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The Lagrangian density of a canonical scalar field is $$ L=-\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)-V(\phi) $$ if we use a $(−,+,+,+)$ sign convention. If the sign of the kinetic term is inverted: $$ L=\color{red}{+}\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)-V(\phi). $$ then $\phi$ is called a ghost. Is the presence of such ghost field (with arbitrary potential) necessarily ruled out in quantum field theory?

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hao123
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1 Answers1

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Physical/bad ghosts$^1$ with a negative kinetic term means that the Hamiltonian is unbounded from below, which usually clashes with unitarity and the existence of a ground/vacuum state. Usually, one therefore assumes non-negative kinetic terms, cf. e.g. this & this Phys.SE posts.

Nevertheless, theories with negative kinetic terms are an active research field, cf. e.g. Refs. 1 & 2.

References:

  1. S.W. Hawking, Who's afraid of (higher derivative) ghosts?, 1985.

  2. S. Chatterjee, Rigorous results for timelike Liouville field theory, arXiv:2504.02348.

$^1$ In contrast, unphysical/good ghosts (such as, e.g., Faddeev-Popov ghosts and Goldstone bosons) are allowed.

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