In one dimension there is no Lorentz group and therefore all bosonic and fermionic fields have no space-time indices. The simplest free action for one bosonic field φ and one fermionic field ψ reads

S = γ∫dt [φ^{.2} – i/2ψψ^{.}] —– (1)

We treat the scalar field as dimensionless and assign dimension cm^{−1/2} to fermions. Therefore, all our actions will contain the parameter γ with the dimension [γ] = cm. (1) provides the first example of a supersymmetric invariant action and is invariant w.r.t. the following transformations:

δφ = −iεψ, δψ = −εφ ̇ —– (2)

The infinitesimal parameter ε anticommutes with fermionic fields and with itself. What is really important about transformations (2) is their commutator

δ_{2}δ_{1}φ = δ_{2}(−ε_{1}ψ) = iε_{1}ε_{2}φ ̇

δ_{1}δ_{2}φ = iε_{2}ε_{1}φ ̇ ⇒ [δ_{2}, δ_{1}] φ = 2iε_{1}ε_{2}φ ̇ —– (3)

Thus, from (3) we may see the main property of supersymmetry transformations: they commute on translations. In our simplest one-dimensional framework this is the time translation. This property has the followin form in terms of the supersymmetry generator Q:

{Q, Q} = −2P —– (4)

The anticommutator (4), together with

[Q, P] = 0 —– (5)

describe N = 1 super-Poincaré algebra in d = 1. The structure of N-extended super-Poincaré algebra includes N real super-charges Q^{A} , A = 1, . . . , N with the following anti commutators:

{Q^{A}, Q^{B} }= −2δ^{AB}P, [Q^{A}, P] = 0 —– (6)

Let us stress that the reality of the supercharges is very important, as well as having the same sign in the r.h.s. of Q^{A}, Q^{B} ∀ Q^{A}.

From (2) we see that the minimal N = 1 supermultiplet includes one bosonic and one fermionic field. A natural question arises: how many components do we need, in order to realize the N-extended superalgebra (6)? In order to mimic the transformations (2) for all N supertranslations

δφ_{i} = −iε^{A}(L_{A})^{iˆ}_{i} ψ_{iˆ}, δψ_{iˆ} = −ε^{A} (R_{A})^{iˆ}_{i}φ ̇_{i} —– (7)

Here the indices i = 1,…,d_{b} and iˆ = 1,…,d_{f} count the numbers of bosonic and fermionic components, while (L_{A})^{iˆ}_{i} and (R_{A})^{i}_{iˆ} are N arbitrary, for the time being, matrices. The additional conditions one should impose on the transformations (7) are

- they should form the N-extended superalgebra (6)
- they should leave invariant the free action constructed from the involved fields.

When N > 8 the minimal dimension of the supermultiplets rapidly increases and the analysis of the corresponding theories becomes very complicated. For many reasons, the most interesting case seems to be the N = 8 supersymmetric mechanics. Being the highest N case of minimal N-extended supersymmetric mechanics admitting realization on N bosons (physical and auxiliary) and N fermions, the systems with eight supercharges are the highest N ones, among the extended supersymmetric systems, which still possess a nontrivial geometry in the bosonic sector. When the number of supercharges exceeds 8, the target spaces are restricted to be symmetric spaces.