Pauli matrices

The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. Usually indicated by the Greek letter 'sigma' (σ), they are occasionally denoted with a 'tau' (τ) when used in connection with isospin symmetries. They are::$$sigma_1 = sigma_x =egin{pmatrix}0&1\1&0end{pmatrix}

:$$sigma_2 = sigma_y =egin{pmatrix}0&-i\i&0end{pmatrix}

:$$sigma_3 = sigma_z =egin{pmatrix}1&0\0&-1end{pmatrix}.

The name refers to Wolfgang Pauli.

Algebraic properties

:$$sigma_1^2 = sigma_2^2 = sigma_3^2 = egin{pmatrix} 1&0\0&1end{pmatrix} = Iwhere "I" is the identity matrix.

*The determinants and traces of the Pauli matrices are:

:$$egin{matrix}det (sigma_i) &=& -1 & \ [1ex] operatorname{Tr} (sigma_i) &=& 0 & quad hbox{for} i = 1, 2, 3end{matrix}

From above we can deduce that the eigenvalues of each σ_"i" are ±1.

*Together with the identity matrix I (which is sometimes written as σ₀), the Pauli matrices form an orthogonal basis, in the sense of Hilbert-Schmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.

Commutation relations

The Pauli matrices obey the following commutation and anticommutation relations:

:$$egin{matrix} [sigma_a, sigma_b] &=& 2 i sum_c varepsilon_{a b c},sigma_c \ [1ex] {sigma_a, sigma_b} &=& 2 delta_{a b} cdot Iend{matrix}

where $$varepsilon_{abc} is the Levi-Civita symbol, $$delta_{ab} is the Kronecker delta, and I is the identity matrix.

The above two relations can be summarized as:

:$$sigma_a sigma_b = delta_{ab} cdot I + i sum_c varepsilon_{abc} sigma_c ,.

For example, :$$egin{matrix}sigma_1sigma_2 &=& isigma_3,\sigma_2sigma_3 &=& isigma_1,\sigma_2sigma_1 &=& -isigma_3,\sigma_1sigma_1 &=& I.\end{matrix}

The Pauli vector is defined by:$$vec{sigma} = sigma_1 hat{x} + sigma_2 hat{y} + sigma_3 hat{z} ,and the summary equation for the commutation relations can be used to prove:$$(vec{a} cdot vec{sigma})(vec{b} cdot vec{sigma}) = vec{a} cdot vec{b} + i vec{sigma} cdot ( vec{a} imes vec{b} ) quad quad quad quad (1) ,:(as long as the vectors "a" and "b" commute with the pauli matrices)as well as:$$e^{i (vec{a} cdot vec{sigma})} = cos{a} + i (hat{n} cdot vec{sigma}) sin{a} quad quad quad quad quad quad (2) ,for $$vec{a} = a hat{n} .

Proof of (1)

:

Proof of (2)

First notice that for even powers:$$(hat{n} cdot vec{sigma})^{2n} = I , but for odd powers:$$(hat{n} cdot vec{sigma})^{2n+1} = hat{n} cdot vec{sigma} ,

Combine these two facts with the knowledge of the relation of the exponential to sine and cosine::Which, when we use $$x = a (hat{n} cdot vec{sigma}) ,gives us::$$= sum_{n=0}^infty{frac{(-1)^n (ahat{n}cdot vec{sigma})^{2n{(2n)! + isum_{n=0}^infty{frac{(-1)^n (ahat{n}cdot vec{sigma})^{2n+1{(2n+1)! ,::$$= sum_{n=0}^infty{frac{(-1)^n a^{2n{(2n)! + i (hat{n}cdot vec{sigma}) sum_{n=0}^infty{frac{(-1)^n a^{2n+1{(2n+1)! ,The sum on the left is cosine, and the sum on the right is sine so finally,:$$e^{i a(hat{n} cdot vec{sigma})} = cos{a} + i (hat{n} cdot vec{sigma}) sin{a} ,

SU(2)

The matrix group SU(2) is a Lie group, and its Lie algebra is the set of the anti-Hermitian 2×2 matrices with trace 0. Direct calculation shows that the Lie algebra su(2) is the 3 dimensional real algebra spanned by the set {"i" σ_"j"}. In symbols,

:$;$operatorname{su}(2) = operatorname{span} { i sigma_1, i sigma_2 , i sigma_3 }.

As a result, "i" σ_"j"s can be seen as infinitesimal generators of SU(2).

A Cartan decomposition of SU(2)

This relationship between the Pauli matrices and SU(2) can be explored further, as can be seen from the following simple example. We can write

:$;$operatorname{su}(2) = operatorname{span} {i sigma_2} oplus operatorname{span} { i sigma_1, i sigma_3}.

We put

:$;$mathfrak{k} = operatorname{span} {i sigma_3},

and

:$;$mathfrak{p} = operatorname{span} { i sigma_1, i sigma_2}

Using the algebraic identities listed in the previous section, it can be verified that$$mathfrak{k} and $$mathfrak{p} form a Cartan pair of the Lie algebra SU(2). Furthermore,

:$;$mathfrak{a} = operatorname{span} { i sigma_2}

is a maximal abelian subalgebra of $$mathfrak{p}. Now, a version of Cartan decomposition states that any element "U" in the Lie group SU(2) can be expressed in the form

:$$U = e^{k_1} e^a e^{k_2},! where $$k_1, k_2 in mathfrak{k} and $$a in mathfrak{a}.

In other words, any unitary "U" of determinant 1 is of the form

:$$U = e^{i alpha sigma_1} e^{i eta sigma_2} e^{i gamma sigma_3},!

for some real numbers α, β, and γ.

Extending to unitary matrices gives that any unitary 2 × 2 "U" is of the form

:$$U = e^{i delta} e^{i alpha sigma_1} e^{i eta sigma_2} e^{i gamma sigma_3},!

where the additional parameter δ is also real.

SO(3)

The Lie algebra su(2) is isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, one can say that $$i sigma_j's are a realization (and, in fact, the lowest-dimensional realization) of "infinitesimal" rotations in three-dimensional space. However, even though su(2) and so(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a double cover of SO(3), meaning that there is a two-to-one homomorphism from SU(2) to SO(3).

Quaternions

Consider the real linear span "S" of {"I", σ₁ σ₂, σ₂ σ₃, σ_"3" σ_"1"}. "S" is isomorphic to the real algebra of quaternions H. The isomorphism from H to "S" is given by

:$$1 simeq 1, i simeq sigma_1 sigma_2, j simeq sigma_3 sigma_1, k simeq sigma_2 sigma_3.

As the quaternions of unit norm is group-isomorphic to SU(2), this gives yet another way of describing SU(2) via the Pauli matrices. The two-to-one homomorphism from SU(2) to SO(3) can also be explicitly given in terms of the Pauli matrices in this formulation.

Quaternions form a division algebra - there always is an inverse - whereas Pauli matrices do not.

Physics

Quantum mechanics

*In quantum mechanics, each Pauli matrix represents an observable describing the spin of a spin ½ particle in the three spatial directions. Also, as an immediate consequence of the Cartan decomposition mentioned above, $$i sigma_j are the generators of rotation acting on non-relativistic particles with spin ½. The state of the particles are represented as two-component spinors. An interesting property of spin ½ particles is that they must be rotated by an angle of 4$$pi 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 "S"^"2", they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space.

*For a spin Fraction|1|2 particle, the spin operator is given by $$mathbf{J} =frachbar2oldsymbol{sigma}. The Pauli matrices can be generalized to describe higher spin systems in three spatial dimensions. The spin matrices for spin 1 and spin Fraction|3|2 are given below:j=1::$$J_x = frachbarsqrt{2}egin{pmatrix}0&1&0\1&0&1\0&1&0end{pmatrix}

:$$J_y = frachbarsqrt{2}egin{pmatrix}0&-i&0\i&0&-i\0&i&0end{pmatrix}

:$$J_z = hbaregin{pmatrix}1&0&0\0&0&0\0&0&-1end{pmatrix}

j=Fraction|3|2::$$J_x = frachbar2egin{pmatrix}0&sqrt{3}&0&0\sqrt{3}&0&2&0\0&2&0&sqrt{3}\0&0&sqrt{3}&0end{pmatrix}

:$$J_y = frachbar2egin{pmatrix}0&-isqrt{3}&0&0\isqrt{3}&0&-2i&0\0&2i&0&-isqrt{3}\0&0&isqrt{3}&0end{pmatrix}

:$$J_z = frachbar2egin{pmatrix}3&0&0&0\0&1&0&0\0&0&-1&0\0&0&0&-3end{pmatrix}

*Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G_n is defined to consist of all n-fold tensor products of Pauli matrices.

*The fact that any 2 × 2 complex Hermitian matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the Bloch sphere representation of 2 × 2 mixed states (2 × 2 positive semidefinite matrices with trace 1). This can be seen by simply first writing a Hermitian matrix as a real linear combination of {σ₀, σ₁, σ₂, σ₃} then impose the positive semidefinite and trace 1 assumptions.

Quantum information

*In quantum information, single-qubit quantum gates are "2" × "2" unitary matrices. The Pauli matrices are some of the most important single-qubit operations. In that context, the Cartan decomposition given above is called the "Z-Y decomposition of a single-qubit gate". Choosing a different Cartan pair gives a similar "X-Y decomposition of a single-qubit gate".

See also

* Angular momentum
* Gell-Mann matrices
* Generalizations of Pauli matrices
* Poincare group
* Pauli equation

References

*cite book | author=Liboff, Richard L. | title=Introductory Quantum Mechanics | publisher=Addison-Wesley | year=2002 | id=ISBN 0-8053-8714-5

*cite book | author=Schiff, Leonard I. | title=Quantum Mechanics | publisher=McGraw-Hill | year=1968 | id=ISBN 007-Y85643-5