Duality symmetries and phase space quantum theory (2509.13872v1)

Abstract: We argue that duality symmetries can be manifestly realised when theories with these symmetries are quantised using phase space quantum theory. In particular, using background fields and phase space quantum theory, we quantise the bosonic string and show that it has SO(26,26) symmetry, even when the string is not compactified on a torus.

• The paper establishes that phase space quantum theory maintains duality symmetries, notably SO(D,D), even in quantizing the bosonic string.
• It utilizes the Groenewold-Moyal star product to compute observables while preserving symmetry structures lost in conventional operator approaches.
• The study implies that the phase space formulation opens new avenues for integrating duality concepts in string theory and related quantum field theories.

Duality Symmetries in Phase Space Quantum Theory

Introduction

This paper presents a rigorous analysis of duality symmetries within the framework of phase space quantum theory, with a particular focus on the quantization of the bosonic string. The authors demonstrate that duality symmetries, such as SO(D,D) invariance, can be manifestly preserved in the quantum theory when employing the Groenewold-Moyal phase space formalism, in contrast to the conventional operator-based quantization schemes. The work provides both a conceptual and technical foundation for the phase space approach, illustrating its equivalence to standard quantum mechanics and its advantages in maintaining duality symmetries, especially in string theoretic contexts.

Phase Space Quantum Theory: Formalism and Properties

The phase space formulation, originally developed by Groenewold and Moyal, eschews operator algebra in favor of functionals over coordinates and momenta, with multiplication defined via the non-commutative star product. The star product is given by

$$f * g = f(x, p) \exp\left[\frac{i\hbar}{2}\left(\overleftarrow{\partial_x}\overrightarrow{\partial_p} - \overleftarrow{\partial_p}\overrightarrow{\partial_x}\right)\right] g(x, p)$$

This product is associative but non-commutative, and its antisymmetric part yields the Moyal bracket, which replaces the classical Poisson bracket in quantum theory. Observables are represented as real functions on phase space, and the state of the system is encoded in the Wigner function, which is manifestly real for pure states.

The equivalence to standard quantum mechanics is established via the Weyl correspondence and the properties of the Wigner function. The expectation values and probabilities are computed as integrals over phase space, and the formalism naturally accommodates the Heisenberg uncertainty principle.

Duality Symmetries and Their Quantum Realization

Duality symmetries, such as those relating electric and magnetic fields in Maxwell theory or higher-form fields in supergravity, are fundamentally linked to the structure of the Poincaré group and its covariant representations. In conventional quantization, the imposition of commutation relations between conjugate variables (coordinates and momenta) necessitates a choice of representation, which typically breaks the manifest duality symmetry by privileging one variable over its dual.

The phase space approach, by treating coordinates and momenta on equal footing, preserves the symmetry structure inherent in the classical duality relations. This is particularly relevant for theories with multiple embeddings of irreducible representations, leading to infinite families of duality relations.

Quantization of the Bosonic String: SO(D,D) Symmetry

The authors apply the phase space formalism to the quantization of the bosonic string in a background, using a dual formulation with coordinates $x^\mu(\sigma)$ and $y_\mu(\sigma)$, both transforming as vectors under SO(D,D). The classical Hamiltonian and constraints are constructed to be SO(D,D) invariant, and the Poisson brackets are generalized accordingly.

In the conventional operator approach, quantization leads to a wavefunctional dependent on either $x^\mu$ or $y_\mu$, breaking the SO(D,D) symmetry. In contrast, the phase space quantization yields a state functional dependent on both $x^\mu$ and $y_\mu$, maintaining the full SO(D,D) invariance even in the absence of toroidal compactification. The zero mode sector is enlarged, effectively doubling the spacetime dimension in the quantum theory.

The constraints, including the Virasoro conditions, are imposed via the star product, and the oscillator algebra is reformulated in terms of phase space variables. The resulting quantum theory possesses a manifest SO(D,D) symmetry, with the state functional living on a $2D$-dimensional space.

Implications and Extensions

The preservation of duality symmetries in phase space quantum theory has significant implications for string theory and related models. The approach provides a natural setting for double field theory and generalized geometry, where background fields and coordinates transform under extended symmetry groups. The formalism is consistent with Siegel's theory and its extensions, as well as with the non-linear realizations of very extended algebras such as $D_{24}^{+++}$.

The enlargement of spacetime coordinates is interpreted in the context of solitonic objects, such as M2 branes, whose dynamics necessitate additional coordinates beyond the conventional spacetime. The phase space approach thus offers a unified framework for incorporating duality symmetries and solitonic moduli in quantum field theory.

Future Directions

The methodology outlined in this paper can be extended to interacting string theories via vertex operator constructions in phase space, with correlation functions computed using the star product. The formalism is expected to generalize to other brane models and higher-form field theories, potentially providing new insights into the structure of M-theory and E-theory. The authors suggest further exploration of duality symmetries in quantum field theories beyond the bosonic string, including the M2 brane and related objects.

Conclusion

This work establishes that phase space quantum theory provides a robust and technically sound framework for the quantization of theories with duality symmetries, preserving invariance structures that are typically lost in operator-based approaches. The explicit quantization of the bosonic string in this formalism demonstrates the persistence of SO(D,D) symmetry without compactification, with the quantum state functional depending on an enlarged set of coordinates. The results have broad implications for string theory, double field theory, and the quantum description of solitonic objects, and open avenues for further research into the role of duality in quantum field theory.