![]() ![]() ![]() In particular, the connection between microlocal properties of the OPE coefficients and their analyticity properties - guaranteed by the powerful machinery of hyperfunctions, comes in handy. in the text by Streater and Wightman, using however input from a mathematical theory called “microlocal analysis”. It is then treated with the analytic continuation methods developed in that case as described e.g. Each term in the curvature expansion represents a quantity that “lives in the tangent-space” of the reference point in the spacetime manifold, i.e. The proof of this theorem is fairly complicated and works by making a “curvature expansion” of the OPE-coefficients, the terms of which contain increasingly complicated curvature terms at a given reference point. In other dimensions, a similar formula holds, too, but one has to take into account that spinors have somewhat different properties depending on the dimension (`Bott periodicity’). ![]() a continuous choice of future lightcone at each point, or a time function $T: M \to \mathbb$ refers to the opposite orientations. a continuous choice of “right-handed frame” at each point, or alternatively a volume form $vol_M$ on spacetime $M$), together with a choice of time arrow (i.e. $PCT$ is the only fundamental symmetry of quantum field theories on generic space times,Īpart possibly from internal “gauge-type” symmetries, that are unaffected by spacetime curvature! The key observation to make is that we should think of spacetime as being equipped not just with a metric, but additionally with an orientation (i.e. Nevertheless, there still holds a version of the $PCT$ theorem even for quantum fields propagating on a curved spacetime, and in this sense, rotationally invariant on a background that itself does not respect this symmetry. Thus, it would seem misguided, at first sight, to hope that quantum field theories could satisfy a version of $PCT$ in curved spacetime, just as one cannot expect a field theory to be e.g. The same holds with regard to parity, because reflections are also not usually symmetries of a curved spacetime. For example, an expanding universe provides a definite time-direction, and if we change any given time coordinate $t$ to $-t$, this simply is not a symmetry of the spacetime metric. Generic curved spacetimes do not, in general possess any symmetries, including symmetries analogous to $P$ and $T$. The PCT theorem in curved space relates how QFT’s with different assignments of the spacetime orientations are related. in the classic text “PCT and all that” by Streater and Wightman cited below. The proof of the theorem combines in a rather non-trivial way other fundamental properties of QFT, such as the “positivity of energy”, and Poincare invariance, and is an extremely beautiful application of ideas of complex analysis in many variables, distribution theory, and functional analysis. $CP$ is separately not a symmetry in general!) Because one can actually prove this statement in a mathematically rigorous framework of QFT in a rather general setting, this is called a “theorem”. negatively charged particle, see picture. In other words, if we could make a movie of some dynamical process such as particle scattering, and then played it backwards and watched it through a mirror, this would also represent a dynamically allowed process in any quantum field theory - we merely would have to change our minds what we view as a positively- resp. The operational meaning is that if one observes a physical process, then the same observation could, in principle, have been made with the opposite attributions of $PCT$. A fundamental symmetry that any quantum field theory on Minkowski spacetime must possess is invariance under a simultaneous flip of parity $(P)$, i.e. ![]()
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