String Theory

Claimed by Choux Ruby Kim Spring 2025

Short Description of Topic:

String theory is a framework in which the fundamental constituents of the universe are modeled as tiny one-dimensional objects called “strings.” Their different vibrational modes correspond to different particles, including the graviton. By unifying quantum mechanics and gravity, string theory aspires to be a “Theory of Everything,” albeit still awaiting direct experimental validation.

The Main Idea

String theory posits that the fundamental particles of nature (electrons, quarks, photons, etc.) can be seen as different vibrational states of incredibly small strings. Rather than point-like entities, these one-dimensional strings can oscillate in many distinct ways. Each vibrational pattern gives rise to a particle with specific mass, spin, and charge. Notably, one vibrational mode matches the graviton—a spin-2 massless particle—thereby naturally including gravity in a quantum framework.

Modern physics rests on two pillars:

• General Relativity (describing gravity at large scales)
• Quantum Field Theory (describing subatomic particles and the other forces)

These two theories do not unify seamlessly in extreme regimes (e.g., near black hole singularities or the big bang). String theory seeks to reconcile them by treating gravity as an emergent phenomenon arising from string interactions, with additional spatial dimensions compactified to scales far below direct detection.

A Mathematical Model

String theory provides several key equations and relationships:

1) Bekenstein–Hawking Black Hole Entropy

[math]\displaystyle{ S_{\text{BH}} = \frac{k_B \, c^3 \, A}{4\, \hbar\, G} \, , }[/math]

where:

• \(A\) is the horizon area,
• \(G\) is the gravitational constant,
• \(c\) is the speed of light,
• \(k_B\) is Boltzmann’s constant,
• \(\hbar\) is Planck’s constant.

This formula connects gravitational, quantum, and thermodynamic aspects of black holes. In 1996, string theorists reproduced this entropy by counting microstates of D-branes, providing a significant check on string theory’s internal consistency.

2) String Vibrational Modes (e.g., Closed String Mass Formula)

[math]\displaystyle{ M^2 \;=\; \frac{2}{\alpha'} \bigl(N + \tilde{N} - 2\bigr)\, , }[/math]

where \(N\) and \(\tilde{N}\) are the integer excitation numbers (left- and right-moving modes), and \(\alpha'\) relates to the string tension. Different integer combinations yield different particle masses, including a massless spin-2 mode identified with the graviton.

3) Compactification

Many consistent string theories live in 10 spacetime dimensions. To appear 4-dimensional at low energies, the extra 6 dimensions are curled up (compactified) into a tiny geometric shape (e.g., a Calabi–Yau manifold or an orbifold). A simple version is a circle \(S^1\) with radius \(R\), leading to Kaluza–Klein momentum quantization:

[math]\displaystyle{ p_n = \frac{n \hbar}{R}\, , \quad n \in \mathbb{Z}. }[/math]

Such quantization means, from a 4D perspective, that there is an infinite “tower” of increasingly massive excitations, each separated by \(1/R\).

A Computational Model

To visualize the idea of different vibrational modes on a “string,” we can implement a simplified **vibrating string** simulation in GlowScript (VPython). While not a quantum simulation, it neatly illustrates how a single fundamental system (the string) produces multiple harmonic patterns. Each mode corresponds to a distinct standing wave—an analogy for how string theory’s quantum modes yield different particles.

You can embed or link to a Trinket-based GlowScript program. For example:

<syntaxhighlight lang=\"python\"> GlowScript 3.2 VPython

1. A classical 1D string with fixed ends,
2. demonstrating standing wave patterns.

import numpy as np

L = 1.0 # length of the string N = 50 # number of mass points dx = L/(N-1) tension = 10.0 mass = 0.02 k = tension/dx dt = 0.0005 y = [0]*(N) # vertical displacements v = [0]*(N) # velocities

1. Create spheres for visualization

masses = [] for i in range(N):

   ball = sphere(pos=vector(i*dx,0,0), radius=0.01)
   masses.append(ball)

1. Pluck the string in the middle

y[int(N/2)] = 0.1

while True:

   rate(1000)
   # Update the vertical displacements using finite difference
   for i in range(1, N-1):
       a = k*(y[i+1]-2*y[i]+y[i-1])/mass
       v[i] += a*dt
   for i in range(N):
       y[i] += v[i]*dt
   # Enforce fixed ends
   y[0] = 0
   y[N-1] = 0
   v[0] = 0
   v[N-1] = 0

   # Update visuals
   for i in range(N):
       masses[i].pos.y = y[i]

</syntaxhighlight>

When you run this code (e.g., on https://trinket.io/glowscript/), you see a discretized string. Different initial conditions (e.g., plucking at different points or shapes) can excite different standing-wave modes. Each stable pattern is analogous to how quantum strings have distinct vibrational excitations interpreted as different particles.

Examples

Simple: Compactification to

Scenario: A 5D spacetime with coordinates \((t, x, y, z, w)\), where \(w\) is compactified into a circle of circumference \(2\pi R\).

1. Set periodic boundary conditions: \(w \sim w + 2\pi R\). A free particle in the extra dimension sees quantized momentum \(p = \frac{n\hbar}{R}\).
2. View from 4D: Each integer \(n\) is a mode with effective mass \(\frac{n\hbar}{Rc}\). Thus a single 5D particle becomes an infinite Kaluza–Klein tower of 4D particles.
3. Interpretation: If \(R\) is extremely small, these massive modes are too heavy to observe at today’s energies, making extra dimensions effectively hidden.

Middling: Orbifold Compactification ()

Scenario: Identify opposite points on a circle, halving it to an interval \([0, \pi R]\) with reflections at the boundaries.

1. Construction: Start with a circle of length \(2\pi R\). Impose . The points \(0\) and \(\pi R\) become “fixed points” of the reflection symmetry.
2. Field Modes: Even fields satisfy Neumann boundaries at the endpoints, odd fields have Dirichlet (zero at endpoints). This “projection” can remove unwanted modes.
3. Applications in String Theory: Orbifolds create “branes” at fixed points and can break large gauge symmetries into smaller ones—useful for model building (reducing extra particles).

Difficult: Black Hole Entropy Counting

Scenario: Strominger and Vafa (1996) counted microstates of certain extremal black holes (5D with D1-, D5-branes, and momentum).

1. D-brane description at weak coupling: The black hole is modeled by a bound state of D1-branes and D5-branes with momentum excitations. One counts the ways to distribute momentum quanta among vibrational modes of the D-brane system.
2. Statistical Entropy: The result for large charges matches [math]\displaystyle{ S = 2\pi \sqrt{N_1 N_5 N_P} }[/math], exactly reproducing the Bekenstein–Hawking formula in the strong-coupling (black hole) limit.
3. Significance: Demonstrates that string theory has the correct microscopic degrees of freedom to account for black hole entropy, a milestone in quantum gravity research.

Connectedness

String theory’s scope is broad:

• Cosmology and Astrophysics: Explains early universe (brane inflation), cosmic strings, black hole thermodynamics.
• Particle Physics: A candidate for unifying the Standard Model forces with gravity; naturally includes supersymmetry and extra dimensions.
• Mathematics and Computation: Calabi–Yau geometry, mirror symmetry, new insights into enumerative geometry, quantum information (holography, entanglement).
• Quantum Information & Holography: AdS/CFT duality links spacetime geometry to quantum entanglement in strongly coupled systems.
• No Immediate Industrial Application: Yet spin-offs include advanced mathematical tools, HPC algorithms for analyzing large dimensional “landscapes,” and cross-pollination with quantum computing.

History

Below is a brief historical timeline:

1968–1970

Dual Resonance Model

Gabriele Veneziano’s formula fits strong-interaction scattering data. Then Nambu, Nielsen, Susskind interpret it as arising from vibrating strings.

1974

Gravity in String Theory

Schwarz, Scherk, and Yoneya identify a massless spin-2 state—interpreted as the graviton—prompting the claim that string theory can unify quantum physics and gravity.

1984–1985

First Superstring Revolution

Green–Schwarz anomaly cancellation. Witten, Candelas, Horowitz, Strominger show Calabi–Yau compactifications can yield realistic 4D physics. Explosion of interest in superstrings.

1995

Second Superstring Revolution

Witten proposes all five superstring theories + 11D supergravity are unified via M-theory. D-branes (Polchinski) become central to non-perturbative dualities.

1996

Black Hole Entropy

Strominger–Vafa counting of D-brane states reproduces the Bekenstein–Hawking formula. Big success for string theory’s microstate approach to quantum gravity.

1997–1998

AdS/CFT Correspondence (Holography)

Maldacena’s conjecture links a 5D gravitational theory (Type IIB on AdS₅ × S₅) to a 4D gauge theory, opening new vistas in quantum gravity, QCD, and condensed matter.

2000s–2020s

String Landscape & Applications

Many possible “vacua,” possibly or more, complicating direct phenomenological predictions. Meanwhile, string insights drive progress in quantum information, black hole physics, etc.

See also

• M-Theory
• Quantum_Gravity
• Extra_Dimensions
• Supergravity
• Black_Hole_Entropy
• AdS_CFT_Correspondence
• Big_Bang_Theory (cosmology context)
• Elementary_Particles_and_Particle_Physics_Theory
• Quantum_Theory

Further reading

• Greene, B. The Elegant Universe. (W.W. Norton, 1999).
• Zwiebach, B. A First Course in String Theory. (Cambridge University Press, 2009).
• Green, M.B., Schwarz, J.H., & Witten, E. Superstring Theory. Vol. 1 & 2 (Cambridge University Press, 1987).
• Becker, K., Becker, M., & Schwarz, J. String Theory and M-Theory: A Modern Introduction. (Cambridge University Press, 2007).
• Kaku, M. Introduction to Superstrings and M-Theory. (Springer, 1999).
• Polchinski, J. String Theory: Volumes 1 & 2. (Cambridge University Press, 1998).
• Susskind, L. The Cosmic Landscape: String Theory and the Illusion of Intelligent Design. (Little, Brown, 2005).

External links

• superstringtheory.com – The Official String Theory Website
• GlowScript/Trinket – Interactive VPython for classical string oscillations
• MIT OpenCourseWare on String Theory
• Lectures on AdS/CFT and String Theory by Juan Maldacena
• archive – Preprint repository with thousands of papers on string theory, cosmology, quantum gravity, etc.
• Stanford Encyclopedia of Philosophy: String Theory – Philosophical overview.

References

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