138
P. Moylan
As in the Sp(2, R) case any f ∈ H (σ,,,p 0 ) is uniquely specified by its values on
V ∼ = H/M.
From Definitions 1 and 2 it is clear that tachyonic representations of H T 4
are associated, in the sense of Sect. 3, with the representations π (σ,,,ν) of Sp(2, R)
and that these representations should go over into tachyonic representations of
H T 4 in the contraction limit, where by contraction limit we mean in the (global)
sense of Ref. [9]. In fact, the method given in Ref. [9] for the contraction limit
of the principal series unitary representations of a non-compact semisimple Lie
group G into its associated Cartan motion group K V for (G, K) a non-compact
Riemannian symmetric pair should carry over to the case considered here and this
should provide an example of their contraction process for a non-compact semiRiemannian symmetric pair not of rank one, namely, (Sp(2, R), H ).
For the action of the two Casimir operators D 2 and D 4 in the representation
π (σ,,,ν) we obtain the following: (ν = iρ, ρ real and I is the identity)
π
(σ,,,ν) (D 2 ) = −
ρ
2
+ σ (σ + 1) +
9
4
I , π
(σ,,,ν) (D 4 ) =
ρ
2
+
1
4
σ (σ + 1)I.
Substitution of these values into Eqs. (4) and solving for Y 2 (= P 2 ) and W gives the
following possibilities for the action of P 2 and W in this representation
˜
π
(σ,,,ν) (P
2 ) = ρ
2 , ˜
π
(σ,,,ν) (W) = ρ
2 σ (σ + 1)
or
˜
π
(σ,,,ν) (P
2 ) =
σ +
1
2
2
−
1
2
, ˜
π
(σ,,,ν) (W) =
ρ
2
+
1
4
σ +
1
2
2
−
1
2
.
Since the P μ are skew symmetric translation generators, this result implies, for
the first possibly, that representations of H T 4 obtained out of Theorem 4
are tachyonic. A study of the unitary representations of SL(2, R) shows that we
also obtain imaginary mass with the second possibility for the case of Sp(2, R)
representations induced from discrete series unitary representations of SL(2, R). We
leave it to the reader to show that the hypotheses of Theorems 3 and 4 hold true for at
least some of these representations, so that we get skew symmetric representations
of p (cf. Refs. [5, 6] where some special cases are worked out).
Finally, we conclude with a few remarks about possible physical relevance
of such tachyonic representations of the Poincaré group. Some of the tachyonic
representations described in the previous paragraph occur as contractions of his
case 4 representations in Ehrman’s classification of the unitary representations of
the universal covering group of SO 0 (2, 3) [10, 11]. They include contractions of
principal series representations of SO 0 (2, 3) to representations of the Poincaré
group [10]. In addition to calling attention to recent radical proposals for their
possible use in dark matter [12], we think it should also be possible to use
P. Moylan
As in the Sp(2, R) case any f ∈ H (σ,,,p 0 ) is uniquely specified by its values on
V ∼ = H/M.
From Definitions 1 and 2 it is clear that tachyonic representations of H T 4
are associated, in the sense of Sect. 3, with the representations π (σ,,,ν) of Sp(2, R)
and that these representations should go over into tachyonic representations of
H T 4 in the contraction limit, where by contraction limit we mean in the (global)
sense of Ref. [9]. In fact, the method given in Ref. [9] for the contraction limit
of the principal series unitary representations of a non-compact semisimple Lie
group G into its associated Cartan motion group K V for (G, K) a non-compact
Riemannian symmetric pair should carry over to the case considered here and this
should provide an example of their contraction process for a non-compact semiRiemannian symmetric pair not of rank one, namely, (Sp(2, R), H ).
For the action of the two Casimir operators D 2 and D 4 in the representation
π (σ,,,ν) we obtain the following: (ν = iρ, ρ real and I is the identity)
π
(σ,,,ν) (D 2 ) = −
ρ
2
+ σ (σ + 1) +
9
4
I , π
(σ,,,ν) (D 4 ) =
ρ
2
+
1
4
σ (σ + 1)I.
Substitution of these values into Eqs. (4) and solving for Y 2 (= P 2 ) and W gives the
following possibilities for the action of P 2 and W in this representation
˜
π
(σ,,,ν) (P
2 ) = ρ
2 , ˜
π
(σ,,,ν) (W) = ρ
2 σ (σ + 1)
or
˜
π
(σ,,,ν) (P
2 ) =
σ +
1
2
2
−
1
2
, ˜
π
(σ,,,ν) (W) =
ρ
2
+
1
4
σ +
1
2
2
−
1
2
.
Since the P μ are skew symmetric translation generators, this result implies, for
the first possibly, that representations of H T 4 obtained out of Theorem 4
are tachyonic. A study of the unitary representations of SL(2, R) shows that we
also obtain imaginary mass with the second possibility for the case of Sp(2, R)
representations induced from discrete series unitary representations of SL(2, R). We
leave it to the reader to show that the hypotheses of Theorems 3 and 4 hold true for at
least some of these representations, so that we get skew symmetric representations
of p (cf. Refs. [5, 6] where some special cases are worked out).
Finally, we conclude with a few remarks about possible physical relevance
of such tachyonic representations of the Poincaré group. Some of the tachyonic
representations described in the previous paragraph occur as contractions of his
case 4 representations in Ehrman’s classification of the unitary representations of
the universal covering group of SO 0 (2, 3) [10, 11]. They include contractions of
principal series representations of SO 0 (2, 3) to representations of the Poincaré
group [10]. In addition to calling attention to recent radical proposals for their
possible use in dark matter [12], we think it should also be possible to use