# String Theory

Claimed by Julie Heil Spring 2022

## Introduction to String Theory

String theory is a theoretical framework to address the shortcomings of Einstein's theory of general relativity and quantum physics - two primary tools for understanding modern physics. String theory is often referred to as the *theory of everything* - a just identification given its ability to bridge the gap between gravity and quantum physics. The theory of general relativity and quantum mechanics are the primary, established explanations for how the universe works on a macroscopic and microscopic scale, respectively. However, these two physical explanations address different ends of the spectrum of matter in terms of relativistic effects, fundamental forces, and physical properties. String theory provides explanations where modern physics fails. These fields of interest include the early universe, black holes, and atomic nuclei. To understand the purpose of the *theory of everything,* it is vital to have a sound understanding of general relativity's purpose and the role of relativistic quantum mechanics - that is, in terms of gravitational effects on the matter, the fundamental physics of the universe. General relativity serves an essential purpose in determining the gravitational effects on planetary-sized objects and observable particles. However, the "gap" in Einstein's theory lies within its inability to address gravity's impact on elementary particles. General relativity successfully explains macro-scaled particles, given its ability to incorporate gravitational effects in its theoretical and mathematical foundation. However, the same cannot be said for elementary particles. Accordingly, these elementary particles require relativistic quantum mechanics to explain their physical properties, assuming that gravity's effect on this scale is negligible. This is the disconnect between modern physics' primary schools of thought. As a result, the purpose of the string theory is realized in its ability to successfully incorporate gravitational effects, independent of the scale of particles. Thus, the *theory of everything* emerged.

*See reference 3.*

String theory is a theory that utilizes and brings together electromagnetism, the weak force, gravity, and the strong forces. These are the four fundamental forces of nature. The particles that are identified within the string theory all have a particular vibration pattern of a 1D object, a string. String theory falls under the category of quantum theory and is an example of (the quantum) theory of gravity. String theory is highly constrained due to it being confined by the parameters of 1D, that parameter being the string length. While string theory is far from being complete, efforts from modern research in string theory have to advance our current understanding of other related fields of study like black hole physics, algebraic geometry, cosmology, and condensed matter physics.

*See reference 5.*

### Relevant Mathematical Equations (Bekenstein-Hawking Formula)

- [math]\displaystyle{ S= \frac{c^3kA}{4\hbar G} }[/math]

The Bekenstein-Hawking formula for the entropy of a black hole, shown above, provides a theoretical value for the entropy of a black hole. While this equation provides an expected value that correlates to the entropy of a black hole, according to macroscopic features resulting from its microstates, the formula lacks a derivation for a black hole's entropy-based on counting microstates on a quantum scale with consideration of gravitational aspects. In 1996, Andrew Strominger and Cumrun Vafa provided a derivation for a black hole's entropy with respect to the number of microstates of a black hole and in terms of the string theory. The results obtained from these calculations provided solutions matching Berkenstein and Hawking's original formula. Thus, string theory was validated as a mode of quantum gravity.

See *reference 1*.

### Theoretical Foundation

String theory is founded on the basis that elementary particles are components of microscopically small strings - to the degree that technology is not available to visualize. A valid and common depiction of strings lined with elementary particles is a visualization of these identical particles laying on a violin string. String theory advocates propose that elementary particles are believed to be in "excited" states due to vibrations in these strings. This is significant given the bending of spacetime and the existence of black holes and wormholes - solutions to Einstein's equations that are not explained entirely by quantum physics or general relativity in their own right. On a theoretical basis, string theory provides solutions based on gravity acting on elementary particles (theories of quantum gravity), a unification of quantum mechanics, and Einstein's theory of general relativity.

*See reference 2*.
*Oscillations of the closed string.*

## History of String Theory

The origin of string theory came from particle physics back in the 1960s. A series of experiments related to the interaction of the strong forces showed there are infinite hadrons, and their masses and spins become more prominent and higher. They are called resonance states because most of the particles are unstable particles. If an infinite number of particles engage in the interaction, the particle-particle scattering amplitude satisfies a property: duality. Gabriele Veneziano, an Italian physicist, found a simple function that helps duality. This is known as the Veneziano formula. While no experiment could fully satisfy his formula, researchers soon discovered that the formula could be naturally explained as the string and the string's scattering amplitude. So, string theory did not originate from one or a series of experiments but rather from a formula.

The theory of relativity comes into play when talking about string theory. Einstein's theory of relativity can be discussed and considered as string theory's origin. The space-time view of general relativity reformed Newton's space-time view due to the limitations of Newton’s space-time view. Einstein replaced Newton's absolute time and space causality with the law of relativity. Those who believe in the theory of string theory believe gravity has been quantized (successfully). It is debated that there is no successful theory of quantum gravity, but it is still acknowledged to an extent the success of string theory.

String theory was first proposed to describe strong forces. The quark model and quantum chromodynamics were not widely accepted as theories explaining strong forces. Based on current understanding, the meson comprises quarks and antiquarks due to strong forces. However, in string theory, it is characterized as an open string. The ends of the string correspond to quarks and antiquarks, and the string itself corresponds to the strong pulling force created between the two forces. Quantum chromodynamics is relatively prosperous due to string theory’s inability to explain many phenomena of strong forces. Due to this, string theory was abandoned by most physicists for some time.

According to popular saying, the string itself has undergone two revolutions. After the first revolution, string became popular. Some string theory experts, in1985, shortly after the first revolution, believed that the ultimate theory was in sight. Some people say that this is the Theory of Everything (TOE). John Ellis, director of the European Nuclear Center's Theory Department, is a representative of this school. These people were overly optimistic at that time, or that their understanding of strings was more superficial. Schwarz and Green 1980 developed the first superstring theory model. This model dealt with open string vibrations in ten-dimensional space, which can be connected or broken.

The first superstring revolution in 1984 was led by Schwarz and Green’s discovery that there is a symmetry capable of eliminating all distortion and infinity. This brought forth a new candidate for the theory of everything. He brought attention back to string theory and led physicists to pay more attention to string theory.

The second superstring revolution in 1994 brought forth many questions. There were five different versions and explanations revolving around string theory at the time. The traditional methods scientists use could not be used to prove the authenticity of string theory. This led to the desire for new techniques and technologies. The claim that string theory is the theory of everything raises the question of why there are five different versions; they’re like plotholes. Also, suppose point particles can be regarded as string vibrations. What is the reason they cannot be caused by two-dimensional membranes, three-dimensional squares, or higher-dimensional physical vibrations? In the early 1990s, physicists started to understand the duality between versions. Edward Witten made the most notable breakthrough in 1995, where he unified the various dualities under the eleven-dimensional M theory

## Potential Discoveries and Applications

### Wormholes

The potential usage of string theory is to provide the quantum gravitational solutions that Einstein's theory of general relativity fails to recognize at the center of black holes or wormholes. This inability is due to its lack of consideration of quantum forces alongside gravity. String theory is integral to discovering wormholes' (derived solutions to Einstein's equations) level of stability. Specifically, string theory's quantum forces and gravity enables the determination of the radiative effects and stability of this unexplained astrophysical phenomenon. In conclusion, these recently gained understandings of string theory may ultimately yield answers to questions such as...

1. Given Kerr wormholes and their potential to connect distant points in the universe, is it theoretically feasible to use these solutions to travel vast lengths through the universe?

*See reference 2*.

[Wormhole Simulation: Unknown Source]

### Computational Models

**Could not find any good PHET simulations or interactive models other than the wormhole that would help achieve a better understanding on the topic.**

[Computational Exploration of String Theory]

## Examples

**Example One**

Question: Explain how the circle S^1 is equivalent to the real line R with the identification x ~ x + 2πR

Solution: The given identification identifies 0 with 2πR. All points x between 0 and 2πR are identified with points 2πR with a magnitude larger than 2πR. The identification has made the real line periodic with period 2πR. One can think of this as identifying the interval [0,2πR) with the circumference of the circle of radius 1. In this case, the compact space S^1 is found by taking the quotient of the non-compact space R by the discrete symmetry group Z, the group of integers. This corresponds to the fact that points shifted by integer multiples of 2πR are identified.

**Example Two**

Question: Explain how the complex plane C with the identification z∼e^(2πi/3z defines an orbifold compactification.

Solution: The given identification identifies points in the complex plane with themselves rotated by 2π/3 and by 4π/3. The rotation of the complex plane about the origin leaves the origin fixed, so there exists an orbifold singularity at the origin. The resulting space is a cone with an opening angle 2π/3. Since the rotations of the complex plane by 2π/3 correspond to a representation of the cyclic group Zsubscript3 of order three, this space is called the C/Zsubscript3 orbifold.

## Connectedness

- How is this topic connected to something that you are interested in?

String Theory is a particularly intriguing topic due to far-reaching applications that today are widely considered science fiction. (i.e., interstellar space travel utilizing wormholes). This topic is something that not only makes you think but still has the scientific world asking questions. A general understanding of the basics yields a profound knowledge of how the physical world operates

- How is it connected to your major?

String theory falls into the category of Physics, Astrophysics, and Theoretical Physics. My major is Physics with a concentration in Astrophysics, and it mends nicely into my career path.

- Is there an interesting industrial application?

Scientists may not realize industrial applications for string theory in the near future. Still, in the long run, an understanding of string theory could have a profound, yet unforeseen, impact on the world - in a similar revolutionary and unpredicted manner as that of quantum mechanics and the development of modern communication and flow of information.

## Scientific Development Timeline

**1968-1974**: Dual resonance model

**1970**: String theory is proposed to understand the quantum mechanics of oscillating strings.

**1976**: Supergravity is proposed as a means of explaining the interdependence of gravity and sub-atomic particles' spectrum of excitations - an integral component in string theory.

**1974-1984**: Bosonic string theory & superstring theory

**1980**: Initially perceived to discredit string theory as a unification of quantum mechanics, classical physics, and particle physics, the inconsistencies of the theory were found to nullify each other when considered as special cases. This is the year that string theory becomes widely accepted as a potentially unifying explanation in the scientific community of the time.

**1984-1994**: 1st superstring revolution

**1991-1995**: String theory exploration, in terms of black holes, results in development towards the understanding of the different forms of string theory, in terms of their relationship.

**1994-2003**: 2nd supervising revolution

**1996**: String theory provides a microscopic understanding of black hole entropy and the nature of black hole quantum physics.

*See reference 3 & 4*.

*See reference 4 for in-depth explanation*.

## Additional Resources

### Further wiki reading

Elementary Particles and Particle Physics Theory

### External links

#### Articles

String theory beyond the Planck scale

String theory and noncommutative geometry

String Theory, University of Cambridge Part III Mathematical Tripos

#### Videos

Brian Greene: Making Sense of String Theory (YouTube)

String Theory Explained – What is The True Nature of Reality?

ScienceClic English: String Theory

String Theory and the End of Space and Time with Robbert Dijkgraaf

#### Books

Steven Gubser, The Little Book of String Theory

P. Di Francesco, P. Mathieu and D. S´en´echal, Conformal Field Theory

B. Zwiebach, A First Course in String Theory

M. Green, J. Schwarz and E. Witten, Superstring Theory

J. Polchinski, String Theory

## References

**Reference 1**: "SUPERSTRINGS! Black Holes ." *Web.physics.ucsb.edu*. UCSB, n.d. Web. 03 Dec. 2015. <http://web.physics.ucsb.edu/~strings/superstrings/bholes.htm>.

**Reference 2**: Kaku, Micho, Dr. "Blackholes, Wormholes and the Tenth Dimension." *Mkaku.org*. N.p., n.d. Web. 03 Dec. 2015. <http://mkaku.org/home/articles/blackholes-wormholes-and-the-tenth-dimension/>.

**Reference 3**: Shwarz, Patricia, Dr. "The Official String Theory Web Site." *Superstringtheory.com*. N.p., n.d. Web. 03 Dec. 2015. <http://www.superstringtheory.com/index.html>.

**Reference 4**: Beibei Chen 2021 IOP Conf. Ser.: Earth Environ. Sci. 658 012002 *https://iopscience.iop.org/article/10.1088/1755-1315/658/1/012002/pdf . IOP Publishing;.
*

**Reference 5**: Matt DeCross, Christopher Williams, Tejas Suresh, Satyabrata Dash, Eli Ross. String Theory. Brilliant. https://brilliant.org/wiki/string-theory/

**Photo References**:

*Giribet, Gaston & Celis, Emilio & Simeone, Claudio. (2019). Traversable wormholes in five-dimensional Lovelock theory. Physical Review D. 100. 10.1103/PhysRevD.100.044011.
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*Vyas, Rakshit & Joshi, Mihir. (2019). STRING THEORY: A MATHEMATICAL FRAMEWORK BEHIND COMPATIBILITY AND RECONCILEMENT BETWEEN GENERAL RELATIVITY AND QUANTUM MECHANICS. ISBN:9788192952147. 60-65.
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*American Journal of Physics 83, 486 (2015); doi: 10.1119/1.4916949
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