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.

中文翻译：

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从泡利-维拉到四阶量子引力的鬼问题及其解决方法

我们回顾了量子场论中鬼问题的历史，从 Pauli-Villars 调节器理论到目前流行的四阶导数量子引力理论。虽然这些理论似乎都具有负范数的违反统一性的鬼态，但我们表明，实际上这些鬼态的出现只是因为这些理论是在错误的希尔伯特空间中制定的。在这些理论中，哈密顿量不是厄米特，而是具有反对线性对称性。因此，不能使用由状态及其 Hermitian 共轭构成的内积。相反，必须使用由状态及其关于反对线性对称的共轭构成的内积，而这些后一种内积是正的。这样，