Quantum geometry
The term quantum geometry summarizes mathematical concepts with which a common description of phenomena of general relativity and quantum field theory is attempted. Such a concept is required in the research areas of quantum gravity , for example, for the treatment of effects on the Planck scale, i.e. in the range of very short lengths (10 −35 m). This is relevant for some aspects of singularities in general relativity, the properties of black holes and the very early universe .
One problem for a joint treatment of general relativity theory and quantum mechanics is that the usual methods of quantum mechanics presuppose space and time ( summarized in relativity theory as four-dimensional spacetime ) as unchangeable quantities. On the other hand, according to the general theory of relativity, space is dynamic, matter influences space-time through the gravitational field .
A space-time is described in the general theory of relativity by a Lorentz manifold . With regard to the goal of linking the general theory of relativity with quantum mechanics, quantum geometry should not necessarily describe a classical space (or spacetime), but a generalized form of geometry from which the properties of physical spacetime result in special cases. Instead of sets of points , non- commutative quantities are often assumed as basic objects ; quantum geometry is then a non-commutative geometry .
Quantum geometry theories are still in development. An early attempt was made by John Archibald Wheeler , who coined the term quantum geometry for a quantum mechanics of metric quantities which, if possible, should also explain the properties of the elementary particles . With the results of the Yang-Mills theory , the task arose of including the internal degrees of freedom of the particles of the standard model of quantum field theory. In the meantime, various concepts have been developed in theoretical physics , but none has so far got beyond the mathematical description of less specific problems. Examples of such approaches are loop quantum gravity and string theory . The latter is usually based on a “conventional” (continuous) geometry, but with at least 10 space or 11 space and time dimensions , of which only four are observed as space-time.
In many concepts of quantum geometry (e.g. in loop quantum gravity ) the structure of spacetime in the Planck scale is not continuous , but quantized (i.e., discrete ). The hope has not been fulfilled that through the discretization a natural limit of the smallest lengths, the shortest times and thus also the highest energies will come about, which will make the problem of infinite expressions in quantum field theory and the resulting necessity of renormalization disappear.
See also
literature
- Rüdiger Vaas : Tunnel through space and time. From Einstein to Hawking - black holes, time travel and faster than light. 5th updated edition. Franckh-Kosmos, Stuttgart 2012, ISBN 978-3-440-13431-3 .
- John Archibald Wheeler : Geometrodynamics. Acad. Press, New York 1962.