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.
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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 |
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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) |
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Equals chirality because
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Comment |
Probably reason for defining chirality +/ as RH and LH |
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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 |
Comment |
Take care as some authors may use ψL for LH helicity field |
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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 |
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Effect of |
creates LH chiral particle; destroys RH chiral antiparticle |
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Effect of |
creates RH chiral particle; destroys LH chiral antiparticle |
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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 |
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Proof of above |
Take def of ψL,R at top and use gamma matrix algebra |
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Conservation properties (change in time)
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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.” |
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Weak charges |
Only LH chiral particles charged Via SU(2) symmetry (LH sym) Weak charges eL νL + eR and νR = 0. |
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Weak charge vs weak 4-current under Lorentz transformation |
Weak 4-current wjμ is actually Lorentz co-variant. Total weak charge is invariant. |
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comment |
Like electric charge 4-current ejμ 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. |
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