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 γ^{5}, 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 
P^{L}ψ = ψ_{L}; similar for RH Projects out L (or R for P^{R}) chirality component 
Π^{L}ψ = ψ_{L helicity}; similar for RH Projects out L (or R for Π^{R}) helicity component 
Comment 
Take care as some authors may use ψ_{L} for LH helicity field 

Effect of LH field ψ_{L}_{} 
ψ_{L} destroys LH chiral particle; creates RH chiral antiparticle 
ψ_{L helicity} destroys LH helicity particle (& creates LH helicity antiparticle I think) 
Effect of RH field ψ_{R} 
ψ_{R} 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 γ^{5} to weak charge, such that only ψ_{L} “feels” that charge. 
Unrelated to weak charge, unless at v = c, then same as chirality. 
Lorentz transf properties (change to diff frame) 
Lorentz invariant γ^{5} (and thus P^{L,R}) is 4D pseudo scalar 
Not Lorentz invariant (for v ≠ c) if frame velocity > v_{p}, 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 ψ_{L,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 nonconservation 
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^{ MPSetChAttrs('ch0021','ch1',[[4,1,2,0,0],[5,1,2,0,0],[7,1,3,0,0],[],[],[],[16,1,7,0,0]]) MPInlineChar(2) }_{L } ν_{L} + e^{ MPSetChAttrs('ch0026','ch1',[[4,1,2,0,0],[5,1,2,0,0],[7,1,3,0,0],[],[],[],[16,1,7,0,0]]) MPInlineChar(2) }_{R} and ν_{R} = 0. 

Weak charge vs weak 4current under Lorentz transformation 
Weak 4current _{w}j^{μ} is actually Lorentz covariant. Total weak charge is invariant. 

comment 
Like electric charge 4current _{e}j^{μ} in which _{e}j^{0}component 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. 
