Positive central-charge bounds on a massive rest representation fiber

Assume Countable Choice. Let $N\ge 2$. A fixed massive rest representation fiber consists here of a nonzero complex Hilbert space $\mathcal{K}$, a number $M>0$ with $P^0=MI$ and $P^j=0$, and scalar central charges $Z^{IJ}=z^{IJ}I$, where $Z=(z^{IJ})$ is the complex antisymmetric $N\times N$ matrix in the four-dimensional algebra with $\eta =\mathrm{diag}(-1,1,1,1)$ and ${ϵ}_{12}=1$. Suppose the charge operators and their actual Hilbert adjoints have a common dense invariant domain $\mathcal{D}$ on the fiber and every product in the expanded anticommutators is defined there. Use natural units $\mathrm{\hslash }=c=1$, so $M$ and each $z^{IJ}$ have mass dimension one. For $v=(v_1,\dots ,v_N)\in {\mathbb{C}}^N$ define $\mathrm{\parallel }v{\mathrm{\parallel }}_2^2={\sum }_I\mathrm{\mid }{v}_I{\mathrm{\mid }}^2$ and for $u,v\in {\mathbb{C}}^N$ put $u^{\ast }Zv={\sum }_{I,J}{\bar{u}}_I{z}^{IJ}{v}_J$. Define

$\mathrm{\parallel }Z{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}={\sup }_{\mathrm{\parallel }v{\mathrm{\parallel }}_2=1}\mathrm{\parallel }Zv{\mathrm{\parallel }}_2\mathrm{.}$

Then $M\ge \mathrm{\parallel }Z{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. For $N=2$, write $z=z^{12}$, so $Z=\left(\begin{array}{cc}0& z\\ -z& 0\end{array}\right)$ and $\mathrm{\parallel }Z{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=\mathrm{\mid }z\mathrm{\mid }$. If $z\mathrm{\ne }0$, set $\omega =z\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$; if $z=0$, set $\omega =1$. Then $\mathrm{\mid }\omega \mathrm{\mid }=1$ and $\bar{\omega }z=\mathrm{\mid }z\mathrm{\mid }$, as verified in the proof below. Define on $\mathcal{D}$

$A:=Q_1^1-\omega (Q_2^2{)}^{\ast },A^{\mathrm{♯}}:=(Q_1^1{)}^{\ast }-\bar{\omega }{Q}_2^2\mathrm{.}$

The algebraic anticommutator $A{A}^{\mathrm{♯}}+A^{\mathrm{♯}}A$ equals $4(M-\mathrm{\mid }z\mathrm{\mid })I$ on $\mathcal{D}$. Here $A^{\mathrm{♯}}$ is the displayed combination of the individual actual adjoints; it is not asserted to be the adjoint of an unbounded sum on its maximal domain. If $\mathcal{K}$ is finite-dimensional, all charge domains equal $\mathcal{K}$ and $A^{\mathrm{♯}}=A^{\ast }$, so the formula is the ordinary operator anticommutator. At saturation $M=\mathrm{\mid }z\mathrm{\mid }$, for every $\psi \in \mathcal{D}$ one has $A\psi =A^{\mathrm{♯}}\psi =0$. This is a representation-theoretic fiber statement: it does not assert that the fixed-momentum fiber is a normalizable subspace of the full Hilbert space, or that a state with specified charge exists or is stable.

Gates et al., §3.2.

Martin, §1.

Four-dimensional real extended super-Poincare algebra: Four-dimensional real extended super-Poincare algebra.

Positive unitary super-Poincare representation with spectral-domain covariance: Positive unitary super-Poincare representation with spectral-domain covariance.

Real and imaginary parts, complex conjugation, and modulus: Real and imaginary parts, complex conjugation, and modulus.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

For general $N$, the norm inequality does not by itself claim a multiplicity formula or a shortened representation. Those require additional representation-theoretic hypotheses and proofs.