Quantum Information Approach to Bosonization of Supersymmetric Yang-Mills Fields

Abstract: We consider bosonization of supersymmetry in the context of Wess-Zumino quantum mechanics. Our motivation for this investigation is the flexibility the bosonic fock space affords as any classical probability distribution can be realized on it making it a versatile framework to work with for quantum processes. We proceed by constructing a minimal bosonization of a system with one bosonic and two fermionic degrees of freedom. We iterate this process to construct a tower of SUSY systems that is akin to unfolded Adinkras. We then identify an osp(2|2) symmetry of the system constructed. To build an irreducible representation of the system we induce representations across the sectors, a first to our knowledge, as the previous work have focused on induction only within the bosonic sector. First, we start with a fermionic representation using Clifford algebras and then induce a representation to gl(2|2) and restrict it to osp(2|2). In the second method, we induce a representation from that of the bosonic sector. In both cases, our representations are in terms of qubit operators that provide a way to solve SUSY problems using quantum information based approaches. Depending upon the direction of induction the representations are suitable for implementation on a hybrid qubit and fermionic or bosonic quantum computers.

• The paper develops an iterative bosonization program in which Klein-projector compositions and the ν-deformed Heisenberg algebra generate an infinite tower of supersymmetric systems with progressively stronger spontaneous SUSY breaking.
• The authors construct irreducible OSp(2|2) representations through Mackey systems of imprimitivity, inducing representations in both fermionic-to-bosonic and bosonic-to-fermionic directions and identifying highest-weight modules with λ = 1.
• The paper expresses the resulting operators using Pauli, Clifford, and matrix-unit qubit operations, suggesting implementations on hybrid qubit–fermionic and qubit–bosonic platforms such as circuit QED.

Motivation and historical context

The paper by Balu and Gates develops a bosonization program for supersymmetric (SUSY) quantum systems, framed within the long lineage of Klein transformations running from Jordan–Wigner, Tomonaga, Luther–Peschel, and Mattis–Lieb to the earlier work of one author on low-dimensional SUSY field theories. The stated motivation is practical rather than purely structural: a bosonic Fock space can realize any classical probability distribution, making it a flexible carrier for quantum processes. The authors' contribution is threefold: (i) an iterative construction of an infinite tower of SUSY systems in which the $\nu$-deformed Heisenberg algebra controls the degree of spontaneous SUSY breaking; (ii) the construction of irreducible representations (IRRs) of the super Lie group $OSp(2|2)$ via Mackey's systems of imprimitivity, induced in both directions — fermionic-to-bosonic and bosonic-to-fermionic — which the authors claim is new, since prior SUSY applications of induction operated only within the bosonic sector; and (iii) the expression of all constructions in terms of qubit operators, connecting the representation theory to quantum computing architectures.

Bosonization of Wess-Zumino quantum mechanics

The starting point is Wess-Zumino quantum mechanics (WZQM) with one complex bosonic coordinate $\phi(t)$ and two real fermionic degrees of freedom. The Hamiltonian $$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$ with superpotential $W(x) = \phi + g_0\phi^2$ admits supercharges built from the fermions $\psi = a^- K$, $\psi^\dagger = a^+ K$, where $K$ is the Klein operator satisfying $K^2 = 1$ and anticommuting with the oscillator. In the coordinate representation $K$ acts as parity, splitting the Fock space into even and odd sectors.

The key mechanism for generating broken SUSY follows Plyushchay: composing an odd operator with the projectors $OSp(2|2)$0 yields nilpotent supercharges $OSp(2|2)$1. The resulting Hamiltonian acquires an additional commutator term $OSp(2|2)$2. For the modified WZQM system, the authors show explicitly that the extra fermionic term evaluates to $OSp(2|2)$3 at both $OSp(2|2)$4 and $OSp(2|2)$5, producing degeneracy and hence spontaneous SUSY breaking — in contrast to the original WZQM Hamiltonian, whose unique zero-energy ground state realizes exact SUSY. Both choices $OSp(2|2)$6 and $OSp(2|2)$7 break SUSY.

The deformation parameter $OSp(2|2)$8 enters through the deformed Heisenberg relation $OSp(2|2)$9, giving number eigenvalues $\phi(t)$0. Because the construction appends one commutator term per iteration and each iteration can be performed on a deformed oscillator, the authors obtain an infinite tower of SUSY systems with progressively increasing degrees of breaking — a structure they identify with unfolded Adinkras. This generalizes immediately to multiple bosonic or fermionic variables via Klein projectors.

The $\phi(t)$1 superalgebra

Following Plyushchay again, the even generators are built from the deformed oscillator ($\phi(t)$2, $\phi(t)$3, and $\phi(t)$4), while the odd generators $\phi(t)$5 come from the supercharges with $\phi(t)$6. These satisfy the $\phi(t)$7 commutation relations and cover both exact and broken SUSY depending on the sign of $\phi(t)$8. The authors' stated purpose here is representational rather than dynamical: given the spectrum of the smaller (broken-SUSY) sector, branching rules and intertwiners allow computation of all energy levels of the larger system. This motivates the use of Mackey induction as the tool for building IRRs of the larger symmetry from representations of subgroups.

Systems of imprimitivity and induced representations

The paper reviews Mackey machinery in its standard form: for a locally compact group $\phi(t)$9 acting transitively on a Borel space $$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$0, a system of imprimitivity $$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$1 consists of a projection-valued measure and a unitary representation satisfying covariance $$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$2; Mackey's theorem identifies transitive SIs with representations induced from closed subgroups, realized on sections of a homogeneous vector bundle with fiber $$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$3. The authors specialize this to super Hilbert spaces ($$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$4-graded spaces with graded adjoints) and super Harish-Chandra pairs, drawing on Varadarajan's framework. The induction chain proceeds as

$$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$5

exploiting the embedding $$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$6 and the fact that restriction of the fundamental IRR of $$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$7 to $$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$8 remains irreducible because both act on the same carrier space $$H = \tfrac{1}{2}(p_x^2 + p_y^2) + |W(\phi)|^2 + (\text{fermionic coupling})$$9.

Two theorems constitute the main results:

Fermionic-to-bosonic induction. Starting from the Clifford subgroup $W(x) = \phi + g_0\phi^2$0 (with spinor representation $W(x) = \phi + g_0\phi^2$1 acting on the antisymmetric tensor product), the authors build a covariant $W(x) = \phi + g_0\phi^2$2-graded fiber bundle over orbits of fermionic stabilizer states, then iterate once more using the Pauli subgroup generated by $W(x) = \phi + g_0\phi^2$3 over bosonic stabilizer-state orbits. The resulting unitaries furnish a transitive SI for $W(x) = \phi + g_0\phi^2$4 living on $W(x) = \phi + g_0\phi^2$5, hence an IRR of $W(x) = \phi + g_0\phi^2$6 after restriction.

Bosonic-to-fermionic induction (bosonized form). Reversing direction, they start from the Pauli subgroup $W(x) = \phi + g_0\phi^2$7 on the bosonic sector of a finite-dimensional Fock space $W(x) = \phi + g_0\phi^2$8, grade it with Klein operators $W(x) = \phi + g_0\phi^2$9, and induce in a single step $\psi = a^- K$0. The fibers carry the fermionic sector via the Klein operator, yielding a bosonized representation of the full supergroup.

Several structural remarks connect these constructions to standard representation theory: the inducing subgroups are generated by single operators, so stability points are eigenvectors with eigenvalue 1, and the induced IRRs are highest-weight representations with $\psi = a^- K$1 relative to Cartan subalgebras identified with stabilizer subgroups. The authors note that since the Clifford group normalizes the Pauli group, the two theorems can be merged into a single bosonized statement, and that the $\psi = a^- K$2-deformed generalization is straightforward though omitted.

Quantum information implications

All operators appearing in the constructions are qubit operators — Pauli matrices, Clifford gates, matrix units — so the SUSY representations are directly implementable. The direction of induction determines the target architecture: fermionic-sector induction suits hybrid qubit/fermionic quantum computers, while the bosonized version suits hybrid qubit/bosonic platforms such as circuit-QED devices. The authors also note that symmetric/antisymmetric tensor products encode the bosonic/fermionic statistics, hiding Jordan-Wigner strings when no fermionic hardware is available, and that multi-particle IRRs arise from disjoint unions $\psi = a^- K$3.

Limitations and open questions

The paper concedes several points. The cocycle-based SI construction following Varadarajan's lemma is explicitly non-canonical, and the choice of stabilizer points among many possible Pauli subgroups introduces arbitrariness not fully explored. The second induction step from $\psi = a^- K$4 to $\psi = a^- K$5 is asserted but "not detailed," and the $\psi = a^- K$6-deformed version of the representation construction is likewise omitted despite being invoked for the SUSY-breaking tower. The claim that restriction of the $\psi = a^- K$7 fundamental IRR remains irreducible relies on the specific carrier space chosen; irreducibility under restriction is not guaranteed in general. Finally, the connection to Adinkras is asserted at the level of analogy ("akin to unfolded Adinkras") rather than established formally, and the announced program of fermionizing an Adinkra into explicit circuits for a fermionic quantum computer remains future work.

Conclusion

This paper recasts bosonization of supersymmetric quantum mechanics as a problem in induced representations: the infinite tower of SUSY-breaking systems generated by iterated Klein-projector compositions is matched by a Mackey-theoretic construction of $\psi = a^- K$8 IRRs induced across both bosonic and fermionic sectors, expressed entirely in qubit-operator language. The identification of stabilizer subgroups with Cartan subalgebras, and of the induced representations with highest-weight modules at $\psi = a^- K$9, provides a bridge from the extensive literature on reductive Lie algebra representations back toward constructing larger symmetries from smaller ones — and, concretely, toward circuit-level implementations of supersymmetric dynamics on hybrid quantum hardware.