In physics, a symmetry (from ancient Greek σύν syn “together” and μέτρον métron “measure”) is understood to mean the property of a system to remain unchanged (to be invariant ) after a certain change ( transformation , especially coordinate transformations ). If a transformation does not change the state of a physical system , this transformation is called a symmetry transformation . A distinction is made between discrete symmetries (e.g. mirror symmetry), which only have a finite number of symmetry operations, and continuous symmetries (e.g. rotational symmetry), which have an infinite number of symmetry operations.
The mathematical description of symmetries is carried out by group theory .
Symmetries play a major role in modern physical research. If a symmetry is found in an experiment , the associated theory, which is represented by a Lagrangian or an "action functional" , must be invariant under a corresponding symmetry operation. In the gauge theories often used in particle physics , i. H. Theories that are invariant under a gauge transformation , this symmetry largely determines the type and relative strength of the coupling between the particles.
- Findings about symmetries often turned out to be the starting point for entirely new theories. The invariance of the Maxwell equations under Lorentz transformations was a starting point for Albert Einstein to develop the special theory of relativity , and certain patterns in the spectrum of elementary particles led to the development of the Quark model for atomic nuclei (e.g. for the proton ).
- Symmetries are closely related to conservation laws :
The so-called Noether theorem says z. B. that every continuous symmetry can be assigned a conservation quantity. For example, the energy conservation of the system follows from the time translation invariance ; in Hamiltonian mechanics the converse also holds. For a system with energy conservation, the time translation invariance applies as the associated symmetry.
- Not only the symmetries themselves are important, but also symmetry breaking :
Thus, in the theory of the electroweak interaction, the gauge symmetry is broken by the Higgs mechanism , for which the Higgs boson is required. Symmetry breaking processes can also be related to phase transitions, similar to the ferromagnetic phase transition .
The following table gives an overview of important symmetries and their conservation quantities. They are divided into continuous and discrete symmetries.
|Continuous ("flowing") symmetries|
|Translational invariance||pulse||geometric||The total momentum of a closed system is constant. Also called homogeneity of space .|
|Time invariance||energy||geometric||The total energy of a closed system is constant. Also called the homogeneity of time .|
|Rotational invariance||Angular momentum||geometric||The total angular momentum of a closed system is constant. Also called isotropy of space .|
|Gauge transformation invariance||Electric charge||charge||The total electrical charge in a closed system is constant.|
|Discrete (“countable”) symmetries|
|C , charge conjugation||-||charge||If the signs of all charges in a system are reversed, its behavior does not change.|
|P , spatial reflection||-||geometric||If a system is spatially mirrored, its physical behavior does not change. The weak interaction violates this symmetry (see symmetry breaking ).|
|T , reverse time||-||geometric||A system would behave the same way if time ran backwards .|
|CPT||-||geometric||A completely inverse (both spatially and temporally, as well as charge-mirrored) system would behave exactly like the non-mirrored one.|
Like the symmetries themselves, transformations can be continuous or discrete. An example of a continuous transformation is the rotation of a circle by any angle. Examples of a discrete transformation are the mirroring of a bilaterally symmetrical figure, the rotation of a regular polygon or the shifts by integer multiples of grid spacings. The transformations that can be performed determine what type of symmetry it is. While discrete symmetries are described by symmetry groups (such as point groups and space groups ), Lie groups are used to describe continuous symmetries .
Transformations that do not depend on the location are called global transformations . If the transformation parameter can be freely selected at any location (apart from the continuity conditions), one speaks of local transformations or of gauge transformations. Physical theories whose effects are invariant under gauge transformations are called gauge theories . All fundamental interactions, gravitation, the electromagnetic, weak and strong interaction are described by gauge theories according to current knowledge.
Similarly, the weak interaction is not invariant under spatial reflection, as was shown in the Wu experiment in 1956 . The behavior of K mesons and B mesons is not invariant with simultaneous reflection and charge exchange. Without this CP violation , the same amount of matter as antimatter would have been created in the Big Bang and would now be present in the same amount. The baryon asymmetry , which is the predominance of matter today, can only be explained by breaking the CP symmetry .
An example from chemistry are mirror image isomers , which not only look the same (except for the reflection), but also have the same energy levels and transition states. They arise from a prochiral molecule with the same probability or reaction kinetics . Due to autocatalytic reaction mechanisms , i.e. at the latest with the emergence of life , the mirror image symmetry is broken spontaneously , see chirality (chemistry) #biochemistry .
An important example of symmetry is a spherically symmetrical or rotationally symmetrical potential, such as the electrical potential of a point charge (e.g. an electron) or the gravitational potential of a mass (e.g. a star). The potential only depends on the distance to the charge or to the mass, but not on the angle to a selected axis. It does not matter which reference system is chosen for the description, as long as there is charge or mass in its origin. As a consequence of the symmetry, the conservation of angular momentum applies to a particle in a spherically symmetric potential. Because of the lack of translational symmetry, the momentum of the particle in this example is not a conserved quantity.
- Louis Michel: Symmetry defects and broken symmetry. Configurations Hidden Symmetry. In: Reviews of Modern Physics. 52, 1980, pp. 617-651, doi : 10.1103 / RevModPhys.52.617 .
- Werner Hahn: Symmetry as a development principle in nature and art . With a foreword by Rupert Riedl. Koenigstein i. Ts. ( Langewiesche publishing house ) 1989
- Mouchet, A. "Reflections on the four facets of symmetry: how physics exemplifies rational thinking" . European Physical Journal H 38 (2013) 661.
- Symmetry and Symmetry Breaking. Entry in Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
- Walter Greiner , Berndt Mueller : Quantum Mechanics. Symmetries. 3. Edition. German, Frankfurt am Main 1990, ISBN 3-8171-1142-8 . ( Theoretical Physics. Volume 5)
References and comments
- More precisely, the measurement probabilities of the system, which remain invariant, which is also the case with mere time-reversal transformations .
- Michael E. Peskin, Daniel V. Schroeder: An Introduction to Quantum Fields . Westview Press, 1995, ISBN 0-201-50397-2 .
- Horst Stöcker: Taschenbuch der Physik . 6th edition. Harri Deutsch, ISBN 978-3-8171-1860-1 , pp. 811 .