Frontiers in Physics

On Pauli matrices and the implied spacetime geometry

Gregory L. Light

Published May 25, 2026• 0 citations

Abstract

The commutator identity, 1 2 σ z , σ x ≡ i σ y , is shown to be equivalent to P ≡ m o m e n t u m , X ≡ p o s i t i o n = i ℏ I over the eigenspace of σ y spanned by 1 i , where 1 ≡ 1 , 0 , 0 ≡ e X ∈ R 3 , i ≡ 0 , 1 ∈ C 1 ≡ R y , z 2 , and 1 , i ≡ e X ⊕ e z so that the particle travels along e X

DOI: 10.3389/fphy.2026.1789064

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Frontiers in Physics

On Pauli matrices and the implied spacetime geometry

Gregory L. Light

Published May 25, 2026• 0 citations

Publisher

Abstract

DOI: 10.3389/fphy.2026.1789064

Related papers

Explicit derivation of the Pauli spin matrices from the Jones vector

Jairo Alonso Mendoza Suárez, Hernando González Sierra, Francis Segovia-Chaves

Using dyadic representations elaborated from vectors of Jones, and calculating relations of anti-commutation of these tensorial forms, we obtain in shape explicit the Pauli spin matrices.

arXiv:1905.00090Apr 27, 20191 citation

Tensor C*-algebra on Two Qubit Spin-1/2 System

Rivani Adistia Dewi, Imam Nugraha Albania, Rizky Rosjanuardi +1 more

Spin-1/2 system is one of the most important phenomenon in quantum mechanics.Such a system is known to be expressed as Pauli matrices, that isand 𝜎 3 = ( 1 0 0 -1 ) .By inserting the identity matrix 𝐼, we can construct a C*-algebra 𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }) with |{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }⨂{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }| = 16 = |{𝐸 𝑖𝑗 : 𝑖, 𝑗 = 1,2,3,4} |, where 𝐸 𝑖𝑗 is the canonical basis of C*-algebra 𝑀 4 (ℂ).In this study, by using the linear independence concept, we got 𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 })⨂𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }) ≅ 𝑀 4 (ℂ).This means, that for any observable in the composite of two spin-1/2 systems, by self-adjointness of Pauli matrices, it can be expressed by the field reduction from ℂ to ℝ in 𝑠𝑝𝑎𝑛{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }⨂{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 } where the structure itself is nonassociative algebra.Furthermore, given a Hamiltonian 𝐻(𝑡) = (𝜎 1 ⨂𝜎 1 ) + 𝒥 𝑠𝑖𝑛(𝜔𝑡) (𝜎 3 ⨂𝐼) + 𝒥 𝑠𝑖𝑛 (𝜔𝑡 + 𝜋 2 ) (𝐼⨂𝜎 3 ) and the initial state which is a linear combination of Bell basis, by using the integration factor method and the commutation fact of the exponential operator, we got the following timedependent state |𝜓(𝑡)⟩ = 1 √2 (𝑒 ( 𝑖𝑡 ℏ )( 𝑖𝒥 ℏ𝜔 𝑐𝑜𝑠(𝜔𝑡))( 𝑖𝒥 ℏ𝜔 𝑠𝑖𝑛(𝜔𝑡))𝜈 𝑖(5-𝑖) ) 𝑖=1 4 with 𝜈 𝑖𝑗 = 𝛿 (𝑖+𝑗)5 (-1) 𝑚𝑎𝑘𝑠{𝑖,𝑗} , where 𝛿 𝑖𝑗 is Kronecker delta.Spin-1/2 system is one of the most important phenomenon in quantum mechanics.Such a system is known to be expressed as Pauli matrices, that isand 𝜎 3 = ( 1 0 0 -1 ) .By inserting the identity matrix 𝐼, we can construct a C*-algebra 𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }) with |{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }⨂{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }| = 16 = |{𝐸 𝑖𝑗 : 𝑖, 𝑗 = 1,2,3,4} |, where 𝐸 𝑖𝑗 is the canonical basis of C*-algebra 𝑀 4 (ℂ).In this study, by using the linear independence concept, we got 𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 })⨂𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }) ≅ 𝑀 4 (ℂ).This means, that for any observable in the composite of two spin-1/2 systems, by self-adjointness of Pauli matrices, it can be expressed by the field reduction from ℂ to ℝ in 𝑠𝑝𝑎𝑛{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }⨂{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 } where the structure itself is nonassociative algebra.Furthermore, given a Hamiltonian 𝐻(𝑡) = (𝜎 1 ⨂𝜎 1 ) + 𝒥 𝑠𝑖𝑛(𝜔𝑡) (𝜎 3 ⨂𝐼) + 𝒥 𝑠𝑖𝑛 (𝜔𝑡 + 𝜋 2 ) (𝐼⨂𝜎 3 ) and the initial state which is a linear combination of Bell basis, by using the integration factor method and the commutation fact of the exponential operator, we got the following timedependent state |𝜓(𝑡)⟩ = 1 √2 (𝑒 ( 𝑖𝑡 ℏ )( 𝑖𝒥 ℏ𝜔 𝑐𝑜𝑠(𝜔𝑡))( 𝑖𝒥 ℏ𝜔 𝑠𝑖𝑛(𝜔𝑡))𝜈 𝑖(5-𝑖) ) 𝑖=1 4 with 𝜈 𝑖𝑗 = 𝛿 (𝑖+𝑗)5 (-1) 𝑚𝑎𝑘𝑠{𝑖,𝑗} , where 𝛿 𝑖𝑗 is Kronecker delta.

Advances in Social Science, Education and Humanities Research/Advances in social science, education and humanities researchJan 1, 20240 citations

Spin Without Commutation Rules

Saul T. Epstein

The Pauli spin matrices are derived in a way which makes no appeal to the usual apparatus of abstract operators, commutation rules, etc.

American Journal of PhysicsSep 1, 19661 citation

On the probability amplitude of quantum entanglement and the Pauli matrices

F.J. Duarte, T S Taylor, J. C. Slaten

Abstract The probability amplitudes for quantum entanglement, also known as Bell sates, are utilized to arrive explicitly at the identity matrix I and the $\sigma_{x}$ σx , $\sigma_{y}$ σy , and $\sigma_{z}$ σz Pauli matrices, via a straight-forward $2 \times 2$ 2×2 matrix representation that utilizes the vector direct product. It is also indicated that this approach is completely equivalent to the utilization of the Kronecker product $\otimes$ ⊗ to multiply the relevant ket vectors. Furthermore, it is shown that the polarization rotation matrix R, operating on the various versions of the probability amplitude for quantum entanglement, yields four identities elegantly interconnecting these probability amplitudes.Abstract The probability amplitudes for quantum entanglement, also known as Bell sates, are utilized to arrive explicitly at the identity matrix I and the $$\sigma_{x}$$ σx , $$\sigma_{y}$$ σy , and $$\sigma_{z}$$ σz Pauli matrices, via a straight-forward $$2 \times 2$$ 2×2 matrix representation that utilizes the vector direct product. It is also indicated that this approach is completely equivalent to the utilization of the Kronecker product $$\otimes$$ ⊗ to multiply the relevant ket vectors. Furthermore, it is shown that the polarization rotation matrix R, operating on the various versions of the probability amplitude for quantum entanglement, yields four identities elegantly interconnecting these probability amplitudes.

Optical and Quantum ElectronicsFeb 1, 202011 citations

Pauli spin matrices

S. M. Blinder

Jan 1, 20080 citations

5∗ On quantum electrodynamics

L.D. Faddeev, L.A. Khalfin, I.V. Komarov

CRC Press eBooksMay 21, 20041 citation

Abstract The probability amplitudes for quantum entanglement, also known as Bell sates, are utilized to arrive explicitly at the identity matrix I and the

\sigma_{x}

σx ,

\sigma_{y}

σy , and

\sigma_{z}

σz Pauli matrices, via a straight-forward

2 \times 2

2×2 matrix representation that utilizes the vector direct product. It is also indicated that this approach is completely equivalent to the utilization of the Kronecker product

\otimes

⊗ to multiply the relevant ket vectors. Furthermore, it is shown that the polarization rotation matrix R, operating on the various versions of the probability amplitude for quantum entanglement, yields four identities elegantly interconnecting these probability amplitudes.