# Parity (physics)

In physics, parity denotes a symmetry property that a physical system can have compared to a spatial reflection .

## description

The parity is based on a spatial reflection which, after selecting a point as the origin of the coordinates, is represented by a change in sign in each of the three spatial coordinates. The time remains unchanged: ${\ displaystyle t}$

${\ displaystyle (t, x, y, z) \ mapsto (+ t, -x, -y, -z) \.}$

Every place merges into the place which, to put it clearly, lies “exactly across from the origin”. For the spatial representation of this transformation of the coordinates, it is often helpful that it can be composed of a reflection on a plane mirror and a subsequent 180 ° rotation around the direction perpendicular to the mirror. ${\ displaystyle {\ vec {r}}}$${\ displaystyle - {\ vec {r}}}$

A physical question is how a physical system behaves in a certain state when it is spatially mirrored. For the answer, it does not matter whether the above coordinate transformation is only carried out in the description of the system or whether a second system is built as a mirrored copy of the first instead. If a physical variable of the system retains its value, then the system is mirror-symmetrical with regard to this variable, it has positive parity . If a physical variable only changes its sign while the amount remains the same, then the system has negative parity for this variable . (It then has positive parity with regard to the amount.) In all other cases, there is no specific parity. Such systems appear "asymmetrical", at least in relation to the coordinate origin currently used.

### Examples

An electrical point charge at the origin has positive parity in terms of its potential , because . With regard to its electric field, however, it has negative parity, because . With regard to a differently selected coordinate origin, there is no parity at all. ${\ displaystyle Q}$${\ displaystyle V ({\ vec {r}}) = {\ tfrac {Q} {4 \ pi \ varepsilon _ {0} | {\ vec {r}} |}}}$${\ displaystyle V ({\ vec {r}}) = V (- {\ vec {r}})}$${\ displaystyle {\ vec {E}} (- {\ vec {r}}) = - {\ vec {E}} ({\ vec {r}})}$

## Parity maintenance

In all processes that are caused by gravity or electromagnetism, the parity of the initial state, if it has one, is preserved. This preservation of parity therefore applies in the whole of Classical Physics . This clearly means z. B. that a symmetrical state cannot emerge from an asymmetrical one. This statement may seem wrong at times, e.g. B. if after the explosion of a completely symmetrically constructed fireworks the oppositely spaced apart chunks have different sizes. Or when a glowing iron rod spontaneously magnetizes itself in an asymmetrical manner when it cools down. According to classical physics, the cause of such a break in symmetry must be that the initial state was not completely symmetrical, which has remained undetected because of the small size of the disturbance. Everything else contradicts the immediate intuition, because a mechanical device, which in a mirror-image replica would not work exactly as the original, is beyond our imagination. For example, one should be able to imagine what happens between the thread and the wood with a normal wood screw if it violates parity, i.e. comes out when turning it in . The view, however, is in harmony with all practical experiences in the macroscopic world, which are completely determined by the parity-maintaining interactions of gravity and electromagnetism.

Another characteristic of parity maintenance is that simply observing a physical process cannot, in principle, decide whether to observe it directly or after a reflection. Because if a system, be it symmetrical or asymmetrical, changes from an initial state to another state according to the laws of classical physics, then a mirrored initial state of the mirrored system changes at the same time into the mirror image of the final state. The two cases can only be distinguished by showing the presence or absence of a reflection in the observation process.

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

## Parity violation

On the basis of all practical experience and physical knowledge, a violation of the maintenance of parity was considered to be ruled out, until in 1956 a certain observation from elementary particle physics could no longer be interpreted otherwise. Tsung-Dao Lee and Chen Ning Yang suggested this way out of solving the “ τ-θ puzzle ” (pronounced “tau-theta puzzle”) when the kaon disintegrated . In the same year, this was confirmed by Chien-Shiung Wu and Leon Max Lederman in two independent experiments.

The cause of the parity violation lies in the weak interaction with which z. B. 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 invariant to the parity transformation. Fermionic particles such as the electron have a property called chirality with two possible characteristics, which are referred to as left-handed and right -handed and which mutually merge through space reflection. This is comparable with the polarization of light or with the difference between the left and right hand. As is possible in quantum physics, an electron is generally in a kind of superposition of left and right handedness. A parity-preserving interaction must affect both chiralities equally. The weak interaction only affects the left-handed component of the electronic state. As a result, the weak interaction is not symmetrical under the parity transformation and violates parity conservation.

## Theoretical description in quantum mechanics

### Parity operator and eigenvalues

In quantum mechanics , the state of a physical system consisting of a particle is described in the simplest case by a wave function . This is a function . The behavior of such wave functions under the parity transformation is described by an operator , called parity transformation or parity operator , which assigns the associated wave function in the mirrored coordinate system to each wave function . It is defined by the equation ${\ displaystyle \ psi \ colon \ mathbb {R} ^ {3} \ to \ mathbb {C}}$${\ displaystyle {\ hat {P}}}$${\ displaystyle \ psi}$${\ displaystyle {\ hat {P}} \ psi}$

${\ displaystyle ({\ hat {P}} \ psi) ({\ vec {r}}) = \ psi (- {\ vec {r}})}$for every wave function and every position vector .${\ displaystyle \ psi}$${\ displaystyle {\ vec {r}}}$

For Dirac wave functions , the parity operator is not just a space reflection of the wave function. A transformation occurs in the 4-dimensional Dirac space, which is caused by multiplication with the Dirac matrix : ${\ displaystyle \ gamma ^ {0}}$

${\ displaystyle ({\ hat {P}} \ psi) ({\ vec {r}}) = \ gamma ^ {0} \ psi (- {\ vec {r}}) \.}$

The parity operator has simple mathematical properties:

• Linearity
• It is about an involution (mathematics) : By applying twice the original wave function is obtained , so is invertible and .${\ displaystyle {\ hat {P}} {\ hat {P}} \ psi = \ psi}$${\ displaystyle {\ hat {P}}}$ ${\ displaystyle {\ hat {P}} ^ {- 1} = {\ hat {P}}}$
• The operator receives the norm ; since it is linear and invertible, a unitary operator is common , as is the case with symmetry transformations in quantum physics.${\ displaystyle {\ hat {P}}}$${\ displaystyle {\ hat {P}}}$
• Due to the unitarity, is equal to its adjoint , thus it is self-adjoint .${\ displaystyle {\ hat {P}} ^ {- 1}}$ ${\ displaystyle {\ hat {P}} ^ {\ dagger}}$${\ displaystyle {\ hat {P}} = {\ hat {P}} ^ {\ dagger}}$

As a self-adjoint operator it only has real eigenvalues ​​and can be understood as an observable . However, there is no direct classical counterpart to this observable from which it can be derived (e.g. via a functional calculus ). Since the parity operator is unitary, all of its eigenvalues ​​have magnitude . Thus it has at most the eigenvalues and , also referred to as parity quantum number . The eigenfunctions for the eigenvalue fulfill the equation and thus belong to the straight (also: symmetrical ) functions (such as a bell curve ). The eigenvalue includes odd (also: skew-symmetric ) wave functions, because it applies . Each state can be clearly represented as the sum of an eigenstate for eigenvalue and one for eigenvalue , that is, divided into an even and an odd part, as is easy to calculate and also follows from the spectral theorem. ${\ displaystyle {\ hat {P}}}$ ${\ displaystyle 1}$${\ displaystyle {\ hat {P}}}$${\ displaystyle +1}$${\ displaystyle -1}$${\ displaystyle +1}$${\ displaystyle \ psi ({\ vec {r}}) = \ psi (- {\ vec {r}})}$${\ displaystyle -1}$${\ displaystyle \ psi ({\ vec {r}}) = - \ psi (- {\ vec {r}})}$${\ displaystyle +1}$${\ displaystyle -1}$

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

${\ displaystyle {\ hat {P}} (\ psi _ {1} \ otimes \ ldots \ otimes \ psi _ {n}) ({\ vec {r_ {1}}}, \ ldots, {\ vec {r_ {n}}}) = (\ psi _ {1} \ otimes \ ldots \ otimes \ psi _ {n}) (- {\ vec {r_ {1}}}, \ ldots, - {\ vec {r_ { n}}})}$ (to be continued linearly over the entire product area)

The parity operator can also be characterized algebraically by the transformation behavior of the components of the position operator : ${\ displaystyle {\ hat {\ vec {r}}} = ({\ hat {x}} _ {1}, {\ hat {x}} _ {2}, {\ hat {x}} _ {3 })}$

${\ displaystyle {\ hat {P}} {\ hat {x}} _ {i} {\ hat {P}} ^ {- 1} = {\ hat {P}} {\ hat {x}} _ { i} {\ hat {P}} = - {\ hat {x}} _ {i}}$

Or in other words:

${\ displaystyle {\ hat {P}} {\ hat {x}} _ {i} = - {\ hat {x}} _ {i} {\ hat {P}}}$

The parity operator anti-swaps with the position operator:

${\ displaystyle \ left \ {{\ hat {P}}, {\ hat {x}} _ {i} \ right \} = 0}$

The same applies to the components of the momentum operator ${\ displaystyle {\ hat {\ vec {p}}} = ({\ hat {p}} _ {1}, {\ hat {p}} _ {2}, {\ hat {p}} _ {3 })}$

${\ displaystyle \ left \ {{\ hat {P}}, {\ hat {p}} _ {i} \ right \} = 0}$.

### Parity preservation and parity violation

Of parity is guaranteed if the Hamiltonian with the parity operator commutes: . As a result, an intrinsic parity value, once present, is retained for all time. Furthermore, there is a common, complete system of eigenstates for and , with the result that, with the exception of random possible exceptions in the case of energy degeneracy, all energy eigenstates have a well-defined parity. ${\ displaystyle {\ hat {H}}}$${\ displaystyle \ left [{\ hat {P}}, {\ hat {H}} \ right] = {\ hat {P}} {\ hat {H}} - {\ hat {H}} {\ hat {P}} = 0}$${\ displaystyle {\ hat {H}}}$${\ displaystyle {\ hat {P}}}$

Due to the observed parity violation , the Hamilton operator valid for the relevant processes must contain a term that cannot be interchanged with the parity operator. It follows that there are processes in which the initial parity is not preserved and that the energy eigenstates are, strictly speaking, superpositions of two states of opposite parity. Since this parity-violating term only occurs in the weak interaction , the actually observable effects are mostly minor, albeit theoretically significant.

## Other dimensions

Considering physical theories other than three dimensions, it should be noted that in even-numbered dimension of space a reversal of all coordinate nothing but a rotation (the determinant is ). For this reason, the parity transformation is defined as the reversal of a coordinate for a general number of dimensions and otherwise proceeds analogously. This has the practical disadvantage that it is not possible to define such a fixed matrix as a parity transformation independent of the reference system. ${\ displaystyle 1}$