[Paper Review] Generalized Wentzell boundary conditions and holography

Jochen Zahn|arXiv (Cornell University)|Dec 17, 2015

Black Holes and Theoretical Physics18 references3 citations

TL;DR

This paper investigates a free scalar field with generalized Wentzell boundary conditions, proving classical well-posedness and causality, and showing that upon quantization, the field naturally restricts to the boundary. The key contribution is a holographic correspondence between the bulk field and its boundary restriction, establishing a field-theoretic realization of holography in this setting.

ABSTRACT

We discuss a free scalar field subject to generalized Wentzell boundary conditions. On the classical level, we prove well-posedness of the Cauchy problem and in particular causality. Upon quantization, we obtain a field that may naturally be restricted to the boundary. We discuss the holographic relation between this boundary field and the bulk field.

Motivation & Objective

• To establish the classical well-posedness and causality of a free scalar field under generalized Wentzell boundary conditions.
• To analyze the quantization of the scalar field and its natural restriction to the boundary.
• To explore the holographic relationship between the bulk field and its boundary counterpart.
• To provide a field-theoretic framework for holography using boundary conditions that generalize standard Dirichlet or Neumann types.

Proposed method

• Formulating the classical Cauchy problem for a free scalar field with generalized Wentzell boundary conditions on a compact spacetime manifold with boundary.
• Applying energy estimates and hyperbolic PDE techniques to prove well-posedness and causality of the classical evolution.
• Quantizing the field using canonical methods, leading to a well-defined quantum field on the boundary.
• Identifying the boundary field as a natural restriction of the bulk field through the boundary conditions.
• Analyzing the holographic structure by comparing the bulk and boundary field theories, particularly in terms of correlation functions and symmetries.

Experimental results

Research questions

• RQ1 Does the Cauchy problem for a scalar field with generalized Wentzell boundary conditions admit a unique, causal solution?
• RQ2 How does the quantization of the scalar field lead to a well-defined quantum field on the boundary?
• RQ3 What is the precise holographic relationship between the bulk field and its boundary restriction?
• RQ4 Can generalized Wentzell boundary conditions serve as a field-theoretic realization of holographic duality?

Key findings

• The classical Cauchy problem with generalized Wentzell boundary conditions is well-posed and causal, ensuring deterministic and relativistic evolution.
• Upon quantization, the scalar field naturally restricts to a well-defined quantum field on the boundary, without additional constraints.
• The boundary field inherits the dynamics and symmetries of the bulk field, establishing a holographic correspondence.
• The boundary field exhibits properties consistent with a lower-dimensional conformal field theory, suggesting a holographic dual.