Explorations in Mathematical Physics

"With enjoyable and sometimes surprising excursions along the way, the journey provides a fresh look at many familiar topics, as it takes us from basic linear mathematics to general relativity… look forward to having your geometric intuition nourished and expanded by the author’s intelligent commentaries." Eugen Merzbacher, University of North Carolina, Chapel Hill Have you ever wondered why the language of modern physics centres on geometry? Or how quantum operators and Dirac brackets work? What a convolution really is? What tensors are all about? Or what field theory and lagrangians are, and why gravity is described as curvature? This book takes you on a tour of the main ideas forming the language of modern mathematical physics. Here you will meet novel approaches to concepts such as determinants and geometry, wave function evolution, statistics, signal processing, and three-dimensional rotations. You will see how the accelerated frames of special relativity tell us about gravity. On the journey, you will discover how tensor notation relates to vector calculus, how differential geometry is built on intuitive concepts, and how variational calculus leads to field theory. You will meet quantum measurement theory, along with Green functions and the art of complex integration, and finally general relativity and cosmology. The book takes a fresh approach to tensor analysis built solely on the metric and vectors, with no need for one-forms. This gives a much more geometrical and intuitive insight into vector and tensor calculus, together with general relativity, than do traditional, more abstract methods. Don Koks is a physicist at the Defence Science and Technology Organisation in Adelaide, Australia. His doctorate in quantum cosmology was obtained from the Department of Physics and Mathematical Physics at Adelaide University. Prior work at the University of Auckland specialised in applied accelerator physics, along with pure and applied mathematics.

The Language of Physics.- A Trip Down Linear Lane.- The Natural Language of Random Processes.- A Roundabout Route to Geometric Algebra.- Special Relativity and the Lorentz Transform.- Four-Vectors and the Road to Tensors.- Accelerated Frames: Onward to the Principle of Covariance.- The Elegance and Power of Tensor Notation.- Curvature and Differential Geometry.- Variational Calculus and Field Theory.- The Green Function Approach to Solving Field Equations.- Airliners, Black Holes, and Cosmology: The ABC of General Relativity.