Explicit symmetry breaking
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In
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, explicit symmetry breaking is the breaking of a
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
of a
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
by terms in its defining
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
(most typically, to the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
or the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
) 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 The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel prize ...
, due to a magnetic interaction perturbation in the Hamiltonian of the atoms involved. Explicit symmetry breaking differs from
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the ...
. In the latter, the defining equations respect the symmetry but the
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
(
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 dis ...
) of the theory breaks it. Explicit symmetry breaking is also associated with electromagnetic radiation. A system of accelerated charges results in electromagnetic radiation when the geometric symmetry of the electric field in free space is explicitly broken by the associated electrodynamic structure under time varying excitation of the given system. This is quite evident in an antenna where the electric lines of field curl around or have rotational geometry around the radiating terminals in contrast to linear geometric orientation within a pair of transmission lines which does not radiate even under time varying excitation.Sinha & Amaratunga (2016) "Explicit Symmetry Breaking in Electrodynamic Systems and Electromagnetic Radiation" Morgan Claypool, Institute of Physics, UK


Perturbation theory in quantum mechanics

A common setting for explicit symmetry breaking is perturbation theory in quantum mechanics. The symmetry is evident in a base Hamiltonian H_0. This H_0 is often an
integrable 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 ...
Hamiltonian, admitting symmetries which in some sense ''make'' the Hamiltonian integrable. The base Hamiltonian might be chosen to provide a starting point close to the system being modelled. Mathematically, the symmetries can be described by a smooth symmetry group G. Under the action of this group, H_0 is invariant. The explicit symmetry breaking then comes from a second term in the Hamiltonian, H_, which is not invariant under the action of G. This is sometimes interpreted as an interaction of the system with itself or possibly with an externally applied field. It is often chosen to contain a factor of a small interaction parameter. The Hamiltonian can then be written :H = H_0 + H_ where H_ is the term which explicitly breaks the symmetry. The resulting equations of motion will also not have G-symmetry. A typical question in perturbation theory might then be to determine the spectrum of the system to first order in the perturbative interaction parameter.


See also

*
Symmetry breaking 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 observe ...


References

Symmetry {{quantum-stub