Superposition induced topology changes in quantum gravity

Abstract

We show that superpositions of classical states in quantum gravity with fixed topology can lead to new classical states with a different topology. We study this phenomenon in a particular limit of the LLM geometries. In this limit, the UV complete minisuperspace of allowed quantum states is exactly given by the Hilbert space of a free chiral boson in two dimensions. We construct this chiral boson purely in terms of combinatorial objects associated with the permutation group. As a byproduct of this analysis, we rederive the Murnaghan-Nakayama rule for characters of the permutation group. We are able to express this rule in terms of operator relations for raising and lowering operators on the Hilbert space of states in a free fermion basis. Our construction provides a preferred notion of bulk locality by studying an appropriate notion of D-brane state generating functions. We describe how multi-droplet LLM geometries with different topologies give new classical limits of the free chiral boson, even though they can be written as superpositions of coherent states with trivial topology. As a consequence, topology cannot be accessed by a single operator measurement in this quantum system. We study other non-linear measurements in the quantum wave-function, based on uncertainty and entanglement between modes of the chiral boson, that can be used as order parameters to measure the topology of such states.

Gauged fermionic matrix quantum mechanics

“…This sector alone has led to numerous additional insights in the theory of quantum gravity. Making the combinatorial problem of relating the traces and Schur functions more precise, it has been noted that the topology of spacetime can be changed by superposing states of a fixed topology [10,11].…”

Can Topology and Geometry be Measured by an Operator Measurement in Quantum Gravity?

“…We take these to be (6) and similar for c, with all other commutators vanishing. These assertions are proven in the companion work to this paper [7], where the details on the cutoff and the applicability of these commutation relations are deduced from first principles. The nearby states form a small Hilbert space in their own right.…”

“…In this limit, the oscillators give a rise to a free Fock space, with a free mode for every n. We will take the existence of this limit as a statement of fact and it is in this limit that our statements can be made rigorously. Many of the technical details that are required to prove some claims in this paper will appear in a forthcoming paper by the authors [7]. This paper makes the claim that topology cannot be measured by operators.…”

“…These new modes only extend to values of order n << M, L. Beyond that they do not exist as independent operators [7]. This is a type of stringy exclusion principle of the same type as the one implemented in [16].…”

“…Similar limits for folded setups have been considered in the c = 1 matrix model [13], where it is found that in general h(θ) 2 − h(θ) 2 is large, but does not measure the number of edges directly. In folded configurations the mixed correlator (8) will be a non-diagonal matrix, that encodes the Fourier transform of the local number of edges [7].…”