# Reps of the SUSY algebra: raising and lowering operators

• hyperkahler
In summary: Clifford algebra with 2N generators, a^I as rising operator and (a^I)^\dagger as lowering operator for helicity of massless states. Spinors with ##\alpha## index have positive helicity while those with ##\dot{\alpha}## have negative helicity, shown explicitly through the relation between angular momentum and spinor operators.
hyperkahler
I'm reading Alvarez-Gaume review on Seiberg-Witten theory: http://arxiv.org/abs/hep-th/9701069.
Around page 23 you can find the following claim:

"This is a Clifford algebra with 2N generators and has a 2N-dimensional representation. From the point of view of the angular momentum algebra, $a^I$ is a rising operator and $(a^I)^\dagger$ is a lowering operator for the helicity of massless states"

The definitions of $a^I$ and $(a^I)^\dagger$ are given a few lines above. How to see that the first raise helicity while the second lowers it?

It's just because of the convention that the spinors with the ##\alpha## index are taken to be of positive helicity, while those with the ##\dot{\alpha}## have negative helicity. You can probably show this explicitly from the relation

$$[M_{\mu\nu},\bar{Q}_{\dot{\alpha}}] = \frac{1}{2} {(\bar{\sigma}_{\mu\nu})_{\dot{\alpha}}}^{\dot{\beta}} \bar{Q}_{\dot{\beta}}.$$

hyperkahler said:
I'm reading Alvarez-Gaume review on Seiberg-Witten theory: http://arxiv.org/abs/hep-th/9701069.
Around page 23 you can find the following claim:

"This is a Clifford algebra with 2N generators and has a 2N-dimensional representation. From the point of view of the angular momentum algebra, $a^I$ is a rising operator and $(a^I)^\dagger$ is a lowering operator for the helicity of massless states"

The definitions of $a^I$ and $(a^I)^\dagger$ are given a few lines above. How to see that the first raise helicity while the second lowers it?
Fix: 2N-dimensional

## 1. What is the SUSY algebra?

The SUSY algebra is a mathematical framework used in theoretical physics to describe the relationship between bosons (particles with integer spin) and fermions (particles with half-integer spin).

## 2. What are raising and lowering operators in the SUSY algebra?

Raising and lowering operators are operators that act on the states in a SUSY algebra to change their spin. They are used to construct new states with different spin quantum numbers from existing ones.

## 3. How do raising and lowering operators work?

Raising operators increase the spin quantum number of a state by half-integer values, while lowering operators decrease the spin quantum number by half-integer values. These operators allow for the creation and annihilation of particles with different spin states.

## 4. What is the significance of raising and lowering operators in SUSY?

The use of raising and lowering operators in SUSY allows for the construction of supermultiplets, which are representations of the SUSY algebra that contain both bosonic and fermionic states. This is important in understanding the relationship between different types of particles and their symmetries.

## 5. How are raising and lowering operators related to supersymmetry breaking?

Raising and lowering operators play a crucial role in the study of supersymmetry breaking, which is the phenomenon where the symmetry between bosons and fermions is broken. This breaking can be understood through the use of raising and lowering operators, which allow for the construction of supermultiplets with different spin states.

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