  We study the physical property of pair density wave (PDW) and fluctuating
PDW, and use it to build an effective theory of the strongly interacting
pseudogap phase in cuprate high temperature superconductors. In Chapter2, we
study how Fulde-Ferrell state, the simplest form of PDW, responds to incident
light. The collective motion of the condensate plays a key role; gauge
invariance guides us to the correct result. From Chapter 3 to Chapter 7, we
construct a pseudogap metallic state by considering quantum fluctuating PDW. We
analyze a recent scanning tunneling microscope (STM) discovery of period-8
density waves in the vortex halo of the d-wave superconductor. We put it in the
context of the broader pseudogap phenomenology, and compare the experimental
results with various PDW-driven models and a charge density wave (CDW) driven
model. We propose experiments to distinguish these different models. We present
the Bogoliubov bands of PDW. We discuss fluctuating PDW from the general
perspective of fluctuating superconductivity. We discuss how Bogoliubov bands
evolve when the superconducting order parameter is fluctuating. We compare
theoretical predictions with existing experiments on angle-resolved
photoemission spectroscopy (ARPES), infrared conductivity, diamagnetism, and
lattice symmetry breaking. The material presented here is based on
Ref.[38,40,41].
