[Paper Review] From Dual Models to String Theory

Peter Goddard|ArXiv.org|Feb 22, 2008

Black Holes and Theoretical Physics60 references3 citations

TL;DR

This paper recounts Peter Goddard's personal journey from dual models to string theory, emphasizing the development of the No-Ghost Theorem, quantization of the relativistic string, and the role of conformal symmetry in eliminating unphysical states. The key contribution is the proof that Virasoro constraints at α₀ = 1 eliminate all ghost states, establishing a consistent quantum theory of the relativistic string despite the presence of a tachyon in the spectrum.

ABSTRACT

A personal view is given of the development of string theory out of dual models, including the analysis of the structure of the physical states and the proof of the No-Ghost Theorem, the quantization of the relativistic string, and the calculation of fermion-fermion scattering.

Motivation & Objective

• To trace the evolution of dual models into a consistent quantum theory of relativistic strings.
• To resolve the problem of negative-norm (ghost) states in the dual model through the imposition of Virasoro constraints.
• To establish the physical consistency of string theory by proving the No-Ghost Theorem, ensuring unitarity and probabilistic consistency.
• To explore the geometric and dynamical foundations of string theory, including the worldsheet action and conformal invariance.
• To connect developments in dual models with emerging insights in gauge theory, monopoles, and dual groups, later found to be relevant to string compactifications.

Proposed method

• Utilized the oscillator representation of the dual model, with creation and annihilation operators $ a_n^ u $ satisfying canonical commutation relations and acting on a vacuum state $ |0\rangle $.
• Applied the Möbius invariance of the Koba-Nielsen variables to constrain the spectrum and remove some ghost states.
• Introduced the Virasoro algebra with $ \alpha_0 = 1 $, leading to an infinite set of constraints that could potentially eliminate all ghost states.
• Used the worldsheet action $ \mathcal{A}_O \propto \int \mathcal{L}_O \, d\sigma d\tau $ with $ \mathcal{L}_O = \frac{1}{2} \left( \left( \frac{\partial x}{\partial \sigma} \right)^2 - \left( \frac{\partial x}{\partial \tau} \right)^2 \right) $ to describe the string's dynamics.
• Employed Dirac bracket formalism and DDF operators to quantize the string in a time-like gauge, particularly in collaboration with Tullio Regge.
• Explored compactification on group manifolds and later on tori to generate non-abelian gauge symmetries via affine Kac-Moody algebras, leading to the identification of $ E_8 \times E_8 $ and $ SO(32) $ as preferred compactifications.

Experimental results

Research questions

• RQ1 How can the ghost states in the dual model be systematically removed to ensure a unitary quantum theory?
• RQ2 What is the role of conformal invariance in the quantization of the relativistic string and how does it relate to the Virasoro algebra?
• RQ3 Can a geometric action principle be formulated for the relativistic string that leads to the correct dual model amplitudes?
• RQ4 How do compactifications of extra dimensions in string theory lead to non-abelian gauge groups?
• RQ5 What is the connection between dual groups in monopole theory and those emerging from string compactifications?

Key findings

• The No-Ghost Theorem was proven: when $ \alpha_0 = 1 $, the Virasoro constraints eliminate all negative-norm states, ensuring a unitary quantum theory.
• The lowest-lying state in the spectrum has $ M^2 = -2 $, confirming the presence of a tachyon, a consequence of the ghost-free condition at $ \alpha_0 = 1 $.
• The infinite-dimensional conformal symmetry of the worldsheet at $ \alpha_0 = 1 $ provides the necessary constraints to remove all ghost states, establishing a consistent quantum framework.
• The use of DDF operators and Dirac brackets enabled the successful quantization of the string in a time-like gauge, resolving earlier technical obstacles.
• Compactification on group manifolds and later on tori revealed that affine Kac-Moody algebras generate non-abelian gauge symmetries, with $ E_8 \times E_8 $ and $ SO(32) $ emerging as the only consistent choices.
• The dual group structure in monopole theory, later linked to Langlands duality, was found to be deeply connected to the gauge groups arising from string compactifications, suggesting a unifying mathematical structure.