Symmetry (physics)

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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.

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.

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 .

Some symmetries are explored in theoretical physics without proof that they occur in nature. One such hypothetical symmetry is supersymmetry , which predicts an equal number of fermions and bosons .


The following table gives an overview of important symmetries and their conservation quantities. They are divided into continuous and discrete symmetries.

symmetry Conservation size Type meaning
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.

Symmetry breaking

The thermodynamics is not time-invariant, since "reverse heat flow" (from cold to hot) do not exist and the increase in entropy characterizes a time direction.

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 .

With the transition from classical to quantum theories, additional symmetry breaks can occur. Examples are the Higgs mechanism as a dynamic symmetry break and the chiral anomaly .

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 .

Symmetrical potential

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.


References and comments

  1. More precisely, the measurement probabilities of the system, which remain invariant, which is also the case with mere time-reversal transformations .
  2. ^ A b Michael E. Peskin, Daniel V. Schroeder: An Introduction to Quantum Fields . Westview Press, 1995, ISBN 0-201-50397-2 .
  3. Horst Stöcker: Taschenbuch der Physik . 6th edition. Harri Deutsch, ISBN 978-3-8171-1860-1 , pp. 811 .