Holographic and Wilsonian Renormalization Groups (1010.1264v2)

Abstract: We develop parallels between the holographic renormalization group in the bulk and the Wilsonian renormalization group in the dual field theory. Our philosophy differs from most previous work on the holographic RG; the most notable feature is the key role of multi-trace operators. We work out the forms of various single- and double-trace flows. The key question, `what cutoff on the field theory corresponds to a radial cutoff in the bulk?' is left unanswered, but by sharpening the analogy between the two sides we identify possible directions.

• The paper establishes parallels between holographic renormalization in AdS/CFT and Wilsonian RG in QFT by contrasting non-local observables with explicit momentum cutoffs.
• It reveals that multi-trace operators play a pivotal role in the Wilsonian approach, enriching the dynamics often overlooked in traditional HRG analyses.
• The study compares Hamiltonian evolution in both frameworks, suggesting novel boundary conditions that may unify holographic and quantum field perspectives.

Holographic and Wilsonian Renormalization Groups: Exploring the Parallels

The paper "Holographic and Wilsonian Renormalization Groups" by Idse Heemskerk and Joseph Polchinski investigates the intriguing relationship between two seemingly distinct approaches to renormalization: the holographic renormalization group (HRG) within the context of Anti-de Sitter/Conformal Field Theory (AdS/CFT) duality, and the conventional Wilsonian renormalization group (WRG) as developed for quantum field theories (QFTs). A central focus of the paper is understanding how these frameworks, stemming from different perspectives—gravity for HRG and quantum field theory for WRG—can inform and enhance each other when analyzed together.

Key Insights

1. Analogies and Differences: The authors construct parallels between HRG and WRG while respectful of their foundations: HRG is inherently non-local, addressing observables in quantum gravity, whereas WRG emphasizes locality and the systematic integration of momentum shells. Notably, a key distinction lies in the WRG's explicit cutoff—a tool not yet concretely defined in holographic scenarios.
2. Role of Multi-Trace Operators: A pivotal difference articulated in the paper is the role of multi-trace operators within these frameworks. While single-trace operators are often prominent in traditional HRG analyses, the Wilsonian approach, with its inherent flexibility, naturally produces and is influenced by multi-trace operators. The dynamics of these operators reveal additional layers to the RG flow that may not be adequately captured in strict single-trace contexts.
3. Hamiltonian Evolution: The equations governing the evolution of amplitudes in both HRG and WRG are compared. In the WRG, the flow dictated by a Hamilton-Jacobi structure suggests a novel boundary condition solution through which the holographic principle and scale invariance meet. This relationship underscores evolving our understanding of the duality and opens prospects for identifying the holographic analogs of familiar quantum field processes.
4. Scale and Convergence: The paper emphasizes the necessity of understanding what a cutoff signifies in HRG, especially when contrasting it with the more intuitive cutoff principles in WRG. By establishing a deeper analogy between the two, the authors hope for clearer guidelines toward effectively computing local observables in AdS/CFT.

Implications and Future Directions

The implications of this research stretch both practically and theoretically. On the theoretical side, establishing precise correspondences offers a clearer path for translating insights from one domain (quantum field theories) into another (string theories and gravity), potentially illuminating unresolved questions in quantum gravity. Practically, should this framework achieve a seamless integration, it could greatly enhance computational techniques for evaluating physical processes across these realms, such as thermalization in strongly-coupled systems or spotting precise roles of traced composite structures in boundary conformal field theories (CFTs).

The paper leaves several open questions that invite future exploration. What becomes of the detailed stringy nature of fields in the ultra-high-energy regions of the bulk when tackled under a holographic Wilsonian technique? Needed is further refinement of the double-trace flows especially with insight into their stable or unstable nature in diverse holographic schemes—ultimately assisting efforts to unify bulk-boundary understandings fully.

In conclusion, the analogies explored in this work between the holographic and Wilsonian renormalization groups establish a fundamental part of the ongoing discourse in reconciling differences between gravitational and quantum field theoretical notions. The compelling path to clarifying these relations reflects broader efforts to understand how space, time, and quantum processes holographically coexist and influence one another in our universe's foundational structure.