We present a new understanding of the unstable ghost-like resonance which appears in theories such as quadratic gravity and Lee-Wick type theories. Quantum corrections make this resonance unstable, such that it does not appear in the asymptotic spectrum. We prove that these theories are unitary to all orders. Unitarity is satisfied by the inclusion of only cuts from stable states in the unitarity sum. This removes the need to consider this as a ghost state in the unitarity sum. However, we often use a narrow-width approximation where we do include cuts through unstable states, and ignore cuts through the stable decay products. If we do this with the unstable ghost resonance at one loop, we get the correct answer only by using a contour which was originally defined by Lee and Wick. The quantum effects also provide damping in both the Feynman and the retarded propagators, leading to stability under perturbations. * Electronic address: donoghue@physics.umass.edu † Electronic address: gabrielmenezes@ufrrj.br arXiv:1908.02416v2 [hep-th] 21 Aug 2019 1.1. Unitarity with normal resonances Unitarity describes the conservation of probability for the S-matrix. It stateswhere we used the definition of the transfer matrix T , namely S = 1 + iT . Here the associated states are the asymptotic single and multiparticle states of the theory. In processes that involve loop diagrams, the sum over real intermediate states can by accomplished by the Cutkosky cutting rules [30] which project out the on-shell states. Procedurally we often look first at the free field theory to identify the free particles. Then when we include interactions, some of these particles become unstable and no longer appear as the asymptotic states of the theory. As far as the S-matrix is concerned, this is a significant change. The particles were originally needed in the Hilbert space for completeness, but then are no longer present in the interacting theory. The question then arises of how to treat such unstable particles in unitarity relations. Should one include them in the sums over states required for unitarity?The answer was provided by Veltman in 1963 [31], see also [32][33][34][35]. He showed that unitarity is indeed satisfied by the inclusion of only the asymptotically stable states. Cuts are not to be taken through the unstable particles, and unstable particles are not to be included in unitarity sums.However, there is a corollary which is useful in practice. In the narrow-width approximation, where the coupling to the decay products is taken to be very small, the off-resonance production becomes small and only resonance production is important. As we demonstrate in Sec. 6, in this limit a cut taken through the unstable particle with its width set to zero reproduces the same result as a cut through the decay products.This combination reinforces our intuition. The full calculation only requires the stable states, as unitarity demands. But when particles are nearly stable, we may approximate them as being stable in practical calculations. Unstable ghos...

Causality in quantum field theory is defined by the vanishing of field commutators for space-like separations. However, this does not imply a direction for causal effects. Hidden in our conventions for quantization is a connection to the definition of an arrow of causality, i.e. what is the past and what is the future. If we mix quantization conventions within the same theory, we get a violation of microcausality. In such a theory with mixed conventions the dominant definition of the arrow of causality is determined by the stable states. In some quantum gravity theories, such as quadratic gravity and possibly asymptotic safety, such a mixed causality condition occurs. We discuss some of the implications. PACS numbers:When caught making a sign mistake in a phase, a colleague would say: "Physics does not depend on whether we use +i or −i", and happily change the sign. At first sight, this phrase seems true. Classical fields are real. The probabilities of quantum mechanics are absolute values squared. Measurements in physics do not seem to care if we defined √ −1 as +i or −i. On second thought, the sign in front of i often does make a major difference. We define time development byThis results in "positive energy" being defined via e −iEt/ . Canonical quantization is defined viaThe path integral treatment of quantum physics is defined using e iS , not e −iS (in units of = c = 1), with S being the action. The Feynman propagator has very important sign conventions in both the numerator and the denominator, withwith infinitesimal and positive. We see that the specific signs in front of i are important in the formalism of quantum mechanics. On third thought, we can see that these signs are a convention, although they do tie in with another feature of our physical description -in particular the time direction (arrow) of causality.In somewhat colloquial language, we would describe causality as "there is no effect before the cause". In relativistic quantum theories of course, one must be careful about what one means by "before". As a simple example, consider the Feynman diagram shown in Figure 1. The Feynman propagator in coordinate space includes both forward and backward propagation in timeFIG. 1: The simple Feynman diagram on the left is decomposed into two time ordered diagrams. In one of the time orderings the final particles emerge before the initial particles have annihilated.withwith E q = q 2 + m 2 and D back F (x) = (D for F (x)) * . What we commonly refer to as positive frequency, or e −iEt , is propagated forward in time and negative frequency backwards in time. However, we see that in one time ordering the final state particles (the effect) emerge before the initial particles have interacted (the cause). So our colloquial notion of causality is inadequate.We note in passing that the time advance of the final state vertex is not observable -due to the uncertainty principle. The time difference of the vertices is of order ∆t ∼ 1/E, where E is the center of mass energy. The time localization of the initial state ...

We discuss a variation of quadratic gravity in which the gravitational interaction remains weakly coupled at all energies, but is assisted by a Yang-Mills gauge theory which becomes strong at the Planck scale. The Yang-Mills interaction is used to induce the usual Einstein-Hilbert term, which was taken to be small or absent in the original action. We study the spin-two propagator in detail, with a focus on the high mass resonance which is shifted off the real axis by the coupling to real decay channels. We calculate scattering in the J ¼ 2 partial wave and show explicitly that unitarity is satisfied. The theory will in general have a large cosmological constant and we study possible solutions to this, including a unimodular version of the theory. Overall, the theory satisfies our present tests for being a ultraviolet completion of quantum gravity.