# Symmetry breaking

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In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations (or "noise"), the choice will appear arbitrary. This process is called symmetry "breaking", because such transitions usually bring the system from a symmetric but disorderly state into one or more definite states. Symmetry breaking is thought to play a major role in pattern formation.

## Contents

In his 1972 Science paper titled "More is different" [1] Nobel laureate P.W. Anderson used the idea of symmetry breaking to show that even if reductionism is true, its converse, constructionism, which is the idea that scientists can easily predict complex phenomena given theories describing their components, is not.

Symmetry breaking can be distinguished into two types, explicit symmetry breaking and spontaneous symmetry breaking, characterized by whether the equations of motion fail to be invariant or the ground state fails to be invariant.

## Explicit symmetry breaking

In explicit symmetry breaking, the equations of motion describing a system are variant under the broken symmetry. In Hamiltonian mechanics or Lagrangian Mechanics, this happens when there is at least one term in the Hamiltonian (or Lagrangian) that explicitly breaks the given symmetry.

## Spontaneous symmetry breaking

In spontaneous symmetry breaking, the equations of motion of the system are invariant, but the system is not. This is because the background (spacetime) of the system, its vacuum, is non-invariant. Such a symmetry breaking is parametrized by an order parameter. A special case of this type of symmetry breaking is dynamical symmetry breaking.

## Examples

Symmetry breaking can cover any of the following scenarios: [2]

• The breaking of an exact symmetry of the underlying laws of physics by the apparently random formation of some structure;
• A situation in physics in which a minimal energy state has less symmetry than the system itself;
• Situations where the actual state of the system does not reflect the underlying symmetries of the dynamics because the manifestly symmetric state is unstable (stability is gained at the cost of local asymmetry);
• Situations where the equations of a theory may have certain symmetries, though their solutions may not (the symmetries are "hidden").

One of the first cases of broken symmetry discussed in the physics literature is related to the form taken by a uniformly rotating body of incompressible fluid in gravitational and hydrostatic equilibrium. Jacobi [3] and soon later Liouville, [4] in 1834, discussed the fact that a tri-axial ellipsoid was an equilibrium solution for this problem when the kinetic energy compared to the gravitational energy of the rotating body exceeded a certain critical value. The axial symmetry presented by the McLaurin spheroids is broken at this bifurcation point. Furthermore, above this bifurcation point, and for constant angular momentum, the solutions that minimize the kinetic energy are the non-axially symmetric Jacobi ellipsoids instead of the Maclaurin spheroids.

## Related Research Articles

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The following is a timeline of classical mechanics:

Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat and F. Cosserat in 1909. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

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Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.

In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability.

In physics, action is a numerical description of part of how a physical system has changed over time. It is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple case of a single particle moving with a specified velocity, the action is the momentum of the particle times the distance it moves, added up along its path, or equivalently, twice its kinetic energy times the length of time for which it has that amount of energy, added up over the period of time under consideration. For more complicated systems, all such quantities are added together. More formally, action is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has dimensions of energy × time or momentum × length, and its SI unit is joule-second.

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In theoretical physics, explicit symmetry breaking is the breaking of a symmetry of a theory by terms in its defining equations of motion that do not respect the symmetry. Usually this term is used in situations where these symmetry-breaking terms are small, so that the symmetry is approximately respected by the theory. An example is the spectral line splitting in the Zeeman effect, due to a magnetic interaction perturbation in the Hamiltonian of the atoms involved.

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In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space.

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Gennadi Sardanashvily was a theoretical physicist, a principal research scientist of Moscow State University.

Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the hypothesis that the laws of physics are unchanged under such a transformation. Time translation symmetry is a rigorous way to formulate the idea that the laws of physics are the same throughout history. Time translation symmetry is closely connected, via the Noether theorem, to conservation of energy. In mathematics, the set of all time translations on a given system form a Lie group.

## References

1. Anderson, P.W. (1972). "More is Different" (PDF). Science. 177 (4047): 393–396. Bibcode:1972Sci...177..393A. doi:10.1126/science.177.4047.393. PMID   17796623.
2. "Astronomical Glossary". www.angelfire.com.
3. Jacobi, C.G.J. (1834). "Über die figur des gleichgewichts". Annalen der Physik und Chemie . 109 (33): 229–238. Bibcode:1834AnP...109..229J. doi:10.1002/andp.18341090808.
4. Liouville, J. (1834). "Sur la figure d'une masse fluide homogène, en équilibre et douée d'un mouvement de rotation". Journal de l'École Polytechnique (14): 289–296.