Chirality (physics)

Chirality ( handedness , made-up word , derived from the Greek χειρ ~, ch [e] ir ~ - hand ~ ), describes an abstract concept in physics within the framework of relativistic quantum mechanics and quantum field theory .

The chirality of a particle is crucial in processes of weak interaction , since W bosons only couple to particles with negative (left-handed) chirality and to antiparticles with positive (right-handed) chirality.

The chirality has to be distinguished from the helicity . In contrast to chirality in chemistry , the chirality of physical quantities can not be illustrated by reflection on a plane mirror. Instead, it describes the decomposition of Dirac spinors into orthogonal states that merge into one another under parity operations.

definition

The fifth gamma matrix is called the chirality operator ; he is hermitian and self-inverse . Its eigenvalues are therefore : ${\ displaystyle \ gamma ^ {5} = \ mathrm {i} \ gamma ^ {0} \ gamma ^ {1} \ gamma ^ {2} \ gamma ^ {3}}$ ${\ displaystyle \ pm 1}$

the eigenstate belonging to the eigenvalue +1 is called the state of positive / right-handed chirality
the eigenstate belonging to the eigenvalue −1 is called the state of negative / left-handed chirality.

Massless fermions

In the limit of massless fermions like neutrinos, the Weyl equation can be obtained from the Dirac equation . In the context of the Weyl equation, it is advisable to write down the Dirac matrices not in Dirac but in Weyl representation so that only block matrices appear on the off-diagonal . Due to the lack of the mass term, the four components of the Dirac spinors decouple into two independent two-spinors: ${\ displaystyle \ mathrm {i} \ gamma ^ {\ mu} \ partial _ {\ mu} \ psi = 0}$

{\ displaystyle \ partial _ {0} {\ begin {pmatrix} \ psi _ {L} \\\ psi _ {R} \ end {pmatrix}} = - \ mathrm {i} {\ vec {\ sigma}} \ cdot {\ vec {\ nabla}} {\ begin {pmatrix} I_ {2} & 0 \\ 0 & -I_ {2} \ end {pmatrix}} {\ begin {pmatrix} \ psi _ {L} \\\ psi _ {R} \ end {pmatrix}}}

.

The chirality operator commutes with the Weyl- Hamilton operator so that a set of common energy and chirality eigenstates can be found. Due to the diagonal nature of the chirality operator in the Weyl representation

{\ displaystyle \ gamma ^ {5} = {\ begin {pmatrix} -I_ {2} & 0 \\ 0 & I_ {2} \ end {pmatrix}}}

,

it follows directly that the upper two -spinor can be interpreted as a left-handed part and the lower spinor as a right-handed part. Since neutrinos only interact weakly, right-handed neutrinos or left-handed antineutrinos are hypothetical sterile particles . In the Standard Model, however, all neutrinos are negative chirality and antineutrinos are positive chirality. ${\ displaystyle \ psi _ {\ text {L}}}$ ${\ displaystyle \ psi _ {\ text {R}}}$

Bulky fermions

Since the Dirac Hamilton operator has a mass term, it does not commute with the chirality operator ; therefore no common eigen-states can be constructed. In particular, it also follows from this that the chirality of a massive object does not represent a conserved quantity, since the chirality operator also does not commute with the time evolution operator as the exponential of the Hamilton operator.

From the property of the chirality operator or the fifth gamma matrix to be its self-inverse, however, it follows that the operators and form a complete set of projection operators. They project the parts of positive or negative chirality out of the Dirac spinor: ${\ displaystyle P _ {\ text {R}} = {\ frac {1+ \ gamma ^ {5}} {2}}}$ ${\ displaystyle P _ {\ text {L}} = {\ frac {1- \ gamma ^ {5}} {2}}}$

{\ displaystyle \ gamma ^ {5} P _ {\ text {R / L}} \ psi \ equiv \ gamma ^ {5} \ psi _ {\ text {R / L}} = \ pm \ psi _ {\ text {R / L}}}

.

In this way, each Dirac spinor can be broken down into a part of right- or left-handed chirality.

Chirality and weak interaction

The concept of chirality plays a crucial role in the weak interaction . In the context of the historical VA theory , the charged currents of the weak interaction project only the left-handed portion of the fermions out, so that only this part participates in the interaction.

In the Glashow-Salam-Weinberg theory of the electroweak union , the left-handed components of a generation of particles are combined to form doublets under the weak isospin (e.g. or ), while the right-handed components are regarded as singlets ( ). As a result, the covariant derivative acts differently on the left and right-handed components, so that ${\ displaystyle (\ nu _ {e}, e) _ {L}}$ ${\ displaystyle (u, d) _ {L}}$ ${\ displaystyle e_ {R}, u_ {R}, d_ {R}}$

the charged weak currents in the form of the W bosons only affect the left-handed parts
the neutral weak current in the form of the Z boson couples to right and left-handed parts to different degrees
the electromagnetic current in the form of the photon does not differentiate between right-handed and left-handed components.

Relation to other concepts

Helicity

The helicity operator considers the projection of the spin in the direction of motion of a particle and is therefore not Lorentz invariant in contrast to the chirality operator . In contrast to the chirality operator, however, the helicity operator commutes with the Dirac Hamilton operator , so that the helicity is a conserved quantity.

In the case of massless fermions, helicity and chirality agree except for one (spin) factor.

CP invariance

The chirality of the particles is due to the fact that the Chiralitätsoperator with the gamma matrices antikommutiert, not invariant under P aritätsoperationen ( parity violation ): ${\ displaystyle {\ mathcal {P}}}$

{\ displaystyle {\ mathcal {P}} \ psi _ {\ text {R / L}} (x) = \ gamma ^ {0} {\ frac {1 \ pm \ gamma ^ {5}} {2}} \ psi ({\ mathcal {P}} x) = {\ frac {1 \ mp \ gamma ^ {5}} {2}} \ gamma ^ {0} \ psi ({\ mathcal {P}} x) = ({\ mathcal {P}} \ psi (x)) _ {\ text {L / R}}}

Likewise, changes charge conjugation ( C harge conjugation) , the chirality as the Chiralitätsoperator also equal to its complex conjugate is: ${\ displaystyle {\ mathcal {C}}}$

{\ displaystyle {\ mathcal {C}} \ psi _ {\ text {R / L}} (x) = \ mathrm {i} \ gamma ^ {2} {\ frac {1 \ pm \ gamma ^ {5 ^ {*}}} {2}} \ psi ^ {*} (x) = {\ frac {1 \ mp \ gamma ^ {5}} {2}} \ mathrm {i} \ gamma ^ {2} \ psi ^ {*} (x) = ({\ mathcal {C}} \ psi (x)) _ {\ text {L / R}}}

Since the parity operation and charge conjugation both reverse the chirality, the chirality is retained when both operations are carried out one after the other. This fact is called CP invariance .

Individual references, comments

↑ In the original version of the standard model , neutrinos are massless. Experiments on neutrino oscillation have shown that they have a non-vanishing mass; however, the description of neutrinos as massive objects requires further physical models .

Web links

Helicity and chirality.

[1] In the original version of the standard model , neutrinos are massless. Experiments on neutrino oscillation have shown that they have a non-vanishing mass; however, the description of neutrinos as massive objects requires further physical models .