# Degree of freedom of gravitino

• shooride
In summary, the Vielbein's DOF's are easy to obtain, as it is a matrix with D^2 components in D dimensions. However, local Lorentz transformations subtract 1/2*D*(D-1) components, leaving 1/2*D*(D+1) components for the metric. The gravitino has 2^[d/2](d-1) components in the off-shell formalism, but in the on-shell formalism, with the equation of motion ##\gamma^\mu\psi_\mu=0##, the answer is 2^[d/2]/2(d-3). In addition, in 4 dimensions, the supervielbein has 8*

#### shooride

Please tell me how to count the degree of freedom of gravitino on the mass-shell? I read http://arxiv.org/abs/1112.3502, but I can't understand it. How about supervielbein?

Take a look at Van Proeyen's online notes on SUGRA, or his book with Freedman. There it is really explicitly explained, both of-shell and on-shell :)

The Vielbein's DOF's are easy (of-shell): it's a matrix, so it has D^2 components in D dimensions. Local Lorentz transformations make you subtract 1/2*D*(D-1) components from it, leaving you with 1/2*D*(D+1) components for the metric. Which is the right amount for a symmetric tensor like the metric.

haushofer said:
Take a look at Van Proeyen's online notes on SUGRA, or his book with Freedman. There it is really explicitly explained, both of-shell and on-shell :)

The Vielbein's DOF's are easy (of-shell): it's a matrix, so it has D^2 components in D dimensions. Local Lorentz transformations make you subtract 1/2*D*(D-1) components from it, leaving you with 1/2*D*(D+1) components for the metric. Which is the right amount for a symmetric tensor like the metric.

dear haushofer,I want to obtain the DOF of gravitino, but I just found the final answer at Supergravity by Freedman and Van Proeyen..I know gravitino has 2^[d/2](d-1) components in the off-shell formalism (i use local susy gauge invariance)..but I don't understand how to obtain DOF of gravitino in the on-shell formalism? (with E.O.M ##\gamma^\mu\psi_\mu=0## ) The answer is 2^[d/2]/2(d-3). Moreover I find at the book of SUGRA by West witch superveilbein (not vielbein) has 8*8*8 DOF (and general coordinate transformations and super local lorentz transformations subtract 8*8+8*6 components ) in 4-dim, but how to count these DOF?

What is it exactly that you don't understand of the treatment of Van Proeyen? He does it very explicitly on-shell, in section 5.1. It's basically a survey of the amount of initial conditions to be specified if you rewrite the EOM in terms of an antisymmetric derivative. :)

## 1. What is the degree of freedom of a gravitino?

The gravitino is a theoretical particle that is predicted by supersymmetric theories. It has a spin of 3/2, which gives it four possible degrees of freedom. However, in some supersymmetric theories, the gravitino's spin may be reduced to 1/2, resulting in two degrees of freedom.

## 2. How does the degree of freedom of a gravitino affect its interactions with other particles?

The degree of freedom of a gravitino determines its interactions with other particles. For example, a gravitino with four degrees of freedom can interact with all other particles, while a gravitino with two degrees of freedom may only interact with particles that have spin 1/2 or 1.

## 3. Can the degree of freedom of a gravitino change?

Yes, the degree of freedom of a gravitino can change depending on the specific supersymmetric theory being considered. In some theories, the gravitino's spin may be reduced to 1/2, resulting in a decrease in its degrees of freedom.

## 4. How does the degree of freedom of a gravitino relate to supersymmetry?

The degree of freedom of a gravitino is closely related to supersymmetry, as it is a key component of supersymmetric theories. In fact, the gravitino is the supersymmetric partner of the graviton, the particle that mediates the gravitational force.

## 5. What implications does the degree of freedom of a gravitino have for particle physics?

The degree of freedom of a gravitino is important for understanding the fundamental particles and forces of the universe. It is also relevant for theories such as string theory, which attempts to reconcile gravity with quantum mechanics. By studying the degree of freedom of a gravitino, scientists hope to gain a better understanding of the underlying principles of the universe.