On operator growth and emergent Poincaré symmetries

Abstract

We consider operator growth for generic large-N gauge theories at finite temperature. Our analysis is performed in terms of Fourier modes, which do not mix with other operators as time evolves, and whose correlation functions are determined by their two-point functions alone, at leading order in the large-N limit. The algebra of these modes allows for a simple analysis of the operators with whom the initial operator mixes over time, and guarantees the existence of boundary CFT operators closing the bulk Poincaré algebra, describing the experience of infalling observers. We discuss several existing approaches to operator growth, such as number operators, proper energies, the many-body recursion method, quantum circuit complexity, and comment on its relation to classical chaos in black hole dynamics. The analysis evades the bulk vs boundary dichotomy and shows that all such approaches are the same at both sides of the holographic duality, a statement that simply rests on the equality between operator evolution itself. In the way, we show all these approaches have a natural formulation in terms of the Gelfand-Naimark-Segal (GNS) construction, which maps operator evolution to a more conventional quantum state evolution, and provides an extension of the notion of operator growth to QFT.

Random matrix theory for complexity growth and black hole interiors

“…The growing momentum results in gravitational backreaction which, in turn, imprints this exponential growth on the time-dependence of the volume of the maximal volume slices, and by extension on the particle's distance from the boundary along these slices. This characteristic behavior of momentum has been compellingly linked to the dual operator "size" which increases at the same exponential rate due to scrambling [1][2][3][4][5]; similarly, the corresponding maximal volume growth is expected to reflect the evolution of operator complexity [1,[6][7][8][9] -assuming an appropriate definition of the latter. This early-time chaotic behavior ceases around t ∼ β log S BH when the CFT operator's size saturates at its maximal O(S BH ) value and the bulk particle is swallowed up by the shifted horizon of the backreacted geometry.…”

“…In recent works [4,[13][14][15][16], a different definition of operator complexity was proposed, one attuned to the study of complexity generated by time evolution, that depends only on the system's Hamiltonian and a reference quantum state. The idea capitalizes on the intuition that acting with the Hamiltonian on an initially simple operator O via commutators generally help us ascend the complexity ladder.…”

Soft photon theorems from CFT Ward identites in the flat limit of AdS/CFT

“…From the point of view of AdS/CFT and in particular HKLL bulk reconstruction, the existence of such operators is not very surprising and can be explained in terms of error-correcting codes [53]. For a different construction of CFT operators obeying canonical commutation relations, see [54,55]. Light ray operators [56] are another class of non-local operators which again obey a simple but non-trivial algebra and it was shown in [34] that in a Lorentz invariant CFT, certain operators reproduce the BMS algebra.…”

Operator growth in 2d CFT

“…The linear growth of b n ∼ an translates to exponential growth of the Kcomplexity, with characteristic Lyapunov exponent λ L = 2a. This operator growth hypothesis has been further studied and verified numerically in various examples [3,20,22,[34][35][36][37][38][39][40][41][42].…”

“…The precise map between operators and these states will not be relevant for us, but interested readers can consult e.g. [34,35] for details using the GNS construction.…”