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Matrices important in quantum mechanics and the study of spin
mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 {\displaystyle 2\times 2} complex matrices that are traceless, Hermitian, involutory
Pauli_matrices
Families of matrices in mathematics, physics, and quantum information
term generalized Pauli matrices refers to families of matrices which generalize the (linear algebraic) properties of the Pauli matrices. Here, a few classes
Generalizations of Pauli matrices
Generalizations_of_Pauli_matrices
16-element matrix group
group theory, the Pauli group is a group formed by tensor products of Pauli matrices, including the identity. The single-qubit Pauli group is a 16-element
Pauli_group
Austrian physicist (1900–1958)
introduced the 2×2 Pauli matrices as a basis of spin operators, thus solving the nonrelativistic theory of spin. This work, including the Pauli equation, is
Wolfgang_Pauli
Intrinsic quantum property of particles
general Pauli group Gn is defined to consist of all n-fold tensor products of Pauli matrices. The analog formula of Euler's formula in terms of the Pauli matrices
Spin_(physics)
Generators of the Clifford algebra for relativistic quantum mechanics
\gamma ^{2},\gamma ^{3}\right\}\ ,} also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they
Gamma_matrices
Basis for the SU(3) Lie algebra
The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the
Gell-Mann_matrices
Definition of quantum circuits
which normalize the n-qubit Pauli group, i.e., map tensor products of Pauli matrices to tensor products of Pauli matrices through conjugation. The notion
Clifford_gate
Topics referred to by the same term
of Pauli matrices Gamma matrices, which can be represented in terms of the Pauli matrices. Higher-dimensional gamma matrices In pure mathematics and physics:
Spin_matrix
Mathematical description of fermions
{\displaystyle \sigma _{i}} are the Pauli matrices and α {\displaystyle \alpha } is the vector made of gamma matrices α = γ t ( γ x , γ y , γ z ) {\displaystyle
Dirac_spinor
Group of unitary complex matrices with determinant of 1
degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants
Special_unitary_group
Relativistic quantum mechanical wave equation
three Pauli matrices for i ∈ { 1 , 2 , 3 } {\displaystyle i\in \{1,2,3\}} . There are two other common representations for the gamma matrices. The first
Dirac_equation
Complex four-component spinor
}&0_{2}\\0_{2}&-I_{2}\end{bmatrix}}} These two 4×4 matrices are related to the Dirac gamma matrices. Note that 02 is the 2x2 zero matrix and I2 is the
Plane-wave solutions to the Dirac equation
Plane-wave_solutions_to_the_Dirac_equation
Four-dimensional number system
Transparent dry-erase sphere used to teach spherical geometry Pauli matrices – Matrices important in quantum mechanics and the study of spin Quaternionic
Quaternion
article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular
List_of_named_matrices
Model in statistical mechanics
Δt: U = e i H Δ t {\displaystyle U=e^{iH\Delta t}} The product of the U matrices, one after the other, is the total time evolution operator, which is the
Square_lattice_Ising_model
Mathematical model of magnetism
{\displaystyle Z_{j}} are representations of elements of the spin algebra (Pauli matrices, in the case of spin 1/2) acting on the spin variables of the corresponding
Transverse-field_Ising_model
Spin representations of the SO(3) group
}}\equiv (\sigma _{1},\sigma _{2},\sigma _{3})} is the vector form of Pauli matrices. Matrices of this form have the following properties, which relate them intrinsically
Spinors_in_three_dimensions
Array of numbers
matrix of dimension 2 × 3. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric transformations (for example
Matrix_(mathematics)
Matrix representing a Euclidean rotation
through Pauli matrices; see the 2 × 2 derivation for SU(2). For the general n × n case, one might use Ref. The Lie group of n × n rotation matrices, SO(n)
Rotation_matrix
Square matrix with ones on the main diagonal and zeros elsewhere
Elementary matrix Exchange matrix Matrix of ones Pauli matrices (the identity matrix is the zeroth Pauli matrix) Householder transformation (the Householder
Identity_matrix
Set of quantum operations
set of n-fold Pauli group products into itself. It is most famously studied for its use in quantum error correction. The Pauli matrices, σ 0 = I = [ 1
Clifford_group
Representation of a quantum mechanical system
ρ can be expanded using the identity I and the Hermitian, traceless Pauli matrices σ → {\displaystyle {\vec {\sigma }}} , ρ = 1 2 ( I + a → ⋅ σ → ) = 1
Bloch_sphere
Matrix equal to its conjugate-transpose
Hermitian matrices include the Pauli matrices, the Gell-Mann matrices, and their generalizations. In theoretical physics, such Hermitian matrices are often
Hermitian_matrix
Basic circuit in quantum computing
omitted. All real exponents of unitary matrices are also unitary matrices, and all quantum gates are unitary matrices. Positive integer exponents are equivalent
Quantum_logic_gate
Vectors representing a particle spin state
single spin 1/2 particle, they can be defined as the eigenvectors of the Pauli matrices. As such, they are vectors mathematically but physics convention distinguishes
Eigenspinor
Gamma matrices for arbitrary Clifford algebras
mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic
Higher-dimensional gamma matrices
Higher-dimensional_gamma_matrices
Elementary particles with a spin of 1/2
spin operators can be represented as simple 2 × 2 matrices. These matrices are called the Pauli matrices. Creation and annihilation operators can be constructed
Spin_1/2
Spin of an electron
gamma matrices (known as Dirac matrices) and i is the imaginary unit. A second application of the Dirac operator will now reproduce the Pauli term exactly
Electron_magnetic_moment
Interaction of a quantum system with a classical observer
linear combination of the Pauli matrices, which provide a basis for 2 × 2 {\displaystyle 2\times 2} self-adjoint matrices: ρ = 1 2 ( I + r x σ x + r
Measurement in quantum mechanics
Measurement_in_quantum_mechanics
Mathematical model of ferromagnetism in statistical mechanics
description of spins, we replace the spin variables with their respective Pauli matrices. However, depending on the direction of the magnetic field, we can create
Ising_model
Eighteenth letter of the Greek alphabet
are also used instead). In quantum mechanics, σ is used to indicate Pauli matrices. In astronomy, σ represents velocity dispersion. In astronomy, the prefix
Sigma
Mathematical ring whose elements are matrices
a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set of all n × n matrices with entries
Matrix_ring
Algebra of 4D spacetime
Clifford algebra Cl3,0(R) has a faithful representation, generated by Pauli matrices, on the spin representation C2; further, Cl3,0(R) is isomorphic to the
Algebra_of_physical_space
Mathematical object
matrix representation of S3. One convenient choice is given by the Pauli matrices: x 1 + x 2 i + x 3 j + x 4 k ↦ ( x 1 + i x 2 x 3 + i x 4 − x 3 + i x
3-sphere
Group of rotations in 3 dimensions
identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3)
3D_rotation_group
Testable implication of local hidden-variable theories
2 , σ 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} are the Pauli matrices. Then we find the eigenvalues and eigenvectors of the real symmetric
CHSH_inequality
Statistical model in quantum mechanics of magnetic materials
{\displaystyle S^{x}} and S z {\displaystyle S^{z}} are also Pauli matrices which obey the Pauli matrix algebra. Under periodic boundary conditions, the transformed
Quantum_Heisenberg_model
Square matrices satisfy their characteristic equation
{\displaystyle 2\times 2} complex matrices. Cayley in 1858 stated the result for 3 × 3 {\displaystyle 3\times 3} and smaller matrices, but only published a proof
Cayley–Hamilton_theorem
Lie group of Lorentz transformations
elsewhere in this article we know this space of matrices can be viewed as 4-vectors. The space of matrices coming from turning each projective vector in
Lorentz_group
Quantum mechanics principle
= 0 when x = y, which is Pauli exclusion. It is true in any basis since local changes of basis keep antisymmetric matrices antisymmetric. Conversely
Pauli_exclusion_principle
Mathematical tool in quantum physics
combination of the Pauli matrices, which together with the identity matrix provide a basis for 2 × 2 {\displaystyle 2\times 2} self-adjoint matrices: ρ = 1 2 (
Density_matrix
Theorem in physics
| 1 ⟩ {\displaystyle |1\rangle } are the eigenstates of one of the Pauli matrices, σ z = ( 1 0 0 − 1 ) . {\displaystyle \sigma _{z}={\begin{pmatrix}1
Bell's_theorem
Matrix realization of the Clifford algebra
Weyl–Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra of 2⌊n/2⌋ × 2⌊n/2⌋ matrices. They generalize the Pauli matrices to n
Weyl–Brauer_matrices
Unit of quantum information
unitary matrices and gates that act on registers of n {\displaystyle n} qutrits are 3 n × 3 n {\displaystyle 3^{n}\times 3^{n}} unitary matrices (the elements
Qutrit
First-order differential linear operator on spinor bundle, whose square is the Laplacian
{\displaystyle \sigma _{i}} are the Pauli matrices. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining
Dirac_operator
Wave equations respecting special and general relativity
The first two-dimensional spin matrices (better known as the Pauli matrices) were introduced by Pauli in the Pauli equation; the Schrödinger equation
Relativistic_wave_equations
Topics referred to by the same term
(quantum), a model where the spins are treated quantum mechanically using Pauli matrices This disambiguation page lists articles associated with the title Heisenberg
Heisenberg_model
Quantum state of a system
[citation needed] Singlet state Doublet state Diradical Helium atom Pauli matrices Spin isomers of hydrogen Spin multiplicity Spin quantum number Spin
Triplet_state
These matrices, V and U, normally referred to as "shift and clock matrices", were introduced by J. J. Sylvester in the 1880s. (Note that the matrices V are
Generalized_Clifford_algebra
Process in quantum computing
is either a bit flip, or a phase flip, or both (corresponding to the Pauli matrices X {\displaystyle X} , Y {\displaystyle Y} , and Z {\displaystyle Z}
Quantum_error_correction
Linearized quantum-mechanical equation
{\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})} is the vector of Pauli matrices, which is proportional to the spin operator S = 1 2 ℏ σ {\displaystyle
Lévy-Leblond_equation
Simple quantum mechanical system
identity matrix and the matrices σ k {\displaystyle \sigma _{k}} with k = 1 , 2 , 3 {\displaystyle k=1,2,3} are the Pauli matrices. This decomposition simplifies
Two-state_quantum_system
Theory of quantum gravity merging quantum mechanics and general relativity
^{ijk}T_{k}.} The Pauli matrices satisfy the above relation. It turns out that there are infinitely many more examples of sets of matrices that satisfy these
Loop_quantum_gravity
Coefficients of an algebra over a field
generators of 𝔰𝔲(N), based on a generalisation of the Pauli matrices and of the Gell-Mann matrices (using the bra-ket notation where | m ⟩ ⟨ n | {\displaystyle
Structure_constants
Quantum mechanical equation of motion of charged particles in magnetic field
In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-1/2 particles, which takes
Pauli_equation
Relativistic wave description of fermions
\langle u,v\rangle =\langle Su,Sv\rangle } The skew matrix takes the Pauli matrices to minus their transpose: ω σ k ω − 1 = − σ k T {\displaystyle \omega
Majorana_equation
Force acting on charged particles in electric and magnetic fields
\mathbf {B} ,} where σ {\displaystyle {\boldsymbol {\sigma }}} are the Pauli matrices. This term leads to spin-dependent forces absent in the classical theory
Lorentz_force
Element in a ring whose some power is 0
transform from one state to another, for example the raising and lowering Pauli matrices σ ± = ( σ x ± i σ y ) / 2 {\displaystyle \sigma _{\pm }=(\sigma _{x}\pm
Nilpotent
Non-tensorial representation of the spin group
needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the Pauli sigma matrices gives rise to the familiar
Spinor
Quantum number related to the weak interaction
spaces. Namely, for the spin-1/2 case, components of I are equal to Pauli matrices divided by 2, and so Iz = 1/2 τ3, where τ 3 = ( 1 0 0 − 1 ) . {\displaystyle
Isospin
analog rotation matrices in three Cartesian axes of SO(3), along the x, y or z axes of the Bloch sphere projection. As Pauli matrices are related to the
List_of_quantum_logic_gates
Setting of relativistic physics in geometric algebra
}\gamma _{\mu }} The Dirac matrices share these properties, and STA is equivalent to the algebra generated by the Dirac matrices over the field of real numbers;
Spacetime_algebra
}}=(\sigma _{x},\sigma _{y},\sigma _{z})} is the vector of the Pauli matrices. Since Pauli matrices are traceless, it still holds that tr(ρ) = 1. However, by
Purity_(quantum_mechanics)
Matrix decomposition
\end{aligned}}} The matrices U {\displaystyle \mathbf {U} } and V ∗ {\displaystyle \mathbf {V} ^{*}} are unitary (and, as real-valued matrices, orthogonal)
Singular_value_decomposition
classical Stokes parameters. These matrix operators are identical to the Pauli matrices. "Lecture 9: Polarization in quantum optics" (PDF). mpl.mpg.de. Max
Stokes_operators
Four-dimensional associative algebra over the reals
isomorphic to the ring of the 2×2 real matrices. So the study of split-quaternions can be reduced to the study of real matrices, and this may explain why there
Split-quaternion
Classification in abstract algebra
generator to be γ1 = (1, −1). One also needs Cl2(C) ≅ End(C2). The Pauli matrices give a concrete realization: if one sets γ1 = σ1 and γ2 = σ2, then these
Classification of Clifford algebras
Classification_of_Clifford_algebras
Topics referred to by the same term
material's electrical conductivity ability Cross section σ in physics Pauli matrices in quantum physics Sigma baryon in particle physics Sigma receptor,
Sigma_(disambiguation)
satisfy the same commutation relations as the matrices M. For example, the image of the Pauli matrices of SU(2) in this map, J → ≡ a † ⋅ σ → 2 ⋅ a
Jordan_map
Type of error correction in quantum computing
logical qubit. With X {\displaystyle X} and Z {\displaystyle Z} being Pauli matrices and I {\displaystyle I} the Identity matrix, this code's generators
Five-qubit error correcting code
Five-qubit_error_correcting_code
Model for quantum noise in quantum systems
}{4}}}Z} and { I , X , Y , Z } {\displaystyle \{I,X,Y,Z\}} are the Pauli matrices. The trace preserving condition is satisfied by the fact that ∑ i K
Quantum_depolarizing_channel
Function acting on the space of physical states in physics
evolution operator should be unitary, and the operators can be represented as matrices. Any other symmetry, mapping a physical state into another, should keep
Operator_(physics)
Quaternions with complex number coefficients
isomorphic to the Pauli group, the central product of a cyclic group of order 4 and the dihedral group of order 8. Concretely, the Pauli matrices X = ( 0 1 1
Biquaternion
Model for physics of semiconductors
at the impurity site ( σ {\displaystyle \mathbf {\sigma } } are the Pauli matrices). In the Kondo problem, J < 0 {\displaystyle J<0} , i.e. the exchange
Kondo_model
Concept in linear algebra
anti-Hermitian or neither, spectrum and eigenvalues, and more. The Pauli matrices, which represent the spin operator when transforming the spin eigenstates
Coordinate_vector
Matrix representing the effect of scattering on a physical system
Interaction picture Levinson's theorem Initial and final state radiation Pauli Matrices S-symplectomorphism Scattering generator This is not true if an open
S-matrix
Class of quantum error correcting codes
composed of tensor products of Pauli matrices such that each stabilizer contains either only Pauli X operations or only Pauli Z operations. The Shor code
CSS_code
Type of model in quantum statistical physics
{\displaystyle i} , and where σ i {\displaystyle \sigma _{i}} are the Pauli matrices. The Hamiltonian has s l 2 {\displaystyle {\mathfrak {sl}}_{2}} symmetry
Spin_chain
Theory of forces and subatomic particles
Dirac matrices, Ga μ is the 8-component ( a = 1 , 2 , … , 8 {\displaystyle a=1,2,\dots ,8} ) SU(3) gauge field, λa are the 3 × 3 Gell-Mann matrices, generators
Standard_Model
Properties underlying modern physics
J(n). The three J(m) matrices are each (2m + 1)×(2m + 1) square matrices, and the three J(n) are each (2n + 1)×(2n + 1) square matrices. The integers or half-integers
Symmetry_in_quantum_mechanics
Generalization of the Dirac equation
Lorentz group SO ( 1 , 3 ) {\displaystyle {\text{SO}}(1,3)} , just as the Pauli matrices generate a representation of the rotation algebra s o ( 3 ) {\displaystyle
Dirac equation in curved spacetime
Dirac_equation_in_curved_spacetime
Model of quantum optics
ISSN 0031-899X. Note that the spin operators are often represented by Pauli matrices σ ~ α {\displaystyle {\tilde {\sigma }}^{\alpha }} , through the relation
Dicke_model
Apparent lack of definite state before measurement of quantum systems
the surface of a sphere, as shown in the figure on the right. The Pauli spin matrices σ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 − i i 0 ) , σ 3 = ( 1 0 0 − 1 ) {\displaystyle
Quantum_indeterminacy
Function in discrete mathematics
transform FFTPACK Fastest Fourier Transform in the West Generalizations of Pauli matrices Least-squares spectral analysis List of Fourier-related transforms Multidimensional
Discrete_Fourier_transform
Element of a unital algebra over the field of real numbers
real matrices were found isomorphic to coquaternions. Soon the matrix paradigm began to explain several others as they were represented by matrices and
Hypercomplex_number
Concept in physics
\mathbf {B} ,} where σ {\displaystyle \mathbf {\sigma } } denote the Pauli matrices, μ {\displaystyle \mu } is the magnetic moment, and B is the magnetic
Berry connection and curvature
Berry_connection_and_curvature
Supersymmetric generalization of the Poincaré algebra
four components. The Dirac matrices are thus also four-dimensional, and can be expressed as direct sums of the Pauli matrices. The tensor product then gives
Super-Poincaré_algebra
Mathematical game
one also has the Pauli matrices σ 1 , σ 2 , σ 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} ; these are 2x2 complex matrices that have the Lie
Tangloids
Wave equation for arbitrary spin particles
Dirac equation Generalizations of Pauli matrices Wigner D-matrix Weyl–Brauer matrices Higher-dimensional gamma matrices Joos–Weinberg equation, alternative
Bargmann–Wigner_equations
Momentum-dependent division of spin bands in two-dimensional condensed matter systems
{\sigma }}} where σ {\displaystyle {\boldsymbol {\sigma }}} are the Pauli matrices and − g μ B σ / 2 {\displaystyle -g\mu _{\mathrm {B} }{\boldsymbol {\sigma
Rashba_effect
1922 physical experiment demonstrating that atomic spin is quantized
correct this problem Wolfgang Pauli considered a spin-1/2 version of the Schrödinger equation using the 3 Pauli matrices which now bear his name, which
Stern–Gerlach_experiment
Regularization technique in quantum field theory
In theoretical physics, Pauli–Villars regularization (P–V) is a procedure that isolates divergent terms from finite parts in loop calculations in field
Pauli–Villars_regularization
Q8, D4 (3), Z4 × Z2 (3), Z4 (4), Z22 (3), Z2 (7) The Pauli group generated by the Pauli matrices. Nilpotent. 18 44 G181 D9 ≅ Z9 ⋊ {\displaystyle \rtimes
List_of_small_groups
Quantum state of multiple particles represented as complex matrices
\end{bmatrix}}.} This notation uses matrices with entries being state vectors (instead of complex numbers), and when multiplying matrices using tensor product for
Matrix_product_state
Quantum operator for the sum of energies of a system
\mathbf {S} } is the spin operator vector, whose components are the Pauli matrices, hence H ^ = g s e 2 m S ⋅ B {\displaystyle {\hat {H}}={\frac {g_{s}e}{2m}}\mathbf
Hamiltonian (quantum mechanics)
Hamiltonian_(quantum_mechanics)
Concept in quantum entanglement
Observables for these components can be written as products of the Pauli matrices: X = [ 0 1 1 0 ] , Y = [ 0 − i i 0 ] , Z = [ 1 0 0 − 1 ] {\displaystyle
Quantum_pseudo-telepathy
Space in mathematics and theoretical physics
{\vec {\sigma }})} are the Pauli matrices, with A , A ′ = 1 , 2 {\displaystyle A,A^{\prime }=1,2} the indexes on the matrices. This twistor space is a four-dimensional
Twistor_space
Quantum Merlin Arthur
_{i<j}K^{ij}X_{i}X_{j}} where Z , X {\displaystyle Z,X} represent the Pauli matrices σ z , σ x {\displaystyle \sigma _{z},\sigma _{x}} . Such models are
QMA
State of matter with insulating bulk but conductive boundary
2\times 2} hermitian matrix can be written as a linear combination of Pauli matrices, σ 1 , σ 2 {\displaystyle \sigma ^{1},\sigma ^{2}} and σ 3 {\displaystyle
Topological_insulator
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PAULI MATRICES
PAULI MATRICES
PAULI MATRICES
PAULI MATRICES
PAULI MATRICES
PAULI MATRICES
PAULI MATRICES
PAULI MATRICES
PAULI MATRICES
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