Constructing Supersymmetric Lagrangians

Question

It is a very trivial doubt but somehow I am not able to figure it out. While constructing a supersymmetric lagrangian we always even number of fermionic fields.

1. One reason is of course the product of $\psi_1 \psi_2$ behaves as a bosonic variable, but still I would like to know what are the other reasons?

2. If we have terms with odd number of fermions will the theory be non-supersymmetric or is there some fundamental property of an action/lagrangian that it restricts such kind of terms?

Answer 1

The Lagrangian is Lorentz-invariant, which means it cannot contain fermionic terms.

Answer 2

I interpret OP's question as

Why does the action $S$ carry definite Grassmann parity, and in particular, why is $S$ Grassmann-even?

This question is actually not restricted to just supersymmetric models, but could be asked for any theory with Grassmann-odd variables.

It is true that the overwhelming majority of published literature have $S$ manifestly Grassmann-even.

1. At the classical level ($\hbar=0$), there is in principle nothing wrong with an action $S$ with indefinite or odd Grassmann parity, as long as the corresponding Euler-Lagrange equations reproduce the classical equations of motion.

2. At the quantum mechanical level, the situation changes. For instance, in the path integral formalism, it would be rather weird to consider a Boltzmann-factor $\exp(\frac{i}{\hbar}S)$ where the action $S$ has indefinite or odd Grassmann parity. (If one would like to consider such a construction, one would have to insert appropriate projection operators to ensure that the overlaps and amplitudes remain Grassmann-even.) Recall that no measuring device in an experiment is going to measure a Grassmann-odd number. A measuring device can only produce real outputs $\subseteq\mathbb{R}$.

Answer 3

There is a formulation of supersymmetry which based on odd structures. In mathematics it is called odd symplectic geometry. In this framework one can formulate supersymmetric models with odd Hamiltonians and Lagrandians. Please see the following article by: Soroka, Soroka and Wess. In this work they constructed the supersymmetric harmonic oscillator using an odd Lagrangian. Please see also the following article by Soroka.