  The enigmatic pseudogap phase in underdoped cuprate high T_c superconductors
has long been recognized as a central puzzle of the T_c problem. Recent data
show that the pseudogap is likely a distinct phase, characterized by a medium
range and quasi-static charge ordering. However, the origin of the ordering
wavevector and the mechanism of the charge order is unknown. At the same time,
earlier data show that precursive superconducting fluctuations are also
associated with this phase. We propose that the pseudogap phase is a novel
pairing state where electrons on the same side of the Fermi surface are paired,
in strong contrast with conventional BCS theory which pair electrons on
opposite sides of the Fermi surface. In this state the Cooper pair carries a
net momentum and belong to a general class called pair density wave (PDW). The
microscopic pairing mechanism comes from a gauge theory formulation of the
resonating valence bond (RVB) picture, where electrons traveling in the same
direction feel an attractive force in analogy with Ampere's effects in
electromagnetism. We call this Amperean pairing. Charge order automatically
appears as a subsidiary order parameter. Our theory gives a prediction of the
ordering wavevector which is in good agreement with experiment. Furthermore,
the quasiparticle spectrum from our model explains many of the unusual features
reported in photoemission experiments. The Fermi arc and the unusual way the
tip of the arc terminates also come out naturally. We also discuss how the
onset of the Kerr effect in this state can be accommodated. Finally, we propose
an experiment which can directly test the notion of Amperean pairing.
