical rotation) to which can be
described a rotational sense (similar to clockwise and anticlockwise spin) they can be depicted as tracing out
helical paths as they move. Some particles are left-handed, others are right-handed (in a similar way that a
corkscrew or any other kind of helix can be left-handed or right-handed). Helicity is important in electroweak
interactions and so to account for this we can have one pair of left-handed electroweak branes and one pair
of right-handed electroweak branes intersecting the three color branes at right-angles.
The picture below depicts only the two left-handed branes for simplicity. The arrows represent open strings
connecting one electroweak to one color brane and each is a left-handed quark. According to which color
brane it attaches to, each quark possesses one of the three color charges, red, green or blue; so we have
for example a red quark pointing to the red brane. This red brane will carry the field-lines for red charge.
The pair of electroweak lines corresponds to the two possible values of the third component of isopsin, I3, for
a quark, 1/2 (u quark) and -1/2 (d quark) (see QCD and symmetry). The isospin label in the diagram below
corresponds to the value of particles that end on (point to) the left D-brane line. Thus, the u quark with I3 =
1/2, begins on the I3 = -1/2 line. Remember, we can obtain antiparticles by reversing the direction of the
Note that open strings have polarity - they have a
definite front end and a back end and swapping
ends produces a different string. As strings carry
charge, one polarity corresponds to a particle,
e.g. the electron, e-, and the opposite polarity to
the anti-particle, e.g. the positron, e+. Although
we draw branes as lines or planes they can be
hyperplanes (planes existing in more than 3
dimensions of space).
Things become still more complicated when we include the three generations of quarks. The u and quark
belong to the first generation, but there is also the second generation (s and c quarks) and the third
generation (b and t quarks) - see the article on the electroweak theory (EWT) for a description of particle
generations. Instead of adding more branes, we can wrap the branes around our compacted T6 torus such
that the two left-branes (and the right-branes) overlap the three colour lines three times, giving rise to the
three quark generations.
Strings and Particle Interactions
Particles interact! They exchange force bosons and react with one-another in very specific ways. Strings can
also interact in String Theory. One such interaction is illustrated below:
In this interaction, the end of one string interacts with the beginning of a second string, since these ends
exist on the same D-brane and the two strings join into one new string. The three D-branes are labelled, i, j
and k. We might observe this as two point-particles interacting.
Particles and Antiparticles
One the one hand we can convert an open string representing a particle into its anti-particle by reversing its
polarity, that is by changing the direction of the arrow. Changing a particle into its antiparticle is equivalent to
changing the sign of its electric charge, and since the ends of the string are oppositely charged, we are
changing the signs of these point-like charges. As a string behaves as a pair of opposite charges, we can
also model a string as connecting a particle with an antiparticle, such as a quark to an anti-quark. The string
then represents the gluons exchanged between the quark and anti-quark which bind them together. This
explains color confinement - quarks can only be observed in states of no net colour (all white or all black
depending which set of primary colours are used to designate the three color charges). Reversing the
polarity of the string might then be seen as exchanging the quark for its anti-quark and vice versa. In reality,
though, more than one gluon will be exchanged by the quark/anti-quark pair. Is this multiple gluon state a
single excited string state or a bundle of strings? Indeed, in the standard model we can combine states by
superposition (like adding waves together to produce a new wave), so perhaps these views are equivalent -
several superposed strings may connect the particle pair.
Effect of the Compact Dimensions on the Physics
If a space-time with 26 dimensions (25 spatial, one temporal) is mathematically constructed we can compact
one or more of these dimensions into an N-dimensional torus. We can compact just one dimension by
causing it to fold around itself into a circle (or cylinder) which is a 1D-torus. If this cylinder is inhabited by
strings, then some of the strings may wind around the cylinder. The degree of winding is obtained from the
string wave function for a given state by the winding operator. This winding behaves as a kind of
momentum (in addition to momentum due to movement of the string through space) and this generates
additional string states, some of which behave like particles. Interestingly, with just closed strings, a number
of gauge boson fields appears, giving a Yang-Mills symmetry set which recreates a number of particles, in
particular we have three interacting gauge bosons (as needed in the Standard Model for the electroweak
force) similar to the scheme for intersecting D-branes, accept this time we have no intersecting D-branes
and closed strings. In other words, compact dimensions can also generate particles and the forces
governing their interactions.
The Validity of String Theory
String Theory has been criticised on the grounds that there is little empirical evidence to support it, and this
is a fair criticism. Nobody fully understands the nature of mathematics, it is uncanny how a set of equations
and mathematical rules that fit one set of empirical data can make predictions that future experiments verify.
(For example, the prediction and later discovery of the top quark by the Standard Model). However, such
theories also tend to fall short and fail when extended to very unfamiliar situations, such as the failure of
Newton's laws of gravity to account for the detailed orbit of Mercury, which is accurately predicted by General
Relativity. Thus, in the absence of empirical evidence one can never be sure that a theory is valid. For this
reason, String Theory is currently something of a mathematical abstraction or even a philosophical theory
rather than a scientific theory, which has solid foundations in empirical observations, like the theory of the
atom or quantum mechanics. However, String Theory does predict the graviton and if experiments find the
graviton then that certainly would be an empirical success for String Theory.
I first studied String Theory as something of a skeptic. However, the theory has merits and raises intriguing
possibilities. It is my opinion, therefore, that string theories warrant further study, and maybe one day
experiments will be available to test more thoroughly the theories distinguishing predictions. Certainly, in the
quest for knowledge, such theories can not be ruled out and must be considered in full. Who knows where
such theories will lead, time will tell!
More on string theory coming soon...
Fun with strings!
Although the string graphics shown above are only 3D impressions of multidimensional strings, they are
nevertheless useful visualisation aids. The Pov-Ray code for these models (Pov-Ray v3.6) is given below, so
you can make your own strings!