The Two-Measure Theory and an Overview of Some of its Manifestations

The Two-Measure theory (TMT) has been developing since 1998 and has yielded a number of highly interesting results, including those not realized in traditional field theory models. The most important advantage of TMT as an alternative theory is that, under the conditions under which all classical tests of general relativity are performed, TMT models are able to accurately reproduce Einstein's general relativity. Despite this, TMT is still often perceived as something too exotic to be relevant to reality. In fact, the fundamental idea underlying TMT seems undeniable: if we truly believe in the effectiveness of mathematics in studying nature, we must agree that there must be a correspondence between the fundamental laws of nature and the structure of the mathematical apparatus necessary to adequately describe them. It then turns out that there is no reason to ignore the volume measure existing on the differentiable manifold on which the theory of gravity and matter fields is built. This idea has far-reaching implications. The goals of this paper are: 1) to provide a clear mathematical and conceptual justification for TMT; 2) to collect in a single article some of the main results of TMT obtained over the past 25 years.