Manu Paranjape, author of The Theory and Applications of Instanton Calculations, discusses the science behind Stephen Hawking’s recent interview on Star Talk.

Recently, it has been reported in the news that Professor Stephan Hawking, of the University of Cambridge, has been talking about the absence of the notion of time before the big bang. In an interview with Neil deGrasse Tyson, Hawking stated that there was nothing before the big bang.

 

 

Hawking says that there was no notion of time before the appearance of the universe in its very early stages of existence, the so-called “big bang moment”. In the interview, Hawking makes an analogy between time in the universe and the notion of being south on the surface of the earth. It is evident fact that on the surface of the earth there is nothing further south than the South Pole, similarly, there is no earlier time before the beginning of the universe. This video, and multiple news articles, have since appeared in the popular science media. Professor Hawking explains that he will use the “Euclidean approach” to quantum gravity and the “No Boundary Proposal” for the wave function of the universe. The concepts encompassed in these two ideas stem from a renowned article, written in 1983, by Professor Hawking with co-author Professor James Hartle of the University of California at Santa Barbara.

 

The no boundary proposal of Hawking and Hartle corresponds to the idea that the universe tunnelled out and appeared in its initial state from a Euclidean space-time configuration (instanton) that has no initial point (hence no start of time). All dimensions are spatial and there is no singularity anywhere. The ideas of using the Euclidean functional integral (equivalently called the imaginary time functional integral) to extract physical information about a quantum mechanical system, is exactly the same nexus of ideas that are used in Instanton calculations, which can be found in my book, The Theory and Applications of Instanton Calculations. The wave function of the universe should be a functional that, in principle, gives the amplitude that a given universe exists. A given universe is defined by a set of field configurations, specifying the metric and the matter fields corresponding to that universe. Thus, the wave function of the universe is in fact a functional of all the possible the field configurations that could exist, i.e. all possible universes. Its value for a given universe (field configuration) is extracted through the Euclidean functional integral, where the functional integration is done over all possible Euclidean field configurations that agree on a space-like three dimensional surface which corresponds to the given universe of interest. In the No Boundary Proposal, the functional integration is done over all possible compact four geometries that agree on the given space-like surface, and as they are compact, they have no boundaries and which boundary conditions would have to be specified. The Euclidean functional integral is then dominated by space-time configurations which are correctly called instantons.

 

My book explains how the Euclidean function integral can be used to extract the wave function and energy differences of the low lying excitations of general quantum systems, and how instantons can come to dominate the functional integral.  Many examples are treated in particle quantum mechanics and in quantum field theory and many detailed calculations are done to show how the Euclidean functional integral can be used to extract information about tunnelling amplitudes, wave functions and energy differences. Examples of vacuum decay, including gravitational corrections in first order phase transitions in the early universe are most in line with the calculations required to understand the work of Hawking and Hartle.

Leave a reply

Your email address will not be published. Required fields are marked *