A cordial introduction to double scaled SYK.

Area of Science:

• Quantum mechanics
• Quantum gravity
• Statistical physics

Background:

• Sachdev-Ye-Kitaev (SYK) models and p-local quantum Hamiltonians are key to understanding complex quantum systems.
• These models offer insights into quantum chaos and the emergence of gravity.

Purpose of the Study:

• To review recent advancements in the double scaled Sachdev-Ye-Kitaev model and related p-local random Hamiltonians.
• To explore the connection between these models and quantum gravity in Anti-de Sitter space (AdS2).

Main Methods:

• Utilizing expansions with chord diagrams, solvable by combinatorial methods.
• Deriving exact results for the spectrum, correlation functions, and Lyapunov exponent.
• Analyzing the connection to Schwarzian quantum mechanics and quantum groups.

Main Results:

• Exact solutions for the spectrum, correlation functions, and Lyapunov exponent were obtained.
• A link to Schwarzian quantum mechanics, a theory of quantum gravity in AdS2, was established in a specific limit.
• The theory is controlled by quantum group algebraic structures, suggesting quantum gravity on non-commutative q-deformed AdS2.

Conclusions:

• The study provides a comprehensive overview of exact results in SYK and p-local models.
• It highlights the profound connection between these quantum mechanical models and theories of quantum gravity.
• Future research directions and related universality classes are discussed.