The Foldy-Wouthuysen transformation (FW) is one of several methods used to investigate the low-energy limit of the relativistic Dirac equation (low-speed); due to a series of sequential unitary transformations , it has proven to be the favorite method to meaningfully obtain the nonrelativistic limit of the Dirac equation in which it gives the Schrödinger-Pauli equation ; in the FW representation for relativistic particles in external fields, the operators have the same form as in the nonrelativistic quantum theory, this is mainly because of the fact of the hamiltonian and all operators in this representation are block-diagonal, furthermore the basic characteristics of the FW representation are obviously described in .
In the present work, the basic properties of the Dirac hamiltonian in the FW representation in the noncommutative phase-space are investigated and the Schrödinger-Pauli equation is found, where the common methods for extracting the full phase-space noncommutative Dirac equation are both of the linear translation method, which known as Bopp-Shift translation in which it matches between the commutative quantum mechanics and the noncommutative quantum mechanics (NCQM) , and the Moyal-Weyl product (*-product) .
2. Phase-Space Noncommutativity
In the two-dimensional commutative phase-space, the coordinates
and the kinetic momentum
satisfy the usual canonical commutation relations
In the recent study results on the phase-space noncommutativity (PSNC) is shown that at very tiny scales (string scales) the space may not commute anymore, let us consider the operators of coordinates and kinetic momentum in a two-dimensional noncommutative phase-space
respectively, where the noncommutative phase-space operators satisfy the commutation relations
with the effective Plank constant being
are noncommutative parameters, they are real-valued and antisymmetric constant matrices with dimension of (length)2 and (momentum)2, respectively.
The noncommutativity in phase-space can be realized in terms of Moyal-Weyl product (*-product) which means that the noncommutativity information is encoded in the Moyal product, defined as
The noncommutative phase-space operators are related to the commutative phase-space one, due to the so-called Bopp-shift linear transformation , knowing that the latter induced from the *-product, and it is given by
the noncommutative phase-space algebra reduces to the commutative one.
3. Nonrelativistic Limit of the Dirac Equation in Noncommutative Phase-Space
3.1. The Dirac Equation in Noncommutative Phase-Space
As it known by the use of the *-product, we obtain the Dirac equation for the noncommutative quantum mechanics 
knowing that the Dirac equation in interaction with the electromagnetic four-potential
in commutative phase-space is
is the wave function (bi-spinor) in the Dirac representation.
At first we achieve the noncommutativity in space, by the mapping between the noncommutative coordinates
and the commutative coordinates x using the *-product, with the help of Equation (4) we find
Consider the electromagnetic potential
, where h is a constant, the derivations in the Equation (4) roughly turned off in the first order, then Equation (8) can be written as follows
, Equation (9) reduced to
Anew we achieve the noncommutativity in phase by the mapping between the noncommutative kinetic momentum
and the commutative one p, using Equation (5) to get the following full noncommutative phase-space Dirac equation
rewrite the Equation (11) in a more compact form (see Appendix A for the simplification):
is the wave function in noncommutative phase-space.
3.2. Foldy-Wouthuysen Transformation in Noncommutative Phase-Space
Deriving the Schrödinger-Pauli equation in noncommutative phase-space, which is the nonrelativistic limit of the Dirac equation in a simple way using the Foldy-Wouthuysen transformation, this one achieved by a series of successive unitary transformations performed on the phase-space noncommutative Dirac hamiltonian in Equation (12), knowing that it is only applicable to weak fields.
The Dirac hamiltonian in PSNC is given by
in order to perform the FW transformation, we have to rewrite the Dirac hamiltonian Equation (13) to the form:
1Odd operators (off diagonal in Pauli-Dirac basis):
even operators (diagonal in Pauli-Dirac basis):
suffice at the 4rd order of
where the Dirac hamiltonian is divided into block diagonal and off diagonal parts denoted even operator
and odd operator
these are defined to satisfy
Because of the presence of the odd operators (of
matrices), the Dirac hamiltonian is not block diagonalized, so that we try to eliminate odd operators from the Dirac hamiltonian, by applying FW transformation
is Hermitian and it is of the form
Using the Maclaurin series expansion of
in the Equation (17) yields a transformed Dirac hamiltonian (Campbell-Baker-Hausdorff expansion) 
Writing our hamiltonian, restricting ourselves to terms up to order
, thus we suffice at the 4rd order of
Using the properties mentioned in Equation (16) and with Equation (14) and Equation (18), knowing that
we calculate the various commutators of
with the same manner we continue, with taking into account only terms of the order
, we get
by collecting the terms of
Equations ((22), (25)), we find
as it shown in the Equation (27) the new hamiltonian is not free of the odd operator, the odd part not omit, so By further FW transformation we reduce the odd part of the transformed hamiltonian, so we perform a second transformation, remembering that the product of two even or odd operators is an even operator.
To reduce the odd part of the transformed hamiltonian, thus we chose
We restrict ourselves to terms up to order
, as in the first FW transformation (knowing that
), so that the new hamiltonian is
The terms proportional to
contain large powers of
, therefore they can be neglected, more precisely we take into account only terms of order that we restrict ourselves in the expansion, and thus Equation (34) is given by
is not yet free of the odd operators, we apply a third FW transformation to eliminate
Finally the transformed hamiltonian is completely free of odd operators, next we calculate the various terms of the Equation (38), according to Equation (15), we make use of the following known relations for three arbitrary vectors
the Equation (15) becomes
using the following relation2
and using Equation (45) we have
using Equation (43) we simplify Equation (46) (see Appendix B), to have
Adding the various contributions Equations ((44), (47)), in Equation (38), with assuming
, and For
the Schrödinger-Pauli equation is
The reason of the explicit noncommutative terms entangled in the obtained Schrödinger-Pauli Equation (48), (in FW representation) is that the effect of the noncommutativity in the Dirac equation appears as a kind of potential which depends on noncommutativity parameters (
), then after applying the nonrelativistic limit that potential is the responsible on generating a new terms and a modified known terms, which contain the noncommutative parameters (reduced in
), in Equation (48), where terms in the first parenthesis describe the NC nonrelativistic kinetic energy, and its first NC nonrelativistic correction (at least to the order of approximation we have considered), this is manifested as a terms contain phase-space NC parameters added to the known terms, then successive two terms describe the electrostatic energy and the magnetic dipole
, represents the Θ-modified spin-orbit interaction term, in a spherically symmetric potential, with
Next term in Equation (48),
, represents the Θη-modified Darwin term (attributed to the Zitterbewegung ), with
The other terms in the Equation (48) represent the NC Schrödinger-Pauli equation corrections.
Essentially the intriguing part of our result is the fact that noncommutative effects grant a Θη-modified terms entangled in the obtained NC Schrödinger-Pauli equation.
Under the condition that space-space and momentum-momentum are all commutative (namely,
) the results return to that of usual quantum mechanics.
In conclusion, the phase-space noncommutativity effect is introduced in the Dirac equation and subsequently the Foldy-Wouthuysen transformation is exploited to reduce the system in presence of electromagnetic field to a nonrelativistic regime, which gives the Schrödinger-Pauli equation.
Knowing that the phase-space noncommutativity effect is introduced by applying both of the Bopp-shift linear translation method, and the Moyal-Weyl product.
The usage of the FW representation in most cases allows one to reduce the problem of finding a classical limit of relativistic quantum mechanical equations to the replacement of operators in the hamiltonian of the quantum mechanical equations of motion by the respective classical quantities, even with noncommutativity in phase and space, and the effects of the latter are manifested in the various terms of the obtained hamiltonian.
Appendix A: Moving from Equation (11) to Equation (12)
The simplification to move from Equation (11) to Equation (12):
, we find:
, knowing that
with the same manner we prove that
Appendix B: The Elimination of
from the Two Last Terms in Equation (46)
Using The Equation (43) we find
And for the second term
we start with the 1st term of the above Equation (55)
using Equation (39), we continue with the 2nd term,
finally we find