Chirality vs Helicity (original) (raw)
Chirality vs Helicity Chart
By Robert D. Klauber
There is much confusion over the difference between chirality and helicity. This chart compares and contrasts their respective properties.
| | Chirality | Helicity | |
| -------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------- |
| Physical description | Related to weak charge | Related to handedness: thumb in velocity direction, fingers in spin direction. No direct relation to weak charge. |
| Operator form | |
|
| Plus vs minus |
= LH ; + = RH | same as at left |
| Interpretation of RH/LH | Only a label, not real handedness. | Physical handedness via right hand rule |
| Verbal explanation | Function of γ5, i.e., function of a spinor space entity | Function of spin σ component along lin mom direct, i.e., function of phys space entities |
| Components | 4X4 matrix in spinor space | same as at left |
| In limit v → c (or m = 0) | | Equals chirality because
|
| Comment | Probably reason for defining chirality +/
as RH and LH | |
| Operation on a spinor field | PL_ψ = ψL; similar for RH Projects out L (or R for PR) chirality component | Π_L_ψ = ψL helicity; similar for RH Projects out L (or R for Π_R) helicity component |
| Comment | Take care as some authors may use ψL for LH helicity field | |
| Effect of LH field ψL | ψL destroys LH chiral particle; creates RH chiral antiparticle | ψL helicity destroys LH helicity particle (& creates LH helicity antiparticle I think) |
| Effect of RH field ψR | ψR destroys RH chiral particle; creates LH chiral antiparticle | |
| Effect of
|
creates LH chiral particle; destroys RH chiral antiparticle | |
| Effect of
|
creates RH chiral particle; destroys LH chiral antiparticle | |
| Weak charge relation | Somehow nature has chosen to relate γ5 to weak charge, such that only ψL “feels” that charge. | Unrelated to weak charge, unless at v = c, then same as chirality. |
| Lorentz transf properties (change to diff frame) | Lorentz invariant γ5 (and thus PL,R) is 4D pseudo scalar | Not Lorentz invariant (for v ≠ c) if frame velocity > vp, reverses p direction, but not spin. |
| Parity reversal | Changes sign, i.e., RH ↔ LH | Changes sign, i.e., RH↔ LH |
| Mass term in Lagrangian |
| |
| Proof of above | Take def of ψL,R at top and use gamma matrix algebra | |
| Conservation properties (change in time) | Not conserved.
term will destroy a LH (with weak charge) particle and create a RH (zero weak charge) particle. | Conserved for free particles. No external forces or toques leave linear momentum and angular momentum unchanged. |
| Weak charge non-conservation | Weak charge is chiral charge. “Vacuum eats weak charge.” | |
| Weak charges | Only LH chiral particles charged Via SU(2) symmetry (LH sym) Weak charges e
L
νL +
e
R and νR = 0. | |
| Weak charge vs weak 4-current under Lorentz transformation | Weak 4-current w_j_μ is actually Lorentz co-variant. Total weak charge is invariant. | |
| comment | Like electric charge 4-current e_j_μ in which e_j_0component is elec charge density ρ. ρ changes by γ factor under Lor transf, but vol V changes by 1/γ. The product ρV, tot charge q, is invariant. | |