So what's the deal with Majorana, Weyl, and Dirac particles in N dimensions?

[arivero]

Amusingly, a search on these three words here in PF does not show a lot of postings, so I am creating this thread so you can ask all your doubts about N-dimensional Majorana, Weyl and Dirac particles, their representations, their Lagragians, masses, and whatever you have always wanted to know

We have spoken previously of it early this year here
https://www.physicsforums.com/showthread.php?t=378170&highlight=majorana+Weyl+Dirac
A couple years ago here, relating to SUSY!
https://www.physicsforums.com/showthread.php?t=242313&highlight=Majorana+Weyl+Dirac
and five years ago here
https://www.physicsforums.com/showthread.php?t=89846&highlight=Majorana+Weyl+Dirac

 

[arivero]

A first question I am interested on, is when do we start to need Grassman Variables. It seems that we don't need it to write explicitly the Dirac equation, nor the Majorana ones. But reading Peskin, it invokes them already when using a Lagrangian for Majorana.

 

[arivero]

Other issue to address: In Dirac fermions, we are usually told about the limits when c goes to infinity (so classical quantum mechanics with spin 1/2 particles), when mass goes to zero (so it dividis in Weyl spinors) and even when mass is, if not infinite, a lot greater than the energy eigenvalue, and then we see the distinction between two "particle" degrees of freedom and two "antiparticle" states.

What about all of these in Majorana particles. In fact, do Majorana particles have negative energy eigenstates?

 

[Hans de Vries]

Maybe nice to post some of the relations I found, for example this
one yielding the symmetric electromagnetic stress energy tensor:

If we apply the generator of general Lorentz transformations to
a Dirac field, using the definitions:

$$\varphi' ~=~ {\cal F}^\mu_{~\nu}\,\varphi ~=~ \left(\,\vec{E}\cdot\hat{\mathbb{K}} + \vec{B}\cdot\hat{\mathbb{J}}\,\right)\varphi$$


Where J is the rotation generator, K is the boost generator, E and
B are the electromagnetic field and F is the spinor version of the
electromagnetic field tensor.

The (five) Dirac bilinear fields all yield the stress energy tensor T:

$$\begin{aligned}<br /> &\bar{\varphi}'\gamma^\mu\varphi' &=~~~ &\tfrac12\,T^\mu_{~\nu}~\bar{\varphi}\,\gamma^\nu\varphi \\<br /> &\bar{\varphi}'\gamma^5\gamma^\mu\varphi' &=~~~ &\tfrac12\,T^\mu_{~\nu}~\bar{\varphi}\,\gamma^5\gamma^\nu\varphi \\<br /> &\bar{\varphi}'~\mathbb{K}^\mu\,\varphi' &=~~~ &\tfrac12\,T^\mu_{~\nu}~\bar{\varphi}\,\mathbb{K}^\nu\,\varphi \\<br /> &\bar{\varphi}'~\mathbb{J}^\mu\,\varphi' &=~~~ &\tfrac12\,T^\mu_{~\nu}~\bar{\varphi}\,\mathbb{J}^\nu\,\varphi \\<br /> \end{aligned}$$

The first two are the vector and axial vector representing the
four-current and four-spin of the transformed field. The third
and fourth are mixed quantities which contain the remaining fields.
We extend K and J to 4 components for convenience as follows:

$$\mathbb{K}^\mu ~=~ -\tfrac12\,\gamma^\mu\gamma^o, ~~~~~~~~~~ \mathbb{J}^\mu ~=~ \tfrac{i}{2}\,\gamma^5\gamma^\mu\gamma^o$$


The zero'th component of K yields the scalar and the zero'th
component of J yields the pseudo scalar. the spatial components
of J produce the half of the tensor which represents the magneti-
zation of the Dirac field while those of K yield the other half of the
tensor which represents the polarization of the field (zero at rest).


Regards, Hans

 

[arivero]

Hello Hans,

I was not thinking -yet- on the Electromagnetic field, nor its duality properties etc. But your comment remembered me of a footnote in Sakurai's "Advanced Quantum Mechanics"

 

[fermi]

Other issue to address: In Dirac fermions, we are usually told about the limits when c goes to infinity (so classical quantum mechanics with spin 1/2 particles), when mass goes to zero (so it dividis in Weyl spinors) and even when mass is, if not infinite, a lot greater than the energy eigenvalue, and then we see the distinction between two "particle" degrees of freedom and two "antiparticle" states...

All three types of fermions transform as definite representations of the Lorentz group SO(3,1). Both Weyl and Majorana fermions take advantage of a very peculiar and strange property of the Lorentz group. This is because the Complex extension of the Lie algebra associated with SO(3,1) is not simple. It decomposes into a direct sum of the algebras of of SO(3) and SO(2,1). But if you keep everything real and don't do the complex extension, the algebra of SO(3.1) is simple (and the real SO(3.1) is a simple group.) This creates the following unusual situation. The representations of SO(3,1) can be labelled by two indices (corresponding to the Complex direct sum.) The smallest representation of SO(3) and SO(2,1) are both two dimensional. It turns out that both Weyl and Majorana Fermions transform by the (chiral) representation (1/2, 0) (say for left handed fermions.) This is a representation of the Lorentz group, but it is not a faithful representation. The same thing can be said for the other chiral representation (0, 1/2). But then the amazing (to me) fact takes over: the direct sum of two unfaithful representation is a faithful one namely : (1/2, 0) + (0, 1/2). That's what the Dirac Fermions transform as. Of course you need terms in the Lagrangian that couple (1/2, 0) and (0, 1/2) to each other. These require either scalar (or pseudo scalar) or second order antisymmetric tensor couplings for Dirac and Weyl fermions (the vector and axial vector couplings allowed by Lorentz group never mix the two.) The mass term in Dirac equation is the scalar term that does that. This is why we can view (in some sense) that in the limit of of zero mass, Dirac fermions reduce to a pair of Weyl (or Majorana) fermions. Because the Lagrangian does not couple the two components of the representation to each other. (Assuming no tensor couplings which seem not to be present. If there were tensor couplings, it would be a different story.) For Majorana fermions, the (real restricted) representation (1/2, 0) can couple to itself to form a scalar, and thus provide a mass term (which however violates Fermion number.)

As a side note, I will mention the point that the defining (real) four dimensional representation of the Lorentz group is (1/2, 1/2) and it is irreducible. It can couple (as vector or axial vector) to Weyl fermions, or to either (or both) component of the Dirac Fermions (but not to Majorana.)

 

[TriTertButoxy]

SO(3) and SO(2,1)?! I thought we usually dealt with SU(2) x SU(2)? And depending on which of the SU(2) the field transforms non-trivially under, we get either a left handed or right handed chiral field.

 

[fermi]

SO(3) and SO(2,1)?! I thought we usually dealt with SU(2) x SU(2)? And depending on which of the SU(2) the field transforms non-trivially under, we get either a left handed or right handed chiral field.

The SO(3) algebra is isomorphic to the SU(2) algebra. (But the groups are not isomorphic differing by a factor of Z2.) Non-compact SO(2,1) and SU(1,1) algebras are also the same, if you use the complex extensions. So, once you allow the complex algebras, everything is the same. I prefer SO(3) and SO(2,1) because you can keep everything real for whatever it is worth. Likewise working with SU(2) is just fine.