The components of Dirac Equation -- Bosonic Lagrangian?

[arivero]

The four components of Dirac equations obey the Klein-Gordon equation for a particle of mass m. This is always explained when introducing Dirac equation, but it is never exploited further. I am wondering:

• Can we then write a bosonic lagrangian for these four "particles"?
• Is this related to the existence of supersymmetry? We can extract the components for a Dirac equation in any number of dimensions, but supersymmetry only exists for D=10, 6, 4, 3 if I recall correctly.
• Does the process generalise to charged fermions?

 

[Demystifier]

Can we then write a bosonic lagrangian for these four "particles"?

Yes, we can write it down. But it's not equivalent to the usual Dirac Lagrangian, because the Dirac equation is not equivalent to the Klein-Gordon equation. The former implies the latter, but the latter does not imply the former.

Is this related to the existence of supersymmetry?

Not directly, but see my https://arxiv.org/abs/hep-th/0702060 where I study Klein-Gordon current for spin-1/2 particles, as well as for superstrings.

Does the process generalise to charged fermions?

No. If you solve Klein-Gordon equation in Coulomb potential, you get wrong predictions for the spectrum of the hydrogen atom, see the QM textbook by Schiff.

 

[arivero]

Not directly, but see my https://arxiv.org/abs/hep-th/0702060 where I study Klein-Gordon current for spin-1/2 particles, as well as for superstrings.

I like that eq 9 already encodes the same trick that susy infinitesimal variations.

I am not sure that implication only runs in one direction... susy practitioner get to engineer a variation of the scalar field by combining the fermion components, that fulfill klein-gordon equation. But they also get to engineer a variation of the fermion field out of the gradient of the scalar field, so the later should be similar to a solution of dirac equation, I guess.

 

[arivero]

My thinking here is that a didactical introduction to supersymmetry could start by asking the students to try two exercises.

1. Given a two-component solution of Weyl equation, combine it into solutions of Klein Gordon equation. Easy.
2. Given a pair of solutions of Klein Gordon equation, or a complex solution, produce the two components of a Weyl equation.

Solving point 2 is also a good introduction to equation (9) of Demystifier article.

 

[billtodd]

The four components of Dirac equations obey the Klein-Gordon equation for a particle of mass m. This is always explained when introducing Dirac equation, but it is never exploited further. I am wondering:

• Can we then write a bosonic lagrangian for these four "particles"?
• Is this related to the existence of supersymmetry? We can extract the components for a Dirac equation in any number of dimensions, but supersymmetry only exists for D=10, 6, 4, 3 if I recall correctly.
• Does the process generalise to charged fermions?

Any citation for a proof of the claim that SUSY exists only for D=10,6,4,3?

 

[arivero]

Any citation for a proof of the claim that SUSY exists only for D=10,6,4,3?

Yeah, I guess it sounds very fishy as SUSY QM (arguably D=1 or even D=0) is a thing. The exact results are Evans (https://inspirehep.net/literature/22536) and Kugo-Townsend (https://inspirehep.net/literature/181889).
Tong's lecture notes:

The number of polarisation states of a photon is d - 2. So the question really is:in what dimensions does a spinor have d - 2 degrees of freedom? We will see that we can have a supersymmetric theory in which a photon pairs with a single fermion in d = 3, 4, 6 and 10 Lorentzian spacetime dimensions.

A recent slide by Townsend:


Baez and Huerta state the result "Nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green-Schwarz superstring." (https://arxiv.org/abs/0909.0551) which is far from "susy only exists in..." so I guess there are ways to evade the limitation.

 

[Demystifier]

Any citation for a proof of the claim that SUSY exists only for D=10,6,4,3?

From string theory we know that susy exists also for D=2 (world-sheet susy) and D=11 (M-theory).

 

[arivero]

Good point (not sure for D=11 although). Is it because the D=2 theory is not interacting? If so, do free susy theories exist in any number of dimensions, or is there some limitation too?

 

[billtodd]

From string theory we know that susy exists also for D=2 (world-sheet susy) and D=11 (M-theory).

Does SUSY exist for F-theory?

Yeah, I guess it sounds very fishy as SUSY QM (arguably D=1 or even D=0) is a thing. The exact results are Evans (https://inspirehep.net/literature/22536) and Kugo-Townsend (https://inspirehep.net/literature/181889).
Tong's lecture notes:


A recent slide by Townsend:
View attachment 350046

Baez and Huerta state the result "Nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green-Schwarz superstring." (https://arxiv.org/abs/0909.0551) which is far from "susy only exists in..." so I guess there are ways to evade the limitation.

Interesting slides.