Understanding Helicity in Quantum Mechanics: Explained by a Scientist

[Mk]

It is quite difficult to understand, can someone explain what is helicity?

 

[dextercioby]

It is quite difficult to understand, can someone explain what is helicity?

It is defined as the proportionality constant between the operators:4momentum and Pauli-Lubanski 4 vector in the case of a massless irreductible representation of the Poincaré group.
It has the physical meaning of the projection of the total angular momentum on the direction of the linear momemntum of the particle.It is denoted by "lambda":
$$\lambda=:\frac{\vec{J}\cdot\vec{P}}{P^{0}}$$
It can take only positive/negative integer/semiinteger values.
For the photon:$\lambda=\pm 1$;for the scalar boson it is zero,for the graviton it is [itex] \pm 2 [/tex]

Daniel.

 

[marlon]

It is quite difficult to understand, can someone explain what is helicity?

Helicity is used in QM and it expresses the connection between the direction in which a particle rotates around some axis through that particle (expressed by the spin quantum number) and the direction of propagation of a particle (expressed by its momentum p = mv).

In QM you can measure the spin of some particle by using the spin quantum operator S' and the momentum is expressed by some vector p. Now if you want to measure the helicity of some particle you just apply a new operator (the helicity-operator) onto the particle's wavefunction. This new operator is defined as p.S' and it measures the components of the spin operator along the direction of momentum p (this is the direction along which the particle moves). p.S' is the scalar product of some vector with an operator and suppose for example that p is along the z-axis then the helicity operator is nothing else then the length of vector p (this is the momentum-value of the particle) multiplyed with the z-component of the spin operator S'. So you measure the spin along the z-axis.

Positive helicity means that the rotation-axis of the spin is in the same direction as the direction in which the particle moves. If helicity is negative, it is the other way around


regards
marlon

these concepts are very important in QFT, and more specifically the weak interactions because particles of different helicity behave totally different under such interactions. Only left handed fermions will feel the weak force and therefore are fundamentally different as right handed fermions...

 

[marlon]

It is defined as the proportionality constant between the operators:

This is wrong...
What you are referring to is the eigenvalue of this helicity operator, not the operator itself... GROUPTHEORY dextercioby...


Besides, i don't think that this definition will bring much clarity...

And besides, also i this case the direction of the spin can be defined in terms of the direction of momentum...


marlon

 

[dextercioby]

Yes,it's not a fortunate exprimation.That relation should include "hats",of course.It is the operator:

$$\hat{\lambda} =\frac{\hat{\vec{J}}\cdot\hat{\vec{P}}}{P^{0}}$$
$$\hat{W}^{\mu}=:\hat{\lambda}\hat{P}^{\mu}$$
Yields the comutation relations with the generators of the Poincaré group
$$[\hat{\lambda},\hat{M}_{\mu\nu}]_{-}=\hat{0}$$
$$[\hat{\lambda},\hat{W}_{\mu}]_{-}=\hat{0}$$
$$[\hat{\lambda},\hat{P}_{\mu}]_{-}=\hat{0}$$

Daniel.

PS.What i said about its eigenvalues is correct.

 

[dextercioby]

Helicity is used in QM and it expresses the connection between the direction in which a particle rotates around some axis through that particle (expressed by the spin quantum number) and the direction of propagation of a particle (expressed by its momentum p = mv).

regards
marlon

Wow,Marlon,that's the second thread with the bunch of nonsense like "the direction in which a particle rotates around some axis through that particle".
Are u inventing new physics and i don't see it...?

Daniel.

PS.Since these particles are pointlike,how would u define their rotation ??

 

[marlon]

Wow,Marlon,that's the second thread with the bunch of nonsense like "the direction in which a particle rotates around some axis through that particle".
Are u inventing new physics and i don't see it...?

Daniel.

PS.Since these particles are pointlike,how would u define their rotation ??

Please review your group theory... Ever heard of rotations in spin space or is this some new physics?

And yes, your definition is wrong in this sense that you are referring to the actual eigenvalues of the helicity operator, not the operator itself...

Tell me, what do you think this operator expresses, HMMM??


marlon

 

[dextercioby]

Please review your group theory... Ever heard of rotations in spin space or is this some new physics?


Tell me, what do you think this operator expresses, HMMM??

marlon

I don't need group theory.Rotations in spin space would not apply to particles,but to quantum states,WHICH ARE VECTORS.They are rotations of vectors/quanutm states,not of particles.Read first principle of QM.It will enlighten you.The particle is pointlike and,hopefully,u know it.So speaking about "axis through that particle" and rotating round an axis through the particle makes me laugh...Really...
I said what it represents.The projection of the total angular momentum on the direction of movement.If u come up with another definiton,u're free to do so,as long as it is correct.

Daniel.

 

[marlon]

I don't need group theory

wrong again...err i could have sworn i said that before to you. Spin-rotations are defined using the principles of group theory, which you don't seem to grasp.

Rotations in spin space would not apply to particles,but to quantum states,WHICH ARE VECTORS.

Newsflash : particles are the same as quantumstates man, they are represented by it...this remark really is useless...useless arguing...

They are rotations of vectors/quanutm states,not of particles.Read first principle of QM.

this was my entire point...really...thank you...don't keep on whinning about particles vs quantumstates.

The particle is pointlike and,hopefully,u know it.So speaking about "axis through that particle" and rotating round an axis through the particle makes me laugh...Really...

it is really not that difficult to envision. think of a rotating sphere...

I said what it represents.The projection of the total angular momentum on the direction of movement.If u come up with another definiton,u're free to do so,as long as it is correct.

Daniel.

i gave the correct definition, you did not...


marlon

 

[Mk]

I'll just wait till you guys figure it out...

 

[dextercioby]

I'll just wait till you guys figure it out...

Which is never...

My advice:take Sakurai "Modern Quantum Mechanics" for helicity in QM and Bailin & Love "Weak Interactions" for helicity in QFT.
Learn as much math is possible.

Daniel.

 

[marlon]

I'll just wait till you guys figure it out...

Hi, MK, just read my first post and do feel free to ask more questions if something is not clear. All my other posts here, you must disregard, because they are waiste of time...

Besides, don't do any QFT not yet. That is a stupid advice. First start out with any thorough book on QM and there you will also find some explanation for the helicity-operator. I always used Bransden and Joachain "QUANTMMECHANICS"

regards
marlon