The conformal group, its casimir operators, and a four-position operator

Gravity and Nonlinear Symmetry Realization

“…This, in turn, makes it impossible to define states with definite mass and spin and ruins a possibility to justify the standard quantum field theory formalism. An explicit example of such a phenomenon is given by the conformal symmetry, as the conformal group has a different number of Casimir operators [32]. Therefor, in order to preserve the conventional quantum field theory one must preserve a linear realisation of the Poincare group in the gravitational sector.…”

Conjugate variables in quantum field theory and a refinement of Pauli’s theorem

“…Here it is crucial to rely on the presence of polarization vectors. The fact that Q (eff ) 0 = 0 can however be seen already when looking at the off-shell quantities [22]-type Q µ , s. (171), which originate from group theoretic considerations. Going on-shell there confirms the vanishing of Q(∇) 0 .…”

“…Clearly, this formula makes sense only if P 2 does not vanish. In [22] it has been argued by counting number of unknowns and number of equations that one can solve for Q. We note however and discuss in more detail below that K and Q are contracted with the transverse projection operator η µν − P µ P ν /P 2 , hence their relation might be determined only up to a longitudinal term, proportional to P/P 2 .…”

“…It further became clear that the r.h.s. of the commutator equation may be interpreted like in gauge theories: The "pure" η µν form corresponds to Lorentz gauge and is naturally realized off-shell: in the ad hoc version as Fourier transform (no realization of Q as function of other operators of the theory), in the [22]-version Q = Q(K) and in the [22]-type construction in subsection 3.2. On-shell, i.e.…”

“…From a different point of view this has also been studied in [19][20][21]. In detail we will discuss below [22]. Eventually one would have to consider the covering group SU(2, 2) which however is out of reach for the time being.…”