[Paper Review] Pohlmeyer invariants are expressible in terms of DDF invariants

Urs Schreiber|ArXiv.org|Apr 2, 2004

Black Holes and Theoretical Physics8 references3 citations

TL;DR

This paper demonstrates that the classical Pohlmeyer invariants of the bosonic string are a proper subset of the DDF invariants, enabling their consistent quantization via the well-established DDF operator formalism. The key result is that Pohlmeyer invariants remain invariant under the DDF redefinition of oscillators, allowing a direct generalization to the superstring and a transparent quantization without requiring the critical dimension D=26.

ABSTRACT

It is shown that the Pohlmeyer invariants of the classical bosonic string are a proper subset of the classical DDF invariants. This makes the quantization of the Pohlmeyer invariants particularly transparent and allows to generalize them to the superstring.

Motivation & Objective

• To clarify the relationship between Pohlmeyer invariants and DDF invariants in the classical bosonic string.
• To provide a consistent quantization procedure for Pohlmeyer invariants by expressing them in terms of DDF invariants.
• To generalize the Pohlmeyer program to the superstring using the DDF framework.
• To demonstrate that the Pohlmeyer invariants are a proper subset of the DDF invariants, ensuring their quantization is transparent and closed.

Proposed method

• The paper defines DDF invariants $ A_m^\mu $ and $ \tilde{A}_m^\mu $ using lightlike vector fields and phase-space reparameterizations via $ R_\pm(\sigma) $.
• It introduces truncated DDF invariants $ a_m^\mu $ and $ \tilde{a}_m^\mu $ by removing the $ k\cdot x $-dependent phase factor, ensuring level-matching.
• It shows that reparameterization-invariant functionals like Wilson lines—whose Taylor coefficients are the Pohlmeyer invariants—remain invariant under the substitution $ \mathcal{P}_\pm \to \mathcal{P}^R_\pm $.
• It establishes that $ Z^{\mu_1\cdots\mu_n}(\mathcal{P}_\pm) = Z^{\mu_1\cdots\mu_n}(\mathcal{P}^R_\pm) $, proving Pohlmeyer invariants are expressible in terms of DDF invariants.
• It uses the known consistent quantization of DDF invariants to induce a consistent quantization of Pohlmeyer invariants, valid in any spacetime dimension.
• It extends the construction to the superstring by leveraging the known DDF generalization to supersymmetry.

Experimental results

Research questions

• RQ1 Are the Pohlmeyer invariants a proper subset of the DDF invariants in the classical bosonic string?
• RQ2 Can the Pohlmeyer invariants be consistently quantized via the DDF formalism?
• RQ3 Does the DDF-based quantization of Pohlmeyer invariants avoid the need for the critical dimension D=26?
• RQ4 Can the Pohlmeyer invariants be generalized to the superstring using the DDF framework?
• RQ5 Is the Poisson algebra of Pohlmeyer invariants closed under quantization when expressed in terms of DDF operators?

Key findings

• The Pohlmeyer invariants are a proper subset of the DDF invariants, as shown by the invariance of reparameterization-invariant functionals like Wilson lines under the DDF redefinition.
• The Pohlmeyer invariants remain unchanged when the standard oscillators $ \alpha_m^\mu $ and $ \tilde{\alpha}_m^\mu $ are replaced by DDF invariants $ A_m^\mu $ and $ \tilde{A}_m^\mu $, due to the invariance of $ Z^{\mu_1\cdots\mu_n}(\mathcal{P}_\pm) $ under $ \mathcal{P}_\pm \to \mathcal{P}^R_\pm $.
• The consistent quantization of DDF invariants induces a consistent quantization of the Pohlmeyer invariants, with their commutators remaining in the algebra of DDF invariants.
• The quantization of Pohlmeyer invariants is valid in any spacetime dimension D, independent of the no-ghost theorem, which requires D=26 for unitary Hilbert space representations.
• The DDF formalism allows a direct generalization of Pohlmeyer invariants to the superstring, as DDF invariants are already known to generalize to supersymmetry.
• The result resolves a key obstacle in the Pohlmeyer program by showing that its invariants can be consistently quantized without requiring the unproven quadratic generation hypothesis.