Tachyons and Representations of Sp(2, R)
137
Let H be the lift-up of SO 0 (1, 3) specified by the exact sequence in Eq. (2) and
concretely determined by the Lie algebra homomorphism ϕ given in Sect. 1. Then,
according to Sekiguchi, we have Sp(2, R) ∼ = H AN as a C ∞ isomorphism of H AN
onto a dense open subset of G = Sp(2, R). Furthermore, the Cartan decomposition
H ∼ = V M where V ∼ = H/M gives us the isomorphism Sp(2, R) ∼ = V MAN again
onto a dense open subset of Sp(2, R). From this it follows that any f in H (σ,,,ν) is
essentially uniquely specified by its values on V ∼ = H/M. Thus the representation
space H (σ,,,ν) can equivalently be viewed essentially as the space of C ∞ functions
on V with values in V (σ,,) . I do not state the required asymptotic conditions on
f ∈ H (σ,,,ν) . They are determined in a way analogous to that described in Ref. [7]
for SU(2, 2) and in Ref. [8] for S0 0 (2, 4) ∼ = SU(2, 2)/Z 2 .)
Now we come to a description of tachyonic representations of the Poincaré
group. Consider the double cover H T 4 of the Poincaré group, SO 0 (1, 3) T 4 .
The translation subgroup T 4 = {e a μ P μ |a μ ∈ R 4 }. T 4 is an additive vector group
and so every unitary irreducible representation (UIR) of T 4 is one-dimensional and
of the form [15, 16]
χ p : T
4
→ C, a → χ p (a) = exp{(ip · a)},
where p, a ∈ R 4 and p · a is the SO 0 (1, 3) invariant scalar product of the two
vectors p and a. It follows that we can characterize the equivalence classes of the
UIR’s of T 4 by elements p of the vector space dual ˆ
T 4 to T 4 . The coadjoint action
of SO 0 (1, 3) on ˆ
T 4 is given by p → Λ −1 p. Let O p 0 be the orbit in ˆ
T 4 of a point
p 0 ∈ ˆ
T 4 under the action of H and let M p 0 be the isotropy subgroup (stabilizer
subgroup) of the point p 0 . Clearly M p 0 is a closed subgroup of H and so O p 0
∼ =
H/M p 0 . Let γ : O p 0 → H be a smooth cross-section such that for any point
p ∈ O p 0 we have γ (p)p 0 = p. For tachyonic representations of H T 4 it suffices
to consider p 0 of the form p 0 = (0, 0, 0, μ) where μ ∈ R. For such p 0 we have
M p 0
∼ = M and O p 0 = H/M ∼ = {p ∈ T 4 | p · p = −|μ| 2 < 0}. We use the
same representation π (σ,,) of M on the space V (σ,,) as for Sp(2, R), and we extend
it to a representation of the semidirect product B = M T 4 by requiring π (σ,,) ⊗
χ p 0 (m, a) = π (σ,,) (m) χ p 0 (a) where (m, a) ∈ B with m ∈ M and a ∈ T 4 . The
representation of H T 4 induced from π (σ,,) ⊗ χ p 0 is defined as follows:
Definition 2 Let p 0 = (0, 0, 0, μ) where μ ∈ R and let
H
(σ,,,p 0 )
:=I nd
H T 4
B
(π
(σ,,)
⊗ χ p 0 )=
f : H T
4
→ V
(σ,,)
| f ∈ C
∞ (H T
4 )
f (gmp) = π
(σ,,) (m)χ p 0 (a)f (g) f or g ∈ H T
4 , m ∈ M, a ∈ T
4
.
For f ∈ I nd
H 4
B
(π (σ,,) ⊗ χ p 0 ) and g ∈ H T 4 we define the representation
π (σ,,,p 0 ) of H T 4 on H (σ,,,p 0 ) by π (σ,,,ν) (g)f (g ) = f (gg ) with g ∈ H T 4 .
137
Let H be the lift-up of SO 0 (1, 3) specified by the exact sequence in Eq. (2) and
concretely determined by the Lie algebra homomorphism ϕ given in Sect. 1. Then,
according to Sekiguchi, we have Sp(2, R) ∼ = H AN as a C ∞ isomorphism of H AN
onto a dense open subset of G = Sp(2, R). Furthermore, the Cartan decomposition
H ∼ = V M where V ∼ = H/M gives us the isomorphism Sp(2, R) ∼ = V MAN again
onto a dense open subset of Sp(2, R). From this it follows that any f in H (σ,,,ν) is
essentially uniquely specified by its values on V ∼ = H/M. Thus the representation
space H (σ,,,ν) can equivalently be viewed essentially as the space of C ∞ functions
on V with values in V (σ,,) . I do not state the required asymptotic conditions on
f ∈ H (σ,,,ν) . They are determined in a way analogous to that described in Ref. [7]
for SU(2, 2) and in Ref. [8] for S0 0 (2, 4) ∼ = SU(2, 2)/Z 2 .)
Now we come to a description of tachyonic representations of the Poincaré
group. Consider the double cover H T 4 of the Poincaré group, SO 0 (1, 3) T 4 .
The translation subgroup T 4 = {e a μ P μ |a μ ∈ R 4 }. T 4 is an additive vector group
and so every unitary irreducible representation (UIR) of T 4 is one-dimensional and
of the form [15, 16]
χ p : T
4
→ C, a → χ p (a) = exp{(ip · a)},
where p, a ∈ R 4 and p · a is the SO 0 (1, 3) invariant scalar product of the two
vectors p and a. It follows that we can characterize the equivalence classes of the
UIR’s of T 4 by elements p of the vector space dual ˆ
T 4 to T 4 . The coadjoint action
of SO 0 (1, 3) on ˆ
T 4 is given by p → Λ −1 p. Let O p 0 be the orbit in ˆ
T 4 of a point
p 0 ∈ ˆ
T 4 under the action of H and let M p 0 be the isotropy subgroup (stabilizer
subgroup) of the point p 0 . Clearly M p 0 is a closed subgroup of H and so O p 0
∼ =
H/M p 0 . Let γ : O p 0 → H be a smooth cross-section such that for any point
p ∈ O p 0 we have γ (p)p 0 = p. For tachyonic representations of H T 4 it suffices
to consider p 0 of the form p 0 = (0, 0, 0, μ) where μ ∈ R. For such p 0 we have
M p 0
∼ = M and O p 0 = H/M ∼ = {p ∈ T 4 | p · p = −|μ| 2 < 0}. We use the
same representation π (σ,,) of M on the space V (σ,,) as for Sp(2, R), and we extend
it to a representation of the semidirect product B = M T 4 by requiring π (σ,,) ⊗
χ p 0 (m, a) = π (σ,,) (m) χ p 0 (a) where (m, a) ∈ B with m ∈ M and a ∈ T 4 . The
representation of H T 4 induced from π (σ,,) ⊗ χ p 0 is defined as follows:
Definition 2 Let p 0 = (0, 0, 0, μ) where μ ∈ R and let
H
(σ,,,p 0 )
:=I nd
H T 4
B
(π
(σ,,)
⊗ χ p 0 )=
f : H T
4
→ V
(σ,,)
| f ∈ C
∞ (H T
4 )
f (gmp) = π
(σ,,) (m)χ p 0 (a)f (g) f or g ∈ H T
4 , m ∈ M, a ∈ T
4
.
For f ∈ I nd
H 4
B
(π (σ,,) ⊗ χ p 0 ) and g ∈ H T 4 we define the representation
π (σ,,,p 0 ) of H T 4 on H (σ,,,p 0 ) by π (σ,,,ν) (g)f (g ) = f (gg ) with g ∈ H T 4 .