 | This article needs additional citations for verification .(February 2020) |
| | This article needs additional citations for verification .(February 2020) |
In physics, particularly in renormalization theory, an accidental symmetry is a symmetry which is present in an effective field theory because the operators in the Lagrangian that violate this symmetry are irrelevant operators. [1] Since the contribution by irrelevant operators at low energies is small, the low energy theory appears to have this symmetry.
In the Standard Model, the lepton number and the baryon number are accidental symmetries, while in lattice models, rotational invariance is accidental.
The connection between symmetry and degeneracy (that is, the fact that apparently unrelated quantities turn out to be equal) is familiar in every day experience. Consider a simple example, where we draw three points on a plane, and calculate the distance between each of the three points. If the points are placed randomly, then in general all of these distances will be different. However, if the points are arranged so that a rotation by 120 degrees leaves the picture invariant, then the distances between them will all be equal (as this situation obviously describes an equilateral triangle). The observed degeneracy boils down to the fact that the system has a D3 symmetry.
In quantum mechanics, calculations (at least formally) boil down to the diagonalization of Hermitian matrices - in particular, the Hamiltonian, or in the continuous case, the solution of linear differential equations. Again, observed degeneracies in the eigenspectrum are a consequence of discrete (or continuous) symmetries. In the latter case, Noether's theorem also guarantees a conserved current. "Accidental" symmetry is the name given to observed degeneracies that are apparently not a consequence of symmetry.
The term is misleading as often the observed degeneracy is not accidental at all, and is a consequence of a 'hidden' symmetry which is not immediately obvious from the Hamiltonian in a given basis. The non relativistic Hydrogen atoms a good example of this - by construction, its Hamiltonian is invariant under the full rotation group in 3 dimensions, SO(3). A less obvious feature is that the Hamiltonian is also invariant under SO(4), the extension of SO(3) to 4D, of which SO(3) is a subgroup (another way of saying this is that all possible rotations in 3D are also possible in 4D - we just don't rotate about the additional axis). This gives rise to the 'accidental' degeneracy observed in the Hydrogenic eigenspectrum.
As a more palatable example, consider the Hermitian matrix:
Although there is already some suggestive relationships between the matrix elements, it is not clear what the symmetry of this matrix is at first glance. However, it is easy to demonstrate that by a unitary transformation, this matrix is equivalent to:
Which can be verified directly by numerically (or for purists, analytically - see Chebyshev polynomials for some clues) diagonalising the sub-matrix formed by removing the first row and column. Rotating the basis defining this sub matrix using the resulting unitary brings the original matrix into the originally stated form. This matrix has a P4 permutation symmetry, which in this basis is much easier to see, and could constitute a 'hidden' symmetry. In this case, there are no degeneracies in the eigenspectrum. The technical reason for this is that each eigenstate transforms with respect to a different irreducible representation of P4. If one encountered a case where some group of eigenstates correspond to the same irreducible representation of the 'hidden' symmetry group, a degeneracy would be observed.
Although for this simple 4x4 matrix the symmetry could have been guessed (it was after all, always there to begin with), if the matrix was larger, it would have been more difficult to spot.
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum jumps supplanted the Bohr model's electron orbits. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, as manifest in Dirac's bra–ket notation.
In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of spectral theory.
In nuclear physics and particle physics, isospin (I) is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions of baryons and mesons.
The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators in the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch–Gordan coefficient. The name derives from physicists Eugene Wigner and Carl Eckart, who developed the formalism as a link between the symmetry transformation groups of space and the laws of conservation of energy, momentum, and angular momentum.
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates :
In quantum physics and quantum chemistry, an avoided crossing is the phenomenon where two eigenvalues of a Hermitian matrix representing a quantum observable and depending on N continuous real parameters cannot become equal in value ("cross") except on a manifold of N-3 dimensions. The phenomenon is also known as the von Neumann–Wigner theorem. In the case of a diatomic molecule, this means that the eigenvalues cannot cross at all. In the case of a triatomic molecule, this means that the eigenvalues can coincide only at a single point.
The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman.
In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.
In quantum mechanics, the Kramers' degeneracy theorem states that for every energy eigenstate of a time-reversal symmetric system with half-integer total spin, there is another eigenstate with the same energy related by time-reversal. In other words, the degeneracy of every energy level is an even number if it has half-integer spin. The theorem is named after Dutch physicist H. A. Kramers.
In theoretical physics, the BRST formalism, or BRST quantization denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier quantum field theory (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs, especially in non-abelian QFT, where the use of "ghost fields" with superficially bizarre properties is almost unavoidable for technical reasons related to renormalization and anomaly cancellation.
In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).
In chemistry, molecular symmetry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain many of a molecule's chemical properties, such as whether or not it has a dipole moment, as well as its allowed spectroscopic transitions. To do this it is necessary to use group theory. This involves classifying the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Symmetry is useful in the study of molecular orbitals, with applications to the Hückel method, to ligand field theory, and to the Woodward-Hoffmann rules. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry discuss symmetry. Another framework on a larger scale is the use of crystal systems to describe crystallographic symmetry in bulk materials.
The light-front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to a relativistic description of bound systems in terms of quantum-mechanical wave functions. The quantization is based on the choice of light-front coordinates, where plays the role of time and the corresponding spatial coordinate is . Here, is the ordinary time, is one Cartesian coordinate, and is the speed of light. The other two Cartesian coordinates, and , are untouched and often called transverse or perpendicular, denoted by symbols of the type . The choice of the frame of reference where the time and -axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others.
In particle physics, CP violation is a violation of CP-symmetry : the combination of C-symmetry and P-symmetry. CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C-symmetry) while its spatial coordinates are inverted. The discovery of CP violation in 1964 in the decays of neutral kaons resulted in the Nobel Prize in Physics in 1980 for its discoverers James Cronin and Val Fitch.
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems.
Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in the application of Quantum Mechanics in physics and chemistry, for example it can be used to predict or explain many of a molecule's properties, such as its dipole moment and its allowed spectroscopic transitions, without doing the exact rigorous calculations. To do this it is necessary to classify the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Among all the molecular symmetries, diatomic molecules show some distinct features and they are relatively easier to analyze.
In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are known abstractly. The name derives from the German mathematicians Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912), who encountered an equivalent problem in invariant theory.
 | This quantum mechanics-related article is a stub. You can help Wikipedia by expanding it. |
