HOME

TheInfoList



OR:

In the study of the
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of
Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
, the study of representations of
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
is fundamental to the study of representations of
semisimple Lie group In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
s. It is the first case of a Lie group that is both a
compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
and a
non-abelian group In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ' ...
. The first condition implies the representation theory is discrete: representations are
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
s of a collection of basic
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s (governed by the Peter–Weyl theorem). The second means that there will be irreducible representations in dimensions greater than 1. SU(2) is the
universal covering group In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. ...
of
SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is ...
, and so its representation theory includes that of the latter, by dint of a surjective homomorphism to it. This underlies the significance of SU(2) for the description of non-relativistic
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 b ...
in
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimen ...
; see below for other physical and historical context. As shown below, the finite-dimensional irreducible representations of SU(2) are indexed by a non-negative integer m and have dimension m + 1. In the physics literature, the representations are labeled by the quantity l = m/2, where l is then either an integer or a half-integer, and the dimension is 2l + 1.


Lie algebra representations

The representations of the group are found by considering representations of \mathfrak(2), the Lie algebra of SU(2). Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation; we will give an explicit construction of the representations at the group level below.


Real and complexified Lie algebras

The real Lie algebra \mathfrak(2) has a basis given by :u_1 = \begin 0 & i\\ i & 0 \end ,\qquad u_2 = \begin 0 & -1\\ 1 & ~~0 \end ,\qquad u_3 = \begin i & ~~0\\ 0 & -i \end~, (These basis matrices are related to the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
by u_1 = +i\ \sigma_1 \;, \, u_2 = -i\ \sigma_2 \;, and u_3 = +i\ \sigma_3 ~.) The matrices are a representation of the
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 qua ...
s: : u_1\,u_1 = -I\, , ~~\quad u_2\,u_2 = -I \, , ~~\quad u_3\,u_3 = -I\, , : u_1\,u_2 = +u_3\, , \quad u_2\,u_3 = +u_1\, , \quad u_3\,u_1 = +u_2\, , : u_2\,u_1 = -u_3\, , \quad u_3\,u_2 = -u_1\, , \quad u_1\,u_3 = -u_2 ~. where is the conventional 2×2 identity matrix:~~I = \begin 1 & 0\\ 0 & 1 \end ~. Consequently, the commutator brackets of the matrices satisfy : _1, u_2= 2 u_3\, ,\quad _2, u_3= 2 u_1\, ,\quad _3, u_1= 2 u_2 ~. It is then convenient to pass to the complexified Lie algebra :\mathrm(2) + i\,\mathrm(2) = \mathrm(2;\mathbb C) ~. (Skew self-adjoint matrices with trace zero plus self-adjoint matrices with trace zero gives all matrices with trace zero.) As long as we are working with representations over \mathbb C this passage from real to complexified Lie algebra is harmless. The reason for passing to the complexification is that it allows us to construct a nice basis of a type that does not exist in the real Lie algebra \mathfrak(2). The complexified Lie algebra is spanned by three elements X, Y, and H, given by : H = \fracu_3, \qquad X = \frac\left(u_1 - iu_2\right), \qquad Y = \frac(u_1 + iu_2) ~; or, explicitly, : H = \begin 1 & ~~0\\ 0 & -1 \end, \qquad X = \begin 0 & 1\\ 0 & 0 \end, \qquad Y = \begin 0 & 0\\ 1 & 0 \end ~. The non-trivial/non-identical part of the group's multiplication table is : H \, X ~=~~~~X ,\qquad H \, Y ~= -Y ,\qquad X \, Y ~=~ \tfrac\left(I + H \right), : X \, H ~= -X ,\qquad Y \, H ~=~~~~Y ,\qquad Y \, X ~=~ \tfrac\left(I - H \right), : H \, H ~=~~~I~ ,\qquad X \, X ~=~~~~O ,\qquad Y \, Y ~=~ ~O ~~; where is the 2×2 all-zero matrix. Hence their commutation relations are :
, X The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= 2\,X \, , \qquad
, Y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
= -2\,Y \, , \qquad
, Y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
= H ~. Up to a factor of 2, the elements H, X and Y may be identified with the angular momentum operators J_z, J_+, and J_-, respectively. The factor of 2 is a discrepancy between conventions in math and physics; we will attempt to mention both conventions in the results that follow.


Weights and the structure of the representation

In this setting, the eigenvalues for H are referred to as the weights of the representation. The following elementary result is a key step in the analysis. Suppose that v is an
eigenvector 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 denoted ...
for H with eigenvalue \; \alpha \; ; that is, that \; H \, v = \alpha\, v ~. Then :\begin H \, (X \, v) ~=~ (\, X \, H + ,X\,)\, v ~&=~ (\alpha + 2)\, X \, v \; ,\\ pt H \, (Y \, v) ~=~ (\, Y \, H +\, ,Y\,)\, v ~&=~ (\alpha - 2)\, Y \, v ~. \end In other words, \, X\, v \, is either the zero vector or an eigenvector for \, H \, with eigenvalue \, \alpha + 2 \, and \, Y \, v \, is either zero or an eigenvector for \, H \, with eigenvalue \, \alpha - 2 ~. Thus, the operator \, X \, acts as a raising operator, increasing the weight by 2, while \, Y \, acts as a lowering operator. Suppose now that \, V \, is an irreducible, finite-dimensional representation of the complexified Lie algebra. Then \, H \, can have only finitely many eigenvalues. In particular, there must be some final eigenvalue \; \lambda \in \mathbb \; with the property that \, \lambda + 2 \, is ''not'' an eigenvalue. Let \, v_0 \, be an eigenvector for \, H \, with that eigenvalue \, \lambda \; : :H \, v_0 = \lambda \, v_0 \; , then we must have :X \, v_0 = 0 \; , or else the above identity would tell us that \, X \, v_0 \, is an eigenvector with eigenvalue \,\lambda + 2 \; . Now define a "chain" of vectors v_0, v_1, \ldots by :v_k = Y^k \, v_0. A simple argument by induction then shows that :X \, v_k = k\,(\lambda - (k - 1))\,v_ for all k = 1, 2, \ldots ~. Now, if \, v_k \, is not the zero vector, it is an eigenvector for H with eigenvalue \, \lambda - 2k ~. Since, again, \, H \, has only finitely many eigenvectors, we conclude that \, v_\ell \, must be zero for some \, \ell \, (and then v_k = 0 for all \, k > \ell \,). Let v_m be the last nonzero vector in the chain; that is, \; v_m \neq 0 \; but \; v_ = 0 ~. Then of course \; X \, v_ = 0 \; and by the above identity with k = m + 1 \;, we have :\; 0 = X \, v_ = (m + 1)\,(\lambda - m)\,v_m ~. Since \, m + 1 \, is at least one and \, v_m \neq 0 \; , we conclude that \, \lambda \, ''must be equal to the non-negative integer'' \, m \; . We thus obtain a chain of \, m + 1 \, vectors, \; v_0, v_1, \ldots, v_m \; , such that \, Y \, acts as : Y \, v_m = 0, \quad Y \, v_k = v_ \quad (k < m) and \, X \, acts as : X \, v_0 = 0, \quad X \, v_k = k \, (m - (k - 1)) \, v_ \quad (k \ge 1) and \, H \, acts as :H \, v_k = (m - 2k) \, v_k ~. (We have replaced \lambda with its currently known value of \, m \, in the formulas above.) Since the vectors \, v_k \, are eigenvectors for H with distinct eigenvalues, they must be linearly independent. Furthermore, the span of \; v_0, \ldots , v_m \; is clearly invariant under the action of the complexified Lie algebra. Since V is assumed irreducible, this span must be all of \, V \;. We thus obtain a complete description of what an irreducible representation must look like; that is, a basis for the space and a complete description of how the generators of the Lie algebra act. Conversely, for any \; m \geq 0 \; we can construct a representation by simply using the above formulas and checking that the commutation relations hold. This representation can then be shown to be irreducible. Conclusion: For each non-negative integer \, m \,, there is a unique irreducible representation with highest weight \, m \; . Each irreducible representation is equivalent to one of these. The representation with highest weight \, m \, has dimension \, m + 1 \, with weights \; m, m - 2, \ldots, -(m - 2), -m \; , each having multiplicity one.


The Casimir element

We now introduce the (quadratic)
Casimir element In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
, C given by :C = -\left(u_1^2 + u_2^2 + u_3^2\right). We can view C as an element of the
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representat ...
or as an operator in each irreducible representation. Viewing \, C \, as an operator on the representation with highest weight \, m \,, we may easily compute that \, C \, commutes with each \, u_i \; . Thus, by
Schur's lemma In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ...
, \, C \, acts as a scalar multiple c_m of the identity for each \, m \;. We can write C in terms of the \,\ \, basis as follows: :C = (X + Y)^2 - (-X + Y)^2 + H^2 \; , which can be reduced to :C = 4YX + H^2 + 2H ~. The eigenvalue of \, C \, in the representation with highest weight \, m \, can be computed by applying \, C \, to the highest weight vector, which is annihilated by \, X \; ; thus, we get :c_m = m^2 + 2m = m(m + 2) ~. In the physics literature, the Casimir is normalized as \; C' = \frac\,C ~. Labeling things in terms of \; \ell = \frac\,m \; , the eigenvalue \, d_\ell \, of \, C' \, is then computed as : d_\ell = \frac\,(2\ell)\,(2\ell + 2) = \ell \,(\ell + 1) ~.


The group representations


Action on polynomials

Since SU(2) is simply connected, a general result shows that every representation of its (complexified) Lie algebra gives rise to a representation of SU(2) itself. It is desirable, however, to give an explicit realization of the representations at the group level. The group representations can be realized on spaces of polynomials in two complex variables. That is, for each non-negative integer m, we let V_m denote the space of homogeneous polynomials p of degree m in two complex variables. Then the dimension of V_m is m + 1. There is a natural action of SU(2) on each V_m, given by : \cdot pz) = p\left(U^z\right),\quad z\in\mathbb C^2,\, U\in\mathrm(2). The associated Lie algebra representation is simply the one described in the previous section. (See
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a ...
for an explicit formula for the action of the Lie algebra on the space of polynomials.)


The characters

The
character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of a representation \Pi: G \rightarrow \operatorname(V) is the function \Chi: G \rightarrow \mathbb given by :\Chi(g) = \operatorname(\Pi(g)). Characters plays an important role in the representation theory of compact groups. The character is easily seen to be a class function, that is, invariant under conjugation. In the SU(2) case, the fact that the character is a class function means it is determined by its value on the
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefore ...
T consisting of the diagonal matrices in SU(2), since the elements are orthogonally diagonalizable with the spectral theorem. Since the irreducible representation with highest weight m has weights m, m - 2, \ldots, -(m - 2), -m, it is easy to see that the associated character satisfies :\Chi\left(\begin e^ & 0\\ 0 & e^ \end\right) = e^ + e^ + \cdots + e^ + e^. This expression is a finite geometric series that can be simplified to :\Chi\left(\begin e^ & 0\\ 0 & e^ \end\right) = \frac. This last expression is just the statement of the
Weyl character formula In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
for the SU(2) case. Actually, following Weyl's original analysis of the representation theory of compact groups, one can classify the representations entirely from the group perspective, without using Lie algebra representations at all. In this approach, the Weyl character formula plays an essential part in the classification, along with the Peter–Weyl theorem. The SU(2) case of this story is described
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a ...
.


Relation to the representations of SO(3)

Note that either all of the weights of the representation are even (if m is even) or all of the weights are odd (if m is odd). In physical terms, this distinction is important: The representations with even weights correspond to ordinary representations of the
rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is ...
. By contrast, the representations with odd weights correspond to double-valued (spinorial) representation of SO(3), also known as
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where GL ...
s. In the physics conventions, m being even corresponds to l being an integer while m being odd corresponds to l being a half-integer. These two cases are described as integer spin and
half-integer spin In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
, respectively. The representations with odd, positive values of m are faithful representations of SU(2), while the representations of SU(2) with non-negative, even m are not faithful.


Another approach

See under the example for
Borel–Weil–Bott theorem In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, ...
.


Most important irreducible representations and their applications

Representations of SU(2) describe non-relativistic
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 b ...
, due to being a double covering of the
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
group of
Euclidean 3-space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
. Relativistic spin is described by the representation theory of SL2(C), a supergroup of SU(2), which in a similar way covers SO+(1;3), the relativistic version of the rotation group. SU(2) symmetry also supports concepts of isobaric spin and
weak isospin In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It can ...
, collectively known as ''isospin''. The representation with m = 1 (i.e., l = 1/2 in the physics convention) is the 2 representation, the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defi ...
of SU(2). When an element of SU(2) is written as a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, it is simply a
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being a ...
of column 2-vectors. It is known in physics as the
spin-½ In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one ful ...
and, historically, as the multiplication 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 qua ...
s (more precisely, multiplication by a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
quaternion). This representation can also be viewed as a double-valued
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where GL ...
of the rotation group SO(3). The representation with m = 2 (i.e., l = 1) is the 3 representation, the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...
. It describes 3-d
rotations Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
, the standard representation of SO(3), so
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s are sufficient for it. Physicists use it for the description of massive spin-1 particles, such as
vector meson In high energy physics, a vector meson is a meson with total spin 1 and odd parity (usually noted as ). Vector mesons have been seen in experiments since the 1960s, and are well known for their spectroscopic pattern of masses. The vector meso ...
s, but its importance for spin theory is much higher because it anchors spin states to the
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
of the physical
3-space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
. This representation emerged simultaneously with the 2 when
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irel ...
introduced
versor In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
s, his term for elements of SU(2). Note that Hamilton did not use standard
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
terminology since his work preceded Lie group developments. The m = 3 (i.e. l = 3/2) representation is used in
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
for certain
baryon In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classifi ...
s, such as the Δ.


See also

* Rotation operator (vector space) * Rotation operator (quantum mechanics) * Representation theory of SO(3) * Connection between SO(3) and SU(2) * representation theory of SL2(R) *
Electroweak interaction In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
* Rotation group SO(3) § A note on Lie algebra


References

* * Gerard 't Hooft (2007)
''Lie groups in Physics''
Chapter 5 "Ladder operators" * {{DEFAULTSORT:Representation Theory Of Su(2) Representation theory of Lie groups Rotation in three dimensions