EXTENDED LAGRANGE AND HAMILTON FORMALISM FOR POINT MECHANICS AND COVARIANT HAMILTON FIELD THEORY - Hardcover

Struckmeier, Jürgen; Greiner, Walter

Hardcover

ISBN 10: 981457841X ISBN 13: 9789814578417

Publisher: World Scientific Publishing Co Pte Ltd, 2024

Synopsis

This book presents the extended Lagrange and Hamilton formalisms of point mechanics and field theory in the usual tensor language of standard textbooks on classical dynamics. The notion "extended" signifies that the physical time of point dynamics as well as the space-time in field theories are treated as dynamical variables. It thus elaborates on some important questions including: How do we convert the canonical formalisms of Lagrange and Hamilton that are built upon Newton's concept of an absolute time into the appropriate form of the post-Einstein era? How do we devise a Hamiltonian field theory with space-time as a dynamical variable in order to also cover General Relativity?In this book, the authors demonstrate how the canonical transformation formalism enables us to systematically devise gauge theories. With the extended canonical transformation formalism that allows to map the space-time geometry, it is possible to formulate a generalized theory of gauge transformations. For a system that is form-invariant under both a local gauge transformation of the fields and under local variations of the space-time geometry, we will find a formulation of General Relativity to emerge naturally from basic principles rather than being postulated.

About this title

From the Back Cover

This book offers an explicitly covariant canonical formalism that is devised in the usual mathematical language of standard textbooks on classical dynamics. It elaborates on important questions: How do we convert the entire canonical formalism of Lagrange and Hamilton that are built upon Newton's concept of an absolute time into a relativistically correct form that is appropriate to our present knowledge? How do we treat the space-time variables in a Hamiltonian Field Theory on equal footing as in the Lagrangian description of field theory without introducing a new mathematical language? How can a closed covariant canonical gauge theory be obtained from it? To answer the last question, the theory of homogenous and inhomogeneous gauge transformations is worked out in this book on the basis of the canonical transformation theory for fields elaborated before. In analogy to the treatment of time in relativistic point mechanics, the canonical formalism in field theory is further extended to a space-time that is no longer fixed but is also treated as a canonical variable. Applied to a generalized theory of gauge transformations, this opens the door to a new approach to general relativity.

Edition details

• Publisher: World Scientific Publishing Co Pte Ltd
• Publication date: 2024
• Language: English
• ISBN 10: 981457841X
• ISBN 13: 9789814578417
• Binding: Hardcover
• Number of pages: 384