Anti-de Sitterian “massive” elementary systems and their Minkowskian and Newton-Hooke contraction limits Available to Purchase

Here, one may consider a comprehensive program of quantization of functions (or distributions) by using all resources of covariant integral quantization as it is defined, for instance, in Refs. 33, 78, and ^81–83.

Note that, trivially, this exception holds for any 1 + n-dimensional dS and AdS spacetimes (n = 1, 2, …), but, since in the current paper we are interested in (1 + 3) dimension, we merely point out dS₄ and AdS₄ spacetimes throughout the paper.

In the context of the latter category, specifically in the AdS₄ scenario pertinent to our study, there exist particular UIRs of the universal covering group of SO₀(2, 3), which contract to massive yet tachyonic (m² < 0) representations of the Poincaré group.^18,30 These representations are classified as type III_|m| according to the scheme outlined in Ref. 31. Owing to their tachyonic nature and consequently the acausal propagation of their corresponding fields, we categorize them within this latter category, representing AdS₄ UIRs devoid of a physical Poincaré contraction limit. Moreover, the present work solely focuses on the physical Wigner massive representations of the Poincaré group (with positive mass and positive energy). These representations are demonstrated to be contraction limits of the (extended) holomorphic discrete series of the AdS group and its universal covering.

Note that, in the sequel, the Minkowski indices μ, ν, … will be reserved to the subset {0, 1, 2, 3}, the spatial indices i, j, … to the subset {1, 2, 3}, and the indices a, b, … to the subset {1, 2, 3, 5}.

Technically, the connected component of O(2, 3), SO₀(2, 3), containing the identity consists of all linear transformations in $R2,3$ that leave invariant the form of the metric η_αβ, have determinant unity, and also preserve the orientation of the “time” variable t, being defined through the relation $y5±iy0≡(y5)2+(y0)2exp±it$ (−π ⩽ t < π). Remember that O(2, 3) and SO(2, 3) have four and two connected components, respectively.

For the real quaternions and the related discussions, readers can go to Ref. 14.

For a real quaternion $x=(x4,x)∈H$⁠, the norm ‖x‖ is given by: $‖x‖2=detx=xx̃=(x1)2+(x2)2+(x3)2+(x4)2∈R+$⁠. It is zero if and only if all the components are zero.

A full justification of this action will be provided in the next subsection.

Note that the selection of the origin is only a matter of choice because all the points on the AdS₄ manifold are equivalent [recall that AdS₄ is actually a homogeneous space of Sp(4, R), as we have seen above]. Of course, if one was to deal with, for instance, the unit-sphere $S4$ as representing a four-dimensional manifold in $R5$⁠, one has no hesitation to acknowledge such a property. Dealing with the representation of the AdS₄ hyperboloid embedded in $R2,3$ might be however misleading in this respect due to its deformed shape.

Note that the form (3.6) is proportional to the Killing form for $sp(4,R)$⁠,¹⁴ that is, K(X, X′) ≡ tr(ad_Xad_X′), where ad_X stands for the adjoint action of $sp(4,R)$ on itself: $sp(4,R)∋X′↦adX(X′)≡X,X′.$ This action is nothing but the derivative of the respective adjoint action of Sp$(4,R)$ on $sp(4,R)$ [as given by (3.7)]. See more details in Ref. 14.

Note that the space-rotations generators Y_is commute with the time-translations one X₀ [see (2.56)].

One notices here that the generic element $X(k0,ς,α,β)$ is kind of reminiscent of the form assumed by the elements of Sp$(4,R)$⁠, and subsequently, the identities given in (3.12) of the identities already given in (2.26). Having this point in mind, one can show that the determinant of the generic element $X(k0,ς,α,β)$ is fixed and equal to $(ℓ2)2$⁠. This is indeed the result already expected from the fact that the (co-)adjoint action (3.7) is determinant preserving since $det(X)=det(2ℓX0)=(ℓ2)2$⁠. Another interesting point to be noticed here is that the first identity in (3.12) is consistent with the constraint issued from the Killing form of the algebra for the generic element $X$ of the (co-)adjoint orbit class, i.e., $k(X,X)∝tr(XX)=2kk*−(ik0+ς)(ik0+ς)*s=−2ℓ2$⁠, where we have used the notation (z)_s for the scalar part of the complex quaternion z and the fact that $(ik0+ς)*=−(ik0+ς)$ and that $k*=−k̄$⁠.

The co-adjoint orbit class describing massive elementary systems for the Poincaré kinematical symmetry is described in Ref. 17.

In this context, we encourage readers to see  Appendix B.

For Newton-Hooke groups and its relevant discussions, see  Appendix C.

We will show in the sequel that when ℓ = ȷ, it corresponds to the situations known as the “massless” “spin” cases.

Again, the first identity is consistent with the constraint issued from the Killing form of the algebra for the generic element of the (co-)adjoint orbit class $O2ℓX0+2ȷY3$ and the second one with the fact that the (co-)adjoint action (3.7) is determinant preserving.

The representations $P(m,s)$ are described in  Appendix B.

The UIRs of the Newton-Hooke group are described in  Appendix C.

Here, for later use, it is worthwhile noting that, for negative values of $λ∈Z$⁠, the left-hand side of Eq. (F25) turns to a finite sum as: $(1+u2−2ut)−λ=∑l=02|λ|ulClλ(t)$⁠.