Complexity of operators generated by quantum mechanical Hamiltonians
Abstract
We propose how to compute the complexity of operators generated by Hamiltonians in quantum field theory (QFT) and quantum mechanics (QM). The Hamiltonians in QFT/QM and quantum circuit have a few essential differences, for which we introduce new principles and methods for complexity. We show that the complexity geometry corresponding to onedimensional quadratic Hamiltonians is equivalent to AdS_{3} spacetime. Here, the requirement that the complexity is nonnegative corresponds to the fact that the Hamiltonian is lower bounded and the speed of a particle is not superluminal. Our proposal proves the complexity of the operator generated by a free Hamiltonian is zero, as expected. By studying a nonrelativistic particle in compact Riemannian manifolds we find the complexity is given by the global geometric property of the space. In particular, we show that in low energy limit the critical spacetime dimension to ensure the `nonnegative' complexity is the 3+1 dimension.
 Publication:

Journal of High Energy Physics
 Pub Date:
 March 2019
 DOI:
 10.1007/JHEP03(2019)010
 arXiv:
 arXiv:1810.09405
 Bibcode:
 2019JHEP...03..010Y
 Keywords:

 Gaugegravity correspondence;
 Holography and condensed matter physics (AdS/CMT);
 High Energy Physics  Theory;
 Quantum Physics
 EPrint:
 A revised version and gives more detailed examples on why "complexity'' in quantum mechanics and quantum field theory should be biinvariant