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

P^Lψ = ψ_L; similar for RH

Projects out L (or R for P^R) chirality component

Π^Lψ = 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 γ⁵ 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

γ⁵ (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 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    

ν_L   +  

e_R and  ν_R = 0.  

Weak charge vs weak 4-current under Lorentz transformation

Weak 4-current _wj^μ is actually Lorentz co-variant.  Total weak charge is invariant.

comment

Like electric charge 4-current _ej^μ in which _ej⁰component 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.