Tachyons and Representations
of Sp(2, R)
P. Moylan
Abstract Lacking in the mathematical physics literature is a detailed treatment of
tachyonic representations of the Poincaré group along lines similar to that for its real
mass, positive and negative energy representations. Such representations Wigner
did not consider in any detail in his 1939 paper on the unitary representations
of the inhomogeneous Lorentz group (Wigner, Ann Math 40:149–204, 1939),
and Bargmann and Wigner in their paper on the group theoretical classification
of relativistic wave equations did not consider them either because “they are
. . . unlikely to have a simple physical interpretation” (Bargmann and Wigner, Proc
Nat Acad Sci (USA) 34(5):211–223, 1948). We are making a detailed study of
tachyonic representations of the Poincaré group in four space-time dimensions
and we describe some of our results here. In particular, we relate tachyonic
representations of the Poincaré group to representations of the anti-de Sitter group,
in a way analogous to the way in which positive energy, real mass representations
of the Poincaré group are related to unitary principal series representations of the de
Sitter group via group contraction and deformation.
Keywords Tachyons · Representation theory · Poincaré group · de Sitter groups
1 Introduction
The connection between the unitary representation theory of the universal covering
groups of the de Sitter group, SO 0 (1, 4), and the Poincaré group is well understood
and much of it even in explicit detail. Unfortunately, the same cannot be said
for the connection between the unitary representations of the anti-de Sitter group,
SO 0 (2, 3), and the Poincaré group, even though SO 0 (2, 3) is more interesting from
the point of view of physical applications.
P. Moylan ()
Physics Department, Pennsylvania State University, The Abington College, Abington, PA, USA
e-mail: pjm11@psu.edu
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_12
131
of Sp(2, R)
P. Moylan
Abstract Lacking in the mathematical physics literature is a detailed treatment of
tachyonic representations of the Poincaré group along lines similar to that for its real
mass, positive and negative energy representations. Such representations Wigner
did not consider in any detail in his 1939 paper on the unitary representations
of the inhomogeneous Lorentz group (Wigner, Ann Math 40:149–204, 1939),
and Bargmann and Wigner in their paper on the group theoretical classification
of relativistic wave equations did not consider them either because “they are
. . . unlikely to have a simple physical interpretation” (Bargmann and Wigner, Proc
Nat Acad Sci (USA) 34(5):211–223, 1948). We are making a detailed study of
tachyonic representations of the Poincaré group in four space-time dimensions
and we describe some of our results here. In particular, we relate tachyonic
representations of the Poincaré group to representations of the anti-de Sitter group,
in a way analogous to the way in which positive energy, real mass representations
of the Poincaré group are related to unitary principal series representations of the de
Sitter group via group contraction and deformation.
Keywords Tachyons · Representation theory · Poincaré group · de Sitter groups
1 Introduction
The connection between the unitary representation theory of the universal covering
groups of the de Sitter group, SO 0 (1, 4), and the Poincaré group is well understood
and much of it even in explicit detail. Unfortunately, the same cannot be said
for the connection between the unitary representations of the anti-de Sitter group,
SO 0 (2, 3), and the Poincaré group, even though SO 0 (2, 3) is more interesting from
the point of view of physical applications.
P. Moylan ()
Physics Department, Pennsylvania State University, The Abington College, Abington, PA, USA
e-mail: pjm11@psu.edu
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_12
131