[Paper Review] Physical States in the $W_3$ String

C.N. Pope|ArXiv.org|Nov 11, 1992

Computational Physics and Python Applications3 references3 citations

TL;DR

This paper investigates the physical state spectrum of the $W_3$ string, a higher-spin extension of bosonic string theory, focusing on states with non-standard ghost structures. Using covariant quantization and BRST cohomology, it identifies discrete and continuous momentum states that include excitations of the ghost fields, revealing a richer physical spectrum than standard strings.

ABSTRACT

We present a review of some of the recent developments in the study of the $W_3$ string. One of the interesting features of the theory is that the physical spectrum includes states with non-standard ghost structure, such as excitations of the ghost fields, both for discrete-momentum and continuous-momentum states. (Contribution to the Proceedings of the 16'th Johns Hopkins Workshop on Current Problems in Particle Physics, Gothenburg, Sweden, June 1992)

Motivation & Objective

• To understand the physical state spectrum of the $W_3$ string, a higher-spin extension of string theory.
• To analyze how the inclusion of $W_3$ algebra symmetries alters the structure of physical states compared to standard bosonic strings.
• To investigate the role of ghost fields in the physical spectrum, particularly excitations beyond the usual Fock vacuum.
• To determine whether continuous-momentum states and discrete-momentum states both contribute to the physical spectrum in a consistent way.
• To clarify the implications of non-standard ghost structures for the unitarity and consistency of the $W_3$ string theory.

Proposed method

• Employing covariant quantization techniques adapted to the $W_3$ algebra, which extends the Virasoro algebra by including a spin-3 current.
• Applying BRST cohomology to identify physical states as those annihilated by the BRST charge and not exact with respect to it.
• Analyzing the ghost sector using the $W_3$ ghost system, which includes fields with non-trivial conformal dimensions and statistics.
• Deriving conditions on momentum states (both discrete and continuous) to ensure they lie in the physical spectrum via BRST invariance.
• Using the operator product expansion (OPE) of the $W_3$ current algebra to constrain the allowed physical states.
• Classifying physical states by their ghost number and momentum quantum numbers, identifying states with non-standard ghost excitations.

Experimental results

Research questions

• RQ1 How does the $W_3$ algebra symmetry modify the physical state spectrum compared to the standard $W_1 = \text{Virasoro}$ string?
• RQ2 What is the role of ghost field excitations in the physical spectrum of the $W_3$ string?
• RQ3 Do both discrete-momentum and continuous-momentum states contribute to the physical spectrum in the $W_3$ string?
• RQ4 Can physical states with non-standard ghost structures (e.g., excited ghost Fock states) be consistently included in the BRST cohomology?
• RQ5 What are the implications of these findings for the unitarity and consistency of $W_3$ string theory?

Key findings

• The physical spectrum of the $W_3$ string includes states with non-standard ghost structures, such as excited ghost Fock states.
• Both discrete-momentum and continuous-momentum states are found to contribute to the physical spectrum, extending the standard string state content.
• Excitations of the ghost fields are shown to be physical in the BRST cohomology, indicating that the ghost system supports non-trivial physical degrees of freedom.
• The BRST cohomology analysis reveals that the $W_3$ string's physical state space is richer than that of the standard bosonic string due to the extended symmetry algebra.
• The presence of continuous-momentum states suggests a more complex structure in the spectrum, potentially linked to non-perturbative or higher-spin effects.
• The results indicate that the $W_3$ string theory allows for physical states outside the standard Fock vacuum sector, challenging conventional assumptions about ghost content in string theories.