Similarly as in the relativistic case, one can study the NR N = 4, D = 3+1 supersym-
metric theories by following several paths:
i) One can start with NR N = 4, d = 3 Galilean superalgebras (for arbitrary N see [12])
and then construct their superspace and superfield realizations. In this way we obtain
the universal tool for constructing NR supersymmetric field theories.
ii) The NR field theories can be reproduced by performing the non-relativistic contraction
c → ∞ (c is the speed of light) in the known relativistic non-supersymmetric, as well
as supersymmetric field theory models (see, e.g., [4, 13, 14]). One of the advantages
of such a method is the possibility to derive the proper NR contractions of relativistic
action integrals.
iii) For a definite type of (super)symmetric framework one can consider the dynamics as-
sociated with particles, fields, string, p-branes, etc. An important role in such a list is
played by the free classical and first-quantized (super)particle models, with the prop-
erty that their first quantization leads to the classical (super)field realizations (see,
e.g., [15, 16]). In our case, we will look for the free superparticle models invariant
under N = 4, d = 3 Galilean supersymmetry. One can mention that in the rela-
tivistic case this way of deriving free superfields from the classical and first-quantized
superparticles with extended N = 4, D = 4 Poincar´e supersymmetry was already
proposed in [17, 18].
2
In this paper we will follow the path iii). We will consider the most general NR
N = 4, d = 3 Galilean superalgebra, introduce the corresponding N = 4 Galilean super-
group and its cosets, construct the relevant nonlinear realizations and use the associated
Maurer-Cartan (MC) one-forms to build NR N = 4 superparticle models. They will
be subsequently quantized to obtain the NR superfields providing realizations of N = 4
Galilean supersymmetries. Note that in such a setting the original coset parameters are
treated as the D=1 world-line fields. However, the whole formalism could be equally ap-
plied along the lines of path i), with the coset parameters treated as independent NR
superspace coordinates.
As a prelude to our considerations, we will describe the Galilean symmetries and their
supersymmetrizations in a short historical survey.
The Galilean theories describe the low energy, non-relativistic dynamical systems,
3
which can also be obtained as non-relativistic limit (c → ∞) of the corresponding relativis-
tic theories (see, e.g., [28–32]). Such a contraction limit, applied to D = 4 Poincar´e algebra
(P
µ
, M
µν
; P
µ
= (P
0
, P
i
), M
µν
= (M
i
=
1
2
ε
ijk
M
jk
, N
i
= M
i0
)), after shifting and rescaling
P
0
= m
0
c +
H
c
, P
i
= P
i
, N
i
= c B
i
, M
i
= J
i
, (1.1)
2
Here we will deal only with those Galilean supersymmetries which, after quantization, leads to the
superfields depending on spinorial Grassmann variables. One could introduce as well relativistic [19–22]
and Galilean [23, 24] world-line supersymmetries with vectorial Grassmann variables. However, this option
is out of the scope of our study.
3
The Galilean symmetries are used also in the description of light cone quantization of relativistic
theories [25–27]. In this paper, we shall not deal with this application of Galilean symmetries.
– 2 –
