What Does CPT-Conjugate Mean in Supersymmetry?

[302021895]

When building the massless supermultiplets in N=1 supersymmetry one needs to add the CPT conjugate states to render the theory CPT invariant.

My question is, what do they mean by "CPT-conjugate"? Is it just the state with opposite helicity? Is it the state with opposite C, P and T? Or is it something else?

I wonder about this because when working with a massless chiral supermultiplet, it turns out that the real part of the complex scalar field is a real scalar field, but the imaginary part of the complex field is a real pseudoscalar field: the supersymmetry transformation in Weyl spinor notation is

$\delta\phi=\epsilon\psi$

which implies in Majorana notation that

$\delta a = \frac{1}{\sqrt{2}}\overline{\epsilon}\Psi$
$\delta b = \frac{i}{\sqrt{2}}\overline{\epsilon}\gamma_{5} \Psi$

where $\phi=(a+ib)/\sqrt{2}$

 

[AndyW]

and \Psi is the fermionic component of the supermultiplet.


The term "CPT-conjugate" refers to a state that has opposite charge, parity, and time-reversal properties as the original state. In supersymmetry, this means that the CPT-conjugate state will have opposite helicity, or chirality, as the original state. In other words, if the original state has positive helicity, the CPT-conjugate state will have negative helicity.

However, it is important to note that the CPT-conjugate state is not simply the state with opposite C, P, and T properties. In fact, the CPT transformation is a combination of these three operations, and it involves complex conjugation as well. Therefore, the CPT-conjugate state will also have a complex conjugate of its wavefunction.

In the context of supersymmetry, the CPT-conjugate state is necessary to ensure that the theory is CPT-invariant. This means that the theory will have the same properties under the combined operation of charge conjugation (C), parity transformation (P), and time reversal (T), as it does under the CPT transformation. This is important because CPT invariance is a fundamental symmetry of quantum field theories.

In the case of a massless chiral supermultiplet, the CPT-conjugate state will have opposite chirality as the original state, as well as a complex conjugate of its wavefunction. This is necessary to ensure that the supersymmetry transformation remains consistent and the theory is CPT-invariant.

Overall, the term "CPT-conjugate" refers to a state that is related to the original state by the CPT transformation, which involves a combination of charge conjugation, parity transformation, time reversal, and complex conjugation.