Arrow of Causality and Quantum Gravity

Abstract

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 ...

Unitarity, stability, and loops of unstable ghosts

“…This feature is a crucial modification as it removes the ghost from the asymptotic spectrum. Past experience with Lee-Wick theories indicates that they can be stable and unitary, although causality does seem to be violated on microscopic scales of order the width of the resonance [20,24,26,29]. In this paper we describe further the unitarity and stability of such theories and come to an understanding of how unitarity is satisfied in the presence of unstable ghosts.…”

A Critique of the Asymptotic Safety Program

“…For the purposes of quadriatic gravity, this is an arguably acceptable result. The resulting theory is unitary and stable near Minkowski space [16], but violates microcausality on timescales of order the width [17,44], which is proportional to the inverse Planck scale. A look at the underlying calculations shows that this would appear to continue to happen if the propagator was defined with yet higher order dependence even if there were other unstable ghosts induced, as long as there were no tachyonic states allowed.…”

“…The a = +1 ghost is non-traditional in QFT, but seems to be more manageable. When treated properly, it can lead to a unitary theory [16], but one which violate microcausality [17,44]. However, these options are ones which any truncation of AS will be forced to confront.…”

“…The location of the poles in the propagator has been explored in the quadratic gravity literature. I am particularly biased towards my own recent work with G. Menezes [16,17], which is representative of the present status. The heavy ghost state will necessarily be unstable due to the coupling with the light gravitons and other light degrees of freedom.…”

“…rather than usual resonances which occur below the real axis. In Ref [17] we have labeled ghost resonances with this pole location as Merlin modes as they propagate backwards in time. We note that this construction would also work for higher order ghosts in the spin two channel.…”