Quantum Mechanics for Mathematicians: The Rotation and ...€¦ · Quantum Mechanics for Mathematicians: The Rotation and Spin Groups in 3 and 4 Dimensions Peter Woit Department of

Quantum Mechanics for Mathematicians: The

Rotation and Spin Groups in 3 and 4 Dimensions

Peter WoitDepartment of Mathematics, Columbia University

[email protected]

October 25, 2012

Among the basic symmetry groups of the physical world is the orthogonalgroup SO(3) of rotations about a point in three-dimensional space. The observ-ables one gets from this group are the components of angular momentum, andunderstanding how the state space of a quantum system behaves as a representa-tion of this group is a crucial part of the analysis of atomic physics examples andmany others. Unlike some of the material covered in this course, this is a topicyou will find in some version or other in every quantum mechanics textbook.

Remarkably, it turns out that the quantum systems in nature are oftenrepresentations not of SO(3), but of a larger group called Spin(3), one thathas two elements corresponding to every element of SO(3). This is called the“Spin” group, and in any dimension n it exists, always as a “doubled” versionof the orthogonal group SO(n), one that is needed to understand some of themore subtle aspects of geometry in n dimensions. In the n = 3 case it turnsout that Spin(3) ' SU(2) and we will study in detail the relationship of SO(3)and SU(2). This appearance of the unitary group SU(2) is special to 3 (and tosome extent 4) dimensions.

1 The Rotation Group in Three Dimensions

In R2 rotations about the origin are given by elements of SO(2), with a counter-clockwise rotation by an angle θ given by the matrix

R(θ) =

(cos θ − sin θsin θ cos θ

)This can be written as an exponential, R(θ) = eθL = cos θ + L sin θ for

L =

(0 −11 0

)Here SO(2) is a commutative Lie group with Lie algebra R (the Lie bracket istrivial, all elements of the Lie algebra commute). Note that we have a represen-tation on V = R2 here, but it is a real representation, not one of the complex

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ones we have when we have a representation on a quantum mechanical statespace.

In three dimensions the group SO(3) is 3-dimensional and non-commutative.One now has three independent directions one can rotate about, which one cantake to be the x, y and z-axes, with rotations about these axes given by

Rx(θ) =

1 0 00 cos θ − sin θ0 sin θ cos θ

Ry(θ) =

cos θ 0 sin θ0 1 0

− sin θ 0 cos θ

Rz(θ) =

cos θ − sin θ 0sin θ cos θ 0

0 0 1

A standard parametrization for elements of SO(3) is in terms of 3 “Euler angles”φ, θ, ψ with a general rotation given by

R(φ, θ, ψ) = Rz(ψ)Rx(θ)Rz(φ)

i.e. first a rotation about the z-axis by an angle φ, then a rotation by anangle θ about the new x-axis, followed by a rotation by ψ about the new z-axis.Multiplying out the matrices gives a rather complicated expression for a rotationin terms of the three angles, and one needs to figure out what the proper rangeto choose for the angles to avoid multiple counting.

The infinitesimal picture near the identity of the group, given by the Liealgebra structure on so(3) = R3 is much easier to understand. Recall thatfor orthogonal groups the Lie algebra can be identified with the space of anti-symmetric matrices, so one in this case has a basis

l1 =

0 0 00 0 −10 1 0

l2 =

0 0 10 0 0−1 0 0

l3 =

0 −1 01 0 00 0 0

which satisfy the commutation relations

[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

Note that these are exactly the relations satisfied by the cross-product of stan-dard basis vectors ei in R3:

e1 × e2 = e3, e2 × e3 = e1, e3 × e1 = e2

so the Lie bracket operation

(X,Y ) ∈ R3 ×R3 → [X,Y ] ∈ R3

that makes R3 a Lie algebra so(3) is just the cross-product on vectors in R3.Something very special that happens only in 3-dimensions is that the vector

representation is isomorphic to the adjoint representation. Recall that any Lie

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group G has a representation (Ad, g) on its Lie algebra g. so(n) can be identified

with the anti-symmetric n by n matrices, so is of (real) dimension n2−n2 . Only

for n = 3 is this equal to n, the dimension of the representation on vectors in Rn.This corresponds to the geometrical fact that only in 3 dimensions is a plane (inall dimensions rotations are built out of rotations in various planes) determineduniquely by a vector (the vector perpendicular to the plane). Equivalently,only in 3 dimensions is there a cross-product v × w which takes two vectorsdetermining a plane to a vector perpendicular to the plane.

To see explicitlly what is going on, let’s look at the vector and adjoint Liealgebra representations. The commutation relations for the li determine the Liealgebra representation (ad, so(n)) using (ad(X))(Y ) = [X,Y ]. For instance, forinfinitesimal rotations about the x-axis, one has for the adjoint representation

(ad(l1))(l1) = 0, (ad(l1))(l2) = l3, (ad(l1))(l3) = −l2On vectors, such infinitesimal rotations act on the standard basis ei of Rn bymatrix multiplication, giving

l1e1 = 0, l1e2 = e3, l1e3 = −e2from which one can see that one has the same Lie algebra representation, withthe isomorphism identifying li = ei. At the level of the group, rotations aboutthe x-axis by an angle θ correspond to matrices

eθl1 =

1 0 00 cos θ − sin θ0 sin θ cos θ

which act by conjugation on anti-symmetric real matrices and in the usual wayon column vectors

Our two isomorphic representations are on column vectors (“vector” repre-sentation on R3) and on antisymmetric real matrices (“adjoint” representationon so(3)), with the isomorphism given byv1v2

v3

↔ 0 −v3 v2v3 0 −v1−v2 v1 0

On the column vectors both the Lie algebra and the Lie group representationare given just by matrix multiplication. On the antisymmetric matrices the Liegroup representation is given by

Ad(g)

0 −v3 v2v3 0 −v1−v2 v1 0

= g

0 −v3 v2v3 0 −v1−v2 v1 0

g−1

where g is a 3 by 3 orthogonal matrix. On the antisymmetric matrices the Liealgebra representation is given by

ad(X)

0 −v3 v2v3 0 −v1−v2 v1 0

= [X,

0 −v3 v2v3 0 −v1−v2 v1 0

]

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where X is a 3 by 3 antisymmetric matrix.

2 Spin Groups in Three and Four Dimensions

A remarkable property of the orthogonal groups SO(n) is that they come with anassociated group, called Spin(n), with every element of SO(n) correspondingto two distinct elements of Spin(n). If you have seen some topology, whatis at work here is that the fundamental group of SO(n) is non-trivial, withπ1(SO(n)) = Z2. Spin(n) is topologically the simply-connected double-coverof SO(n), and one can choose the covering map Φ : Spin(n) → SO(n) to bea group homomorphism. Spin(n) is a Lie group of the same dimension, withan isomorphic tangent space at the identity, so the Lie algebras of the twogroups are isomorphic so(n) ' spin(n). We will construct Spin(n) and thecovering map Φ only for the cases n = 3 and n = 4, with higher dimensionalcases requiring more sophisticated techniques. For n = 3 it turns out thatSpin(n) = SU(2), and for n = 4, Spin(4) = SU(2) × SU(2). For perhaps thesimplest way to see how this works, it is best to not use just real and complexnumbers, but also bring in a third number system, the quaternions.

2.1 Quaternions

The quaternions are a number system (denoted by H) generalizing the complexnumber system, with elements q ∈ H that can be written as

q = q0 + q1i + q2j + q3k, qi ∈ R

with i, j,k ∈ H satisfying

i2 = j2 = k2 = −1, ij = k,ki = j, jk = i

and a conjugation operation that takes

q → q = q0 − q1i− q2j− q3k

As a vector space over R, H is isomorphic with R4. The length-squaredfunction on this R4 can be written in terms of quaternions as

|q|2 = qq = q20 + q21 + q22 + q23

Sinceqq

|q|2= 1

one has a formula for the inverse of a quaternion

q−1 =q

|q|2

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From this one can also see that the conjugation operation satisfies (for u, v ∈ H)

uv = vu

and that the length-squared function is multiplicative:

|uv|2 = uvuv = uvvu = |u|2|v|2

The length one quaternions thus form a group under multiplication, called Sp(1).There are also Lie groups called Sp(n), consisting of invertible matrices withquaternionic entries that act on quaternionic vectors preserving the quaternioniclength-squared, but these play no role in quantum mechanics so we won’t studythem further. Sp(1) can be identified with the three-dimensional sphere sincethe length one condition on q is

q20 + q21 + q22 + q23 = 1

the equation of the unit sphere S3 ⊂ R4.

2.2 Rotations and Spin Groups in Four Dimensions

Pairs (u, v) of unit quaternions give the product group Sp(1) × Sp(1). Anelement of this group acts on H = R4 by

q → uqv

and this action preserves lengths of vectors and is linear in q, so it must corre-spond to an element of the group SO(4). One can easily see that pairs (u, v) and(−u,−v) give the same orthogonal transformation of R4, so the same elementof SO(4). One can show that SO(4) is the group Sp(1)× Sp(1), with elements(u, v) and (−u,−v) identified. The name Spin(4) is given to the Lie group that“double covers” SO(4), so here Spin(4) = Sp(1)× Sp(1).

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2.3 Rotations and Spin Groups in Three Dimensions

Later on in the course we’ll encounter Spin(4) and SO(4) again, but for nowwe’re interested in its subgroup Spin(3) that only acts on 3 of the dimensions,and double-covers not SO(4) but SO(3). To find this, consider the subgroup ofSpin(4) consisting of pairs (u, v) of the form (u, u−1) (a subgroup isomorphicto Sp(1), since elements correspond to a single unit length quaternion u). Thissubgroup acts on quaternions by conjugation

q → uqu−1

an action which is trivial on the real quaternions, but nontrivial on the “pureimaginary” quaternions of the form

q = ~v = v1i + v2j + v3k

An element u ∈ Sp(1) acts on ~v ∈ R3 ⊂ H as

~v → u~vu−1

This is a linear action, preserving the length |~v|, so corresponds to an elementof SO(3). We thus have a map (which can be checked to be a homomorphism)

Φ : u ∈ Sp(1)→ {~v → u~vu−1} ∈ SO(3)

Both u and −u act in the same way on |~v|, so we have two elements inSp(1) corresponding to the same element in SO(3). One can show that Φ is a

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surjective map (one can get any element of SO(3) this way), so it is what is calleda “covering” map, specifically a two-fold cover. It makes Sp(1) a double-coverof SO(3), and we give this the name “Spin(3)”. This also tells us exactly which3-dimensional space SO(3) is. It is S3 = Sp(1) = Spin(3) with opposite pointson the three-sphere identified. This space is known as RP(3), real projective3-space, which can also be thought of as the space of lines through the origin inR4 (each such line intersects S3 in two opposite points).

For those who have seen some topology, note that the covering map Φ is anexample of a topological non-trivial cover. It is just not true that topologicallyS3 ' RP3 × (+1,−1). S3 is a connected space, not two disconnected pieces.This topological non-triviality implies that globally there is no possible homo-morphism going in the opposite direction from Φ (i.e. SO(3)→ Spin(3)). Onecan do this locally, picking a local patch in SO(3) and taking the inverse of Φto a local patch in Spin(3), but this won’t work if we try and extend it globallyto all of SO(3).

The relationship between rotations of R3 and unit quaternions is quite sim-ple: for

~w = w1i + w2j + w3k

a unit vector in R3 ⊂ H, conjugation by the unit quaternion

u(θ, ~w) = cos θ + ~w sin θ

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gives a rotation about the ~w axis by an angle 2θ. The factor of 2 here reflectsthe fact that unit quaternions double-cover the rotation group SO(3): takingθ → θ + π gives a different unit quaternion that implements the same rotationin 3-space. To see how this works, one can for example take as axis of rotationthe z-axis by choosing ~w = k. The unit quaternion

uθ = cos θ + k sin θ

has inverseu−1θ = cos θ − k sin θ

and acts on ~v = v1i + v2j + v3k as

~v → uθ~vu−1θ =(cos θ + k sin θ)(v1i + v2j + v3k)(cos θ − k sin θ)

=(v1(cos2 θ − sin2 θ)− v2(2 sin θ cos θ))i

+ (2v1 sin θ cos θ + v2(cos2 θ − sin2 θ))j + v3k

=(v1 cos 2θ − v2 sin 2θ)i + (v1 sin 2θ + v2 cos 2θ)j + v3k

=

cos 2θ − sin 2θ 0sin 2θ cos 2θ 0

0 0 1

v1v2v3

Here one sees explicitly for rotations about the z-axis the double-covering map

Φ : uθ = cos θ + k sin θ ∈ Sp(1) = Spin(3)→

cos 2θ − sin 2θ 0sin 2θ cos 2θ 0

0 0 1

∈ SO(3)

As θ goes from 0 to 2π, uθ traces out a circle in Sp(1). The double-coveringhomomorphism Φ takes this to a circle in SO(3), one that gets traced out twiceas θ goes from 0 to 2π.

In the case of U(1), the unit vector in R2, one can take the identity tobe in the real direction, and then the tangent space at the identity (the Liealgebra u(1)) is iR, with basis i. For the case of Sp(1), one can again takethe identity to be in the real direction, and the tangent space (the Lie algebrasp(1)) is isomorphic to R3, with basis vectors i, j,k. The Lie bracket is just thecommutator, e.g.

[i, j] = ij− ji = 2k

Linear combinations of these basis vectors are precisely the velocity vectors onegets for the paths u(θ, ~w) = cos θ+ ~w sin θ, which are parametrized by θ and gothrough the identity at θ = 0, since

d

dθu(θ, ~w)|θ=0 = (sin θ + ~w cos θ)|θ=0 = ~w = w1i + w2j + w3k

The derivative of the map Φ will be a linear map

Φ′ : sp(1)→ so(3)

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It takes pure imaginary quaternions to 3 by 3 antisymmetric real matrices. Oncan compute it easily on basis vectors, using for instance the formula above

Φ(cos θ + k sin θ) =

cos 2θ − sin 2θ 0sin 2θ cos 2θ 0

0 0 1

Using the definition as a derivative with respect to the parameter of a path,evaluated at the identity, one finds

Φ′(k) =d

dθΦ(cos θ + k sin θ)|θ=0

=

−2 sin 2θ −2 cos 2θ 02 cos 2θ −2 sin 2θ 0

0 0 0

|θ=0

=

0 −2 02 0 00 0 0

= 2l3

Repeating this on other basis vectors one finds that

Φ′(i) = 2l1,Φ′(j) = 2l2,Φ

′(k) = 2l3

Thus Φ′ is an isomorphism of sp(1) and so(3) identifying the bases

i

2,j

2,k

2and l1, l2, l3

Note that it is the i2 ,

j2 ,

k2 that satisfy simple commutation relations

[i

2,j

2] =

k

2, [

j

2,k

2] =

i

2, [

k

2,i

2] =

j

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2.4 The Spin Group and SU(2)

Instead of working with quaternions and their non-commutativity and specialmultiplication laws, it is more conventional to choose an isomorphism betweenquaternions H and a space of 2 by 2 complex matrices, and work just withmatrix multiplication and complex numbers. The Pauli matrices can be usedto gives such an isomorphism, taking

1→ 1 =

(1 00 1

), i→ −iσ1 =

(0 −i−i 0

), j→ −iσ2 =

(0 −11 0

)

k→ −iσ3 =

(−i 00 i

)The correspondence between H and 2 by 2 complex matrices is then given

by

q = q0 + q1i + q2j + q3k↔(q0 − iq3 −q2 − iq1q2 − iq1 q0 + iq3

)

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Since

det

(q0 − iq3 −q2 − iq1q2 − iq1 q0 + iq3

)= q20 + q21 + q22 + q23

we see that the length-squared function on quaternions corresponds to the de-terminant function on 2 by 2 complex matrices. Taking q ∈ Sp(1), so of lengthone, corresponds both to the matrix having determinant one, and its rows andcolumns having length one, so the complex matrix is in SU(2).

Recall that any SU(2) matrix can be written in the form(α β

−β α

)with α, β ∈ C arbitrary complex numbers satisfying |α|2 + |β|2 = 1. Theisomorphism with unit vectors in H is given by

α = q0 − iq3, β = −q2 − iq1

We see that Sp(1), Spin(3) and SU(2) are all names for the same group, geo-metrically S3, the unit sphere in R4.

Under our identification of H with 2 by 2 complex matrices, we have anidentification of Lie algebras sp(1) = su(2) between pure imaginary quaternionsand skew-Hermitian trace-zero 2 by 2 complex matrices

~w = w1i+w2j+w3k↔(−iw3 −w2 − iw1

w2 − iw1 iw3

)= −i

(w3 w1 − iw2

w1 + iw2 −w3

)The basis i

2 ,j2 ,

k2 gets identified with a basis for the Lie algebra su(2) which

written in terms of the Pauli matrices is

sa = −iσa2

with the sa satisfying the commutation relations

[s1, s2] = s3, [s2, s3] = s1, [s3, s1] = s2

which are precisely the same commutation relations as for so(3)

[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

We now have no less than three isomorphic Lie algebras sp(1) = su(2) =so(3) on which we have the adjoint representation, with bases that get identifiedunder the following . These are identified as follows:

(w1

i2 + w2

j2 + w3

k2

)↔ − i

2

(w3 w1 − iw2

w1 + iw2 −w3

)↔

0 −w3 w2

w3 0 −w1

−w2 w1 0

with this isomorphism identifying basis vectors as

i

2↔ −iσ1

2↔ l1

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etc. The first of these identifications comes from the way we identify H with2 by 2 complex matrices, these are Lie algebras of isomorphic groups. Thesecond identification is Φ′, the derivative at the identity of the covering map Φ :Sp(1) = SU(2)→ SO(3) that is a homomorphism between two non-isomorphicgroups.

On each of these we have adjoint Lie group (Ad) and Lie algebra (ad) repre-sentations, with Ad given by conjugation with the corresponding group elementsin Sp(1), SU(2) and SO(3), and ad given by taking commutators in the respec-tive Lie algebras of pure imaginary quaternions, skew-Hermitian trace-zero 2 by2 complex matrices and 3 by 3 real antisymmetric matrices.

Note that these three Lie algebras are all three-dimensional real vectorspaces, so these are real representations. If one wants a complex representa-tion, one can complexify and take complex linear combinations of elements.This is less confusing in the case of su(2) than for sp(1) since taking complexlinear combinations of skew-Hermitian trace-zero 2 by 2 complex matrices justgives all trace-zero 2 by 2 matrices.

In addition, recall from earlier that there is a fourth isomorphic version ofthis representation, the representation of SO(3) on column vectors. This is alsoa real representation, but can straightforwardly be complexified.

At the level of Lie groups we have seen that our identification of H and 2by 2 matrices identifies Sp(1) with SU(2), taking

u(θ, ~w)→ cos θ1− i(~w · ~σ) sin θ

=

(cos θ − iw3 sin θ (−iw1 − w2) sin θ

(−iw1 + w2) sin θ cos θ + iw3 sin θ

)The relation to SO(3) rotations is that this is an SU(2) element such that,

if one identifies vectors (v1, v2, v3) ∈ R3 with complex matrices(v3 v1 − iv2

v1 + iv2 −v3

)(this is the same identification as used before, up to an irrelevant overall scalar),then

(cos θ1− i(~w · ~σ) sin θ)

(v3 v1 − iv2

v1 + iv2 −v3

)(cos θ1− i(~w · ~σ) sin θ)−1

is the same vector, rotated by an angle 2θ about the axis given by ~w.We will define

R(θ, ~w) = eθ ~w·~s

= cos(θ

2)1− i(~w · ~σ) sin(

θ

2)

and then it is conjugation by R(θ, ~w) that rotates vectors by an angle θ about~w.

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In particular, rotation about the z-axis by an angle θ is given by conjugationby

R(θ,

001

) =

(e−i

θ2 0

0 eiθ2

)

In terms of the group SU(2), the double covering map Φ thus acts on diag-onalized matrices as

Φ :

(e−iθ 0

0 eiθ

)∈ SU(2)→

cos 2θ − sin 2θ 0sin 2θ cos 2θ 0

0 0 1

One can write down a somewhat unenlightening formula for the map Φ :

SU(2)→ SO(3) in general, getting

Φ(

(α β

−β α

)) =

Re(α2 − β2) −2Im(α2 + β2) −2Re(αβ)Im(β2 − α2) i2Re(α2 + β2) 2Im(αβ)

2αβ 2Im(αβ) |α|2 − |β|2

See [2], page 123-4, for a derivation (I’ve redone the calculation myself using myconventions, seem to have introduced an error, which the class will fix for mewhen they complete the third problem set).

3 Spin groups in higher dimension

In this course we won’t encounter again orthogonal groups above 3 or 4 di-mensions, but the phenomenon of spin groups occurs in every dimension. Foreach n > 2, the orthogonal group SO(n) is double-covered by a group Spin(n)with an isomorphic Lie algebra. Special phenomena relating these Spin groupsoccur for n < 7 (it turns out that Spin(5) = Sp(2) (2 by 2 norm-preservingquaternionic matrices), and Spin(6) = SU(4), but in higher dimensions thesegroups have no relation to quaternions or unitary groups. The construction ofthe double-covering map Spin(n)→ SO(n) for n > 4 requires use of a new classof algebras that generalize the quaternion algebra, known as Clifford algebras.

4 For Further Reading

For another discussion of the relationship of SO(3) and SU(2) as well as aconstruction of the map Π, see [2], sections 4.2 and 4.3, as well as [1], chapter8, and [3] Chapters 2 and 4.

References

[1] Artin, M., Algebra, Prentice-Hall, 1991.

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[2] Singer, S., Linearity, Symmetry, and Prediction in the Hydrogen Atom,Springer-Verlag, 2005.

[3] Stillwell, J., Naive Lie Theory Springer-Verlag, 2010.http://www.springerlink.com/content/978-0-387-78214-0

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