Abstract
Previous work has explored the connections between three concepts — operator size, complexity, and the bulk radial momentum of an infalling object — in the context of JT gravity and the SYK model. In this paper we investigate the higher dimensional generalizations of these connections. We use a toy model to study the growth of an operator when perturbing the vacuum of a CFT. From circuit analysis we relate the operator growth to the rate of increase of complexity and check it by complexity-volume duality. We further give an empirical formula relating complexity and the bulk radial momentum that works from the time that the perturbation just comes in from the cutoff boundary, to after the scrambling time.
Highlights
Are thermal scale quanta [6]
From the bulk point of view, the particle falls into a deeper and deeper radial location and gets a higher and higher momentum. We study this process by a toy model of gluon-spitting and see that the operator size grows linearly in time in this regime, and this parallels the linear growth of momentum
In this paper we studied the connections between operator size growth, complexity increase, and bulk radial momentum in spacetime dimension D ≥ 3
Summary
The growth of a simple operator under time evolution in SYK was studied in [3, 11]. Starting from one fermion ψ1, the average number of fermions making up the operator ψ1(t) increases as time increases. In [1, 2] it was pointed out the growth of the operator corresponds to the increase of the particle momentum as it falls in. The operator ψ(t) produces an infalling particle in the dual bulk geometry. 1 βis the local energy scale depending on the radial location of the particle [2]. From another point of view, one can look for the SL(2) symmetry generators of AdS2 in JT gravity [4, 5]. One can relate the operator size to the complexity of the perturbed state [9, 10]. We will generalize the above discussions to bulk dimensions D ≥ 3
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