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::

:

:

The name refers to Wolfgang Pauli.

Algebraic properties

:where "I" is the identity matrix.

*The determinants and traces of the Pauli matrices are:

:

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:

:

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, :

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

:

for some real numbers α, β, and γ.

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

:

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 . 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=Fraction|3|2::

:

:

*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