Pauli matrices

Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries.

In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field.

This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.

In the context of Pauli's work, σk represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space

[1] and the (unital) associative algebra generated by iσ1, iσ2, iσ3 functions identically (is isomorphic) to that of quaternions (

All three of the Pauli matrices can be compacted into a single expression: where the solution to i2 = −1 is the "imaginary unit", and δjk is the Kronecker delta, which equals +1 if j = k and 0 otherwise.

The determinants and traces of the Pauli matrices are from which we can deduce that each matrix σj has eigenvalues +1 and −1.

With the inclusion of the identity matrix I (sometimes denoted σ0), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the Hilbert space

The Pauli matrices obey the following commutation relations: where the Levi-Civita symbol εjkl is used.

These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra

These anti-commutation relations make the Pauli matrices the generators of a representation of the Clifford algebra for

using the Clifford algebra recovers the commutation relations above, up to unimportant numerical factors.

More abstractly, without computing the determinant, which requires explicit properties of the Pauli matrices, this follows from

to the vector space of Hermitian matrices, which also encodes the Minkowski metric (with mostly minus convention) in its determinant: This 4-vector also has a completeness relation.

It is convenient to define a second Pauli 4-vector and allow raising and lowering using the Minkowski metric tensor.

Contracting each side of the equation with components of two 3-vectors ap and bq (which commute with the Pauli matrices, i.e., apσq = σqap) for each matrix σq and vector component ap (and likewise with bq) yields Finally, translating the index notation for the dot product and cross product results in

where Greek indices α, β, γ and μ assume values from {0, x, y, z} and the notation

Note that while the determinant of the exponential itself is just 1, which makes it the generic group element of SU(2).

Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to[4]

It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle

Taking the dot product of any unit vector with the above formula generates the expression of any single qubit operator under any rotation.

An alternative notation that is commonly used for the Pauli matrices is to write the vector index k in the superscript, and the matrix indices as subscripts, so that the element in row α and column β of the k-th Pauli matrix is σ kαβ.

This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of {σ0, σ1, σ2, σ3} as above, and then imposing the positive-semidefinite and trace 1 conditions.

Let Pjk be the transposition (also known as a permutation) between two spins σj and σk living in the tensor product space ⁠

It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.

The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector.

The conventional normalization is λ = ⁠1/2⁠, so that As SU(2) is a compact group, its Cartan decomposition is trivial.

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,[5] As the set of versors U ⊂

An interesting property of spin 1⁄2 particles is that they must be rotated by an angle of 4π in order to return to their original configuration.

This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north–south pole on the 2-sphere S2, they are actually represented by orthogonal vectors in the two-dimensional complex Hilbert space.