In
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
and
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Pauli matrices are a set of three
complex matrices which are
Hermitian,
involutory and
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
.
Usually indicated by the
Greek letter
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used a ...
(), they are occasionally denoted by
tau () when used in connection with
isospin symmetries.
These matrices are named after the physicist
Wolfgang Pauli
Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics ...
. In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, they occur in the
Pauli equation
In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic f ...
which takes into account the interaction of the
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...
of a particle with an external
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left).
Each Pauli matrix is
Hermitian, and together with the identity matrix (sometimes considered as the zeroth Pauli matrix ), the Pauli matrices form a
basis for the real
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of Hermitian matrices.
This means that any
Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.
Hermitian operators represent
observables in quantum mechanics, so the Pauli matrices span the space of observables of the
complex -dimensional
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. In the context of Pauli's work, represents the observable corresponding to spin along the th coordinate axis in three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
The Pauli matrices (after multiplication by to make them
anti-Hermitian) also generate transformations in the sense of
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s: the matrices form a basis for the real Lie algebra
, which
exponentiates to the special unitary group
SU(2). The
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
generated by the three matrices is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
of , and the (unital associative) algebra generated by is effectively identical (isomorphic) to that of
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s (
).
Algebraic properties
All three of the Pauli matrices can be compacted into a single expression:
:
where the solution to is the "
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
", and is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
, which equals +1 if and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of , in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations.
The matrices are
''involutory'':
:
where is the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
.
The
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
s and
traces of the Pauli matrices are:
:
From which, we can deduce that each matrix has
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
+1 and −1.
With the inclusion of the identity matrix, (sometimes denoted ), the Pauli matrices form an orthogonal basis (in the sense of
Hilbert–Schmidt) of the
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
of Hermitian matrices,
over
, and the Hilbert space of all
complex matrices,
.
Eigenvectors and eigenvalues
Each of the (
Hermitian) Pauli matrices has two
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
, and . The corresponding
normalized eigenvectors are:
:
Pauli vector
The Pauli vector is defined by
where
,
, and
are an equivalent notation for the more familiar
,
, and
; the subscripted notation
is more compact than the old
form.
The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows,
using
Einstein's summation convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
.
More formally, this defines a map from
to the vector space of traceless Hermitian
matrices. This map encodes structures of
as a normed vector space and as a Lie algebra (with the
cross-product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
as its Lie bracket) via functions of matrices, making the map an isomorphism of Lie algebras. This makes the Pauli matrices intertwiners from the point of view of representation theory.
Another way to view the Pauli vector is as a
Hermitian traceless matrix-valued dual vector, that is, an element of
which maps
.
Completeness relation
Each component of
can be recovered from the matrix (see
completeness relation below)
This constitutes an inverse to the map
, making it manifest that the map is a bijection.
Determinant
The norm is given by the determinant (up to a minus sign)
Then considering the conjugation action of an
matrix
on this space of matrices,
:
,
we find
, and that
is Hermitian and traceless. It then makes sense to define
where
has the same norm as
, and therefore interpret
as a rotation of 3-dimensional space. In fact, it turns out that the ''special'' restriction on
implies that the rotation is orientation preserving. This allows the definition of a map
given by
:
,
where
. This map is the concrete realization of the double cover of
by
, and therefore shows that
. The components of
can be recovered using the tracing process above:
:
Cross-product
The cross-product is given by the matrix commutator (up to a factor of
)
In fact, the existence of a norm follows from the fact that
is a Lie algebra: see
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) sho ...
.
This cross-product can be used to prove the orientation-preserving property of the map above.
Eigenvalues and eigenvectors
The eigenvalues of
are
. This follows immediately from tracelessness and explicitly computing the determinant.
More abstractly, without computing the determinant which requires explicit properties of the Pauli matrices, this follows from
, since this can be factorised into
. A standard result in linear algebra (a linear map which satisfies a polynomial equation written in distinct linear factors is diagonal) means this implies
is diagonal with possible eigenvalues
. The tracelessness of
means it has exactly one of each eigenvalue.
Its normalized eigenvectors are
Pauli 4-vector
The Pauli 4-vector, used in spinor theory, is written
with components
:
This defines a map 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. The relation can then be written
Similarly to the Pauli 3-vector case, we can find a matrix group which acts as isometries on
; in this case the matrix group is
, and this shows
Similarly to above, this can be explicitly realized for
with components
:
In fact, the determinant property follows abstractly from trace properties of the
. For
matrices, the following identity holds:
:
That is, the 'cross-terms' can be written as traces. When
are chosen to be different
, the cross-terms vanish. It then follows, now showing summation explicitly,
Since the matrices are
, this is equal to
(Anti-)Commutation relations
The Pauli matrices obey the following
commutation
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
relations:
:
where the
structure constant
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting pro ...
is the
Levi-Civita symbol and Einstein summation notation is used.
These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra
They also satisfy the
anticommutation relations:
:
where is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
, and is the identity matrix.
These anti-commutation relations make the Pauli matrices the generators of a representation of the
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
for
, denoted
The usual construction of generators