tailieunhanh - The Philosophy of Vacuum Part 30

The Philosophy of Vacuum Part 30. Physicists will find it extremely interesting, covering, as it does, technical subjects in an accessible way. For those with the necessary expertise, this book will provide an illuminating and authoritative exposition of a many-sided subject." -John D. Barrow, Times Literary Supplement. | How Empty is the Vacuum 283 information about the 4manifold. So Donaldson s theory is one using classical vacuum to say something about pure space. It is a very powerful theory. It shows for example that even on good old 4-dimensional Euclidean space there are infinitely many inequivalent ways of defining differentiability of functions that is the differential topology of 4-dimensional vector spaces is very non-trivial. Also here Witten 1988 has developed a complicated supersymmetric quantum field theory such that expectation values of certain observables give the relevant information about persisting loops. Atiyah Donaldson and Quillen established that the theory has the flavour of an algebraic topological theory in infinite dimensions and it might be possible to give it a rigorous foundation without having rigorously to construct the much feared Feynmann path integrals on infinite-dimensional spaces. Witten s quantum field theory in its original version depends on a metric on X just like Donaldson s theory. Witten shows that his expectation values do not change if we vary the metric thereby showing that they are topological invariants that is invariants of pure space. However his theory is full of all kinds of fields and superfields. So again a non-vacuous theory gives information about the ultimate vacuum pure space. We shall end this paper with a more technical description of this Donaldson-Witten theory. In gauge theory the issue is to find antiself-dual connections on some bundle over the 4-manifold X. These are precisely the connections for which the self-dual part of the curvature vanishes. Thus we are looking for zeroes of a function s defined on the space A of connections the function assigns to a connection the self-dual part of its curvature which is an element of an infinite-dimensional vector space V. Gauge invariance enters the picture in the following way. If we apply a gauge transformation to the connection then the self-dual part of the curvature