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Ghost Problems from Pauli-Villars to Fourth-Order Quantum Gravity and their Resolution

We review the history of the ghost problem in quantum field theory from the Pauli-Villars regulator theory to currently popular fourth-order derivative quantum gravity theories. While these theories all appear to have unitarity-violating ghost states with negative norm, we show that in fact these ghost states only appear because the theories are being formulated in the wrong Hilbert space. In these theories the Hamiltonians are not Hermitian but instead possess an antilinear symmetry. Consequently, one cannot use inner products that are built out of states and their Hermitian conjugates. Rather, one must use inner products built out of states and their conjugates with respect to the antilinear symmetry, and these latter inner products are positive. In this way one can build quantum theories of gravity in four spacetime dimensions that are unitary.