![]() ![]() ![]() ![]() 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 $PCT$. Institute of Physics Publishing (Bristol).A fundamental symmetry that any quantum field theory on Minkowski spacetime must possess is invariance under a simultaneous flip of parity $(P)$, i.e. 1993, Testing relativistic gravity with binary and millisecond pulsars, in General Relativity and Gravitation 1992, eds. Given by the general relativity (reaction to the gravitational waves emission) ![]() We can "unwrap" a cylinder into a flat surface, we can't unwrap a sphere without distorting it.Ĭonstant metric implies a flat space, but the opp. a cylinder will satisfy (most) Euclidean geometry. How do we know analytically if we are in a curved space? e.g. We can measure the curvature k ≡ 1/R 2 (a physicist would say we are measuring distant-dependence corrections to π!) These are experiments that we can almost do. of large circles, we can determine if we live in a curved space. Specifically, consider a sphere: circles don't satisfy In fact we can carry out tests to decide if we live in a "normal" 3-D space (Euclidean)Īngles of a triangle add up to 180 0 α + β + γ = 180 Hence If a massive body curves space, it almost implies extra dimensions. If you take the example of the 2-D curved surface of the Earth, this is embedded in a 3-D space. Note we have carefully avoided saying what we mean by a curved space If your browser supports JavaScript, be sure it is enabled. Requires JavaScript to process the mathematics on this page. ![]()
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