Is There a Problem with Quantum Gravitation?

We argue that Quantum Gravitation forces us to sum over metrics of all signatures. Since the beginning of General Theory of Relativity the metric has played the dynamical role of the gravitational potential. This has led to an enormously successful classical theory of gravitation. All subsequent efforts at a quantum theory of gravitation have therefore been built around the idea that the graviton is a part of the space-time metric. In this short note we draw the attention towards a usually neglected fact that a usual path integral quantization of the gravitational field perhaps forces us to sum over spacetime metrics of all signatures. The signature of space-time metric is defined by the positive and negative eigenvalues of the diagonalized metric. The usual way of quantization makes us choose a “classical” background metric of a certain signature (usually Lorentzian or Euclidean as the case may be). We then consider the fluctuations given by all symmetric matrices of the same dimensionality as the dimensionality of the space-time under consideration. Actually we must divide the space of all symmetric matrices by the group of General Coordinate Transformations (GCT) as many of these matrices are related by GCT and do not describe metrics differing from each other. However the GCT do not change the signature of the metric. Therefore metrics with different signatures correspond to different points in the quotient space. It is actually the quotient space over which we have to perform the sum. The problem becomes obvious if we consider the simplest case of a two-dimensional world. Let us consider the Euclidean case. In a certain coordinate system the background metric takes the form (