Symmetry Breaking
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
, symmetry breaking is a
phenomenon A phenomenon (plural, : phenomena) is an observable event. The term came into its modern Philosophy, philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be directly observed. Kant was heavily influe ...
in which (infinitesimally) small
fluctuation Fluctuation may refer to: Physics and mathematics * Statistical fluctuations, in statistics, statistical mechanics, and thermodynamics ** Thermal fluctuations, statistical fluctuations in a thermodynamic variable * Quantum fluctuation, arising f ...
s 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 State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * '' Our ...
into one or more definite states. The phenomenon is part of most
theories of everything A theory of everything (TOE or TOE/ToE), final theory, ultimate theory, unified field theory or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all asp ...
. Symmetry breaking is thought to play a major role in
pattern formation The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature. In developmental biology, pattern formation refers to the generation of ...
. In his 1972 ''Science'' paper titled "More is different" Nobel laureate
P.W. Anderson Philip Warren Anderson (December 13, 1923 – March 29, 2020) was an American theoretical physicist and Nobel laureate. Anderson made contributions to the theories of localization, antiferromagnetism, symmetry breaking (including a paper in 1 ...
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 Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momen ...
or Lagrangian Mechanics, this happens when there is at least one term in the Hamiltonian (or Lagrangian) that explicitly breaks the given symmetry. In the Hamiltonian setting, this is most often studied when the Hamiltonian can be written H = H_0 + H_. Here H_0 is a 'base Hamiltonian', which has some manifest symmetry. More explicitly, it is symmetric under the action of a (Lie) group G. Often this is an integrable Hamiltonian. The H_ is a perturbation or interaction Hamiltonian. This is not invariant under the action of G. It is often proportional to a small, perturbative parameter.


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 In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...
) of the system, its
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
, 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. In the Lagrangian setting of quantum field theory, the Lagrangian L is a functional of quantum fields which is invariant under the action of a symmetry group G. However, the ground state configuration (the vacuum expectation value) of the fields may not be invariant under G, but instead partially breaks the symmetry to a subgroup H of G. This is spontaneous symmetry breaking. Outside of gauge symmetry, spontaneous symmetry breaking is associated with phase transitions. For example in the Ising model, as the temperature of the system falls below the critical temperature the \mathbb_2 symmetry of the vacuum is broken, giving a phase transition of the system. Within the context of gauge symmetry, spontaneous symmetry breaking is the mechanism by which gauge fields can 'acquire a mass' despite gauge-invariance enforcing that such fields be massless. This is because spontaneous symmetry breaking of gauge symmetry breaks the gauge-invariance, allowing the gauge fields to be massive. Also, in this context the usage of 'symmetry breaking', while standard, is a misnomer, as gauge 'symmetry' is not really a symmetry but a redundancy in the description of the system. Mathematically, this redundancy is a choice of trivialization, somewhat analogous to redundancy arising from a choice of basis.


Examples

Symmetry breaking can cover any of the following scenarios: :* 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 and soon later
Liouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérès ...
, 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 ellipsoid A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Car ...
s instead of the
Maclaurin spheroid A Maclaurin spheroid is an oblate spheroid which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. This spheroid is named after the Scottish mathematician Colin Maclaurin, who formulated it for t ...
s.


See also

* Higgs mechanism * QCD vacuum * Goldstone boson * 1964 PRL symmetry breaking papers


References


External links

* {{DEFAULTSORT:Symmetry Breaking Symmetry Pattern formation