[Paper Review] Symmetries in particle and string theories

W. Chagas-Filho|ArXiv.org|Sep 23, 2003

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TL;DR

This paper investigates space-time symmetries in bosonic relativistic particles and strings using a constrained Hamiltonian formalism that incorporates Hamiltonian constraints. It shows that massless particles and tensionless strings possess enhanced space-time invariances—specifically scale and conformal symmetries—compared to their massive and tensionful counterparts. The key result is that in the tensionless limit, the string action admits a gauge-fixed description where every point of the string moves like an almost-free massless particle, consistent with gauge-theory/string-duality expectations.

ABSTRACT

We study the space-time invariances of the bosonic relativistic particle and bosonic relativistic string using general formulations obtained by incorporating the Hamiltonian constraints into the formalism. We point out that massless particles and tensionless strings have a larger set of space-time invariances than their massive and tensionful partners, respectively. We also show that it is possible to use the reparametrization invariance of the string formulation we present to reach the classical conformal equations of motion without the use of two-dimensional Weyl scalings of the string world sheet. Finally, we show that it is possible to fix a gauge with an enlarged number of space-time invariances in which every point of the free tensionless string moves as if it were an almost-free massless particle. The existence of such a string motion agrees with what is expected from gauge theory-string duality.

Motivation & Objective

• To understand the space-time invariances of relativistic particles and strings using a general Hamiltonian formulation incorporating constraints.
• To clarify the origin of enhanced symmetries in massless particles and tensionless strings compared to their massive and tensionful counterparts.
• To demonstrate that the classical conformal equations of motion for strings can be derived from reparametrization invariance alone, without relying on 2D Weyl scaling.
• To explore the tensionless limit of string theory and its implications for symmetry enhancement and duality with gauge theories.

Proposed method

• Formulate the relativistic particle and string actions using a constrained Hamiltonian approach with first-class constraints.
• Introduce a Lagrange multiplier field λ to generalize the action, allowing consistent treatment of massless and tensionless limits.
• Use reparametrization invariance to fix the gauge λ = constant, leading to wave-like or particle-like equations of motion depending on tension T.
• Analyze the space-time transformations (Poincaré, scale, conformal) under which the actions are invariant, particularly in the T=0 limit.
• Derive the classical equations of motion from the generalized action and examine their symmetry structure in the tensionless case.
• Identify a new gauge transformation for the T=0 string that extends the massless particle symmetry, preserving the dynamics under arbitrary functions of ẋ².

Experimental results

Research questions

• RQ1 Why do massless particles and tensionless strings exhibit a larger set of space-time invariances than their massive and tensionful counterparts?
• RQ2 Can the classical conformal equations of motion for the string be derived without invoking two-dimensional Weyl invariance?
• RQ3 What is the physical and algebraic origin of the enhanced symmetries in the tensionless string limit?
• RQ4 How does the gauge-fixed dynamics of the tensionless string compare to that of a free massless particle?
• RQ5 Is there a consistent description of the tensionless string that respects both the string structure and the enhanced symmetries?

Key findings

• The action for the tensionless string (T=0) is invariant under scale and conformal transformations, in addition to Poincaré invariance, unlike the T≠0 case.
• In the gauge λ=constant, the tensionless string satisfies ∂²xᵘ/∂τ²=0, meaning each worldsheet point moves like a free massless particle.
• The T=0 string action admits an additional gauge symmetry δxᵘ = exp{(1/3)γ(ẋ²)}xᵘ, δλ = exp{(2/3)γ(ẋ²)}λ, which extends the massless particle symmetry.
• The classical conformal wave equation for the string can be derived using only reparametrization invariance, without requiring 2D Weyl scaling.
• The tensionless string action is consistent only when the constraint p·ẋ≈0 is imposed, which preserves the string-like structure while allowing particle-like motion.
• The enhanced symmetries in the tensionless limit are consistent with expectations from gauge-theory/string-duality, particularly in the weak-coupling regime of gauge theories.