Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. At this point the system no longer appears to behave in a symmetric manner. It is a phenomenon that naturally occurs in many situations. The symmetry group can be discrete, such as the space group of a crystal, or continuous (i.e. a Lie group), such as the rotational symmetry of space. A black hole concept drawing by NASA. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ...

A common example to help explain this phenomenon is a ball sitting on top of a hill. This ball is in a completely symmetric state. However, it is not a stable one: the ball can easily roll down the hill. At some point, the ball will spontaneously roll down the hill in one direction or another. The symmetry has been broken because the direction the ball rolled down in has now been singled out from other directions.

Mathematical example: the Mexican hat potential

Graph of spontaneous symmetry breaking function in equation (2)

In physics, one way of seeing spontaneous symmetry breaking is through the use of Lagrangians. Lagrangians, which essentially dictate how a system will behave, can be split up into kinetic and potential terms Graph of z = -10(x^2 + y^2) + (x^2 + y^2)^2 This image was generated by a program I, the uploader (Jcobb), wrote myself. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

(1)

It is in this potential term (V(φ)) that the action of symmetry breaking occurs. An example of a potential is illustrated in the graph at the right.

V(φ) = − 10 | φ | ² + | φ | ⁴ (2)

This potential has many possible minima (vacuum states) given by The largest and the smallest element of a set are called extreme values, or extreme records. ...

(3)

for any real θ between 0 and 2π. The system also has an unstable vacuum state corresponding to φ = 0. In this state the Lagrangian has a U(1) symmetry. However, once it falls into a specific stable vacuum state (corresponding to a choice of θ) this symmetry will be lost or spontaneously broken. In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. ...

In the Standard Model, spontaneous symmetry breaking is accomplished by using the Higgs boson and is responsible for the masses of the W and Z bosons. A slightly more technical presentation of this mechanism is given in the article on the Yukawa interaction, where it is shown how spontaneous symmetry breaking can be used to give mass to fermions. The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ... The Higgs boson is a hypothetical massive scalar elementary particle predicted to exist by the Standard Model of particle physics. ... Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... In physics, the W and Z bosons are the elementary particles that mediate the weak nuclear force. ... In particle physics, Yukawa interaction, named after Hideki Yukawa, is an interaction between a scalar field and a Dirac field of the type . The Yukawa interaction can be used to describe the strong nuclear force between nucleons (which are fermions), mediated by pions (which are scalar mesons). ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ...

Broader concept

More generally, we can have spontaneous symmetry breaking in nonvacuum situations and for systems not described by actions. The crucial concept here is the order parameter. If there is a field (often a background field) which acquires an expectation value (not necessarily a vacuum expectation value) which is not invariant under the symmetry in question, we say that the system is in the ordered phase and the symmetry is spontaneously broken. This is because other subsystems interact with the order parameter which forms a "frame of reference" to be measured against, so to speak. In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. ... In quantum field theory and statistical mechanics, a system can be in two possible phases: an ordered phase and a disordered phase. ...

Examples

• For ferromagnetic materials, the laws describing it are invariant under spatial rotations. Here, the order parameter is the magnetization, which measures the magnetic dipole density. Above the Curie temperature, the order parameter is zero, which is spatially invariant and there is no symmetry breaking. Below the Curie temperature, however, the magnetization acquires a constant (in the idealized situation where we have full equilibrium; otherwise, translational symmetry gets broken as well) nonzero value which points in a certain direction. The residual rotational symmetries which leaves the orientation of this vector invariant remain unbroken but the other rotations get spontaneously broken.
• The laws describing a solid are invariant under the full Euclidean group, but the solid itself spontaneously breaks this group down to a space group. The displacement and the orientation are the order parameters.
• The laws of physics are spatially invariant, but here on the surface of the Earth, we have a background gravitational field (which plays the role of the order parameter here) which points downwards, breaking the full rotational symmetry. This explains why up, down and the horizontal directions are all "different" but all the horizontal directions are still isotropic.
• General relativity has a Lorentz gauge symmetry, but in FRW cosmological models, the mean 4-velocity field defined by averaging over the velocities of the galaxies (the galaxies act like gas particles at cosmological scales) acts as an order parameter breaking this Lorentz symmetry. Similar comments can be made about the cosmic microwave background.
• Here on Earth, Galilean invariance (in the nonrelativistic approximation) is broken by the velocity field of the Earth/atmosphere, which acts as the order parameter here. This explains why people thought moving bodies tend towards rest before Galileo. We tend not to be aware of broken symmetries.
• For the electroweak model, as explained earlier, the Higgs field acts as the order parameter breaking the electroweak gauge symmetry to the electromagnetic gauge symmetry. Like the ferromagnetic example, there is a phase transition at the electroweak temperature. The same comment about us not tending to notice broken symmetries explains why it took so long for us to discover electroweak unification.
• For superconductors, there is a collective condensed matter field ψ which acts as the order parameter breaking the electromagnetic gauge symmetry.
• In general relativity, diffeomorphism covariance is broken by the nonzero order parameter, the metric tensor field.
• Take a flat plastic ruler which is identical on both sides and push both ends together. Before buckling, the system is symmetric under the reflection about the plane of the ruler. But after buckling, it either buckles upwards or downwards.
• Consider a uniform layer of fluid over an infinite horizontal plane. This system has all the symmetries of the Euclidean plane. But now heat the bottom surface uniformly so that it becomes much hotter than the upper surface. When the temperature gradient becomes large enough, convection cells will form, breaking the Euclidean symmetry.

A ferromagnet is a piece of ferromagnetic material, in which the microscopic magnetized regions, called domains, have been aligned by an external magnetic field (e. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In physics, the Curie point, or Curie temperature, is the temperature above which a ferromagnet loses its ferromagnetic ability to possess a net (spontaneous) magnetization in the absence of an external magnetic field. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the... Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ... Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ... In physics, Lorentz symmetry is the invariance of physical laws under the Lorentz transformations. ... Galilean invariance is a principle which states that the fundamental laws of physics are the same in all inertial (uniform-velocity) frames of reference. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... In theoretical physics, general covariance (also known as diffeomorphism invariance) is the invariance of physical laws (for example, the equations of general relativity) under arbitrary coordinate transformations. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... A ruler is an instrument used in geometry and technical drawing to measure short distances and/or to rule straight lines. ... Look up reflection in Wiktionary, the free dictionary. ... A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ... A convection cell is a phenomenon of fluid dynamics which occurs in situations where there are temperature differences within a body of liquid or gas. ...

See also

• Mermin-Wagner theorem
• Catastrophe theory
• Vacuum fluctuation
• Second-order phase transition
• Goldstone boson
• Electroweak
• Grand unified theory
• Explicit symmetry breaking

Catastrophe theory is a branch of mathematics that deals with dynamical systems and was originated with the work of the French mathematician René Thom in the 1960s. ... In the description of the interaction between elementary particles in quantum field theory, a virtual particle is a temporary elementary particle, used to describe an intermediate stage in the interaction. ... In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. ... In particle physics, Goldstone bosons are bosons that appear in models with spontaneously broken symmetry. ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... Grand unification, grand unified theory, or GUT is a theory in physics that unifies the strong interaction and electroweak interaction. ... In theoretical physics, explicit symmetry breaking is the act of breaking symmetry of a theory by adding terms to its defining equations of motion (most typically, to the Lagrangian or the Hamiltonian) that do not respect the symmetry. ...

External links

• Spontaneous symmetry breaking