String Theory |

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

forthcoming article)

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.

*D-branes*

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

this model.

*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!).

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

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

bosons

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

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

forthcoming article)

There is currently no experimental evidence demonstrating

the existence of the

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

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

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.

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

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:

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.

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.

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

this model.

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

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

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).

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.

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

tachyon.

D-branes and vibrating at the lowest possible

energy/frequency. Such a string would also be a

tachyon.

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.

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

well-defined frequencies.

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

well-defined frequencies.

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).

*Quantisation*

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

frequencies persist.

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).

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

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

frequencies persist.

Left: the lowest mode of vibration of both closed

strings, shown here, and open strings represent

tachyons.

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!)

strings, shown here, and open strings represent

tachyons.

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):

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

energetic) state.

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**.

mechanical

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

cone coordinates

beams in space-time) it happens that the result is a wave function satisfying

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).

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

energetic) state.

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

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

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.

more advanced theories that incorporate

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!).

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

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

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

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

scale?).

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

dimensions

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

D-branes!

attached to the same D-brane.

Note: closed strings can exist independently of

D-branes!

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

fascinating!

*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

exchanging gluons!

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

arrows.

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

fascinating!

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,

strings behave like charged point particles

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

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

exchanging gluons!

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

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

arrows.

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).

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

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:

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.

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!

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.

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.

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

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

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.

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...

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!

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