Parity (physics)

The parity indicated in physics, a symmetry property, may have a physical system with respect to a three-dimensional mirror.

• 4.1 parity operator and eigenvalues
• 4.2 conservation of parity and parity violation

Description

Parity is a spatial reflection based, which is represented on the selection of a point as the coordinate origin by a change of sign in each of the three spatial coordinates. The time remains unchanged:

Every place goes beyond the place of spoken clearly " opposite the origin " is. For the spatial representation of this transformation of the coordinates is often helpful that they can be composed of a reflection in a plane mirror and a subsequent 180 ° rotation around the vertical direction to the mirror.

A physical question is now how a physical system behaves in a certain state when it is mirrored space. For the answer, it 's all the same whether the above coordinate transformation is performed only in the description of the system or whether, instead, a second system is set up as a mirrored copy of the first. Maintains a physical size of the system while its value, then the system is mirror symmetric with respect to this size, it has positive parity. Moves a physical quantity with a constant amount only its sign, then the system has in terms of this size negative parity. (In relation to the amount so then it has positive parity. ) In all other cases, there is no definite parity before. Such systems appear " unbalanced ", at least in relation to the currently used coordinate origin.

Examples

One located at the origin of electric point charge has positive parity for their potential because. But in terms of their electric field it has negative parity, because. Reference to a coordinate origin chosen otherwise there is no parity.

Parity conservation

In all these processes, which are effected by gravitational or electromagnetism, the parity of the initial state remains as it was obtained. This parity conservation is then valid in the whole classical physics. Intuitively, this means for example that of a symmetrical state can not emerge unbalanced. This statement may sometimes appear incorrectly, such as when the entgegesetzt apart flying chunks have different size after the explosion of a fully symmetrically designed firework. Or if a red hot iron bar is magnetized spontaneously upon cooling in an asymmetrical way. According to classical physics, the cause of such symmetry breaking must be that already the initial state was not completely symmetrical, which has remained undetected because of the smallness of the disorder. Everything else is contrary to the immediate intuition, because a mechanical apparatus, which would not work in mirror image replica exactly like the original, is far beyond imagination possibilities. For example, one would have to imagine what happens in a normal wood screw between the screw and the wood if it violates the parity, so out comes the screwing. The intuition is, however, consistent with all the practical experiences in the macroscopic world, which are completely determined by the parity-conserving interactions of gravity and electromagnetism.

Another characteristic of the parity conservation is that you basically can not decide by merely observing a physical process, whether you observed him directly or after reflection. Because it is a system, whether symmetric or asymmetric, about from an initial state according to the laws of classical physics in a different state, then goes a mirrored initial state of the mirrored system constructed in the same time in the mirror image of the final state over. The two cases are the only way to distinguish that it detects the presence or absence of a mirror in the observation process.

The theoretical justification of both labels the parity conservation based on the fact that the equations of motion for gravitation and electromagnetism remain unchanged if one carries out the above mentioned coordinate transformation. It is said that these equations themselves have mirror symmetry, they are covariant under this transformation.

Parity violation

Because all practical experience and physical knowledge a violation of parity conservation was considered excluded until in 1956 a particular observation from elementary particle physics could not be interpreted otherwise. Tsung- Dao Lee and Chen Ning Yang suggested this way (pronounced " tau -theta puzzle " ) in the decay of the kaon before to solve the " τ - θ puzzle ". In the same year, this could be confirmed by Chien - Shiung Wu and Leon Lederman in two independent experiments.

The cause of parity violation lies in the weak interaction, with which, for example, the beta - radioactivity and the decay of many short-lived elementary particles is described. The formulas of the theoretical formulation of the weak interaction are not covariant with respect to the parity transformation. Fermionic particles such as the electron possess a property called chirality with two possible values ​​, which are referred to as left-handed and right-handed and mutually pass through space mirroring each other. This is similar to the polarization of light, or flush with the difference of left and right hands. An electron is located, as possible in quantum physics, in general, in a kind of superposition of left-and right-handedness condition. A parity-conserving interaction must affect equally both chiralities. However, the weak interaction acts on only the left-handed component of the electron state. Thus, the weak interaction is not symmetric under the parity transformation and violates the parity conservation.

Theoretical description in quantum mechanics

Parity operator and eigenvalues

In quantum mechanics the state of a physical system is described consisting of a particle in the simplest case by a wave function. This is a function. The behavior of such wave functions of the parity transformation is described by an operator or transformation parity parity operator called which assigns each wave function, the wave function in the corresponding mirrored coordinate system. Defined by the equation

For Dirac wave functions of the parity operator is not only a spatial reflection of the wave function. There shall be additional transformation in the 4th dimensional Dirac space, which is caused by multiplication by the Dirac matrix:

The parity operator has a simple mathematical properties:

• Linearity
• There is an involution (mathematics): Twofold application again obtained the initial wave function, thus is invertible and.
• The operator receives the norm; is as linear and invertible, a unitary operator, as in the symmetry transformations in quantum physics is common.
• Due to the unitarity is equal to its adjoint, thus is self-adjoint.

As a self-adjoint operator has only real eigenvalues ​​and can be understood as having observable. But there is no direct classical counterpart from which it arises (eg via a functional calculus ) for this observable. Since the parity operator is unitary, have all its eigenvalues ​​amount. Thus has a maximum of the eigenvalues ​​, and also referred to as parity quantum number. The eigenfunctions corresponding to the eigenvalue satisfy the equation and thus belong to the straight (also symmetric ) functions ( such as a bell curve ). For eigenvalue are odd ( also: skew-symmetric ) wave functions, because it is. Each state can be uniquely as the sum of an eigenstate with eigenvalue and a represent the eigenvalue, that is, in a straight and disassemble an odd part is how easily recalculate and also follows from the spectral theorem.

For Mehrteilchensysteme the parity operator is first defined analogously for the space of each individual particle, and then continued on the tensor product of the spaces:

Algebraically can the parity operator also by the transformation behavior of the components of the position operator characterize:

Or in other words:

So the parity operator antivertauscht with the position operator:

The same applies for the components of the momentum operator

Conservation of parity and parity violation

Conservation of parity is ensured if the Hamiltonian commutes with the parity operator. As a result, a once present parity eigenvalue for all time remains. Furthermore, there's about a common complete system of eigenstates and, with the consequence that, except for random possible exceptions in the case of degeneracy of energy, all energy eigenstates have a well-defined parity.

Due to the observed parity violation of the Hamiltonian must contain a non- interchangeable with the parity operator term. Thus it follows that there are processes in which the initial parity is not conserved, and that, strictly speaking, the energy eigenstates are superpositions of two states of opposite parity. Since this parity-violating term is found only in the weak interaction, the actual observable effects are usually minor, although theoretically significant.

Other dimensions

Considering physical theories other than three dimensions, it should be noted that, in a reversal of even dimension of space coordinates of all no more than one rotation (the determinant ). Therefore, we define for general dimension number parity transformation as the reverse of a coordinate and the procedure is otherwise analogous. This one has the practical disadvantage that it is not possible to respect independent of the system to define a solid matrix such as a parity transformation.