Notes on String Theory

The Problem: Trying to combine the massive and the small together, where General Relativity and Quantum mechanics apply together (Examples: Big Bang, Black Holes)

Development of the Standard Model:

QM – Heisenberg’s Uncertainty implies that turbulence exists even in empty

space. From E=mc², ENERGY ß à MATTER, thus looking at small enough

regions implies big enough uncertainty in energy to create virtual particles,

violently fluctuating fields, and turbulent space at the Planck scale.

QED – Combination of Quantum theory with electromagnetism (helped explain

the optical behavior of light in terms of probability) Most accurate theory in

all of science!

QCD – Combination of Quantum theory with nuclear forces (weak and strong)

Origins of the electroweak force.

Standard Model: Consistent quantum formulation of nuclear and

electromagnetic forces

Forces as Particles:

E&M à Photons (messengers of electromagnetic attraction and repulsion)

Weak à Weak Gauge Bosons (W and Z) – responsible for radioactive decay.

Strong à Gluons (gluing quarks together)

Gauge Symmetry: Like in relativity, where all observers are on equal footing and

equivalence exists between acceleration and gravity and there is a necessity to

define a gravitational force, particles have an associated color which has similar

symmetry and invariance. Interactions between all like and non-like colors are

identical, and all interactions are invariant under shifts in color (as a sphere is

invariant to rotation). This necessitates the symmetric description of other forces,

governing electromagnetic, nuclear weak and nuclear strong interactions.

General Relativity vs. Quantum Mechanics: On a very small scale, gravitational fields

fluctuate so violently as to create Planck scale fluctuations in curvature. This is

often referred to as “Quantum Foam” (John Wheeler) Since the curvature of

space-time is fluctuating at the Planck scale, notions of direction and time are

destroyed at this scale. Hence General Relativity cannot be applied at this scale.

(Planck length = 10^-33m)

Formulation of String Theory: First formulations by Green and Schwarz at Caltech

used 1D strings as fundamental building blocks. The main advantage to this

model was that everything at its most fundamental level, was made from the

same stuff.

1984-86 – The first string theory revolution – as physicists began to believe this

theory might eventually unite the four forces. (Became a popular research area)

The central problem: The equations themselves were too difficult to formulate,

hence their predictions could not be made and hence tested against experiment.

(analogy: simply hypothesizing that a die behaves randomly does not make the

predictive outcomes of a complicated dice game easy to ascertain)

String compositions and advantages: Fundamental strings are on the Planck scale in

size (hence impossible to see with any technology). As they are truly

fundamental, they are uncuttable (as letters are fundamental to written language)

The standard model cannot explain the origins of its constituent particles (there is

no intuition). Worse, it is a malleable theory, easily reshaped to match

experimental data. String Theory is unique and totally inflexible. Hence if it is

right, it will not be arbitrary or based on a seemingly arbitrary assortment of

fundamental constituents.

Strings contain vibrational resonances, like a violin: Different patterns correspond to

different forces and masses.

Bigger amplitude and Shorter wavelength ßà Greater Energy

From special relativity, mass is equivalent to energy, hence shape defines mass.

**Strings well describe photons, gauge bosons, gluons and (!) the graviton

**Beauty: Every string is identical, like a single string that is capable of producing

every type of imaginable music.

Tensions in Strings: Force carried in a string is inversely proportional to the tension

(it’s harder to propagate signals on tense strings). A calculation can be made for

the string that describes the graviton. Tension = 10³⁹ tons! This is the Planck

tension.

Enormous Tension implies: 1. Contraction of the string to the Planck scale.

2. Enormous energy

Hence, Energy depends on Tension and Pattern (more franticà more energy)

A Serious Problem emerges: Because of quantum mechanics, energy is quantized (just

like with the photoelectric effect). Strings have a minimum energy equal to the

Planck Energy. (Otherwise, a loose string with no vibrational pattern would have

no energy or mass). Unfortunately, the Planck energy corresponds to a huge

mass (the size of dust, or 10¹⁹ protons!)

Important Question: How can massive strings be the building blocks of

much lighter particles?

Strange Answer: Strings contain their own quantum vibrations. These random

fluctuations (recall nothing can be entirely still) cancel out most of the energy

associated with the strings vibrations. In fact, in the case of the graviton,

cancellation has been shown to be PERFECT, implying no energy, and hence no

mass – exactly as predicted for the particle from the standard model! Because the

difference between the quantum (negative) energy and the vibrational (positive)

energy is so small compared to the energies themselves, the difference is very

difficult to calculate.

Conclusion: While there technically could be an infinite number of particles

(corresponding to multiples of the Planck energy), all higher energy modes, even

after subtracting off negative quantum jitter energy, would correspond to particles

so massive they would be impossible for us to detect, or create (with our current

particle accelerators). But maybe they existed in the Big Bang, and there just

might be one floating around somewhere!!

Back to the original problem: How does String Theory unite QM and GR?

Measuring fine structure: In order to measure the shape of an object we need to

interact with it. If we fire tiny particles and watch how they deflect, we can

get structural accuracy to within the size of the probe itself. (marbles won’t

show the detailed structure of a peach pit) To measure the horrible effects

of fluctuations in Space-Time, we need a probe on the Planck scale.

Why can’t strings be our probe into the sub-Planck world?

If we increase the energy of strings, their corresponding vibrational wavelengths

become smaller and hence they become capable of measuring smaller and smaller

things. BUT, if we continue to pump energy into them, they begin to grow in

size and hence can no longer probe small structure. Thus there is a theoretical

limit to the size of what strings as probes can see. And this size is………….

……drum roll……..bigger than the Planck scale!!!!!!!!

What we can’t see, can’t hurt us.

If elementary strings can’t probe sub-Planck space then anything made from

strings (which are Planck size) can’t be affected by quantum fluctuations

below the Planck scale (it is impossible for them ever to be noticed!!!!)

A more precise picture: The central problem with the standard model of particles is

that the particles are treated as points. Interactions between particles and hence

the transfer of force must occur in a space of zero extent. This is where many

problems had emerged (often leading to infinite energies and probabilities and

other incompatibilities with QM and GR) But now the strings are extended

bodies, and the interactions themselves naturally smear out quantum fluctuations

in space time. Take a look!

Below are two diagrams, one showing a standard model particle/antiparticle interaction, and the other showing the same interaction using strings. The horizontal axis represents time.

To analyze the interaction, we need to know precisely where and when it takes place. With point particles, different relativistic viewpoints lead to the same time and place (see below left). But with strings, the interaction time and place differ for different observers, hence the location and time are “smeared out” (can’t be known) to within the Planck length, and hence are protected (even for graviton interactions) from quantum fluctuations!