Global supersymmetry vacuum state and invariance

Let a positive unitary representation of the four-dimensional extended super-Poincare algebra act on a complex Hilbert space $\mathcal{H}$, with common invariant product domain $\mathcal{D}$, translation generators $P^{\mu }$, supercharges $Q_{\alpha }^I$, and actual Hilbert adjoints $(Q_{\alpha }^I{)}^{\ast }$, using metric $\eta =\mathrm{diag}(-1,1,1,1)$, ${\sigma }^0=I_2$, and natural units $\mathrm{\hslash }=c=1$. A rest-frame ground-state vector is a vector $\mathrm{\Omega }\in \mathcal{D}$ such that $\mathrm{\parallel }\mathrm{\Omega }\mathrm{\parallel }=1$, there is a real number $E_{\mathrm{\Omega }}$ with $P^0\mathrm{\Omega }=E_{\mathrm{\Omega }}\mathrm{\Omega }$ and $P^j\mathrm{\Omega }=0$ for $j=1,2,3$, and

$⟨P^0\psi ,\psi ⟩\ge E_{\mathrm{\Omega }}\mathrm{\parallel }\psi {\mathrm{\parallel }}^2(\psi \in D(P^0))\mathrm{.}$

Such a vector is called a global supersymmetry vacuum state in the representation. Its vacuum energy is the eigenvalue $E_{\mathrm{\Omega }}$, with units of mass in natural units; it is not an energy density. The vector is boundary-neutral when $Z^{IJ}\mathrm{\Omega }=0$ and ${\bar{Z}}_{IJ}\mathrm{\Omega }=0$ for every central generator, supersymmetry-invariant when $Q_{\alpha }^I\mathrm{\Omega }=(Q_{\alpha }^I{)}^{\ast }\mathrm{\Omega }=0$ for every $I,\alpha$, and supersymmetry-noninvariant when at least one of these vectors is nonzero. These are separate predicates: boundary neutrality does not define supersymmetry invariance. The definition does not assert that any such vector exists or that a constant field configuration is allowed by a separate field-model boundary condition.

Stephen P. Martin, A Supersymmetry Primer, §7.1, equations (7.1.1)–(7.1.2).

Adel Bilal, Introduction to Supersymmetry, §6.1.

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

Positive unitary representation, joint spectrum, common product domain, and actual charge adjoints: Positive unitary super-Poincare representation with spectral-domain covariance.

The Axiom of Countable Choice, which is part of the representation's mathematical base: The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).