Basics of String Theory

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Introduction

The development of string theory, a cornerstone of modern theoretical physics, began in the late 1960s as a collaborative effort among several physicists, notably Gabriele Veneziano. Veneziano, in his quest to understand the strong nuclear force that binds the constituents of atomic nuclei, proposed a mathematical formula in 1968 that accurately described the scattering amplitudes of hadrons (particles like protons and neutrons) at high energies. This formula, known as the Veneziano amplitude, surprisingly mirrored the characteristics of Regge theory-a framework developed in the 1950s by Italian physicist Tullio Regge to analyze the angular momentum and properties of particles involved in scattering processes.

Regge theory introduced the concept of Regge poles, which are complex angular momenta corresponding to the resonances observed in particle physics. These poles explained the behavior of scattering amplitudes at high energies and were pivotal in understanding the dynamics of hadrons. However, the physical interpretation of Veneziano’s formula and its connection to Regge poles remained elusive until it was realized that the formula could be derived from a model where particles are not point-like but are instead envisioned as one-dimensional “strings.”

This insight led to the foundation of string theory, initially called dual resonance models, where the fundamental objects are strings whose vibrational modes correspond to different particles. The resonance phenomena, explained by Regge theory, naturally emerged from the vibrational patterns of these strings, providing a unified description of hadrons that could potentially include the graviton-the hypothetical quantum of gravity.

The initial excitement over string theory’s ability to describe the strong force waned as quantum chromodynamics (QCD) emerged as a more accurate theory for that purpose. However, string theory found new life in the 1970s and 1980s as a promising candidate for a quantum theory of gravity and a unified framework for all fundamental interactions. The introduction of supersymmetry, leading to superstring theory, further enriched the theoretical landscape, setting the stage for string theory’s evolution into a leading approach for addressing some of the most profound questions in physics.

Thus, the development of string theory, from its inception to address the issue of Regge poles and the strong nuclear force, to its current status as a potential theory of everything, illustrates a remarkable trajectory of scientific innovation and interdisciplinary collaboration.

Gabriele Veneziano made a groundbreaking contribution to theoretical physics in the late 1960s when he presented the Veneziano formula, which provided a fresh perspective on hadron scattering processes. This formula supplied the groundwork for the development of string theory in addition to fitting in with Regge theory, a framework developed to explain the angular momentum characteristics of particles. We discuss the Veneziano formula in this article, along with how it relates to Regge poles and how it makes the conceptual jump to string theory.

The Veneziano Formula: The Veneziano amplitude provided a solution to the puzzle of hadron scattering amplitudes, encapsulating the duality observed in these processes. It is expressed as:

where

represents the amplitude for scattering,

is the Gamma function, and

denotes the Regge trajectory, a linear function given by:

Here,

is the intercept, and

is the slope of the trajectory.

Let us see what it means with a concrete example.

Recall the Veneziano amplitude:

, where the Regge trajectory

is linear in

, the square of the total energy in the center-of-mass system:

. For this example, let’s assume values for the intercept

and the slope

that are typical in discussions of string theory, though in real-world applications these would be determined by experimental data:

• Let

• , a common intercept for light mesons.
• Let

• , a typical value for the slope of Regge trajectories.
• Example: Calculating the Amplitude for Specific

• and

• . Suppose we want to calculate the amplitude for specific values of s and t to understand the interaction at a certain energy and momentum transfer. Let’s choose:

• (total energy squared),

• (momentum transfer squared, negative because it’s spacelike).
• First, calculate the values of

• and

• :

• Then, substitute these into the Veneziano amplitude:

• Using the Gamma function properties and values (recall that

• for positive integers, and

• ):

• The Gamma function for negative non-integer values would typically require numerical computation, but for illustrative purposes, let’s focus on the structure of the calculation rather than the exact numerical result.

A computation of the Veneziano amplitude for a given set of s and t values is only one piece of a much broader puzzle, and it may not reveal all about particle physics or string theory. Physicists can develop and improve theories that seek to explain the basic principles underlying the world by having a better understanding of the relationship between these computations and physical facts.

Connection to Regge Poles: Regge theory introduces the concept of Regge trajectories and poles, fundamental to understanding the scattering amplitudes at high energies. The Veneziano formula’s incorporation of these trajectories highlights the theory’s predictive power and its elegant mathematical structure. The linear relationship in the Veneziano formula mirrors the linear Regge trajectories observed in particle physics.

In Regge theory, the trajectory of a particle is a function that relates the angular momentum

of the particle to its mass squared

, symbolized as

. This function is called a Regge trajectory, and it is observed to be approximately linear for many families of particles, which can be expressed as:

where

(the intercept) and

(the slope) are constants. The Veneziano amplitude can be written as:

where s and t are the Mandelstam variables representing the square of the total energy and the square of the momentum transfer in the center-of-mass frame, respectively {1}. The function

here represents a Regge trajectory, with

playing the role of

(mass squared of the exchanged particle in the t-channel). By substituting the linear form of the Regge trajectory into the Veneziano amplitude, we explicitly see the connection:

. This implies that the resonances (or poles) in the amplitude, which correspond to physical particles, occur at values of

and

where

and

have poles, which is when

and

are non-negative integers because

has poles for non-negative integers

.

Illustration with an Example: Suppose a Regge trajectory with

and

. For a given resonance with

(angular momentum of the exchanged particle), we can solve for its mass squared:

,

This shows how a specific point on the Regge trajectory corresponds to a resonance with a certain mass and angular momentum.

The Veneziano formula, through its incorporation of Regge trajectories, provides a powerful framework for predicting the properties of resonances in particle physics. The elegance of the formula lies in its ability to encapsulate the duality and linear relationship of Regge trajectories, offering insights into the scattering amplitudes at high energies and the spectral organization of particles.

Link to Strings: The Veneziano formula’s implications extend beyond particle physics, providing a gateway to string theory. The formula can be derived from a model in which fundamental entities are one-dimensional strings, rather than point particles. This realization unveiled a new perspective on the fabric of the universe, where the vibrational states of strings correspond to different particles, including the graviton-a hypothetical quantum of gravity naturally emerging from string theory.

Example: Prediction of the Graviton: A notable example of the Veneziano formula’s impact is its implication for quantum gravity. String theory predicts the existence of a massless spin-2 particle, identified as the graviton, through the quantization of string modes. This discovery illustrates the formula’s far-reaching consequences, bridging the gap between the strong nuclear force and the gravitational force within a unified theoretical framework. The journey from the Veneziano formula to string theory encapsulates a significant chapter in the history of theoretical physics. It illustrates how a mathematical formulation intended to describe hadron scattering led to the development of a theory that promises to unify all fundamental forces of nature, showcasing the intricate beauty and interconnectedness of the universe’s fundamental structures.

String Theory Basics, String Action (Polyakov Action):

The action for a string propagating in spacetime is given by:

where the parameter

is related to the string tension, and

are the spacetime coordinates of the string.

are the parameters of the string worldsheet.

In string theory, the parameter α′ plays a crucial role in the dynamics of strings. It is closely related to the tension of the string, which is a fundamental property determining how the string behaves and interacts. α′ is inversely proportional to the tension of the string, T. The tension can be thought of as the energy per unit length of the string. Higher tension means the string is stiffer, as we know from classical vibrations in a guitar. Mathematically, this relationship is expressed as

.

The value of α ′ affects the mass spectrum of the particles represented by string modes. It determines the scale at which stringy effects become significant. Typically, it’s set to a very small value, ensuring that stringy behavior is evident only at very high energies, far beyond the reach of current particle accelerators.

Example with a Known Particle:

Quantization: By quantizing this action (applying quantum mechanics to string theory), we obtain a spectrum of possible vibrational states of the string. Each state can be interpreted as a different particle. Calculating Particle Properties, Mass Spectrum: The mass of the vibrational modes is determined by the level of excitation of the string. For a closed string, the mass formula in the simplest case is given by:

where

is the number operator, representing the level of string excitation.

Spin and Other Quantum Numbers: The type of vibration (how the string vibrates in different dimensions) determines the spin and other quantum numbers of the particles. For instance, a string vibrating in a particular mode might correspond to a particle with spin-1, like a photon.

Tachyons and Stability: Early models of string theory predicted particles with imaginary mass, known as tachyons, indicating an instability in the theory. This led to the development of superstring theory, which includes fermions and supersymmetry, to address these issues.

General Relativity (Higher-Dimensional Metrics): In higher-dimensional general relativity, the spacetime metric extends beyond four dimensions. For instance, in 5D Kaluza-Klein theory, the metric tensor

(where

run from 0 to 4) combines the 4D spacetime metric with an extra dimension, potentially unifying gravity and electromagnetism.

Topological Quantum Field Theory (TQFT): In TQFT, one often encounters algebraic structures like tensor categories. An example is the partition function

which assigns a complex number to a manifold

:

where

is the action functional of the field

over the manifold

. In higher dimensions, these structures become more complex and are often studied using higher categorical algebra.

Condensed Matter Physics (Topological Insulators): In topological insulators, the Chern number, an integer describing the topology of the band structure, plays a crucial role. It is given by an integral over the Brillouin zone:

where

is the Berry curvature in momentum space. In higher-dimensional systems, these integrals extend over more complex manifolds.

String Theory Quantization and Mass Spectrum

The process of deriving the mass spectrum in string theory from the Polyakov action involves several steps, outlined as follows:

Mode Expansion: The string’s spacetime coordinates,

, are expanded in terms of mode functions.

Canonical Quantization: Apply quantum mechanics principles, promoting classical fields to quantum operators and imposing commutation relations.

Virasoro Operators: From the quantization, we obtain Virasoro operators

, which are central to string theory. These operators are constructed from the energy-momentum tensor derived from the Polyakov action.

The Virasoro operators play a pivotal role in the quantization of string theory, providing an algebraic structure essential for the theory’s consistency. They emerge from the energy-momentum tensor derived from the Polyakov action, encapsulating the dynamics of strings in spacetime.

-Virasoro Operators: The Virasoro algebra consists of an infinite set of operators

and their antiholomorphic counterparts

, where

is an integer. These operators satisfy the following commutation relations:

where

and

are the central charges, and \(\delta_{m+n,0}\) is the Kronecker delta function.

-Physical State Conditions: A state

in string theory is considered physical if it satisfies the Virasoro constraints:

-These conditions ensure the elimination of negative norm states and the proper quantization of the string’s energy levels.

Example: Massless Scalar Field: To illustrate the application of Virasoro operators, consider a bosonic string in flat spacetime. The mode expansion of

leads to quantized oscillatory modes, and the Virasoro operators are expressed as:

,

where

are the oscillator modes, and the normal ordering is denoted by

. For a massless scalar field, consider the state created by acting on the vacuum with

This state represents a massless particle, satisfying the physical state conditions, including the crucial

constraint:

indicating that

, where

is the momentum of the state, identifying it as a massless scalar field in the spectrum of string theory. The Virasoro operators are instrumental in defining the physical states of string theory, ensuring the theory’s consistency by eliminating unphysical states and determining the properties of the particles in the string spectrum.

Level Matching Condition: For closed strings, the left-moving and right-moving modes of vibration must be equal, leading to a level matching condition

.

Mass Formula: The mass-squared operator in string theory is related to these Virasoro operators. For a closed string, the mass-squared is given by:

for the left-moving modes and similarly for the right-moving modes with

.

But the main interest for this text is: can string theory be helpful for the society, for anyone beyond the limited and rarefied atmosphere of the savants? The answer is, yes.

The formalism and ideas of string theory, while fundamentally rooted in the quest to understand the universe’s smallest constituents and forces, also inspire innovative approaches across various fields, including medicine, engineering, and finance. Here are some specific examples where the formalism and ideas of string theory might find application outside of pure physics:

Medicine: Understanding Genetic Diseases

Engineering: Design of Meta-materials

Finance: Modeling Market Dynamics

Cross-disciplinary: Network Theory and Machine Learning

Quantum Computing

While the direct application of string theory’s formalism to these fields may require a creative leap, the underlying mathematical tools and conceptual frameworks offer a rich source of inspiration. The analogies and insights derived from string theory can stimulate innovative approaches to longstanding problems, driving advancements in technology, finance, and healthcare.

Just look at these references to imagine the span of applications of string theory concepts in fields outside of traditional physics:

1. Data Science Applications to String Theory by Fabian Ruehle (2020) discusses how machine learning and data science techniques can be applied to string theory, including example codes. These approaches could potentially be adapted for use in medicine, engineering, and finance .https://www.sciencedirect.com/science/article/pii/S0370157319303072
2. Nonlocality in String Theory by G. Calcagni and L. Modesto (2013) explores the concept of nonlocality in string theory, which could have implications for understanding complex systems in engineering and finance. https://iopscience.iop.org/article/10.1088/1751-8113/47/35/355402/pdf
3. String Field Theory by Harold Erbin provides an introduction to string field theory, which as a field theory, offers a constructive formulation of string theory. This formalism could inspire new computational models in various disciplines. https://arxiv.org/abs/2301.01686
4. String Theory and Particle Physics: An Introduction to String Phenomenology by L. Ibáñez and Á. Uranga (2011) focuses on how string theory is connected to the real world of particle physics, providing models of physics beyond the Standard Model. The methodologies discussed could find applications in computational biology and complex systems engineering. https://www.cambridge.org/core/books/string-theory-and-particle-physics/7D005A97DA657F6675C2A62E449FC62E
5. Introduction to String Theory by Samarth Parekh (2022) offers an overview of the basic concepts of string theory, including its implications for quantum field theory, gravitational physics, and the nature of spacetime. These concepts could potentially influence advanced computational techniques in medicine and finance. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4009963
6. Focus Issue on String Cosmology by V. Balasubramanian and P. Moniz (2011) appraises recent applications of string-theoretic and string-inspired ideas to cosmology. The discussion on alternative models and dynamics could inspire novel approaches in data analysis and predictive modeling in finance. https://iopscience.iop.org/article/10.1088/0264-9381/28/20/200301

POSTSCRIPT:

The strong nuclear force appears to be described by Leonhard Euler’s beta function, as Gabrielle Veneziano inadvertently found in 1968. Because it is a function of the binomiale distribution, the beta function gets its name. As previously mentioned, it may be connected to the gamma function. In 1969, Veneziano discovered that Euler’s beta function was a suitable option when searching for a function that satisfies the basic postulates for the scattering amplitudes of elementary particles in strong contact. String theory was launched by this initial step, despite the fact that Veneziano was searching for a theory of strong interactions.

According to string theory, the cosmos is made up of vibrating energy filaments, or strings, as it presents us with a new understanding of the universe. The presence of additional physical things known as branes is also predicted by this scientific theory. String theory is a quantum field theory that describes the existence of particles and the generation of forces based on the compactification-the process of condensing hidden dimensions into extremely small sizes-at the level of high-energy theoretical physics.

The hypothesis was created in 1968 in an effort to explain hadron behaviour, but it was quickly shelved until the middle of the 1980s since it required more hidden dimensions. It developed into the M-theory, a more complex theory, in the 1990s [FN1]. Not only does string theory bring all particles together, but it also brings together the forces that govern their interactions. The electromagnetic field is a force that kind of spreads over space, enabling object interaction, and it fits the quantum theory rather well. Gravity doesn’t, though. And why? primarily because space is warped by gravity [FN2].

Additional References:

[FN1] The other leading alternative is known as loop quantum gravity. According to this theory space consists of not ordinary atoms, but extremely small chunks of space (like knots in a carpet).

[FN2] Quantum theory of gravity try to handle with this.

{1} The variables s and t are Mandelstam variables, key components in the description of particle collisions. They are defined in the context of a scattering process involving two incoming particles and two outgoing particles.

• The Mandelstam variable s is defined as the square of the total energy in the center-of-mass frame. It reflects the overall energy available for the interaction. In mathematical terms, for particles with four-momenta p1​ and p2​, it’s defined as

• . It gives an indication of the energy level at which the scattering process occurs.
• The Mandelstam variable t, on the other hand, represents the square of the momentum transfer between the incoming and outgoing particles. It’s defined as

• , where p3​ is the four-momentum of one of the outgoing particles. This variable measures how much the direction of motion of the particles has changed due to the scattering process.

The function α( s) and α( t) are Regge trajectories, which relate the spin of a particle to its mass squared, essentially describing how the properties of exchanged particles in the interaction vary with s and t. These trajectories are central in reggeon field theory, a framework used before QCD to describe the strong interaction at high energies.

The Veneziano amplitude thus combines these elements to predict the likelihood (amplitude) of various outcomes of particle collisions at different energy levels ( s) and angles ( t). Its form, invoking the Gamma function, ensures the amplitude has poles at integer values of − α( s) and − α( t), corresponding to the masses of the resonances involved in the scattering process. This was an attempt to explain the pattern of hadron masses and their interactions in the era before quarks and gluons were established as the fundamental constituents of hadrons.

Acknowledgment: Part of this post was based on the course materials provided by MIT’s course, “String Theory for Undergraduates” available in Spring 2007. More information can be found at: https://ocw.mit.edu/courses/8-251-string-theory-for-undergraduates-spring-2007/

Originally published at http://science2be.wordpress.com on April 12, 2024.