Grassmann Numbers and Clifford-Jordan-Wigner Representation of Supersymmetry

Catto, Sultan; Choun, Yoon Seok; Gürcan, Yasemin; Khalfan, Amish; Kurt, Levent

doi:10.1088/1742-6596/411/1/012009

Cited by 8 publications

(7 citation statements)

References 9 publications

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“…Each Clifford subalgebra generated by the three p k and q k can be represented by the eight by eight matrices with complex entries defined over the field C [41,42] ξ p = 1 2 (q p + ip p ) →σ x ⊗Î ⊗Î + iσ y ⊗Î ⊗Î, (168) ξ r = 1 2 (q r + ip r ) →σ z ⊗σ x ⊗Î + iσ z ⊗σ y ⊗Î, (169) ξ q = 1 2 (q q + ip q ) →σ z ⊗σ z ⊗σ x + iσ z ⊗σ z ⊗σ y ,…”

Section: Acknowledgmentsmentioning

confidence: 99%

Discrete Wigner formalism for qubits and noncontextuality of Clifford gates on qubit stabilizer states

Kocia

Love

2017

Phys. Rev. A

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We show that qubit stabilizer states can be represented by non-negative quasi-probability distributions associated with a Wigner-Weyl-Moyal formalism where Clifford gates are positive stateindependent maps. This is accomplished by generalizing the Wigner-Weyl-Moyal formalism to three generators instead of two-producing an exterior, or Grassmann, algebra-which results in Clifford group gates for qubits that act as a permutation on the finite Weyl phase space points naturally associated with stabilizer states. As a result, a non-negative probability distribution can be associated with each stabilizer state's three-generator Wigner function, and these distributions evolve deterministically to one another under Clifford gates. This corresponds to a hidden variable theory that is non-contextual and local for qubit Clifford gates while Clifford (Pauli) measurements have a context-dependent representation. Equivalently, we show that qubit Clifford gates can be expressed as propagators within the three-generator Wigner-Weyl-Moyal formalism whose semiclassical expansion is truncated at order 0 with a finite number of terms. The T -gate, which extends the Clifford gate set to one capable of universal quantum computation, require a semiclassical expansion of the propagator to order 1 . We compare this approach to previous quasi-probability descriptions of qubits that relied on the two-generator Wigner-Weyl-Moyal formalism and find that the twogenerator Weyl symbols of stabilizer states result in a description of evolution under Clifford gates that is state-dependent, in contrast to the three-generator formalism. We have thus extended Wigner non-negative quasi-probability distributions from the odd d-dimensional case to d = 2 qubits, which describe the non-contextuality of Clifford gates and contextuality of Pauli measurements on qubit stabilizer states.

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Section: Acknowledgmentsmentioning

confidence: 99%

Discrete Wigner formalism for qubits and noncontextuality of Clifford gates on qubit stabilizer states

Kocia

Love

2017

Phys. Rev. A

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“…Consider Maxwell's equations ( 2) with vector of current j ν that satisfies the condition ∂ ν j ν = 0 and the conditions (7) ∂ µ ∂ µ j ν + m 2 j ν = 0.…”

Section: Proca Equationsmentioning

confidence: 99%

“…Consequently solutions of Proca equations can be considered as partial subclass of solutions of Maxwell's equations with right hand side (current j ν ) that satisfies additional conditions (7).…”

Section: Proca Equationsmentioning

confidence: 99%

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Constant Solutions of Yang-Mills Equations and Generalized Proca Equations

Marchuk

Shirokov²

2016

JGSP

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ABSTRACT. In this paper we present some new equations which we call Yang-Mills-Proca equations (or generalized Proca equations). This system of equations is a generalization of Proca equation and Yang-Mills equations and it is not gauge invariant. We present a number of constant solutions of this system of equations in the case of arbitrary Lie algebra. In details we consider the case when this Lie algebra is Clifford algebra or Grassmann algebra. We consider solutions of Yang-Mills equations in the form of perturbation theory series near the constant solution.

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“…The fundamental representation of SU (6/1) is 7-dimensional which decomposes into a sextet and a singlet under the spin-flavor group. There is also a 28-dimensional representation of SU (7). Under the SU (6) subgroup it has the decomposition 28 = 21 + 6 + 1 (1) Hence, this supermultiplet can accommodate the bosonic antidiquark and fermionic quark in it, provided we are willing to add another scalar.…”

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confidence: 99%

Supersymmetry in the Quark-Diquark and the Baryon-Meson Systems

Catto¹,

Choun²

2011

Acta Polytech

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A superalgebra extracted from the Jordan algebra of the 27 and 27 dim. representations of the group E6 is shown to be relevant to the description of the quark-antidiquark system. A bilocal baryon-meson field is constructed from two quark-antiquark fields. In the local approximation the hadron field is shown to exhibit supersymmetry which is then extended to hadronic mother trajectories and inclusion of multiquark states. Solving the spin-free Hamiltonian with light quark masses we develop a new kind of special function theory generalizing all existing mathematical theories of confluent hypergeometric type. The solution produces extra “hidden” quantum numbers relevant for description of supersymmetry and for generating new mass formulas.

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Grassmann Numbers and Clifford-Jordan-Wigner Representation of Supersymmetry

Cited by 8 publications

References 9 publications

Discrete Wigner formalism for qubits and noncontextuality of Clifford gates on qubit stabilizer states

Discrete Wigner formalism for qubits and noncontextuality of Clifford gates on qubit stabilizer states

Constant Solutions of Yang-Mills Equations and Generalized Proca Equations

Supersymmetry in the Quark-Diquark and the Baryon-Meson Systems

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