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Finite $N = 2$ rest-fiber algebra at central-charge saturation

Physical scope

A finite-dimensional N=2 rest-frame supercharge-algebra module in natural units; it demonstrates algebraic saturation only and is not a full Poincare representation, a constructed QFT, or evidence for a physical particle.

Example

Let $M > 0$ and let $K$ have orthonormal basis $∣ 0 ⟩, ∣ 1 ⟩, ∣ 2 ⟩, ∣ 12 ⟩$ . Give $K$ the grading for which $∣ 0 ⟩, ∣ 12 ⟩$ are even and $∣ 1 ⟩, ∣ 2 ⟩$ are odd. Define operators $c_{1}, c_{2}$ by

$c_{1} ∣ 0 ⟩ = 0, c_{1} ∣ 1 ⟩ = ∣ 0 ⟩, c_{1} ∣ 2 ⟩ = 0, c_{1} ∣ 12 ⟩ = ∣ 2 ⟩,$

$c_{2} ∣ 0 ⟩ = 0, c_{2} ∣ 1 ⟩ = 0, c_{2} ∣ 2 ⟩ = ∣ 0 ⟩, c_{2} ∣ 12 ⟩ = - ∣ 1 ⟩,$

and let $c_{1}^{†}, c_{2}^{†}$ be their actual Hilbert adjoints. Set

$Q_{1}^{1} = \sqrt{2 M} c_{1}, Q_{2}^{2} = \sqrt{2 M} c_{1}^{†}, Q_{2}^{1} = \sqrt{2 M} c_{2}, Q_{1}^{2} = - \sqrt{2 M} c_{2}^{†} .$

Take $P^{μ} = (M, 0, 0, 0)$ , $Z^{12} = M I$ , $Z^{21} = - M I$ , $Z^{11} = Z^{22} = 0$ , and ${\overset{ˉ}{Q}}_{\dot{α} I} = (Q_{α}^{I})^{†}$ . On $D = K$ these matrices satisfy the massive $N = 2$ rest-frame supercharge relations with $η = diag (- 1, 1, 1, 1)$ and $ϵ_{12} = 1$ . The central-charge bound is saturated and the corresponding combination $A = Q_{1}^{1} - (Q_{2}^{2})^{†}$ vanishes identically. This is a finite-dimensional module for the rest-frame supercharge algebra; no extension to the full Poincare group or physical particle is asserted.

Example references

BPS state and central-charge saturation

Four-dimensional real extended super-Poincare algebra

Facts & Assumptions

Given: The four-dimensional graded Hilbert space, its stated operators, $M > 0$ , $P^{μ} = (M, 0, 0, 0)$ , and the displayed central charges.

[A1]

Assume ${A C}_{ω}$ , inherited from the selected representation and BPS-bound interface (The Axiom of Countable Choice ( ${A C}_{ω}$ )): every sequence of nonempty sets has a choice function.

[F1]

BPS state and central-charge saturation defines BPS states inside a rest fiber of a selected full unitary super-Poincare representation. This example uses only the scalar equality $M = ∣ z ∣$ and the displayed null combinations as algebraic comparisons; it does not apply that physical-state definition to its module.

[F2]

Four-dimensional real extended super-Poincare algebra: $η = diag (- 1, 1, 1, 1)$ , $ϵ_{12} = 1$ , $\overset{ˉ}{Q} = (Q)^{†}$ , ${Q_{α}^{I}, {\overset{ˉ}{Q}}_{\dot{β} J}} = - 2 σ_{α \dot{β}}^{μ} P_{μ} δ^{I}_{J}$ , and ${Q_{α}^{I}, Q_{β}^{J}} = 2 ϵ_{α β} Z^{I J}$ .

Verification

technique · direct

Given: [A1], [F1], [F2], and the matrices defined in the Example.

1.1givenalgebra

In the ordered basis $∣ 0 ⟩, ∣ 1 ⟩, ∣ 2 ⟩, ∣ 12 ⟩$ , the stated operators are $c_{1} = (\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array})$ and $c_{2} = (\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array})$ . Their conjugate-transpose matrices are the actual adjoints. Multiplication gives ${c_{i}, c_{j}^{†}} = δ_{i j} I_{4}$ and ${c_{i}, c_{j}} = 0$ for $i, j \in {1, 2}$ ; taking adjoints also gives ${c_{i}^{†}, c_{j}^{†}} = 0$ .

2.1F2step 1.1algebra

Scaling these identities by $2 M$ shows ${Q_{α}^{I}, (Q_{β}^{J})^{†}} = 2 M δ^{I}_{J} δ_{α β} I_{4}$ . The only nonzero same-chirality brackets with $I = 1, J = 2$ are ${Q_{1}^{1}, Q_{2}^{2}} = 2 M I_{4}$ and ${Q_{2}^{1}, Q_{1}^{2}} = - 2 M I_{4}$ . They equal $2 ϵ_{12} Z^{12}$ and $2 ϵ_{21} Z^{12}$ , respectively; the reversed flavor pairs give the same values by symmetry of the anticommutator and antisymmetry of both $ϵ$ and $Z$ . Thus the undotted central brackets match [F2], and their adjoints give the dotted relations; all other same-chirality brackets vanish.

3.1F1step 2.1algebra

Here $Z = (\begin{array}{cc} 0 & M \\ - M & 0 \end{array})$ has $Z^{†} Z = M^{2} I_{2}$ , hence $∥ Z ∥_{o p} = M = ∣ Z^{12} ∣$ . Also $A = \sqrt{2 M} c_{1} - \sqrt{2 M} c_{1} = 0$ , and its adjoint combination is zero. Thus the module has $M = ∣ z ∣$ and a vanishing supercharge combination as matrix identities. No BPS-state membership under [F1] follows without its selected full-representation input.

4.1A1givenstep 3.1∎

The construction verifies only the finite rest-frame supercharge algebra. It uses no choice from a family, while [A1] is retained as the inherited interface assumption of the BPS definition. Since the data supply no Lorentz action or full unitary Poincare representation, they do not establish a physical particle or QFT state.

Finite $N = 2$ rest-fiber algebra at central-charge saturation

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

Finite N=2N=2 rest-fiber algebra at central-charge saturation

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

Finite $N = 2$ rest-fiber algebra at central-charge saturation