as creation and annihilation operators respectively. The creation operator transforms according to and similarly for the annihilation operator. The point to be made is that the field operator transforms according to a finite-dimensional non-unitary representation of the Lorentz group, while the creation operator transforms under the infinite-dimensional unitary representation of the Poincare group characterized by the mass and spin of the particle. The connection between the two are the wave functions, also called coefficient functions that carry both the indices operated on by Lorentz transformations and the indices operated on by Poincaré transformations. This may be called the Lorentz–Poincaré connection. To exhibit the connection, subject both sides of equation to a Lorentz transformation resulting in for e.g. , where is the non-unitary Lorentz group representative of and is a unitary representative of the so-called Wigner rotation associated to and that derives from the representation of the Poincaré group, and is the spin of the particle. All of the above formulas, including the definition of the field operator in terms of creation and annihilation operators, as well as the differential equations satisfied by the field operator for a particle with specified mass, spin and the representation under which it is supposed to transform, and also that of the wave function, can be derived from group theoretical considerations alone once the frameworks of quantum mechanics and special relativity is given. Speculative theories In theories in which spacetime can have more than dimensions, the generalized Lorentz groups of the appropriate dimension take the place of . The requirement of Lorentz invariance takes on perhaps its most dramatic effect in string theory. Classical relativistic strings can be handled in the Lagrangian framework by using the Nambu–Goto action. This results in a relativistically invariant theory in any spacetime dimension. But as it turns out, the theory of open and closed bosonic strings (the simplest string theory) is impossible to quantize in such a way that the Lorentz group is represented on the space of states (a Hilbert space) unless the dimension of spacetime is 26. The corresponding result for superstring theory is again deduced demanding Lorentz invariance, but now with supersymmetry. In these theories the Poincaré algebra is replaced by a supersymmetry algebra which is a -graded Lie algebra extending the Poincaré algebra. The structure of such an algebra is to a large degree fixed by the demands of Lorentz invariance. In particular, the fermionic