N=1 SUSY masive supermultiplets

[thatsnotfunny]

N=1 susy with massive supermultiplets; therefore we have two pairs of raising/lowering operators.
Define a1 dagger as raising j3 by half and a2 dagger as lowering j3 by half.
Take vacuum state with j, j3 where j3 has 2J+1 possible values.
If we act with just a1 dagger we get superposition of states, both with j3+1/2 but one has j+1/2 and one has j-1/2 .
If we act just with a2 dagger we get a similar situation but with j3-1/2 .
But if we act with both a1 and a2 dagger we only get a state with j, j3. I understand that the j3 is additive, but surely we should have states with j plus or minus one as well? It seems to be postulated that the spin can only go up or down by half but not by one?

 

[Jhero]

I would like to address your concerns and provide some clarification on the concept of N=1 supersymmetry with massive supermultiplets.

Firstly, let's define what we mean by N=1 supersymmetry. This refers to a type of symmetry in particle physics that relates fermions (particles with half-integer spin) to bosons (particles with integer spin). In this case, we are specifically looking at supersymmetry with massive supermultiplets, which involves particles with non-zero mass.

Now, let's consider the operators a1 dagger and a2 dagger that you have mentioned. These are known as raising and lowering operators, respectively, and they act on the spin state of a particle. In this case, we are looking at a spin state with j, j3 where j is the total spin and j3 is the component of spin along a specific axis.

When we act with just a1 dagger, we are indeed creating a superposition of states with j3+1/2. However, as you correctly pointed out, one of these states has j+1/2 and the other has j-1/2. Similarly, when we act with just a2 dagger, we get a superposition of states with j3-1/2, with one state having j+1/2 and the other having j-1/2.

But when we act with both a1 and a2 dagger, we are actually creating a state with j, j3. This is because the two operators work in a complementary manner, with one raising the spin by half and the other lowering it by half. So, in this case, the net effect is that the spin state remains unchanged.

It is important to note that the values of j and j3 are not restricted to just j+1/2 or j-1/2. In fact, they can take on any value within the allowed range. However, when we act with both a1 and a2 dagger, we are specifically looking at the state with j, j3, which is a unique and well-defined state in the context of N=1 supersymmetry.

In conclusion, while it may seem that the spin can only go up or down by half, this is not the case. The operators a1 dagger and a2 dagger are just two specific ways of manipulating the spin state, and there are other operators that can change the spin by different amounts. But