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Quantum Information Approach to Bosonization of Supersymmetric Yang-Mills Fields

Radhakrishnan Balu, S. James Gates

TL;DR

The paper addresses the problem of bosonizing supersymmetry in the Wess-Zumino quantum mechanics framework by leveraging Mackey’s system of imprimitivity to construct irreducible representations of the supergroup $osp(2|2)$ in two complementary induction directions. The approach yields an explicit tower of SUSY systems and connects induced representations to highest-weight structures, while recasting the constructions in a quantum-information language using qubit, Pauli, and Clifford operators. Key contributions include concrete SI-based realizations on super Hilbert spaces, two pathways to build larger symmetries from smaller subalgebras, and a bosonized representation on bébé Fock spaces that can interface with quantum hardware. The work also notes a straightforward q-deformed generalization and outlines future steps toward fermionizing Adinkras and implementing corresponding circuits, highlighting practical pathways for SUSY problems on quantum devices.

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.

Quantum Information Approach to Bosonization of Supersymmetric Yang-Mills Fields

TL;DR

The paper addresses the problem of bosonizing supersymmetry in the Wess-Zumino quantum mechanics framework by leveraging Mackey’s system of imprimitivity to construct irreducible representations of the supergroup in two complementary induction directions. The approach yields an explicit tower of SUSY systems and connects induced representations to highest-weight structures, while recasting the constructions in a quantum-information language using qubit, Pauli, and Clifford operators. Key contributions include concrete SI-based realizations on super Hilbert spaces, two pathways to build larger symmetries from smaller subalgebras, and a bosonized representation on bébé Fock spaces that can interface with quantum hardware. The work also notes a straightforward q-deformed generalization and outlines future steps toward fermionizing Adinkras and implementing corresponding circuits, highlighting practical pathways for SUSY problems on quantum devices.

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.
Paper Structure (7 sections, 2 theorems, 34 equations)

This paper contains 7 sections, 2 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $OSp(2|2)$ supersymmetric YM systems
  4. Systems of Imprimitivity
  5. Stabilizer subgroups (Little groups)
  6. Summary and conclusions
  7. Appendix

Key Result

Theorem 5.7

Representation of the super group $gl(2|2)$ on the Hilbert space space $\mathbb{C}^2 \otimes_s \mathbb{C}^2 \oplus \mathbb{C}^2 \otimes_a \mathbb{C}^2$ is a transitive system of imprimitivity living on the orbit $\Omega_2 \subset \mathbb{C}^2 \otimes \mathbb{C}^2 \oplus \mathbb{C}^2 \otimes_a \mathb

Theorems & Definitions (10)

  • Definition 5.1
  • Definition 5.2
  • Definition 5.3
  • Definition 5.4
  • Definition 5.5
  • Definition 5.6
  • Theorem 5.7
  • proof
  • Theorem 5.8
  • proof