Deriving Field Theory from String Theory: A Comprehensive Exploration
The interrelation between string theory and field theory is a fascinating area of study in modern theoretical physics. This article aims to explore how field theory can be derived from string theory, highlighting key equations and concepts that bridge these two fundamental theoretical frameworks. By delving into the mathematical details, we will uncover the underlying principles that connect quantum and classical physics.
Introduction to String Theory and Field Theory
String theory posits that particles in the universe are not point-like, but rather one-dimensional 'strings' vibrating at specific frequencies. These strings interact and 'play' the physical laws we observe in the universe. On the other hand, field theory, such as Einstein's theory of general relativity, describes how forces and particles interact through fields. The goal of this article is to demonstrate how string theory can lead to the derivation of field theory, focusing on the interactions between gravitons and other fundamental particles.
Graviton Interaction and Witten's Knot Theory
Gravitons, the hypothetical particles mediators of gravitational forces, can be described through string theory. They interact indirectly through self-interaction terms, which are derived from Witten's knot theory. Consider the equation self-interact graviton which forms the basis for understanding gravitational interactions at the quantum level.
Quantum Black Hole at Planck Scale
At the Planck scale, where the quantum effects of gravity become significant, a quantum black hole can be modeled using the equation . This equation describes the length scale at which quantum gravity effects are expected to be observable. This quantum black hole can be further analyzed to deduce the chemical potential ch2pigm^28pigmc^2/2^2/c^4, which is a solution to the field equations of general relativity.
Proton Scale and Atomic Scale
Proton Scale
At the proton scale, the equation plgp4pipm/3/c^28.80910^-16 meter describes the length scale associated with the proton. Using this, the chemical potential at the proton scale can be derived as ch2pilmc^2/4pi/3.
Atomic Scale
The atomic scale is determined by the equation A^2gppime/128.4980143c^2. This equation reveals the relationship between gravitational constants and the atomic size, leading to the chemical potential ch2piAmec^2/137.036 at the quantum field scale.
Deriving Strong Force and EM Force
Strong Force
The strong force, which binds quarks and gluons, can be derived using the equation gpgm^2/pm^2gpl/4.1888l^21.1310^28. This force can be related to the chemical potential at the proton scale as ch2pilmc^22piplpmc^2/4pi/3.
EM Force
The electromagnetic force between a proton and an electron can be described using the equation ke^2gppm^2/137.036ch/2pi137.036. This equation unites electromagnetism with quantum field theory, with the permittivity of free space ech/2pi137.036 and the magnetic moment of the electron e-mec/137.036^2A.
Unification of Forces and Gauge Fields
The unified force can be derived from the ratio of proton to electron masses and the inverse fine structure constant, as shown by the equation ke^2gppm^2/137.036. This unification is a critical aspect of gauge theory and can be represented as 0.0011614097251/2pi137.036, the anomalous magnetic moment of the electron's g-2/2 factor.
Vacuum Field and Photon
The energy levels of the vacuum field can be calculated using ch2pigm^22pig137.036upe-/l137.036upe/l, which leads to the emission of photons. The energy levels of the photon are given by rEnch/LchRmec/137.036^2/213.6e[1.60210^-19].
Weak Force and Supersymmetry
The weak force can be unified with the electromagnetic force using the mass ratios between particles. The equation 2pi0.001161409725me/pm^2me/pm^2/137.036me/pm^2/gm^2/ke^2128.4980143A/3.14159262.1610^-9 demonstrates this unification, revealing the discrepancy in the muon's magnetic moment of g-2/2 factor. This discrepancy is further explored through the Fermilab experiment conducted on August 23, 2023.
Conclusion
In conclusion, the derivation of field theory from string theory highlights the intricate connections between quantum mechanics and general relativity. The equations discussed in this article demonstrate how fundamental forces can be derived from the interaction of strings at various scales, from the quantum black hole at the Planck scale to the proton and atomic scales. The unification of forces, particularly through the fine structure constant and gauge fields, is a critical aspect of modern theoretical physics.