On the Poincaré algebra in a complex space-time manifold

We extend the Poincaré group to the complex Minkowski spacetime. Special attention is paid to the corresponding algebra that we achieve through matrices as well as differential operators. We also point out the generalizations of the two Casimir operators.


Introduction
Complex numbers are known to be powerful mathematical tools to describe physical phenomena. But it seems that they are more than that. They are not only used to report reality : they actually could be part of reality.
At our knowledge, complex energies first appeared as intrinsic components of the physical paradigm in the relativistic Kemmer-Duffin-Petiau equation [1], dealing with vector mesons. When they are subject to a sufficiently strong magnetic field (B > m 2 e ) , their energies, whose squares are given by [2] E 2 = m 2 + 2eB(n + 1 2 + s) ; n = 0, 1, 2, ... ; s = 0, ±1 become purely imaginary ones. More recently, some cosmological models [3] used an imaginary time. They all conjecture a join between areas of imaginary time and areas of real time. Somehow, real time emerges from imaginary time, the latest being helpful to remove gravitational singularities. This idea is clearly not unanimous and voices [4] rose to emphasize the inconsistency of such a transition which still remains vague, although some attempts [5] have been performed to specify it.
One of the most promising advances of complex numbers inside physical reality is the concept of Minkowski complex space-time [6]. The key idea involved by this (3+1)-D complex manifold is to extend spatial as well as time coordinates to complex numbers while the (Hermitian) metric is given by with η = diag(1, −1, −1, −1) ; x µ = {t, x, y, z} ; µ = 0, 1, 2, 3 Here the notation x µ refers to the complex conjugate of x µ . Indeed the fact that imaginary coordinates could coexist with their real counterparts gives a theoretical meaning to some observed physical phenomena. Among them, the physics of nonlocality takes an important part. Initiated by Einstein, Podolsky, and Rosen [7] as an eventual defect of quantum theory, nonlocality has been since mathematically proved by Bell [8] and repeatedly experimented. Two quanta given off from a single source and travelling in opposite directions, maintain their connection to one another whatever the distance between them is [9]. Keeping this connection simultaneous regardless of the distance defies the laws of physics if the coordinates are real numbers but could make sense if these coordinates are complex.
Complex coordinates also have consequences on theories other than physical ones. For instance, psychic phenomena [10] could be explained by them. It has been indeed frequently observed that the accuracy of "psi" awareness does not depend on temporal or spatial separation between the percipient and the target. Once again, this can only be explained through the use of complex time and space variables.
So considering a (3+1)-D complex space-time manifold makes sense, especially since the usual (3+1)-D real Minkowski space appears to be kind of a slice of this complex extension.
The purpose of this paper is to investigate this (3+1)-D manifold and, in particular, to see what would the Poincaré group/algebra become when coordinates are complex numbers. To do so, we first have to consider a (6+2)-D real manifold. This is the subject of the next Section. Then, we use the corresponding coadjoint action to put in evidence the transformation laws on momenta which are nothing else than basis operators for the algebra. Sections 4 and 5 are devoted to the differential realization of these momenta in the real as well as the complex cases. The transformations of the group are then available in Section 6. We point out the Casimir operators in Section 7 and finally conclude in Section 8.

The extended Poincaré group/algebra
As well known the Lorentz group O(1, 3) preserves where η has been defined in (2) while (X T refers here to the transposition of X).
If we now consider complex coordinates The Lie group U (1, 3, C) is, by definition, the one leaving the quadratic form (4) invariant. This quadratic form (4) is equivalent to the orthogonal one We obviously have The Lie group preserving (5) is O (2,6). Even if the quadratic forms are formally equivalent, the two groups U (1, 3, C) and O(2, 6) have a different number of parameters (16 for U (1, 3, C) and 29 for O (2,6)). This finding, that might be surprising at first sight, will be explained in a next Section.
Due to the embedding of U (1, 3, C) in O(2, 6), we focus on the larger group.
Consequently, we define the extended Poincaré group on the (6+2)-D real Minkowski space as the set of the following (9 by 9) matrices where the 8-vector is associated with (real) translations in the (6+2)-D manifold and L is the The Lie algebra corresponding to (9) is the vector space of the matrices given by Here the coefficients γ A are real numbers and ω is the matrix of the so(2, 6) algebra defined by In details, we have where J AB are real numbers and j AB are the (28) basis matrices of so (2,6) namely (no summation on repeated indices). In Eq (14), the numbers ϵ(AB)(= ±1) are constrained by while the notation e AB stands for a 8 by 8 matrix made of "0" everywhere except a "1" at the intersection of the (A+1) th line and the (B +1) th column.
Noticing that we come easily to the so(2, 6) commutation relations We will come back to this algebra in Section 4. Let us just conclude this one by mentioning that the adjoint representation of the extended Poincaré group is, by definition, given by This leads to as well as to the following relation

The coadjoint representation : transformation laws of the momenta
Here we follow Souriau's approach [11], one of us having already successfully applied it [12] to Kaluza 5-D space-time. By analogy with what has been done in the so(1, 3) case, we define a torsor µ of the extended Poincaré group by the identity We require the invariance or, on other words where we have used Eqs (19)-(20) of the adjoint representation. The relation (24) implies the following transformation on the momenta P : Thus, Eq (24) reduces to Remembering that the last term is the product of a transposed 8-vector and a 8-vector, we can rewrite this relation as which, after usual manipulations on the trace, leads to or, in an equivalent way We thus come to the conclusion that Eqs (25) and (27) provide the transformation laws of the momenta.

Differential realization of the momenta : the (6+2)-D real Poincaré algebra
Let us take a look at the result (25). If we write the transformations on the coordinates through the group as (cf.Eq (9)) they can be inversed following This implies that where ∇ is the 8D-gradient Comparing with (25), it is obvious to conclude that With the conventions (6) and We recover the momenta of usual Quantum Mechanics if the constant C is fixed as C = ih and A limited to the first four values. However, for simplicity, we fix here C = 1 so that In the same way, the momenta M satisfying (27) can be realized through a matrix similar to ω (see Eq (13)) In other words, we have And it is straightforward to convince ourselves of (27) to be satisfied by using (25) and (28). It is then easy to find out the commutation relations of the (6+2)-D Poincaré algebra by using They read in agreement with Eq (17) and [P A , P B ] = 0 (38)

The (3+1)-D complex Poincaré algebra
By making use of the change of variables (8), we can define linear combinations of the J AB as well as the P A operators in order to restore the extended Poincaré algebra in a (3+1)-D complex manifold. These linear combinations write They lead to the following realizations in terms of complex coordinates x µ and their derivatives It was already evident through Eqs(39) but it is even more obvious here that M µν ,N µν ,N µν I are antisymmetric on their indices and thus there are six of them for each category while M µν I is symmetric leading to ten different operators. These 28 operators are real ones : only the four P µ C are complex and have to be supplemented by their conjugates.

Transformations of the extended Poincaré group
Now that the algebraic content is clear, let us come to the group approach subtended by the transformations (28). We recognize in (28) eight real translations associated with α, or, in an equivalent way, four complex ones. The contributions of the matrix L can be understood as follows : 1. Twelve transformations of "boost" type i.e. "rotations" between a (real or imaginary) time component and three (real or imaginary) space components : with j = 1, 2, 3; a = R, I; b = R, I 2. Twelve rotations between (real or imaginary) space components : with j = 1, 2, 3, j ̸ = k; a = R, I; b = R, I 3. Four rotations between real and imaginary parts of one of the components : Let us rewrite these results within the complex coordinates : These equations are remarkable in the sense that they show that the complexification enables the connection between the Lorentz components. Indeed we know [13] that the Lorentz group has four components : L ↑ + , L ↑ − = P L ↑ + , L ↓ + = P T L ↑ + and L ↓ − = T L ↑ + . Each of the three last ones is related to the first one by acting on it with a discrete symmetry, either the parity operator P or the time-reversal operator T . Two objects in "mirror symmetry" are necessarily linked by a rotation in a higher dimensional space and that's exactly what happens here : the complexification allows these dimensions to exist in order to relate a coordinate and its opposite (which corresponds to θ µµ = π).
Similar discussions can be made with the complex algebra. We will just mention here that if the U (1, 3, C) operators perform transformations such as boosts or rotations on the four complex coordinates x µ , the other 12 ones (corresponding to exponentiations of N µν or N µν I ) realize, in addition, a complex conjugation. It is actually the reason why, despite of the fact that the quadratic forms (4) and (5) are equivalent, the Lie groups U (1, 3, C) and O (2,6) are not isomorphic.

The Casimir operators
Th first Casimir operator is It is indeed invariant under the transformation (25). Another way to be convinced of the form (47) is to rewrite it in details and verify that it commutes with each of the 36 operators of the extended Poincaré algebra by using Eqs (37)-(38) or (42k)-(42o) depending on whether we choose to work with eight real coordinates or four complex ones. By analogy with the real case, we can thus define the mass of a particle living in the complex Minkowski space-time by a relation which simplifies when the system is at rest and gives This is the complex version of the famous Einstein relation E = m.
The second Casimir operator requests more calculations. We indeed need to introduce the following 3-rank tensor It is antisymmetric on the two first indices as well as on the two last ones but is (evidently) symmetric on A and C. It thus gives rise to 56 operators. Let us take a while to mention that, in the real case, these operators reduce to the four well known ones namely the four components of the Pauli-Lubanski pseudo-vector. No need to say that, here, there is no interest to go through the dual to obtain a similar writing, as this will lead to a pseudo-tensor of rank 5. So let us focus on the operators (51). It is easy (even if laborious) to convince ourselves that they are such that The two first relations show that does commute with each of the 36 generators of the extended Poincaré algebra and, consequently, is the second Casimir operator. Its interpretation goes through the so-called "little groups" technics i.e. restricting ourselves to particular momenta P . If the rest is considered, only two P operators do not vanish : (55) We notice that the only J AB that does not appear in (55) is J 04 namely the rotation on time. We also recover the spin interpretation of the real case except that we have here 20 possibilities for one spin algebra (so(3)) to be put in evidence. This is evidently due to the increasing of spatial dimensions which are now 6.

Conclusions
We have extended the Poincaré group as well as its algebra to a complex Minkowski space. Beyond the real operators corresponding to transformations of the real coordinates, our approach shows that supplementary imaginary as well as complex operators appear to form with the previous ones a 36-dimensional real or complex algebra according to choosing 8 real coordinates or 4 complex ones.
Are there other Casimir operators ? What are the unirreps of the extended Poincaré algebra ? What physical implications could have this complex algebra ? What would be a complex version of usual quantum mechanics ?
These questions remain open at this stage and will be the subject of further considerations.