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.





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


4X4 matrix in spinor space

same as at left

In limit vc (or m = 0)


Equals chirality because



Probably reason for defining chirality +/ as RH and LH

Operation on a spinor field

PLψ = ψL; similar for RH

Projects out L (or R for PR) chirality component

ΠLψ = ψL helicity; similar for RH

Projects out L (or R for ΠR) helicity component


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 vc)

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.


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


νL   +  

eR and  νR = 0.  


Weak charge vs weak 4-current under Lorentz transformation

Weak 4-current wjμ is actually Lorentz co-variant.  Total weak charge is invariant.



Like electric charge 4-current ejμ in which ej0component is elec charge density ρρ  changes by γ factor under Lor transf, but vol V changes by 1/γ.  The product ρV, tot charge q, is invariant.