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In 1956, Tsung-Dao Lee and Chen Ning Yang examined the evidence for parity conservation in the weak interaction. Their paper, Question of Parity Conservation in Weak Interactions, distinguished an untested assumption from an established experimental result and proposed comparing electron emission along opposite directions relative to oriented nuclear spins, explicitly considering cobalt-60. It proposed a test; it did not report the later Wu experiment.^[ref]

The conceptual improvement is the choice of observable. An angle between two polar vectors is preserved by spatial reflection; its distribution can conceal parity violation. An angle between an axial vector and a polar vector changes to its supplement. Thus a spin-resolved directional measurement asks a different question from an unpolarized total decay rate.

Under spatial inversion the electric polarization, a polar vector, changes from $\mathbf{P}$ to $-\mathbf{P}$. Neumann's principle requires an equilibrium property of a phase retaining inversion symmetry to be unchanged. Thus $\mathbf{P}=-\mathbf{P}$, forcing $\mathbf{P}=0$: a centrosymmetric phase cannot itself be ferroelectric. A transition can, of course, remove that symmetry.

For an inversion-symmetric parent phase the Landau theory free energy satisfies $F(\mathbf{P})=F(-\mathbf{P})$. Every polynomial term of odd total degree changes sign and must have zero coefficient. Restricting to one fixed direction gives an even scalar expansion in $P$.

For the full prism point group, a threefold rotation eliminates any invariant basal component of a polar vector, while the horizontal mirror eliminates its axial component. Therefore the phase preserving all of $D_{3h}$ is nonpolar, despite lacking inversion. A polar phase would have to break some of these symmetries.

The indicated six vertex directions comprise three basal directions and their horizontal-mirror partners. Crucially, this orbit does not contain the negatives of those vectors. Choose $z$ along the prism and $x$ toward one basal vertex. The cubic polynomialis invariant under the prism's rotations and mirrors. Along a vertex direction $\mathbf{P}=P\hat{\mathbf{n}}$ with a nonzero basal component, it is proportional to $P^3$, so odd scalar terms are symmetry-allowed. Along the prism axis the horizontal mirror instead sends $P_z\mapsto -P_z$, forcing an even free energy. Merely omitting inversion is insufficient to decide either polarity or the allowed invariants.

In mean-field electrodynamics, use Reynolds averaging to write ${\mathbf{U}}_{\mathrm{tot}}=\overline{\mathbf{U}}+\mathbf{u}$ and ${\mathbf{B}}_{\mathrm{tot}}=\overline{\mathbf{B}}+\mathbf{b}$, with zero-mean fluctuations. Assume the averaging operation commutes with the derivatives and that resolved quantities vary little across an averaging region. Averaging the induction equation then givesThis is the mean-field electromotive force, a vector correlation with units of velocity times magnetic field. The closure problem is to express this correlation in terms of the resolved field.

In the kinematic regime the flow statistics are prescribed, so the response is linear in the mean magnetic field, after independently excited fluctuation transients have decayed. If the mean-field scale $L$ is much greater than the turbulent scale $\ell$, and its evolution is slow compared with the response memory, expand that response in the local mean magnetic field and its gradients:Without scale separation one generally has a spatially nonlocal and temporally retarded response rather than constant local coefficients.

For homogeneous turbulence the coefficients are independent of position. Invariance under proper rotations makes the rank-two response proportional to ${\delta }_{ij}$, and the isotropic rank-three gradient response proportional to the alternating tensor. Choosing the diffusivity sign convention givesThe alpha effect can regenerate a mean magnetic field, while $\beta$ supplies turbulent magnetic diffusivity. For constant positive $\beta$, $\nabla \times (-\beta \nabla \times \overline{\mathbf{B}})=\beta {\nabla }^2\overline{\mathbf{B}}$ because the magnetic field is solenoidal. Thus the total diffusivity is $\eta +\beta$.

The electromotive vector is a polar vector, whereas the magnetic field is an axial vector. Therefore $\alpha$ is a pseudoscalar and changes sign under reflection; $\beta$ is an ordinary scalar. A nonzero isotropic alpha effect requires broken reflection symmetry, such as nonzero kinetic helicity. If “isotropic” includes reflection symmetry as well as rotations, $\alpha$ must vanish. The dimensions are $[\alpha ]=L/T$ and $[\beta ]=L^2/T$.

Linearity of Stokes flow and rotational symmetry, including reflection in mathematics invariance of the spherical geometry, determine the possible rigid motions before any calculation. Translation is a polar vector; its linear dependence on a single vector $\mathbf{A}$ must be $c\mathbf{A}$. A traceless second-rank tensor $\mathbf{B}$ cannot produce a polar vector by an isotropic linear map: tensor contraction with the identity matrix vanishes and tensor contraction with the totally antisymmetric tensor vanishes because $\mathbf{B}$ is a symmetric second-rank tensor. Angular velocity is an axial vector. Neither $\mathbf{A}$ nor a symmetric $\mathbf{B}$ can produce one by an isotropic linear map. Products such as $\mathbf{A}\times \mathbf{BA}$ are excluded by Linearity of Stokes flow. Thus $\mathbf{\Omega }=0$.

Use the Unscaled Papkovich–Neuber representation, with harmonic functions $\mathbf{\Phi }$ and $\chi$:Indeed $\nabla \cdot \mathbf{u}=0$ and $\mu {\nabla }^2\mathbf{u}=\nabla p$ because each potential is a harmonic function. Put $r=|\mathbf{x}|$ and $S=\mathbf{x}\cdot \mathbf{Bx}$. The decaying harmonic functions with the necessary angular dependence are $\mathbf{A}\cdot \mathbf{x}/r^3$, $\mathbf{Bx}/r^3$ and $S/r^5$. The last is a harmonic function precisely because $\mathrm{tr}\mathbf{B}=0$. A $1/r$ Stokeslet is excluded by the force-free condition, and a rotlet by the torque-free condition. TryThe resulting Stokes flow isAt $r=a$, the coefficient of $\mathbf{n}(\mathbf{A}\cdot \mathbf{n})$ fixes $c=a^3/3$; the remaining constant vector gives $\mathbf{U}=-2\mathbf{A}/3$. In the $\mathbf{B}$ mode, matching $\mathbf{Bn}-(\mathbf{n}\cdot \mathbf{Bn})\mathbf{n}$ gives $e=a^4/2$ and $d=-a^2/2$. Therefore the complete two-mode tensorial squirmer flow isThe boundary condition is satisfied in the laboratory frame, so the Stokes flow tends to zero at infinity. The potential dipole in the $\mathbf{A}$ mode gives an irrotational flow and decays as $r^{-3}$; the leading $\mathbf{B}$ mode is a stresslet, decaying as $r^{-2}$. Neither carries a net force or torque. Direct surface integral averaging with the surface slip velocity formula gives the same translation, providing an independent check. Uniqueness of Stokes flow then identifies this decaying solution.

An axial vector transforms under an orthogonal matrix $Q$ asFor a proper rotation, $\det Q=1$, so this is the usual polar vector rule. Under an orientation-reversing change, the extra sign distinguishes the two. In particular, spatial inversion leaves an axial vector unchanged while reversing a polar vector.

Let $C_i=(\nabla \times u{)}_i={ϵ}_{ijk}{\partial }_j{u}_k$. The determinant identity for the Levi-Civita symbol isIt follows by multiplying the three-index determinant identity by $Q_{ir}$ and using orthogonality. The derivative transformation from part (ii) then givesHence the curl of a polar vector field is an axial vector field. The Levi-Civita determinant factor is exactly the extra sign required under reflections.

parity reverses spatial coordinates and polar vectors such as momentum, while leaving axial vectors such as angular momentum unchanged. The correlation $\mathbf{J}\cdot {\mathbf{p}}_e$ is therefore parity odd. For polarized beta-emitting nuclei one can measurewhere $\theta$ is measured from nuclear polarization, $\mathcal{P}$ its magnitude and $A$ a decay-dependent coefficient. A nonzero spin–momentum asymmetry is incompatible with parity invariance.

The cobalt-60 experiment polarized nuclei at low temperature and compared beta emission along and opposite the polarization axis. Preferential emission on one side established weak-interaction parity violation. Reversing polarization tests whether the asymmetry follows nuclear orientation; warming the sample removes polarization, while gamma anisotropy monitors its loss. Detector acceptance, backgrounds and magnetic deflection of charged particles must be separated from a true emission asymmetry.^[ref]

In the modern weak charged current, the V-A interaction couples to left-chiral fermion fields. In the massless limit this selects left-helicity neutrinos and right-helicity antineutrinos. Massive charged leptons need not be pure-helicity states, consistent with the factor $v/c$. Strong and electromagnetic interactions conserve parity to the usual experimental approximation; parity violation alone should not be confused with CP violation or a failure of CPT.

Parity is full spatial reflection, $\mathbf{x}\mapsto -\mathbf{x}$, represented by a parity operator $\mathcal{P}$. A polar vector, such as momentum, reverses, whereas an axial vector, such as spin, does not. A state of definite parity has $\mathcal{P}|\psi ⟩={\eta }_P|\psi ⟩$, with ${\eta }_P=\pm 1$ in the usual convention. A multiparticle state's parity includes both the product of its intrinsic parities and its orbital spatial factor, for example $(-1{)}^L$ for relative orbital angular momentum $L$.

An interaction conserves parity when its dynamics are invariant under this transformation, equivalently $[H,\mathcal{P}]=0$ for a time-independent Hamiltonian. Parity violation means that a process and its fully reflected counterpart need not have the same probability. It is more than a left-right asymmetry in an unreflected laboratory apparatus.

The electromagnetic and strong sectors of the Standard Model conserve parity to the accuracy relevant here, taking the strong-sector Yang-Mills theta term to vanish. The weak charged current selects left-chiral fermions and right-chiral antifermions, as defined by fermion chirality; the reflected right-chiral charged current is not equally present. The neutral weak current also generally has different left- and right-chiral couplings. Consequently the weak sector violates parity; this does not mean that every individual weak decay must have opposite initial and final parity. For a massive particle, helicity and fermion chirality should be distinguished.

An axial vector transforms as ${\mathbf{a}}^{\prime }=(\det Q)Q\mathbf{a}$ under an orthogonal matrix $Q$. It agrees with a polar vector under a proper rotation matrix and has the additional determinant sign under an orientation-reversing transformation. Angular momentum and angular velocity are axial vectors.

The weak interaction changes particle species through the charged current and also acts through the neutral current. Familiar examples include beta-minus decay and neutrino scattering. It is short-range and chiral: its W boson and Z boson couplings distinguish chirality. Above the electroweak scale it belongs to the unbroken $SU(2{)}_L\times U(1{)}_Y$ electroweak interaction. At much lower energies, W boson exchange is approximated by the Fermi interaction.

A particularly direct window into the weak interaction is the Wu experiment: a sample with oriented nuclear spins emits electrons preferentially toward one end of its spin axis. The comparison tests parity conservation, because spin is an axial vector and momentum is a polar vector. The polarized beta-decay angular distribution translates that geometric distinction into a measurable counting asymmetry.