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 Permanent Link:
 http://digital.auraria.edu/AA00001842/00001
Material Information
 Title:
 Applications of Lie algebras to physics
 Creator:
 Wood, Laura Ellen
 Place of Publication:
 Denver, Colo.
 Publisher:
 University of Colorado Denver
 Publication Date:
 1990
 Language:
 English
 Physical Description:
 vi, 52 leaves : ; 29 cm
Thesis/Dissertation Information
 Degree:
 Master's ( Master of Arts)
 Degree Grantor:
 University of Colorado Denver
 Degree Divisions:
 Department of Mathematical and Statistical Sciences, CU Denver
 Degree Disciplines:
 Mathematics
 Committee Chair:
 Payne, Stanley E.
 Committee Members:
 Cherowitzo, William E.
Briggs, William L.
Subjects
 Subjects / Keywords:
 Lie algebras ( lcsh )
Physics ( lcsh ) Lie algebras ( fast ) Physics ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Bibliography:
 Includes bibliographical references (leaf 52).
 Thesis:
 Submitted in partial fulfillment of the requirements for the degree, Master of Arts, Department of Mathematical and Statistical Science
 Statement of Responsibility:
 by Laura Ellen Wood.
Record Information
 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 22947560 ( OCLC )
ocm22947560
 Classification:
 LD1190.L62 1990m .W66 ( lcc )

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APPLICATIONS OF LIE ALGEBRAS TO PHYSICS
by
Laura Ellen Wood
B.S., University of South Carolina, 1985
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Arts
Department of Mathematics
1990
1990 by Laura Ellen Wood
All rights reserved.
This thesis for the Master of Arts degree by
Laura Ellen Wood
has been approved for the
Department of Mathematics
by
ALMIP
Date
Wood, Laura Ellen (MA, Mathematics)
Applications of Lie Algebras to Physics
Thesis directed by Professor Stanley E. Payne
A brief review of basic abstract algebra is given, followed by definitions of Lie groups
and Lie algebras and a discussion of their relation to each other. Representations of Lie
algebras are defined and then shown as examples in applications to physics. Various
(
representations of su(2) and the Heisenberg algebra are carried throughout the paper.
Classical Hamiltonian mechanics is developed and shown to have a Lie algebraic
t
representation with the Poisson bracket. The one dimensional quantum mechanic harmonic
oscillator is developed using the Heisenberg Lie algebra and the raising and lowering
operators. This model is then extended to multiple dimensions and used to construct a few
representations of su(2) in the study of angular momentum. Examples are distributed
throughout the paper to aid intuitive understanding.
The form and content of this abstract are approved. I recommend its publication.
Signed
Stanley E. Payne
Contents
1.0 Introduction........................................................................1
2.0 Review of Basic Algebra.............................................................2
2.1 Groups......................................................................2
2.2 Reids.......................................................................2
2.3 Vector Spaces...............................................................3
2.4 Algebras................................................................... 3
3.0 Introduction to Lie Groups..........................................................5
3.1 Topological Groups..........................................................5
3.2 Analysis and Lie Groups.....................................................5
3.2.1 Analyticity........................................................6
3.2.2 Analytic Manifold................................................ 7
3.3 The Linear Lie Group GL(n,F) and SU(2)......................................7
4.0 Introduction to Lie Algebras........................................................10
4.1 {9t3. +v. 0y. x} as an Example of a Lie Algebra..............................10
4.2 Linear Lie Algebras, gl(n,F), and su(n)......................................11
5.0 The Relationship between Lie Algebras and Lie Groups................................13
5.1 SU(2) and su(2) as an Example...............................................13
5.2 One Parameter Groups...................................................... 14
6.0 Representations of Lie Algebras....................................................17
7.0 Application of Lie Algebras Classical............................................19
7.1 Introduction to Generalized Systems........................................19
7.2 Lagrangian Mechanics.......................................................20
vi
7.3 Hamiltonian Mechanics......................................................23
7.4 The Hamiltonian and the Poisson Brackets...................................24
7.5 The Hamiltonian, the Poisson Brackets, and Conservation....................25
7.6 {f,{}} Is a Lie Algebra...................................................26
8.0 Converting to Quantum Mechanics....................................................29
9.0 Application of Lie Algebras Quantum Mechanical Harmonic Oscillator...............32
9.1 The Heisenberg Algebra Revisited.............................................34
9.2 The Heisenberg Representation of the Harmonic Oscillator...................35
9.3 The Raising and Lowering Operators.........................................35
9.4 Dirac Notation.............................................................40
10.0 Extension of the Harmonic Oscillator Boson Operators............................42
11.0 Construction of su(2) Using Boson Operators.......................................44
12.0 Conclusion........................................................................51
Figures
Figure
1.0 The Stereographic Projection......................................................9
I
1
1.0 Introduction
"All physics is the study of Lie groups and algebras." This statement appears
extreme, and it definitely reveals the attitudes of a mathematician more than a physicist, but it
does imply the strength of the connection between Lie theory and the study of physics.
Sophus Lie (18421899) provided the foundation of Lie group theory in his work in
continuous groups. He also reduced local problems in Lie groups to corresponding problems
in Lie algebras. His work in the 19th century laid the groundwork for much advancement in
theoretical physics.
The use of Lie groups and their associated Lie algebras in physics is widespread,
extending from classical physics through to the most advanced representations in modem
physics. Lie theory may be applied in classical physics in operations as simple as the rotation
of space around axes, or in more complicated operations modeling fluid flows. The
applications in modem physics are especially numerous, including the addition and
subtraction of energy from a system, the strong interactions between nucleons, and the weak
and electromagnetic interactions.
This paper gives a brief introduction to Lie theory and applies the theory in areas of
classical and modem physics. The algebra su(2) is carried throughout the paper to show the
power and utility of only one algebra among infinitely many.
2
2.0 Review of Basic Algebra
First, a brief review of abstract algebra [1] is given for completeness.
2.1 Groups
Let G be a nonvacuous set, and let be an associative binary operation taking G x G
intoG. Then, for a, b, c e G,
1. a*beG
2. a(b*c) = (a*b)*c.
Let there exist an element 1 in G, called an identity, such that for all a e G:
3. a 1 = 1 a = a.
Now, for each a e G, let there exist an element b e G such that
4. a*b = b*a = l.
A set {G, *, 1} satisfying the above conditions is called a group.
The binary operation : G x G > G need not be commutative. That is, a* b does not
necessarily equal b a. If a b = b a for all a, b e G, the group is said to be commutative or
abelian. An example of an abelian group is the real numbers 9t, except for the number 0, with
the usual multiplication and the number 1 as the identity {91 {0}, , 1}.
2.2 Fields
The concept of a field is an extension of the concept of a group. A field {F, *,+,1,0}
is a nonvacuous set F together with two associative binary operations, and +, and two
distinct elements of F, 1 and 0, such that
1 {F, +, 0} is an abelian group.
2. {F {0}, , 1} is an abelian group.
3. Fora,b,CEF, a*(b + c) = a*b + a*c
(b+c)*a = ba+c*a.
An example of a field is the real numbers Sit, with the usual addition and multiplication, and
identities the numbers 0 and 1: {SR, *, +, 1, 0}.
2.3 Vector Spaces
A vector space can be described as a combination of a group and a field. A vector
space consists of an abelian group {V, tv, 0V), a field {F, \ +, 1,0}, and a scalar multiplication
taking F x V into V which satisfies
1. a(x+vy) ^ax+vay for a e F; x. y e V.
2. (a+b)x=ax+vbx for a, b e F; x e V.
3. (a b) x = a (bx) for a, b e F; x e V.
4. 1x =x for x e V.
An example of a vector space is the field of the real numbers {SR, , +, 1,0} together
with the abelian group {SRn, +v, 0v} of ntuples of real entries, (xi, %2, X3,..., xn) with xj e SR,
such that x tv y = (xt + yj, X2 + y2.. *n + Vn) and 0v = (0,0.0). The scalar multiplication
applies termwise so that, for a e SR, ax = (axi, ax2, axg,..., axn).
2.4 Algebras
As the concept of a group is extended to make a field, the concept of a vector space
is extended to make an algebra. Let {A, +a, 0a} be a vector space over a field {F, \ +, 1,0}.
Now let *a be a bilinear multiplication taking A x A into A. The bilinear multiplication has the
following properties for x, y, z e A, and a e F.
1 (X+Ay) AZ = (X*AZ)+A(y*AZ)
2. x *a (y +a z) = (x A y) +A (x *A *)
3. a (x A y) = (ax) *A y = x A (ay)
The resulting set, {A, +a. 0a, *a}> together with the field and scalar multiplication from the
vector space, form an algebra.
4
An example of an algebra is the set of n x n matrices with entries in the real numbers,
Mn(9t), forming a vector space over the field of real numbers, {9t, , +, 1,0}, with the usual
matrix product as the bilinear multiplication a
5
3.0 Introduction to Lie Groups
The study of Lie groups unites three branches of mathematics algebra, analysis, and
topology. The basics of algebra have been discussed briefly above. Some concepts in
topology and analysis that apply to Lie groups now follow.
3.1 Topological Groups
A Lie group has a topological group as its base. "A topological group is a group which
is also a topological space (so that ideas such as continuity, connectedness and compactness
apply) in which the group operations are continuous" [2; p. 3], Recall that a topology on a set
establishes the notion of "nearness" of one element of the set to another. If the underlying
set of a group has a topology associated with it such that the group operations are
continuous, it is a topological group.
Let {G, , 1} be a group. There are two group operations the binary operation and
the map that sends an element to its inverse. These operations are continuous if, for xi, X2,
Y1. Y2 e G such that x\ is near to X2 and yi is near to y2,
1. xi yi is near to X2 y2. and
2. xi*1 is near to X2*1.
An example of a topological group is {31, +, 0} with the usual absolute value distance function
establishing the topology.
3.2 Analysis and Lie Groups
A Lie group is a topological group in which the group operations are not only
continuous, but also analytic. P. M. Cohn defines [3; p. 44] a Lie group as follows.
1. {G, *, 1} is a group.
2. G is an analytic manifold.
3. The mapping G x G > G taking (x,y) x y is analytic.
He then proves that the mapping x x'1 is analytic.
6
To understand the concept of a Lie group, the concepts of analyticity and analytic
manifolds must be discussed.
3.2.1 Analyticity
A topological space T is said to be locally Euclidean at a point p if a homeomorphism
can be found from a neighborhood of p into 9tn [3; p. 4,5]. This homeomorphism is called a
chart of p.
A topological space T is said to be locally homeomorphic to 9tn if any nonempty open
subset of T can be mapped by a homeomoiphism to a nonempty open subset of 9tn. Let W
be an open subset of T and let o be a homeomoiphism from W to X, an open subset of 9tn.
Now consider a homeomorphism $from X to Y, another subset of 3tn, and let Y be its
inverse. Then, for x, y e 9tn,
(1a) yi = i M,
(1b) Xi = Yi(y)
The homeomorphisms a and <& can be composed to form another chart
The two charts a and 4>(o), whose coordinates are related by (1), are analytically related if the
functions 4> and Y are analytic functions on their domains X and Y. A function is said to be
analytic at a point p of 9tn if it can be expressed as a convergent power series in (x; p;) for i =
1,2,...,n in some neighborhood of p.
Thus, for group operations to be analytic, homeomorphisms must exist from the
spaces G x G into 9tn and from G into 9tn whose ranges in 9tn can be analytically related as
above for any point in the group. That is, suppose two subsets T and W of G have charts o
and p respectively. Then the group operation mapping T x W into S in G is analytic if there
exists a chart x of S such that there is an analytic map in 9tn taking ctx p into x.
7
3.2.2 Analytic Manifold
A topological space that is locally Euclidean at each point of the space is called a
manifold [3; p.5,6]. An analytic manifold has added constrictions on it.
An analytic manifold has as its base a Hausdorff space, also known as a T2space. A
Hausdorff space is a space M which satisfies the second axiom of countability, which states
that for each pair of distinct points x and y in M, there are neighborhoods n(x) of x and Tj(y) of y
such that t(x) and q(y) have the empty set as their intersection [4; p. 85],
An analytic manifold has a Hausdorff space M as a base, and then builds an analytic
structure on it. An analytic structure is a family $ of charts on M such that
1. At each point of M there is a chart which belongs to
2. Any two charts of if are analytically related.
3. Any chart of M which is analytically related to every chart
of & itself belongs to if.
In summary, a Lie group is a topological group which has nice, Euclideanlike structure
locally, and which has smooth, wellbehaved group operations. An example of a Lie group is
{*, +, 0).
Different authors assume different qualities in their definitions of Lie groups. For
example, some define a Lie group as a differentiable manifold with a group structure, and
state that the set of infinitely differentiable functions on the real line is "the" Lie group. The
less restrictive definitions are used in this paper.
3.3 The Linear Lie Group GL(n,F) and SU(2)
One Lie group is especially useful in physics applications. This group is the set of all
nonsingular n x n matrices together with the usual matrix multiplication. It is called the general
8
linear group, and it is denoted GL(n,9t) or GL(n,Â£) depending on whether the entries are real
or complex.
A subgroup SU(2) of GL(2,<Â£) is the set of all 2 x 2 matrices over the complex field
which are unitary and whose determinant is 1. The notation SU(2) is used instead of SU(2,<Â£)
for simplicity. Thus, for U e SU(2),
U*U = 1 s the identity matrix, a2 + jp2=1,
where denotes the complex conjugate transpose and a bar over a complex number denotes
its complex conjugate.
It can be shown [2; p. 814] that three matrices of SU(2) defined below, U^(a) U^P),
and U^fy), can represent the group of rotations of the Riemann sphere. In fact, if U^(a)
represents a specific rotation, U^(a) represents that same rotation. The representation of
SU(2) by the matrices U is of primary importance in the study of particles with spin of onehalf,
such as the electron. The reason for this is that one rotation in three space can be
represented by two different matrices, U. This forms a double valued map from the rotations
of three space into Sll(2). Particles of spin onehalf transform under a rotation in three
dimensions according to these double valued maps, called spinor representations.
The matrices U^ajjU^tP), and U^(y) are as follows.
U$(a)
Un(P) =
U;(t)
/ a cost i sin
. a a
t sinT cost
V y
^cos  sin^
sin cos
V y
( & 0 >
s e2 Si.
0 6 2
V
9
ix iy iziJ 1
Â£" III2 + 1 T?_ Ul2 + 1 lz2 + l
Figure 1. The Stereographic Projection.
Source: Reprinted from D. H. Sattinger and O. L. Weaver.
Lie groups and algebras with applications to ohvsics.
geometry, and mechanics. SpringerVerlag New York, Inc.,
New York, 1986
This is a picture that shows the axes of rotation of the Riemann sphere by the above matrices
and the relationships between the angles and complex numbers.
A A An
If eA, for A a matrix, is defined e* the matrices U^, U^, and llÂ£ can be written
as
i(a/2)a1
U$(a) = e
i(p/2)o2
Uq(P) = e
i(y/2)a3
U;(y) = e
for cri, 02, and
01 =
r o
o)
02 =
(?
03 =
i
0
0
1
)
)
10
4.0 Introduction to Lie Algebras
Algebras can be either associative or nonassociative, meaning that the bilinear
multiplication of the algebra is either associative or nonassociative. Associative algebras
satisfy
(x*Ay)*AZ = x*A(y*Az)
for x, y, z e A, while nonassociative algebras may not. A nonassociative algebra whose
multiplication satisfies
1. xAy = y'AX
2. (x *A (y A Z)) +A (y *A (z 'A X)) +A (z *A (x *A Y)) A
for x, y, z e A is called a Lie algebra [5],
In a Lie algebra, there can be no identity for the multiplication. To prove this, assume
there exists an identity 1 a. and let x e A. Then 1a *A x = x *A 1A However, Property 1 states
t
that 1 a *a x = x >a 1ai which is a contradiction. Thus, there can be no identity in a Lie algebra.
4.1 (3*3, +V) oVl x} as an Example of a Lie Algebra
An example of a Lie algebra is {9t3, +v, 0V, x}, where the vector space {3t3, +v, 0V} is
the three dimensional case of 9tn described previously, x is the cross product on 3t3, and the
field is the real numbers {% \ +, 1,0}. The only items to inspect to determine if an algebra is a
Lie algebra are properties 1 and 2 above. Let x, y, z e 9t3.
Property 1. xxy = (x2y3 Xgy2)l + (x3y1 x1y3)j + (x1y2x2y1)k
= (xsy2 x2y3)l (x,y3 x3y.)j (x2y, 
= yxx
Property 2. z x (x x y) = [z2(x1y2 x^) z3(x3y1 x1y3)]I +
[z3(x2y3 x^) z1(x1y2 x2y^)l +
11
[Z1 (XgY! X^g) Z2(X2y3 Xgy^Jk
Taking the Rh component as an example,
(y x (z x x))i = y2(z1x2 z2xi) y3(z3Xi zix3)
(x x (y x z))i = x2(yiz2 y2z\) x^y^i yZ3).
Summing the Ith components:
z2
0.
In like manner, the J and k components are 0. Thus. {9t3. +v, 0V, x}, denoted {9t3,x} for
simplicity, is a Lie algebra.
4.2 Linear Lie Algebras, gl(n,F), and su(n)
The commutator [X,Y] = XY YX is a commonly used Lie product. The commutator
satisfies properties 1 and 2, so replacing the multiplication in any associative algebra with the
commutator will result in a Lie algebra [5; p. 6]. If {Lj} is a basis for a linear Lie algebra, the
commutators of the basis elements with themselves can be found [2; p. 22,23].
[4,Lj]^>Ci/
The constants C^ are called structure constants, and the algebra is completely determined
by them. As an example, I, j, and k are a basis for 9t3, denoted ei, e2, and e3. The Lie
algebra {9t3,x} can be described as
ejxej = Â£ijkek
where Ejjk is the antisymmetric tensor. This tensor is 1 if {ijk} is a positive permutation of {123},
1 if {ijk} is a negative permutation of {123}, and 0 otherwise. The structure constants of {9l3,x}
are Ejjk.
The set of n x n matrices with entries in F is the Lie algebra gl(n,F). If the set gl(n,F) is
decreased to the set of skewHermitian (A* = A) matrices of trace 0, the Lie algebra su(n) is
formed. The algebra su(2) is spanned by
12
c 1
El = JICT1
E2= y i02
E3=yio3
*
where the oj's are the Pauli spin matrices as described above. The structure constants can be
found as follows.
[E1>E21 = E1E2E2E1
*(? j)(? ^h(? o')(? j)
Ho )l(o ?,)
i ft ?)
= E3
Also [E2iE3] = Ei and [E3,E] = E2. Thus, su(2) has the antisymmetric tensor as its structure
constants, and su(2) is therefore isomorphic to {5R3,x}.
13
5.0 The Relationship between Lie Algebras and Lie Groups
Lie algebras and Lie groups are directly related to each other. Every Lie group has an
associated Lie algebra, and every Lie algebra has an associated Lie group.
As mentioned above, a Lie group is an analytic manifold. Therefore, the tangent
space to that manifold makes sense. The tangent space of a Lie group at the group identity is
defined to be the Lie algebra [2; p. 21], For linear Lie groups, the Lie algebra can be found
simpty by differentiating the curves in the Lie group through the identity.
A curve in a group is defined to be a differentiable function f(t) taking a closed interval
of the real line into the group. For the application stated here, the closed interval must include
0 and f(0) = 1, the identity of the Lie group. Because the curve is a differentiable function on
the reals, it makes sense to speak of differentiating the curve at the identity.
5.1 SU(2) and su(2) as an Example
As an example, the Lie algebra su(2) can be computed from the Lie group SU(2) by
differentiating the curves in SU(2) through the identity. Recall that SU(2) is the set of 2 x 2
unitary matrices of determinant 1. Let U(t) be a curve in SU(2) passing through the identity at
t=0. Because the curve lives within SU(2), each element U(t) is unitary, and U(t)U*(t) = 1. To
find su(2), this equation is differentiated with respect to t and evaluated at t=0.
U(t)U*(t) = 1
Â£unuio
U(t)U*(t) + U(t)LT(t) = 0
Evaluating at t=0, and recalling that U(0) = 1, so U*(0) = 1,
1) U'(0) + LT(0) = 0.
Also, since the curve lives within SU(2), each element U(t) has determinant 1. Differentiating,
14
det(U(t = 1,
det(U(t)) = 0.
Now, U(t) can be expanded about t=0. First, define U'(0) = ^ Then,
U(t) = 1 + tu'(0) + 0
det(U(t)) = det(1 +tU'(0) + (3
= 1 +at + dt + (J(t2)
= 1 + tTrU'(0) + 0(t2).
So, J~det(U(t)) = 0
= (1+tTrU<(0) + CKt2))
2) = TrU'(O).
So, since su(2) is the derivative of curves in SU(2) evaluated at the origin, it has been
calculated to be the set of 2 x 2 matrices with entries in the complex field with properties
1) u + ir = o
2) Trtl = 0.
Thus, su(2) is the set of skew Hermitian matrices of trace 0, as was defined in a previous
section.
5.2 One Parameter Groups
A smooth homomorphism <> from a matrix group G into a matrix group H is a
homomorphism whose composition with any differentiable curve in the group G is itself
15
differentiable [6; p. 41,42], A one parameter subgroup in a matrix group G is the image of a
smooth homomorphism from the additive real group into G. Thus, for U a smooth
homomorphism, U takes U(t+s) to U(t)U(s) for U(t) a dfferentiable curve in GL(n.F). if G is a
one parameter subgroup of GL(n,F) generated by a smooth homomorphism U, then there
tA
exists a matrix A in Mn(F) such that U(t) = e [6; p. 51,52]. This can be proven [2; p. 32,33] in
a more general case as follows.
Take U(t) a differentiable curve in a one parameter subgroup. Take the derivative of
U(t) with respect to t,
dJ_ Gm U(t+h)U(t)
dt h>0 h
Dm U(t)U(h)U(t)
= h>0 h
= U'(0)U(t)
= LU(t) for L = U'(0).
Thus, the differential equation is ^ => LU(t), and this is solved by eLt. L is defined to be the
infinitesimal generator of the one parameter group eLt. The generator L stays fixed while the
parameter t varies to yield different elements of the group.
Thus, for Lie groups which are also one parameter subgroups, the curves U(t)
through the identity can be represented as e^, where L are elements of the Lie algebra. So
every one parameter Lie group has an associated Lie algebra. The exponentials of every
linear Lie algebra generate a linear Lie group. However, if the curves through the identity in a
Lie group are differentiated to obtain a Lie algebra, and the elements of the Lie algebra are
then exponentiated to obtain a Lie group, the original Lie group may not be regained fully.
This is because the initial curves may represent only a portion of the original Lie group.
As an example of these notions, recall that SU(2) is generated by e^* el^c2 2tT3
and the algebra su(2) is generated by itxi, ^102, ^103. The parameters in SU(2) are a, p,
17
6.Q Representations of Lie Algebras
A representation [2; p. 24] of a Lie algebra g on a vector space V is a mapping p from g
to the linear transformations of V such that
1. p(ax + py) = ap(x) + pp(y)
2. p([x,y]) = [p(x),p(y)] = p(x)p(y) p(y)p(x).
Representing Lie algebras as operators on vector spaces is one of the most commonly used
forms in physics applications.
The Heisenberg algebra [2; p. 25] is an example of representations of Lie algebras.
This algebra has three elements and can take various useful forms. The field that It operates
on is the field of continuously differentiable functions of the real line. The underlying set of
the algebra is the two operators Q and P, where Q is multiplication by x on the left, P is the act
of differentiating once, and the identity operator sends a function to itself. The structure
constants are determined to define the algebra.
[Q,Q] = [P.P] = 0
[P.QlPQQP.(^xx)
To see what [P,Q] means, it must operate on a function.
[P.Q] f(x) = x xj^ )f(x)
= f(x)+x^f(x)x^f(x)
= f(x)
Thus, [P,Q] is the identity function, and the algebra is {Q,P,1}. The algebra is completely
determined since the structure constants are known.
Another variation of the Heisenberg algebra is found by
18
Note that
P+Q QP
a= f and a  .
V 2 V2
[a,a] = [a*,a*] = 0
[a,a*] = aa* a*a
~2 {(P+Q)(QP) (QP)(P+Q)}
(PQ + Q2 P2 QP QP Q2 + P2+PQ)
(2PQ2QP)
= [P.Q]
= 1.
So, {a,a*,1} is another representation of the Heisenberg algebra since the structure constants
are the same as {Q,P,1}.
19
7.0 Application of Lie Algebras Classical
Lie groups and algebras arise often in modem physics as symmetry groups of
dynamical systems. These symmetries are intimately associated with conservation laws.
These symmetry groups correspond to both geometrical symmetries and symmetries
associated with the internal degrees of freedom of particles. The former lead to conservation
of energy, linear momentum, etc. The latter symmetry groups lead to conservation of
quantities such as isospin, strangeness, etc. More will be discussed on this topic in this
section and the section on quantum mechanics applications.
Classical treatment of dynamical systems is a good base for learning more modem
treatments. Hamiltonian mechanics is a general approach to classical physics that lends itself
well to extension to modem physics, so Hamiltonian mechanics will be addressed in this
section. In addition to the preparation for modern physics, the classical treatment will give rise
to discussion of the Poisson brackets and an example of a Lie algebra.
In classical physics, the set of physical observables (position, energy, momentum,
etc.) is a set of differentiable functions on the variables of position and momentum which,
together with the Poisson bracket {f,g}, and the identity function, form a Lie algebra over the
field 9t2n,where n is the number of degrees of freedom in the system. For qj the position in
the ith coordinate and pj the momentum corresponding to the ith coordinate, the Poisson
bracket {f.g} is defined:
This Lie algebra will be developed as Hamiltonian mechanics is developed.
7.1 Introduction to Generalized Systems
Consider a system with n degrees of freedom. It is always possible to choose n
positional coordinates q .q2..dn whose values completely determine the position of every
particle in the system at a given time. For example, two pool balls on a pool table form a
20
system with four degrees of freedom. The coordinates x\, X2, yi, and y2 specify the
positions of balls one and two if the origin and x and y axes are specified.
The coordinates qi,q2,...,qn are not necessarily distances. They may be angles or
some other coordinate. The only requirements on them are that they must be independent
of each other and there must be as many coordinates as there are degrees of freedom.
Therefore, the n coordinates form a basis for the positions of the system. These coordinates
are called generalized coordinates.
The generalized velocities are the time derivatives of the generalized coordinates.
The notation adopted in this paper is that the time derivative of a function f(qi, qa.. dn> t) >s
f'(qi, q2..qn> t). Thus, the velocity of a particle corresponding to the qi coordinate is q'.
The velocity of the ith particle in a generalized system in the x direction can be computed as
follows.
,
*5rq,*a5'*++aq>
7.2 Lagranglan Mechanics
Lagrange formulated [7] a useful approach to classical mechanics that provides a
foundation for Hamiltonian mechanics. He defined a variable L, the Lagrangian, that is the
difference between the kinetic and potential energies: L = T U. The rationale for this choice,
and its usefulness, are described below.
The kinetic energy T of a system of particles is the sum of the kinetic energies of the
indvidual particles. Recall that for a particle of mass mj and vector velocity Vj, the kinetic
energy of the particle is j mjvj2, where the dot product is used for the multiplication of
velocity. Therefore, for a system of n particles, the kinetic energy T of the system is
jrnvi2=T> jni (xj2 + yj2 + zj2).
The force acting on a particle can be written in terms of the kinetic energy. The most
general form of Newton's second law is F = ^ P, where P is the momentum of the particle
21
subjected to the force F. Perhaps the more familar form is F = ma where m is the mass and a
is the acceleration of the mass under the force F. The more familiar form is obtained by
F= ^ P with the momentum P equal to mv.
Because F P, the component of the force in the ith coordinate "direction" can be
found by
Note from the above expression of the kinetic energy that
So,
3T
3xj'
m xf = Pj.
P d_
l = dt 3xj"
This states that the component of the force acting in the ith coordinate is the time derivative of
the partial with respect to the velocity in the ith coordinate of the kinetic energy.
in addition to the kinetic energy, there is a quantity called the potential energy,
denoted U. The potential energy is a spatially dependent quantity. As a particle is moved in
space by a force, its potential energy may change. For example, the earth's gravity
establishes a potential energy. As an object is lifted above the ground, as it is moved in space
away from the center of the earth, its potential energy is increased. As the object falls to the
ground, its potential energy is decreased. To increase the potential energy, work must be
done on a particle by a force. The component of this force in the direction of the movement in
space is given by the following for a velocity independent potential.
c. 3U
F=
For systems that have no energy added or taken away, the total energy, the kinetic
plus the potential energy, is conserved. Therefore, if the potential energy is increased, the
kinetic energy is decreased by the same amount. Therefore, the forces associated with the
22
kinetic and potential energies are always equal and opposite for these systems in each
coordinate.
d. 3T JU
dt 3xj' "dxj
And so
d_ 3T aj .
dt dXj' + dxj= u
The kinetic energy is not dependent directly on the position of a particle. The kinetic
energy of a particle will change with its position, for example an object falling to the ground has
higher kinetic energy the farther it has fallen, but this relation is by way of the potential energy
change. If the particle moves along equipotential lines, its kinetic energy will stay the same.
Therefore, 0.
oXj
The assumption that the potential energy is not velocity dependent means that
Therefore,
dxj u
d_ 3T aj n
dt 3xj + dxj= u
can become
Lagrange defined the term L = T U, so this equation can be written as
dt3xj,L 3xjL u'
3t au
Recall from above that k~: = Pi using generalized coordinates. Because ; = 0, the
dqj dqj
momentum canonically conjugate to the coordinate qj can be defined by pj =
dqi
The physicist is most interested in the movement and distribution of energy in a
system. Therefore, a generalized system of n particles in 3 space can be completely
described by 3n positions and momenta and the energy Lagrangian function L:
L = L(qi, qz....q3n. P1. P2......P3n. t).
dt dqj'
and by the 3n equations
23
7.3 Hamiltonian Mechanics
Hamilton built on the work of Lagrange and defined the Hamiltonian H, similar to the
Lagrangian L, by the sum of the kinetic and potential energies, rather than their difference.
Thus, the Hamiltonian H T + U is the total energy of a system. The Hamiltonian is defined by
H =^> pjq' L, where L is the Lagrangian.
To see that the Hamiltonian as defined above is the total energy of the system, recall
[8; p. 186,187] that the kinetic energy T can be written
T=52mi(xi'2 + yi'2 + Zi,2)
and the velocity conjugate to the ith coordinate can be written
, dxi 3xj dxi
Substituting into the equation for the kinetic energy,
1 ,3xi? .9 3xj 3xj . 3xi, .9 3yi, ,9 3zio .9.
Now note that the momentum canonically conjugate to the ith coordinate was defined to be
3T
pi = Therefore,
, dxj 9 , 3xj 3xj , 3yj 9 , 3zi 9
?m*5^ 1"1" + sqi*'"1
,3xi0 . 3xj 3x . 3yio 3zj0 ..
"i ^ V + +  + *ag <*>
Now form the sum piqt + P2q2 +
3xj 3x;
3qi3q2
^i*qa'
3xj 9 ,9 3xj 3xj .
++ dqf ^ & +
3qi3q2
^prfZT.
i
Thus, H = ^> pjqi,L = 2TL = 2T(TU) = T + U, the total energy of the system.
I
24
7.4 The Hamiltonian and the Poisson Brackets
The Hamiltonian is a function of the position and momentum of the particles in a
system, its value for a given particle is the total eneigy of that particle. Recall the Poisson
brackets take two functions of position and momentum and return another function.
{f q} = ^
1,9f api 3Pi
When one of the functions is the Hamiltonian, and the other function is an arbitrary
function g, the Poisson bracket {H,g} is the time derivative of g minus the partial of g with
respect to time. To prove this, the differential of H must be computed.
h=^pk*'l
ft
Because the momenta and coordinates are independent of each other,
apkqk and apkOk'
3Qk'
3pk
are pk and qi*' respectively. Thus,
dH= ^ pkdqk + qk'dpk dL.
X
Now the differential of the Lagrangian must be computed.
dL=
+**
dL
By definition, the momentum canonically conjugate to the kth coordinate is pk =
ai
and, as seen above, ^ is the component of the force in the kth direction, and this is equal to
dclk
^Pk Therefore,
dL= Pkdqk + Pk'dqk + Jfdt.
Now substituting the equation for dL into the equation for dH,
dH= ^ Pkdtk' + qk'dPkPkdqk'Pk'dqk 3fdt,
By inspection, =qk'. Jj; Pk\ andf=f.
dH= ^ qk'dpkpk'dqk
ft QX
25
Now, for an arbitrary function g, {g,H} can be computed.
i! ifl_ id\
t9, } ^Vaqk 3pk 3pk W
This equation is ^ J, so^=^ + {g,H} for an arbitrary g.
Consider g = qk, {qk,H} = qk'. Taking g = pk, {Pk,H} = pk*.
7.5 The Hamiltonian, the Poisson Brackets, and Conservation
It has been shown above that, for an arbitrary function 9< * Jj* + {9.H}. Now
consider the effects of commutation with H. Two functions f and g are said to commute if {f,g}
= {g,f). However,
(fg} = ^/iL ia.. iL ia\
l,9f ^Vaqi 3Pi api aqj/
and
19,1)^^ 3pj 3pi 3qj.
Therefore, {f,g} = {g,f}. So, for two functions to commute, their Poisson bracket must be 0.
Consider a function g which does not explicitly depend on time and which commutes
with H, {g,H} = 0. Then,^ + {g,H} = 0. Therefore, g is a constant of the dynamical
system. So, functions which are not time dependent and which commute with the
Hamiltonian are constants of the motion. The Hamiltonian obviously commutes with itself, so
energy is conserved.
For another example [2; p. 42] of this, consider a one dimensional system which is
translationally invariant; that is, the system is the same if it is moved in space. Then linear
momentum is conserved. To see this, place two particles on a line. Because the system is
translationally invariant, for any function f on the system,
f(qi,q2. P1, P2) = fA(qiA. q2A. P1A. P2A) = fA(qi X,q2X, p,p2)
for fA denoting the shift of space under the function f.
26
^ (qiA. q^. pi A. P2A)=^ (qi + x, q2+x, pi p2)
at at
= 3qi +3q2
Now note that **?(Â£ t*
so {f,pj} =* Therefore,
37(qiA. q2A. P1A. P2A) =3^7+a^= {f P1 + P2>
If the Hamiltonian is taken for the function f, that is, if the Hamiltonian is translationally
invariant, then ^ = o = {H, pi + p2}. But, the Poisson bracket of the Hamiltonian and a
function is the time derivative of that function. Therefore, the time derivative of pi + P2 is 0
and linear momentum does not change with time. If an entity does not change with time, it is
said to be conserved. Thus, if H is invariant under spatial translations, linear momentum is
conserved.
7.6 {!,{}) Is a Lie Algebra
To this point in this section, Lagrangian and Hamiltonian approaches to classical
physics have been developed, and some examples of the use of the Poisson bracket and its
relationship to the Hamiltonian have been given. Now the notion of a Lie algebra is re
introduced with the proof that the differentiable functions yielding the observables of a
generalized physical system, together with the Poisson bracket, form a Lie algebra.
Take the vector space of the algebra to be the set of infinitely differentiable functions
over 9t2n, where n is the number of degrees of freedom in the system, with the usual function
addition and identity. Take the algebra multiplication to be the Poisson bracket. For this to be
an algebra, the following must hold for F, G in the set.
1. {F,G} e the algebra:
JPr, w3F 3G 3F 3G
apj "aPi 3qj
The sum of products of differentiable functions is back in the space, so 1 holds.
27
2. {F, otGi + pG2} = a{F,G} + p{F,G2}
{F, aGi + PQ2}
cF 3(aGi+pG2) aF 9(aGi+pG2)
3qi 3pj dpj 3qj
3F 3Gi 3F n3Gg JF 3Gi 3F 3Q2
3qiaapi +3qiP3pi '3pja3qj 3piP3qj
(I
F 3Gi
3pj
3F 3GjN q f 3F 3GÂ£ 3 F 3G2A
dPi 3qi ) +^ P (dpi 3pi 3pi dqj J
= 0 {F,Gi} + p{F,G2}
So 2 holds, and this set is an algebra. For the set to also be a Lie algebra, the following
properties must hold.
3. {F,G} = {G.F}
3F 3G _cF 33
3qj 3pj 3pj 3qj
3L ^ 3F 3G
3pi 3qj 3qj 3pi
33 3F 33 3F
dqj 3p 3pj 3qi
= {G.F}
So 3 holds.
4. {F,{G,H}} + {G,{H,F}} + {H,{F,G}} = 0
J
3F_3_ A_3_G 3_H 3_G 3__H\ 3F_3_ / 3J3 3JH 3_G 3H V
dpjdpj \; dqj 3p' 3pj 3qjJ3pj3qj \ T Pi dpj dPi dqj/
3F f 3^G 3 H 3 G 32H 32G 3 H 3 G 32H
3qj [dpjSqj 3pi + 3qj 3pjpj ' 3pj3pj 3qj 3pi 3pi3qj J
3F fd^G 3_H 3_G 32H 32G 3_H 3 J3 32H 'N
3pj ^dqjdqj 3pi + 3qi 3qj3pj 3qj3pj 3qj dpj 3qj3qj J
In like manner,
{G,{H,F} =
3G f 32H 3F 3_H 32F 32H 3F 3_H 32F ^
^ 3qj (3pj3qj 3pi + 3qi 3pjpj 3pj3pi 3qi 3pj 3p3qi J
^ 3G f 32H 3F 3_H 32F 32H 3F 3_H 32F ^
apj ^3qj3qi 3pj + 3qj 3qj3pj 3qj3pj 3q, 3pj 3q3qj J
nl fp ru dH ( d^F 3 G 3F 32G 32F 3 G 3F 32G \
1 1 "" ^ aqj V^apjaqi 3pi + 3qj 3pjpi ' dpjdpi 3qi 3pi 3p3qj J
^ 3H fdzF 3G 3F 32G 32F 3G 3F 3^ >
^ api laqi3qi api + 3q' 3cfi3pj aqjapi 3pi sqisqj J
These sum to zero if the p's and q's are independent in the operators, which has been
assumed throughout, so 4 holds.
Therefore, the functions yielding obervables of a generalized physical system and
the Poisson bracket create a Lie algebra in classical physics.
29
8.0 Converting to Quantum Mechanics
As has been shown above, in classical physics the observables are functions of
positions and momenta. In both classical and quantum physics, the observables are real,
measureable quantities. In classical physics, a particle's position and momentum, and the
system it is in, determine the values of the observables. The particle has a precise position
and momentum so the functions yielding the observables act on precise quantities.
However, in modem physics, a particle does not have a precise position and
momentum. Instead, the particle has an associated probability function y(9t3. t) that describes
the more likely and less likely values for the particle's position and momentum. The functions
yielding the observables, then, do not act on precise quantities, but instead act on functions
of space and time. Therefore, in quantum mechanics, the observables are determined by
operators acting on the particle's associated probability function, also known as the wave
function.
As was shown in the above section, in classical treatments the functions yielding the
observables form a Lie algebra with the Poisson bracket as the Lie product. To convert to
quantum mechanics, the classical functions yielding observables are mapped into operators
on wave functions in such a way that the structure of the algebra is retained. The commutator
is the Lie product in the operator algebra. That is, if f and g are classical functions yielding
observables of a system, they are mapped [2; p. 43] onto operators F and G such that
if {tg} = [F.G],
where It is Planck's constant h divided by Zit. The ifl factor is a convenience in this map. It is a
part of the map so that the units and operators make sense, but it is common in physics to
suggest normalizing a system of units such that Ti = 1 so the constants are then simplified.
However, the imaginary number i cannot be normalized out of existence.
30
Because of the straightforward relationship between classical functions and quantum
mechanical operators, a few basic facts about modem physics are already known by the work
done in section 7.0. For example, consider the classical Hamiltonian. The quantum
mechanical energy operator corresponding to the classical Hamiltonian is also called the
Hamiltonian, sometimes called the quantum mechanical Hamiltonian. Because functions that
commute with the classical Hamiltonian correspond to observables that are conserved, i.e.,
{f,H> = 0 implies the observable computed by f is conserved, when the functions are mapped
onto operators, the relation remains.
Thus, the observable corresponding to the operator F is conserved for F commuting with H.
As before, H commutes with itself, so energy is conserved.
Now consider the observables of position qr and momentum pr. These are the
fundamental observables in classical physics, and they, with the physical system, determine
the rest of the observables. Because the qr's and pr's are independent,
ih {f,H> = 0 = [F,H]
{Qr.Qs} = {Pr.PsJ 0.
{Pr.Ps} = Srs.
The corresponding operators then satisfy:
[Qr.Qs] [Pr.Psl = 0,
[Qr,Ps] = iftSrs1
31
These are the only constraints on the position and momentum operators in this
approach. One set of operators follows which obeys these relations and which is commonly
used [2; p. 43].
Note that the time portion of the wave function is dropped. In quantum mechanics, there is a
common assumption that the spatial and temporal portions of a wave function may be treated
independently. This assumption is used here. Now, because of the independence of qr and
qs, and of pr and ps, [Qy.&s] = t^r.^s] = 0 is a straightforward result. That [Qv.^s] = ih 5rs1 is
proven below. Note that to determine what an operator is, the operator must act on a wave
function. For two operators to be equal, they must have the same effect on any wave
function.
Yta) > drV(qi)
[Qr.?>sl^q)=(inqra^ifla^qr)v(qi)
3 3 3
= n qr^d) + Hi ^r^Vfa) + m VfaOg^cir
= HI 5rs v(qj)
So, = HI 6rs1
32
9.0 Application of Lie Algebras Quantum Mechanical Harmonic
Oscillator
What is meant by "to solve a problem is to find a relation that describes the quantity or
effect that is of interest. In some cases, to solve a problem means to couch it in a useful,
accessible form. To solve a quantum mechanics problem, usually the associated classical
problem is solved in terms of the position and momentum of a particle, and then the operators
for position and momentum are substituted for the variables.
For example, if the classical equation for the total energy of a particle is derived, and
the operators are substituted for the variables of position and momentum, the quantum
mechanical solution for the energy becomes an operator. The energy is important in physics
because one of the primary methods of studying a system is to determine how energy moves
in the system.
The energy of the particle is found when the operator acts on the particle's wave
function. The energy holds an eigenvalue relationship with the energy operator and the wave
function. This approach is illustrated [2; p. 4850] as follows for the harmonic oscillator.
The harmonic oscillator is fundamental to physics. One reason for this is that many
different systems may be represented using Fourier analysis and harmonic oscillators.
Therefore, for many systems, solution of the one dimensional harmonic oscillator basically
solves the entire system. Recall that a particle moving in simple harmonic motion obeys the
law:
x(t) = A sin tot.
Therefore, x"(t) = to^ A sin tot,
x(t) = to? x(t).
By Newton's second law,
F=m a = m x"(t) = mto2 x(t).
33
Recall from section 7.0 that to increase the potential energy of a particle, work must
3U
be done on a particle by a force. The force is given by F= Therefore, the difference in
potential energy of a system between two points is the integral of the dot product of the force
with the line joining the two points. For a one dimensional system,
U = /F(x)dx.
U = J mw2 x dx,
Us^mea2 x2
For a one dimensional generalized system, the total energy is the kinetic plus
potential energies, and it is expressed as follows.
H mv2 me2 q2
H = 2m (P2 + m2(D2 q2)
The operators Gy and 9s as shown above for the multidimensional case will convert
the classical Hamiltonian to the quantum mechanical Hamiltonian. However, it is useful to
normalize the operators to unitless quantities:
P =
?s
Vmfico
Then, substituting the operators for the variables in the energy equation, the quantum
mechanical energy operator is
H = ~ (S42 + m2^)2 (S2),
(D
H = 2m (mficoP2 + m2co2 ~Q2).
H =^flm(P2 + Q2).
Also, from the work done previously, [Qr,iP3 = rfl 1. Also,
[,5P] = QriP 5PQ,
34
= ti[Q,P].
Therefore, [Q.P] = il.
So, with this representation, the energy operator for the harmonic oscillator is given
by (1). Also, the position operator Q and the momentum operator P form a Lie algebra with
the identity {Q,P,<1}. That these form a Lie algebra is known because the commutator is a Lie
product, and the structure constants are completely known. In fact, this is a form of the
Heisenberg algebra.
9.1 The Heisenberg Algebra Revisited
If the operators a and a* are defined by the following,
a=^(PiQ),
a*=p= (P + iQ),
V 2
they will form another representation of the Heisenberg algebra. This representation is a very
important one in modem physics.
Note that a and a* are complex conjugates of each other. Therefore, they are adjoints
of each other, and, for the inner product the integral over all space of the complex conjugate
transpose of the first factor times the second factor, (a*a4>,4>) = (af>.a).
The verification that a and a* form the Heisenberg algebra follows.
fca*j = aa* aa
=Â£{(P4Q)(P+iQ)(P4iQ)(PiQ)}
= j (P2iQP + iPQ + Q2(P2 + iQPiPQ + Q2))
(2iPQ 2iQP)
= i[P,Q]
35
= i H)1
= 1.
Therefore, the structure constants are known, and {a,a*,1} form the Heisenberg algebra.
9.2 The Heisenberg Representation of the Harmonic Oscillator
Now that the operators a and a* have been defined in terms of P and Q, how would
the energy operator look if they replaced the P and Q operators? This is answered below.
Consider aa =^(P + iQ)(P iQ)
=Â£(P2 + iQPiPQ + Q2)
=(P2 + Q2 + i[Q.PJ)
= (P2 + 021)
= 2 (P2 + Q2) ~2
Therefore, flto a*a =^tico(P2 + Q2) Â£ IKd
= H^flco,
and, H =hco(a*a +^).
9.3 The Raising and Lowering Operators
It is also interesting to consider the commutators of the Hamiltonian with the operators
a and a*. First consider [H,a].
[H,a] = fko(a*a +^)a afko(a*a +^)
=tica(a*aa +a) fioo(aa*a +^a)
36
/
=f)CD(a#aa aa*a)
=fico{a*a aa*) a
= ftw(1)a
= tia>a
Now consider [H,a*].
pUI =tico(a*a +^)a* a*fto)(a*a +j)
=fico(a*aa* +^a*) tim(a*a*a+^a*)
=hco(a*aa* a*a*a)
=fitiia* (aa*a*a)
= tico a* (1)
= ficoa*
Therefore, the commutators have simple relations: [H,a] = tlco a and [H,a*] = ttco a*.
The operators a and a* are called the lowering and raising operators, respectively. The reason
for this will become clear below.
Suppose there is an eigenvalue relationship between H and y, a wave function. The
eigenvalue is the energy of the particle whose position and momentum are described by the
wave function.
H = Xfirn iy
Therefore, the energy of the particle is Xfico.
Then, since [H,a] = ft a, [H,a]\y = hm a iy. But also,
37
[H,atof=(HaaH)y
= HayaHy
= Hay aMlcoy.
So, tnDay=HayaMtmy,
and Hay= fitoay+Xfkoay
=tki) (X1) ay.
Therefore, ay is also an eigenvector of H, with eigenvalue Tioi (k1). In like manner,
ay is an eigenvector of H, with eigenvaluehro (\ +1). Thus, a, the lowering operator, lowers
the eigenvalue of the wave function, and thus the particle's energy, by one unit of fioo when it
operates on the wave function. Similarly, the raising operator a* raises the energy by one unit.
As was mentioned above, the movement of energy in a system is of primary interest in
physics. Therefore, these operators, which move energy to and from physical states in
discrete units are fundamental to modern physics.
The raising and lowering operations can be repeatedly applied to a wave function, but
not indefinitely for the lowering operator.
For a fixed y, H y = ftftro y,
Hay = (A. 1)fk(oay,
Ha2y = (X 2)fim a2y,
Hany = (X n) tloa any.
Negative energy makes no sense, so the eigenvalues must be bounded below. To
determine the lower bound on the possible energies of the system, the inner product of Hy
with y is computed.
(H v,V) = (flm(a*a +^)v,Y)
='ftoj(a*av,v) +Â£fico(v,V)
=^cd (av.av) +fi(D(v.v)
=1m[\\a^\\2 +~\M\2)
Thus, the eigenvalues of H, and therefore the energies possible for the system to
attain, are bounded below by ^fico. Therefore, there exists a vo. the ground state, such that
H VO =ficoÂ£ vo,
and H a* vo = fiw (1 + ^) a*vo,
H a*2 V0='f' (2+2) a*ZV0,
H a*n VO (n +2) a"Vo.
From these relations, it is clear that successively operating on the ground state wave function
by a* increases the energy of the system unit by unit. The quantization of energy is obvious,
as energy can only be added or subtracted in units ofTito. This is a result of the commutation
relation [H,a*] =fltoa*.
Defining a set of eigenfunctions vn = an VO would make sense, but, if the ground
state is normalized to 1, the higher states can also be normalized to one by defining:
39
Because the wave functions vn have distinct eigenvalues, they are orthogonal [9;
p.55], and if they are normalized, the result will be a complete, infinite, orthonormal basis for
the function space of the Hamiltonian.
To show that the vn are normalized, several facts are needed.
,. a*n vo  1. a*n vo
H=fico (n +)=
Vn! 2 Vm
so H vn =hm(n+^)vn
hco(a*a +^) Vn = fico (n +j) vn
Note that thefico and the ^ are on both sides of the equation and can be cancelled. Then,
a*ayn = nvn
This relation is one of those that is needed. Now, note
a*n vo
Vn 
a* a*n~1 vo
Vn V(rvlji
Vn
a*
Vn Vn1
Thus, Vn vn = a* vn1 or Vn+1 vn+1 = a* vn This is another needed relation. A similar one
can be found for a.
a*avn nvn
a*avn = VnVnvn
a*avn = Vna*vn1
So a*avn = a'Vnvn1.
avn = Vn vn1
and
40
Now that there are several useful relations, the proof of normalization is
straightforward. Begin with the inner product of yn with itself.
n(Vn.Vn) = (nyn.yn)
= (a*ayn.Vn)
= (ayn,ayn)
= (Vn yn1 vs Â¥nl)
= n(Â¥n1.Â¥nl)
Thus, (yn.yn) = (Â¥n1.Vnl). and since (yo.Â¥0) = 1, all the eigenvectors are normalized.
9.4 Dirac Notation
Dirac established a notation that is widely used. The notation as it applies to the
above development of the quantum mechanical harmonic oscillator is defined as n> is the
normalized eigenvector of a*a with eigenvalue n. Therefore,
. a n Â¥0
n> = Â¥n *
Vn!
The above wort< can be represented using this notation.
H yn=^<(n +^)yn becomes H n> =110) (n +^) n>.
aÂ¥n = Vnyn1 becomes an>= Vn n1>.
a*Â¥n = Vn+1 yn+1 becomes a*n>= Vn+1 n+1>.
In Dirac notation, the dual space of the space spanned by the wave functions is
denoted
respectively, is denoted . Also, = X and = X. Also,
)* = (Vm m1>)* = Vm
)* = (Vm+1 m+1 >)* = Vm+1
The order of operation is not important, just as in the multiplication of a series of
matrices and vectors.
41
= Vm = VmSmi.n if
= Vn+1 a Vn+1 Smin+1 if a*n> is performed.
These are the same, so the order of operation is irrelevant.
42
10.0 Extension of the Harmonic Oscillator Boson Operators
It is useful to consider [2; p. 50] a system of independent harmonic oscillators Oa.
Each one will have its own energy, so it will have its own Hamiltonian operator Ha and
eigenfunction/eigenvalue pairs. Each one will also have its independent raising and lowering
operators aa and aa\ That these operators are independent of one another yields the
following commutation relations.
[a
The total energy of the system is the sum of all the energies of the oscillators, so the
Hamiltonian of the system is the sum of the individual Hamiltonians.
The eigenfunctions of the entire system are indexed 1,2,3,...> where 1 is the
eigenfunction for the first oscillator, 2 is the eigenfunction for the second oscillator, etc. The
ground state is denoted 0,0,0,...> = 0>, and it is the lowest energy state for the entire
system. This means that the ground state for the system is the state in which all the
independent oscillators are at their lowest energy state. There are no cross terms in the
eigenfunctions, so there can be no coupling between oscillators.
The system eigenfunctions are denoted
in no M n (ai*)n1 (a2*)n2 (a3*)n3 (ak)nk
lni, nz. n3,..., nk,...> ..  0>.
Vni! n2l n3l..nitl...
This system eigenfunction has oscillator 1 in state ni> with ni raisings in energy from its
ground state. Oscillator 2 has n2 raisings in energy, etc. The expected relation holds when
the total system Hamiltonian operates on the system eigenfunction.
H ni,n2,n3,...nk,...> = ^ 1foa(na + Â£) ni,n2,n3,...nk,...>
43
So the energy of the system is the sum of all the units of energy added to each
oscillator to raise it to its current state from the ground state. Each energy unit, or excitation, is
called a boson. In electromagnetic field theory, the boson is a photon. In solid state physics,
the boson is a phonon, the vibration of a lattice. The operators a and a* are therefore called
boson operators*
44
11.0 Construction of su(21 Using Boson Operators
In classical physics; angular momentum is defined to be the cross product between
the radius vector and the linear momentum vector: L = r x p. Therefore,
Lj = x< p x pk, for j,k,l cyclic.
Earlier, it was shown that {9t3,x} forms a Lie algebra, su(2), with the antisymmetric
tensor as the structure constants. The angular momentum vectors hold the same relationship
as any basis vectors for 9t3: {Lj.Lk} ejki L. Therefore, when converting from classical to
quantum mechanics [2; p. 51,52], the quantum mechanical operators {Jj} corresponding to
the classical functions {L;} for angular momentum must satisfy
[Jj, J<] = m Ejki J.
If the position operator Qj is selected to be multiplication on the left by position x;, and
the momentum operator Pj is selected to bey ^, then
4Hx,<*rx,aiy ,ori'IOcylic
With these definitions, the structure constants may be verified to be the antisymmetric tensor,
and the quantum mechanical operators form another representation of su(2).
[J, JalJUaJaJi (f)2 (x2 x3 (x3 m ^) 
(*3 3x7' *1 4)(X24 X34)
/fl\2 .33 33
= (r)
3 3 3 3 3 3 3 3
X^3x2 X33x7 + X33iTX23^ + X1^iX23^
. v 3 3 _ 3 3 .
*33xi X33x2 X' d*3 X33x?
9,3 32 32
*2 (X23x7 + X2X377 + X3X13^J'
45
in like manner, [J2, J3} * 32 3 32 X3X23x,3x3 X13x2 X1X33^i> = tl2 (X23xT X1 ) = in (j (X1^2 = ih J3 = ih Ji, and [J3, Ji] = ifi J2. Therefore, the angular momentum
operators form another Lie algebra isomorphic to {9t3,x} and the algebra formed by the Pauli
spin matrices.
Now choose a system of units such that fi = 1. Then [Jj. Jk]3 i Â£jkl J Define
J+=Ji + iJa J = Ji iJ2, Jo = J3.
Then the commutators may be evaluated to see what Ue algebra these operators generate.
The Ue algebra is, in fact, another representation of su(2).
[Jo.J+] = J0J4. J+Jo = J3J1 + U3J2 Ji J3 U2J3 = J3J1 Ji J3 + i (J3J2 J2J3) = IJ3.J1] + [J3.J2] = i J2 >i Ji = Ji + i J2 = J+
tJo.J1 JqJ JJo = J3J1 U3J2 Ji J3 + 1J2J3 = J3J1 Ji J3 + i (J2J3 J3J2)  [J3.J1] + i [J2J3] = i J2 + i i Jt
46
= (JliJ2)
J.
[J+.J] = J+JJJ+
= (Ji + 1J2) (Jl 1J2) (J1 lJ2) (Jl + 1J2)
= Jl^ + jjgjj iJjJ2 +  (Ji^ U2J1 + UlJ2 + J2^)
= i (J2J1 J1J2 + ^2^1 Jl J2)
i (2
= i 2 (i J3) = 2J3 2J0
These structure constants determine the algebra generated by the operators J+, J.t and Jo.
and it is su(2).
Now this same algebra will be generated using the boson operators a and a*. Let
there be two harmonic oscillator states in a system with a, a2, a*. and a2* as boson
operators with the familiar relationships as follows.
Then the structure constants may be verified to be the same as those seen above.
[ai,a2] [ai*,a2T = 0
[aj,aj*] = Sjj1
Now define the J operators in terms of the boson operators. Put
J+ = ai*a2, J. = a2*ai, Jo=g (ai*aia2*a2).
[J0.J+] = [J+.JQ] = [ai*a2. jai*ai ja2*a2]
47
= \ (ai* (a2a2* a2*a2) a2 + ai* (aia* a*ai) a2)
= ^(ai*a2 + ai*a2)
= J+
[J0,J.] = [J.,J0]
= [a2*ai,^(ai*ai a2*a2)]
 Â£ [a2*ai, ai*ai] [a2*ai, a2*a2])
= \ ([a2*ai, a2*a2] [a2*ai, ai*ail)
= 2 (a2*a1a2*a2 a2*a2a2*ai a2*aai*ai + ai*aa2*ai)
=\(a2*(a2*a2 a2a2*)ai + a2*(ai*at ai ai*)ai)
=\ (a2*ai +a2*a)
= J.
[J+iJ.] = J+J. JJ+
= ai*a2a2*ai a2*aiai*a2
= ai*a2a2*a ai*a2*a2a +ai*a2*a2ai aai*a2*a2
= aj* (a2a2* a2*a2)ai + a2* (ai*ai aia*)a2
= ai*ai a2*a2
= 2J0
So, since these representations have the same structure constants as above, they also
represent su(2).
This representation is a useful one in physics. To see this, act on a wave function
ntn2>. Recall an> = "'/n nt> and a*]n>=. Vn+1 n+l>.
J+ni n2> = ai *a2ni n2>
J+ni n2> = Vni+1 Vni" ni+1 ,n21 >
Therefore, the operator J+ takes one boson, or quanta of energy, from state two and puts it in
state one. The operator J. reverses the process.
48
J.nn2> = a2*ai njn2>
J.ni n2> = VniVni+i" ni 1 ,n2+1 >
The operator Jo finds the difference in the number of excitations in the two states.
J0ni n2> = ~ (ai*ai a2*a2) ni n2>
%
=\ ai*ai nin2> ja2*a2nn2>
(ai*Vni~ni,n2> a2*>/n2"ni,n21>)
= ~ (Vn7VnT"nin2>Vn2"Vn2inin2>)
=Â£ (ni n2) nm2>
If the sign in Jo is changed, and the operator is doubled, the operator P = aai +
32*32 is created. This operator counts the number of states!
P nin2> = (nj + n2) njn2>
Sometimes it is more convenient to label the states n and n2 differently, especially
when the interesting aspects of a system include the angular momentum of the particles. This
algebraic representation is based on angular momentum in a fundamental way, so the
relabeling of states makes sense here.
Define m = j (n n2) and j (ni + n2). The quantity j is the absolute value of the
total angular momentum. The quantity m is the z component of the total angular momentum.
In terms of the original boson operators a* and a2*.
(ai*)i+m (a2*)i~m.
im>
V(j+m)l V (jm)l
) 0>,
(ai*)n1(ag*)n2)
VnTl Vnjt!
to>,
= nin2>
49
Then, substituting j and m directly into the equations derived above in ni and n2 with nj = j +
m and n2 = j m,
J+nin2> = Vni+1 Vn2 ni+1, n21>.
J+j m> = Vj+m+1 Vfm j, m+1 >.
The final state j, m+l> is seen by noting the following.
I.wich aMS J!lD2
m = which goes to 1 = = m +1
Similarly, J.ni ,ni> = V ni (n2+1). ni 1, n2+1
goes to J.y m> =WG+m) (jm+1) [j, ml>, Jolni n2> = ^ (n1 n2) nl n2>
goes to J0lim> = m j m>,
and Pnjn2> = (n + n2) nin2>
goes to Pnin2> = 2j j m>.
The operators J+ and J. act as shift operators on the angular momentum. The plus
operator increases the z component of the angular momentum by one unit and the minus
operator decreases it. Thus, these are the "raising" and "lowering" operators for the z
component of angular momentum. The Jq operator pulls out the value of the z component of
the angular momentum, and the P operator pulls out the total angular momentum.
This representation has n+n2+1 = 2j+1 dimensions (including the ground state).
The states are j m> such that m = j, j+1.+j, since m is the z component of j. In fact, this
representation describes each possible state a system may have if it has a total angular
momentum of j. The number of possible states, or configurations, the system may have is
2j+1, so this representation is called a 2j+1 multiplet.
In the most simple case, for j =D302 (n6utron and the proton are
represented as two states of a doublet, the nucleon. This case has some special names
associated with it as it represents no spatial or temporal structure, but instead represents
internal symmetries. The value j is called the isospin, and it is relabelled I. The value m is I3
lz, and is the Z component of the isospin.
51
12J2____Conclusion
The above sections show some of the utility of Ue theory in the study of both classical
and modern physics. The Lie algebras specifically studied in this paper, su(2) and the
Heisenberg algebra, are only two examples of the many Ue structures, both algebras and
groups, that are useful in physics. In fact, the study of nuclear physics uses the Lie group
SU(4), the strong interactions are described by SU(3), and the electroweak interaction is
modelled by SU(2) x U(1). Work done now in advanced topics such as cosmic strings and
grand unification theory uses Ue theory for modeling.
The applicability of Ue groups and algebras to such a vast array of physical problems
leads to questions about the basic structure of the physical world. Because the Ue products
are noncommutative, much of the successful modeling of the physical world has non
commutative, rather than commutative products. The many areas of study in applications of
Ue theory will provide meaning and intuition to this noncommutivity, as well as uncover more
truths about the physical world.
52
References
[1 ] N. Jacobson, Basic algebra 1.2nd ed., W. H. Freeman and Company, New York,
1985. This source was used throughout this section.
[2] D. H. Sattinger, O. L. Weaver, Lie groups and algebras with applications to physics^
geometry, and mechanics. SpringerVerlag New York, Inc., New York, 1986
[3] P. M. Cohn, Lie groups. Cambridge at the University Press, Cambridge, UK, 1968
[4] A. N. Kolmogorov, S. V. Fomin, Introductory real analysis. Dover Publications, Inc.,
New York, 1970
[5] N. Jacobson, Lie algebras. Dover Publications, Inc., New York, 1962
[6] M. L. Curtis, Matrix groups. 2nd ed., SpringerVerlag New York Inc., New York, 1984
[7] F. T, Avignone, III, Lecture notes from Modem Physics class PHYS 503 Fail,
1983, at University of South Carolina, Columbia, SC was a primary source for this
section. This material is in most mechanics books.
[8] F. W. Constant, Theoretical Dhvsics. AddisonWesley Publishing Company, Inc.,
Reading, Massachusetts, 1954. Chapter 10, Advanced dynamics, is an excellent
source for this material.
[9] K. Gottfried, Quantum mechanics Vol 1. W. A. Benjamin, Inc., New York, 1966

Full Text 
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APPLICATIONS OF LIE ALGEBRAS TO PHYSICS by Laura Ellen Wood B.S., University of South Carolina, 1985 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Arts Department of Mathematics 1990
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@1990 by Laura Ellen Wood All rights reserved.
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This thesis for the Master of Arts degree by Laura Ellen Wood has been approved for the Department of Mathematics by
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Wood, Laura Ellen (M.A., Mathematics) Applications of Ue Algebras to Physics Thesis directed by Professor Stanley.E. Payne A brief review of basic abstract algebra Is given, followed by definitions of Ue groups and Ue algebras and a discussion of their relation to each other. Representations of Ue algebras are defined and then shown as examples in applications to physics. Various representations of su(2) and the Heisenberg algebra are carried throughout the paper. Classical Hamiltonian mechanics is developed and st:Jown to have a Ue algebraic representation with the Poisson bracket. The one dimensional quantum mechanic harmonic oscillator is developed using the Heisenberg Ue algebra and the raising and lowering operators. This model is then extended to multiple dimensions and used to construct a few representations of su(2) in the study of angular momentum. Examples are distributed throughout the paper to aid intuitive understanding. The form and content of this abstract are approved. I recommend its publication. Stanley E. Payne
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Contents 1.0 Introduction ................................................................. 1 2.0 Review of Basic Algebra ................................................................................. 2 2.1 Groups ........................................................................................... 2 2.2 Aelds ....................................................................................................... 2 2.3 Vector Spaces ................................................................................................. 3 2.4 Algebras ............................. ............................................................................. 3 3.0 Introduction to Ue Groups ............................................................................................. 5 3.1 Topological Groups .......................................................................................... 5 3.2 Analysis and Lie Groups .................................................................................... 5 3.2.1 Analyticity ......................................................................................... 6 3.2.2 Analytic Manifold ...................... ......................................................... 7 3.3 The Unear Ue Group GL(n,F) and SU(2) ............................................................ 7 4.0 Introduction to Lie .Algebras ...................................... : ................................................... 1 0 4.1 {9t3, +y, Oy, x} as an Example of a Ue Algebra ..................................................... 1 0 4.2 Linear Lie Algebras, gl(n,F), and su(n) ................................................................ 11 5.0 The Relationship between Lie Algebras and Lie Groups ................................................. 13 5.1 SU(2) and su(2) as an Example .......................................................................... 13 5.2 One Parameter .Groups ...................................................................... ............... 1 4 6.0 Representations of Lie Algebras ................................................................................... 1 7 7.0 Application of Lie Algebras Classical ............................................................................. 1 9 7.1 Introduction to Generalized Systems ................................................................. 1 9 7.2 Lagrangian Mechanics ...................................................................................... 20
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vi 7.3 Hamiltonian Mechanics ..................................................................................... 23 7.4 The Hamiltonian and the Poisson Brackets ......................................................... 24 7.5 The Hamiltonian, the Poisson Brackets, and Conservation .................................. 25 7.6 {f,{}} Is a Ue Algebra .......................................................................................... 26 8.0 Converting to Quantum Mechanics ............................................................................ .... 29 9.0 Application of Ue AlgebrasQuantum Mechanical Harmonic Oscillator ............................. 32 9.1 The Heisenberg Algebra Revisited .................................. ............. ................ ... 34 9.2 The Heisenberg Representation of the Harmonic Oscillator ................................. 35 9.3 The Raising and Lowering Operators ................................................................. 35 9.4 Dirac Notation .. ................................................................................................ 40 10.0 Extension of the Harmonic OscillatorBoson Operators ............................................... .42 11.0 Construction of su(2) Using Boson Operators .............................................................. 44 12.0 Conclusion ................................................................................................................ 51 Figures Figure 1.0 The Stereographic Projection ....................................................................................... 9
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1.0 Introduction "All physics is the study of Lie groups and algebras." This statement appears extreme, and it definitely reveals the attitudes of a mathematician more than a physicist, but it does imply the strength of the connection between Lie theory and the study of physics. 1 Sophus lie (18421899) provided the foundation of lie group theory in his work in continuous groups. He also reduced local problems in Lie groups to corresponding problems in Lie algebras. His work in the 19th century laid the groundwork for much advancement in theoretical physics. The use of Lie groups and their associated Lie algebras in physics is widespread, extending from classical physics through to the most advanced representations in modern physics. Lie theory may be applied in classical physics in operations as simple as the rotation of space around axes, or in more complicated operations modeling fluid flows. The applications in modem physics are especially numerous, including the addition and subtraction of energy from a system, the strong interactions between nucleons, and the weak and electromagnetic interactions. This paper gives a brief introduction to Lie theory and applies the theory in areas of classical and modem physics. The algebra su(2) is carried throughout the paper to show the power and utility of only one algebra among infinitely .many.
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2.0 Review of Basic Algebra First, a brief review of abstract algebra [1] is given for completeness. 2.1 Groups Let G be a nonvacuous set, and let be an associative binary operation taking G x G into G. Then, for a, b, c G, 1. abeG 2. a (b c)= (a b) c. Let there exist an element 1 in G, called an identity, such that for all a e G: 3. a=1a=a. Now, for each a e G, let there exist an element b e G such that 4. ab=ba=1. A set {G, , 1} satisfying the above conditions is called a gmug. 2 The binary operation : G x G + G need not be commutative. That is, a b does not necessarily equal b a. If a b = b a for all a, be G, the group is said to be commutative or abelian. An example of an abelian group is the real numbers 91, except for the number 0, with the usual multiplication and the number 1 as the identity {9t {0}, , 1 }. 2.2 Fields The concept of a field is an extension of the concept of a group. A .fiti1 {F, , +, 1, 0} is a. nonvacuous set F together with two associative binary operaijons, and +, and two distinct elements of F, 1 and 0, such that 1. {F, +, 0} is an abelian group. 2. {F{0}, , 1} is an abelian group. 3. Fora, b, ce F, a (b+c) = a b+ac (b+c)a=ba+ca.
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An example of a field is the real numbers 9t, with the usual addition and multiplication, and identities the numbers 0 and 1: {9t, , +, 1, 0}. 2.3 Vector Spaces A vector space can be described as a combination of a group and a field. of an abelian group {V, +y, Ov}. a field {F, , +, 1, 0}, a!ld a scalar multiplication taking F x V into V which satisfies 1. 2. 3. 4. a (x +v y) = ax +v ay (a+b) x=.ax+vbx (a b) x = a (bx) 1x =X for a e F; x, y e V. for a, be F; x e V. for a, b e F; x e V. for x e V. An example of a vector space is the field of the real numbers {9t, , +, 1, 0} together with the abelian group {9t", +y, Ov} of ntuples of real entries, (x1, x2. x3, ... xn) with Xi e 9t, such that x +v y = (x1 + Y1, x2 + Y2 .... Xn + Yn) and Ov = (0, 0, ... 0). The scalar multiplication applies termwise so that, for a e 9t, ax= (ax1, ax2. ax3, ... axn). 2.4 Algebras 3 As the concept of a group is extended to make a field, the concept of a vector space is extended to make an algebra. Let {A, +A, OA} be a vector space over a field {F, , +, 1, 0}. Now let A be a bilinear multiplication taking A x A into A. The bilinear multiplication has the. following properties for x, y, z e A, and a e F. 1. (x +AY) AZ =(X AZ) +A (y AZ). 2. X A (Y+AZ) =(X AY) +A (X AZ). 3. a(XAY)=(ax)AY=XA(ay). The resulting set, {A, +A. OA, A}. together with the field and scalar multiplication from the vector space, form an algebra.
PAGE 10
An example of an algebra is the set of n x n matrices with entries in the real numbers, Mn(9t), forming a vector. space over the field of real numbers, {9t, , +, 1, 0}, with the usual matrix product as the biiinear muHiplication A 4
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5 3.0. Introduction to Lie Groups The study of Lie groups unites three branches of mathematics algebra, analysis, and topology. The basics of algebra have been discussed briefly above. Some concepts in topology arid analysis that apply to Lie groups now follow. 3.1 Topological Groups A Lie group has a topological group as its base. "A topological group is a group which is also a topological space (so that ideas such as continuity, connectedness and compactness apply) In which the group operations are continuous" [2; p. 3]. Recall that a topology on a set establishes the notion of "nearness" of one element of the set to another. If the underlying set of a group has a topology associated with it such that the group operations are continuous, it is a topologjcal group. Let {G, , 1} be a group. There are two group operations the binary operation and the map that sends an element to its inverse. These operations are continuous if, for x1, x2, Y1, Y2 G such that x1 is near to x2 and Y1 is near to Y2 1. x1 Y1 is near to x2 Y2. and 2. x11 is near to x2. An example of a topological group is {9t, +, 0} with the usual absolute value distance function establishing the topology. 3.2 Analysis and Lie Groups A Lie group is a topological group in which the group operations are not only continuous, but also analytic. P.M. Cohn defines (3; p. 44] a Lie group as follows. 1. {G, , 1} is a group. 2. G is an analytic manifold. 3. The mapping G x G G taking (x,y) x y is analytic. He then proves that the mapping x x is analytic.
PAGE 12
To understand the concept of a Ue group, the concepts of analyticity and analytic manifolds must be discussed. 3.2.1 Analyticity A topological space T is said to be locally Euclidean at a point p if a homeomorphism can be found from a neighborhood of p imo 9tn [3; p. 4,5]. This is called a 6 A topological space T is said to be locally homeomorphic to 9tn if any nonempty open subset ofT can be mapped by a homeomorphism to a nonempty open subset of 9tn. Let W be an open subset ofT and let a be a homeomorphism from W to X, an open subset of 9tn. Now .consider a homeomorphism X to Y, another subset of 9tn, and let be its inverse. Then, for x, y t 9tn, (1a) (1b) Yi = clli (x), Xi = 'ljlj (y) The homeomorphisms a can be composed to form another chart on W. The two charts a and whose coordinates are related by (1), are analytically related if the functions and 'are analytic functions on their domains X and Y. A function is said to be analytic at a point p of 9tn if it can be expressed as a convergent power series in (Xi Pi) for i = 1,2, .. ,n in some neighborhood of p. Thus, for group operations to be analytic, homeomorphisms must exist from the spaces G x G into 9tn and from G into 9tn whose ranges in 9tn can be analytically related as above for any point in the group. is, suppose two subsets T and W G have charts a and p respectively. Then the group operation mapping T x W into S in G is analytic if there exists a chart 't of S such that there is an analytic map in 9tn taking ax p into 't.
PAGE 13
3.2.2 Analytic Manifold A topological space that is locally Euclidean at each point of the space is called a manifold [3; p.5,6]. An analytic manifold has added constrictions on it. An analytic manifold has as its base a Hausdorff space, also known as a T 2space. A 7 Hausdorff space is a space M which satisfies the second axiom of countability, which states that for each pair of distinct points x andy in M, there are neighborhoods Tt(X) of x and Tt(Y) of y such that Tt(X) and 'fl(y) have the empty set as their intersection [4; p. 85). An analytic manifold has a Hausdorff space M as a base, and then builds an analytic structure on An analytic structure is a family fl of charts on M such that 1 At each point of M there is a chart which belongs to fl. 2. Any two charts of fl are analytically related. 3. Any chart of M which is analytically related to every chart of fl itself belongs to fl. In summary, a Ue group is a topological group wtiich has nice, Euclideanlike structure locally, and which has smooth, wellbehaved group operations. An example of a Ue group is {9t, +, 0}. Different authors assume different qualities in their definitions of Lie groups. For example, some define a Ue group as a differentiable manifold with a group structure, and state that the set of infinitely differentiable functions on the real line is "the" Ue group. The less restrictive definitions are used in this paper. 3.3 The Linear Lie Group GL(n,F) and SU(2) One Ue group is especially useful in physics applications. This group is the set of all nonsingular n x n matrices together with the usual matrix multiplication. It is called the general
PAGE 14
8 linear group, and it is denoted GL(n,9t) or GL(n,ct) depending on whether the entries are real or complex. A subgroup SU(2) of GL(2, ct) is the set of all2 x 2 matrices over the complex fiel_d which are unitary and whose determinant is 1. The notation SU(2) is used instead of SU(2,ct) for simplicity. Thus, for U e SU(2), U*U = 1 = the identity mabix, where denotes the complex conjugate transpose and a bar over a complex number denotes its complex conjugate. It can be shown [2; p. 814) that three matrices of SU(2) defined below, and can represent the group of rotations of the Riemann sphere. In fact, if represents a specific represents that same rotation. The representation of SU(2) by the matrices U is of primary importance in the study of particles with spin of onehalf, such as the electron. The reason for this is that one rotaticm in three space can be represented by two different matrices, U. This forms a double valued map from the rotations of three space into SU(2). Particles of spin onehalf transform under a rotation in three dimensions according to these double vafued maps, called spinor representations. The matrices U11(p), and are as follows. ( a .. a) I Sln2 = .. a a 1 sm2 lJrt(P) =(cotsin:) s1n2 cos2 u ('Y) = (ei o e2
PAGE 15
. t + iTJ z=x+1v=. 1t 2x 2y !dI t = lzl2 +I T1 = ld +I t = !d +I Figure 1. The Stereographic Projection. Source: Reprinted from D. H. Sattinger and 0. L. Weaver, Lie groups and algebras wjth applications to physjcs. geometry. and mechanjcs, SpringerVerlag New York, Inc., New York, 1986 This is a picture that shows the axes of rotation of the Riemann sphere by the above matrices and the relationships between the angles and complex numbers. as If eA, for A a matrix, is defined eA matrices U1\, and UC can be written i(a/2)0'1 = e i(j3/2)0'2 =e i(y/2)0'3 = e for 0'1, 0'2, and 0'3 the Pauli spin matrices as defined 6) 0'2 = (? oi) 9
PAGE 16
10 4.0 Introduction to Lie Algebras Algebras can beeither associative or nonassociative, meaning that the bilinear multiplication of the algebra is either associative or nonassociative. Associative algebras satisfy for x, y, z A, while nonassociative algebras may not. A nonassociative algebra whose multiplication satisfies for x, y, z e A is called a Ue algebra (5]. In a Ue algebra, there can be no Identity for the multiplication. To prove this, assume there exists an identity 1A, and let x eA. Then 1A A x = x A 1A. However, Property 1 states that 1 A Ax = x A 1 A. which is a contradiction. Thus, there can be no identity in a Ue algebra. 4.1 (9t3, +v, Oy, x} as an Example of a Lie Algebra An example of a Ue algebra is {9t3, +v. Ov, x}, where the veCtor space {9t3, +v. Ov} is the three dimensional case of 9tn described previously, xis the_ cross product on 9t3, and the field is the real numbers {9t, , +, 1, 0}. The only items to inspect to determine if an algebra is a Ue algebra are properties 1 and 2 above. Let x, y, z e 9t3. Property 1. xxy =yxx Property 2. zx (xxy)
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[z1 (X3Y1 X1 Z2(X2Y3 X3Y2)]k Taking the Hh component as an example, (y X (Z X X))l = Y2(Z1X2Z2X1}Y3(Z3X1 Z1X3) (X X (y X Z))l = X2(Y1Z2Y2Z1) X3(Y3Z1 Y1Z3). Summing the lth components: 11 z2(x1Y2X2Y1) z3(X3Y1 x1y3) +Y2Cz1x2 z2x1) y3(Z3X1 z1x3) +X2(Y1Z2 y2z1) X3(Y3Z1 Y1Z3) = 0. In like manner, the J and k components are 0. Thus, {9t3, +v. Ov. x}, denoted {9t3,x) for simpficity, is a Ue algebra. 4.2 Linear Lie Algebras. gl(n.F). and su(n) The commutator [X,Y] = x:f YX is a commonly used Ue product. The commutator satisfies properties 1 and .2. so replacing the muhiplication in any associative algebra with the. commutator will result in a Ue algebra [5; p. 6]. If {Lj} is a basis for a linear Ue algebra, the commutators of the basis elements with themselves can be found [2; p. 22,23]. [lj,Lj] Cijk Lk K The constants cif are called structure constants, and the algebra is completely determined by them. As an example, I, j, and k are a basis for 9t3, denoted e1, e2, and e3. The Ue algebra {9t3 ,x} can be described as 9i x ej = Eijk .ek where Eijk is the antisymmetric tensor. This tensor is 1 if {ijk} is a positive permutation of {123}, 1 if {ijk} is a negative permutation of {123}, and 0 otherwise. The structure constants of {9t3,x} are Eijk The set of n x n matrices with entries in F is the Ue algebra gl(n,F). If the set gl(n,F) is decreased to the set of skewHermitian (A*=A) matrices of trace 0, the Ue algebra su(n) is formed. The algebra su(2) is spanned by
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12 where the aj's are the Paun spin matrices as described above. The structure constants can be found as follows. =t ?) t ?) =EJ Also [E2,EJ] = E1 and [EJ,E1] = E2. Thus, su(2) has the antisymmetric tensor as its structure constants, and su(2) is therefore isomorphic to {9l3,x}.
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13 5.0 The Relationship between Lie Algebras and Lie Groups Ue algebras and Ue groups are directly related to each other. Every Ue group has an associated Ue algebra, and every Ue algebra has an associated Lie group. As.mentioned above, a Lie group is an analytic manifold. Therefore, the tangent space to that manifold makes sense. The tangent space of a Ue group at the group identity is defined to be the Lie algebra [2; p. 21]. For linear Ue groups, the Lie algebra can be found simply by differentiating the curves in the Ue group through the identity. A curve in a group is defined to be a differentiable function f(t) taking a closed interval of the real line into the group. For the application stated here, the closed interval must include 0 and f(O) = 1 the identity of the Ue group. Because the curve is a differentiable function on the reals, it makes sense to speak of differentiating the curve at the identity. 5.1 SU(2) and su(2) as an Example As an example, the Ue algebra su(2) can be computed from the Ue group SU(2) by differentiating the curves in SU(2) through the identity. Recall that SU(2) is the set of 2 x 2 unitary matrices of determinant 1. Let U(t) be a curve in SU(2) passing through the identity at t=O. Because the curve lives within SU(2), each element U(t) is unitary, and U(t)U*(t) = 1. To find su(2), this equation is differentiated with respect to t and evaluated at t=O. U(t)U*(t) = 1 a a at U(t)U*(t) = at 1 = 0 U'(t)U*(t) + U(t)U*'(t) = 0 Evaluating at t=O, and recalling that U(O) = 1, so U*(O) = 1, 1) U'(O) + U*'(O) = 0. Also, since the curve lives within SU(2), each element U(t) has determinant 1. Differentiating,
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det(U(t)) = 1, a at det(U(t)) = o. Now, U(t) can be expanded about t=O. Rrst, define U'(O) = { Then, U(t) = 1 + tU'(O) + G(t2), det(U(t)) = det(1 + tU'(O) + (9(t2)) det { 1 +at bt ) . ct 1 +dt + = 1 + t TrU'(O) + (g(t2). So, det(U(t)) = 0 = :t (1 + t TrU'(O) + (3(t2)) 2) = TrU'(O). 14 So, since su(2) is the derivative of curves in SU(2) evaluated at the origin, it has been calculated to be the set of 2 x 2 matrices with entries in the complex field with properties 1) 2) U+U*=O TrU =0. Thus, su(2) is the set of skew Hermitian matrices of trace 0, as was defined in a previous section. 5.2 One Parameter Groups A smooth homomorphism a matrix group G into a matrix group H is a homomorphism whose composition with any differentiable curve in the group G is itself
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15 differentiable [6; p. 41,42). A one parameter subgroup in a matrix group G is the image of a smooth homomorphism from the additive real group into G. Thus, for U a smooth homomorphism, U takes U(t+S) to U(t)U(s) for U(t) a dfferentiable anve in GL(n,F). If G is a one parameter subgroup of GL(n,F) generated by a smooth homomorphism U, then there exists a matrix A in Mn(F) such that U(t) = etA [6; p. 51,52]. This can be proven [2; p. 32,33] in a more general case as follows. Take U(t) a differentiabie curve in a one parameter subgroup. Take the derivative of U(t) with respect to t, dJ fim U(t+h)U(t) dt h Om U(t)U(h)U(t) h = U(t) = U'(O)U(t) = LU(t) for L = U'(O). Thus, the differential equation LU(t), and this is solved by alt. Lis defined to be the infinitesimal generator of the one parameter group eLt. The generator L stays fixed while the parameter t varies to yield different elements of the group. Thus, for Ue groups which are also one parameter subgroups, the curves U(t) through the identity can be represented as eU, where L are elements of the Ue algebra. So every one parameter Ue Qroup has an associated Lie algebra. The exponentials of every linear Ue algebra generate a linear Ue group. However, if the curves through the identity in a Ue group are differentiated to obtain a Lie algebra, and the elements of the Lie algebra are then exponentiated to obtain a Ue group, the original Ue group may not be regained fully. This is because the initial curves may represent only a portion of the original Ue group.
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16 As an example of these notions, recall that SU(2) is generated by and the algebra is generated by The parameters in SU(2) are andy.
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17 6.0 Representations of Lie Algebras A representation [2; p. 24] of a Ue algebra g on a vector space V is a mapping p from g to thelinear transformations of V such that 1. p(ax + PY> = ap(x) + PP(Y) 2. p([x,y]) = [p(x),p(y)] = p(x)p(y)p(y)p(x). Representing lie algebras as operators on vector spaces is one of the most commonly used forms in physics applications. The Heisenberg algebra [2; p. 25] is an example of representations of lie algebras. This algebra has three elements and can take various useful forms. The field that it operates on is the field of continuously differentiable functions of the real line. The underlying set of the algebra is the two operators Q and P, where a is multiplication by x on the left, P is the act of differentiating once, and the identity operator sends a function to itself. The structure constants are determined to define the algebra. [Q,Q] = [P,P] = 0 a a [P,Q) = PQQP = To see what [P,Q] means, it must operate on a function .. a a [P,Q] f(x) =
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18 P+O QP a={2 anda=.n [a,a] = [a*,a*] = 0 Note that [a,aj = aa aa 1 = 2 {(P+O)(Q.P) (QP)(P+O)} = i (PO + Q2p2 QP QP Q2 + p2+P0) 1 = 2 (2PQ2QP) = [P,Q] = 1. So, {a,a*,1} is another representation of the Heisenberg algebra since the structure constants are the same as {O,P,1}.
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19 7.0 Application of Lie Algebras Classical Ue groups and algebras arise often in modem physics as symmetry groups of dynamical systems. These symmetries are intimately associated with conservation .. laws. These symmetry groups correspond to both geometrical symmetries and symmetries associated with the internal degrees of freedom of particles. The former lead to conservation of energy, linear momentum, etc. The latter symmetry groups lead to conservation of quantities such as isospin, strangeness, etc. More will be discussed on this topic in this section and the section on quantum mechanics applications. Classical treatment of dynamical systems is a good base for learning more modem treatments. Hamiltonian mechanics is a general approach to classical physics that lends itself well to extension to modem physics, so Hamiltonian mechanics will be addressed in this section. In addition to the preparation for modern physics, the classical treatment will give rise to discussion of the Poisson brackets and an example of a Ue algebra. In classical physics, the set of physical observables (position, energy, momentum, etc.) is a set of differentiable functions on the variables of position and momentum which, together with the Poisson bracket {f,g}, and the identity _function, form a Lie algebra over the field 9t2n,where n is the number of degrees of freedom in the system. For qi the position in the ith coordinate and Pi the momentum corresponding to the ith coordinate, the Poisson bracket {f,g} is defined: f = ...::::::::: (j!_ .QQ. j!_ QQ.) I ,g} f aqi api oPi aqi This Ue algebra will be developed as Hamiltonian mechanics is developed. 7.1 Introduction to Generalized Systems Consider a system with n degrees of freedom. It is always possible to choose n positional coordinates q1 ,q2, .. ,qn whose values completely determine the position of every particle in the system at a given time. For example, two pool balls on a pool table form a
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20 system with four degrees of freedom. The coordinates x x2. Y1 and Y2 specify the positions of balls one and two if the origin and x and y axes are specified. The coordinates q1.q2 ... ,qn are not necessarily distances. They may be angles or some other coordinate. The only requirements on them are that they must be independent of each other and there must be as many coordinates as there are degrees of freedom. Therefore, then coordinates form a basis for the positions of the system. The.se coordinates are called generalized coordinates. The generalized velocities are the time derivatives of the generalized coordinates. The notation adopted in this paper is that the time derivative of a function f(q1. q2 qn. t) is f'(q1. q2, ... qn. t). Thus, the velocitY of a particle correspOnding to the q 1 coordinate is q 1. The velocity of the ith particle in a generalized system in the x direction can be computed as follows. ,axi ,axi, axi, XI =a q1 +a + ... + a qn q1 q2 qn 7.2 Lagrangian Mechanics lagrange formulated [7] a useful approach to classical m.echanics that provides a foundation for Hamiltonian mechanics. He defined a variable L, the Lagrangian, that is the difference between the. kinetic and potential energies: L = TU. The rationale for this choice, and its usefulness, are described below. The kinetic energy T of a system of particles is the sum of the kinetic energies of the individual particles. Recall that for a particle of mass rT1i and vector velocity Vi, the kinetic energy of the particle is t miVj2, where the dot product is used for the multiplication.of velocity. Therefore, for a system of n particles, the kinetic energy T of the system is T = i f11Vj2 = f i "1 (Xi'2 + Yi'2 + Zj'2). The force acting on a particle can be written in terms of the kinetic energy. The most general form of Newton's second law is F = P, where Pis the momentum of the particle
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21 subjected to the force F. Perhaps the more familar form is F = ma where m is the mass and a is the acceleration of the mass under the force F. The more familiar form is obtained by F= P with the momentum P equal to mv. Because F = P, the component of the force in the ith coordinate "direction'; can be found by Note from the above expression of the kinetic energy that So, aT . '"'" axi'=mx, =q. d aT Fi = dt axj'" This states that the component of the force acting in the ith coordinate is the time derivative of the partial with respect to the velocity in the ith coordinate of the kinetic energy. In addition to the kinetic energy, there is a quantity called the potential energy, denoted U. The potential energy is a spatially dependent quantity. As a particle is moved in space by a force, its potential energy may change. For example, the earth's gravity establishes a potential energy. As an object is lifted above the ground, as it is moved in space away from the center of the earth, its potential energy is increased. As the object falls to the ground, its potential energy is To increase the potential energy, work must be done on a particle by a force. The component of this force in the direction of the movement in space is given by the following for a velocity independent potential. For systems that have no energy added or taken away, the total energy, the kinetic plus the potential energy, Is conserved. Therefore, if the potential energy is increased, the kinetic energy is decreased by the same amount. Therefore, the forces associated with the
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kinetic and potential energies are alway's equal and opposite for these systems in each coordinate. And so d aT aJ dt axi' = dxi 22 The kinetic energy is not dependent directly on the position of a particle. The kinetic energy of a particle will change with its position, for example an object falling to the ground has higher kinetic energy the farther it has fallen, but this relation is by way of the potential energy change. If the particle moves along equipotential lines, its kinetic energy will stay the same. aT Therefore, = 0. aXj The assumption that the potential energy is not velocity dependent means that au axi'=O. Therefore, can become d aT aJ +0 dt axj' axi.Q.. L(TU) 1._ (TU) = o. dt dxi' dxi Lagrange defined the term L = T U, so this equation can be written as Recall from above that = Pi using generalized coordinates. Because = 0, the momentum canonically cenjugate to the coordinate qi can be defined by Pi = . The physicist is most interested in the movement and distribution of energy in a system. Therefore, a generalized system of n particles in 3 space can be completely described by 3n positions and momenta and the energy Lagrangian function L: L = L(q1, q2, ... Q3no P1o P2, ... P3no t), and by the 3n equations
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23 7.3 Hamiltonian Mechanics Hamilton built on the work of Lagrange and defined the Hamiltonian H, similar to the Lagrangian L, by the sum of the kinetic and potential energies, rather than their difference. Thus, the Hamiltonian H = T + U is the total energy of a system. The Hamiltonian is defined by H PiQi' L, where Lis the Lagrangian. To see that the Hamiltonian as defined above is the total energy of the system, recall [8; p. 186,187] that the kinetic energy T can be written T = mi (Xi'2 + Yi'2 + Zi'2), I and the velocity conjugate to the ith coordinate can be written 1 dxi 1 dxi 1 ax; Xi =aq1 q 1 + aq2 Q2 + ... + aqn Qn' Substituting into the equation for the kinetic energy, "" 1 axi2 2 ax; axi , 2 2 azi2 2 T= 2"1(aq1 q 1 + 2 aq1aq2q 1 q 2+aq2 Q2 + ... +aq1 q 1 + ... +aqn Qn ). I Now note that the momentum canonically conjugate to the ith coordinate was defined to be ar Pi = aqj'" Therefore, axj 2 I axj axj I 2n2 I azj 2 I P 1 = f rTii 'aq1 q 1 + aq1aq2Q2 + ... + aq1. q 1 + ... + aq1 q 1 ), 0Xi2 1 dxi dxf 1 0Yi2 1 0Zj2 1 ITlj (aq2 Q2 + aq1aq2q 1 + ... + aq2 q2 + ... + aq2 q 2 ). Now form the sum P1Q1' + P2Q2' + ... I ( OXi 2 axi OXi I I OXj 2 dxi OXi I I ) .:;:;. PiCI = 4 rT1 aq1 q1 + aq1aq2q 1 q 2 + ... + aq2 q 2 + aq1aq2q 1 q 2 + ... I I i Thus, H = PiQi' L = 2T L = 2T (T U) = T + U, the total energy of the system. i
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7.4 The Hamiltonian and the Poisson Brackets The Hamiltonian is a function of the position and momentum of the particles in a system. Its value for a given particle is the total energy of that particle. Recall the Poisson brackets take two functions of position and momentum and return another function. f (ll .Qg_ lL .Qg_) { ,g} c:;=. aq, ap, api aqi I When one of the functions is the Hamiltonian, and the other function is an arbitrary function g, the Poisson bracket {H,g} is the time derivative of g minus the partial of g with respect to time. To prove this, the differential of H must be computed. H = PJ(CI('L I& Because the momenta and coordinates are independent of each other, a a I Pkqk are Pk and qkl respectively. Thus aqk' apk dH = Pk dqkl + q)(' dPk dl. IC Now the differential of the Lagrangian must be computed. {aL I aL ) a. dL= cf aqk' dqk + aqk dqk +atdt 24 By definition, the momentum canonically. conjugate to the kth coordinate is Pk = aaqL .. k and, as seen above, is the component of the".force in the kth direction, and this is equal to d dt Pk Therefore, a.. dl = Pk dqkl + Pk' dqk + at dt. K Now substituting the equation for dl into the equation for dH, dH = Pk del<' + q)(l dPk Pk dck' Pk1 dqk :t dt, jC. dH = qk'kPk'dqk :.dt. "' B ct aH aH nd aH a.. y enspe 1on, apk = qk, aqk = Pk a at= at"
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25 Now, for an arbitrary function g, {g,H} can be computed. Consider g = qk, {qk,H} = Taking g = Pk. {Pk,H} = Pk' 7.5 The Hamiltonian, the Poisson Brackets, and Conservation It has been shown above that, for an arbitrary function g, = + {g,H}. Now consider the effects of commutation with H. Two functions f and g are said to commute if {f,g} = {g,f}. However, { f } (l.L Qg_ l!_ Qg_) ,g .,=: aqi api a Pi aqi and c::::: l!_ il) {g,f} f oqj ilPi opi Oqi Therefore, {f,g} ={g,f}. So, for two functions to commute, their Poisson bracket must be 0. Consider a function g which does not explicitly depend on time and which commutes with H, {g,H} = 0. =f + {g,H} = 0. Therefore, g is a constant of the dynamical system. So, functions which are not time dependent and which commute with the Hamiltonian are constants of the motion. The Hamiltonian obviously commutes with itself, so energy is conserved. For another example [2; p. 42] of this, consider a one dimensional system which is translationally invariant; that is, the system is the same if it is moved in space. Then linear momentum is conserved. To see this, place two particles on a line. Because the system is translationally invariant, for any function f on the system, f(q1. q2. P1. p2) = f"(q1 q2" P1 P2") = fll(q1 X, q2X, P1. P2) for f" denoting the shift of space under the function f.
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26 at at +aq1 aq2 Now note that f _'..::::::::(.1!.. dPi { ,pj} 4f Clqk ClPk ClPk ClQk df so {f,pj} = oqj' Therefore, If the Hamiltonian is taken for the function f, that is, if the Hamiltonian is translationally invariant, then = 0 = {H, P1 + P2}But, the Poisson bracket of the Hamiltonian and a function is the time derivative of that function. Therefore, the time derivative of P1 + P2 is o and linear momentum does not change with time. If an entity does not change with time, it is said to be conserved. Thus, if H is invariant under spatial translations, linear momentum is conserved. 7.6 {f,{}} Is a Lie Algebra To this point in this section, Lagrangian and Hamiltonian approaches to classical physics have been developed, and some examples of the use of the Poisson bracket and its relationship to the Hamiltonian have been given. Now the notion of a Ue algebra is reintroduced with the proof that the differentiable functions yielding the observables of a generalized physical system, together with the Poisson bracket, form a Ue algebra. Take the vector space of the algebra to be the set of infinitely differentiable functions over 9t2n, where n is the number of degrees of freedom in the system, with the usual function addition and identity. Take the algebra multiplication to be the Poisson bracket. For this to be an algebra, the following must hold for F, Gin the set. 1. {F,G} e: the algebra: CJF (GCJF (G I f dQi dPi dPi dQi The sum of products of differentiable functions is back in the space, so 1 holds.
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2. {F, aG1 + j3G2} = a{F,G1} + 13{F,G2} {F, aG1 + PG2} = iF o(aG1+J3G2) _iF o(aG1+PG2) f aqi aPi api aqi = a {F,G1} + 13{F,G2} So 2 holds, and this set is an algebra. For the set to also be a Ue algebra, the following properties must hold. 3. {F,G} ={G,F} aF CG iF a:; = f Oqj apj Opj aqi =i ={G,F} So 3 holds. 4. {F,{G,H}} + {G,{H,F}} + {H,{F,G}} = 0 en lH CG CJt {F,{G,H}} = {F; oQi opi opj oQi} = a: = In like manner, .27
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{G,{H,F} = aH ( a2F a_g_ aF a2G a2F a G aF a2G ) {H,{F,G}} = aqi apiaqi api + aqi aPjPi apiaPi aqi api apiaqi aH ( a2F a G aF a2G a2F a G aF a2G ) api aqiaqi api + aqi aqiaPi aqiaPi aqi api aqjaqi These sum to zero if the p's and q's are independent in the operators, which has been assumed throughout, so 4 holds. Therefore, the functions yielding obervables of a generalized physical system and the Poisson bracket create a Ue algebra in classical physics. 28
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8.0 Converting to Quantum Mechanics As has been shown above, in classical physics the observables are functions of positions and momenta. In both classical and quantum physics, the observables are real, measureable quantities. In classical physics, a particle's position and momentum, and the system it is in, detennine the values of the observables. The particle has a precise position and momentum so the functions yielding the observables act on precise quantities. 29 However, in modem physics, a particle does not have a precise position and momentum. Instead, the particle has an associated probability function 'lf(9t3, t) that describes the more likely and less likely values for the particle's position and momentum. The functions yielding the observables, then, do not act on precise quantities, but instead act on functions of space and time. Therefore, in quantum mechanics, the observables are detennined by operators acting on the particle's associated probability function, also known as the wave function. As was shown in the above section, in classical treatments the functions yielding the observables form a Lie algebra with the Poisson bracket as the Lie product. To convert to quantum mechanics, the classical functions yielding observables are mapped into operators on wave functions in such a way that the structure of the algebra is retained. The commutator is the Lie product in the operator algebra. That is, iff and g are classical functions yielding observables of a system, they are mapped [2; p. 431 onto operators F and G such that ifl {f,g} = [F,G}, where 1'1 is Planck's constant h divided by 2n. The ifl factor is a convenience in this. map. It is. a part of the map so that the units and operators make sense, but it is common in physics to suggest nonnalizing a system of units such that 1'1 = 1 so the constants are then simplified. However, the imaginary number i cannot be normalized out of existence.
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30 Because of the straightforward relationship between classical functions and quantum mechanical operators, a few basic facts about modem physics are already known by the work done in section 7.0. For example, consider the classical Hamiltonian. The quantum mechanical energy operator corresponding to the classical Hamiltonian is also called the Hamiltonian, sometimes called the quantum mechanical Hamiltonian. Because functions that commute with the classical Hamiltonian correspond to observables that are conserved, i.e., {f,H} = 0 implies the observable computed by f is conserved, when the functions are mapped onto operators, the relation remains. in {f,H} = 0 = [F,H] Thus, the observable corresponding to the operator F is conserved for F commuting with H. As before, H commutes with itself, so energy is conserved . Now consider the observables of position qr and momentum Pr These are the fundamental observables in classical physics, and they, with the physical system, determine the rest of the observables. Because the qr's and Pr1S are independent, {qr,qs} = = 0, = .rEi dQr Ei T aqi api api aqi I aqi api I I The corresponding operators then satisfy: = [Or,Os] = [Pr,Ps] = 0, [Or,Ps] = :n
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31 These are the only constraints on the position and momentum operators in this approach. One set of operators follows which obeys these relations and which is commonly used [2; p. 43]. (tr: 'l'(q;) > qr'l'( inaqs 'l'(qi) Note that the time portion of the wave function is dropped. In quantum mechanics, there is a common assumption that the spatial and temporal portions of a wave function may be treated independently. This assumption is used here. Now, because of the independence of qr and qs. and of Pr and Ps. [{tr,{tsJ = = 0 is a straightforward result. That = ifl Srs1 is proven below. Note that to determine what an operator is, the operator must act on a wave function. For two operators to be equal, they must have the same effect on any wave function. 'll(q) = (ifl Clr_j_in _j_ qr) 'V(q;) aqs aqs =ifl qr + ifl (qr 'V(q;)) = in qr + in qr 'V(q;) + in qr
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32 9.0 Application of Lie Algebras Quantum Mechanical Harmonic Oscillator What is meant by ,o solve a problem" is to find a relation that describes the quantity or effect that is of interest. In some cases, to solve a problem means to couch it in a useful, accessible form. To solve a quantum mechanics problem, usually the associated classical problem is solved in terms of the position and momentum of a particle, and then the operators for position and momentum are substituted for the variables. For example, if the classical equatiQn for the total energy of a particle is derived, and the operators are substituted for the variables of position and momentum, the quantum mechanical solution for the energy becomes an operator. The energy is important in physics because one of the primary methods of studying a system is to determine how energy moves in the system. The energy of the particle is found when the operator acts on the particle's wave function. The energy holds an eigenvalue relationship with the energy operator and the wave function. This approach is illustrated [2; p. 4850] as follows for the harmonic oscillator. The harmonic oscillator is fundamental to physics. One reason for this is that many different systems may be represented using Fourier analysis and harmonic oscillators. Therefore, for many systems, solution of the one dimensional harmonic oscillator basically solves the entire system. Recall that a particle moving in simple harmonic motion obeys the law: Therefore, By Newton's second law, x(t) = A sin IDt. x"(t) = m2 A sin IDt, x"(t) = x(t). F = m a = m x"(t) = x(t).
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33 Recall from section 7.0 that to increase the potential energy of a particle, work must be done on a particle by a force. The force is given by F = Therefore, the difference in potential energy of a system between two points is the integral of Jhe dot product of the force with the line joining the two points. For a one dimensional system, U = j'F(x)dx, U = f mm2 x dx, U = 1 mm2 x2. 2 For a one dimensional generalized system, the total energy is the kinetic plus potential energies, and it is expressed as follows. 1 1 H =mv2 +mm2 q2 2 2 The operators ll'r and as shown above for the multidimensional case will convert the classical Hamiltonian to the quantum mechanical Hamiltonian. However, it is useful to normalize the operators to unitless quantities: Then, substiMing the operators for the variables in the energy equation, the quantum mechanical energy operator is (1) Also, from the work done previously, =in 1. Also,
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Therefore, [Q,P) = i1. = (QPPO), =1\ [Q,P]. 34 So, with this representation, the energy operator for the harmonic oscillator is given by (1). Also, the position operator a and the momentum operator P form a Ue algebra with the identity (Q,P,i1}. That these form a Ue algebra is known because the commutator is a Ue product, and the structure constants are completely known. In fact, this is a form of the Heisenberg algebra. 9.1 The Heisenberg Algebra Revisited If the a and a* are defined by the following, 1 a*= _r= (P +iO), v2 they will form another representation of the Heisenberg algebra. This representation is a very important one in modem physics. Note that a and a* are complex conjugates of each other. Therefore, they are adjoints of eacl') other, and, for the inner product the integral over all space of the complex conjugate transpose of the first factor times the second factor, (a*a41.41) = (a4l,a4l). The verification that a and a* form the Heisenberg algebra follows. [a.a1 = aa* a*a = i ((PiQ)(P+iO) (PHQ)(PiQ)} =i (P2 iQP +iPO +a2(P2 +iOP iPO+a2)) = 1 (2iPQ 2iQP) 2 =I [P,Q]
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35 = i (i)1 = 1. Therefore, the structure constants are known, and {a,a, 1} form the Heisenberg algebra. 9.2 The Heisenberg Representation of the Harmonic Oscillator Now that the operators a and a have been defined in terms of P and Q, how would the energy operator look if they replaced the P and a operators? This is answered below. Consider a* a = i (P + iO)(P iO) = i (P2 + iQP iPQ + a2) = i (p2 + a2 + i[O,P]) Therefore, H ='hiD(a*a 9.3 The Raising and Lowering OperatQrs It is also interesting to consider the commutators of the Hamiltonian with the operators a and a. First consider [H,a]. [H,a] 1 1 = ftiD(a*aa +?) fiCil(aa*a +2B>
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36 =i'lro(a*aaaa*a) =fico( a* aaa*) a = fiCJl(1) a = flCJl a Now conside.r [H,a*]. [H,a1 =i'ICJl(a*aa* +is*) 1'1co(a*a*a +is*) =i'lco(a*aaa*a*a) = fiCJl a (aa a*a) = 1'1CJl a (1) = fiCJl a Therefore, the commutators have simple relations: [H,a] = fiCJl a and [H,a*] = fiCJl a. The operators a and a are called the lowering and raising operators, respectively. The reason for this will become clear below. Suppose there is an eigenvalue relationship between H and 'I' a wave function. The eigenvalue is the energy of the particle whose position and momentum are described by the wave function. Therefore, the energy of the particle is A.i'ICJl. Then, since [H,a] =fiCJl a, [H,a]'l' =tlro a 'I' But also,
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37 [H,a]W= (Ha aH) 'I' = Ha 'I' a 'I' So, tim a 'I'= Ha 'I' a Aflm 'I' and H a 'I' = fico a 'I' + Attco 8'lf = i'lm (A. 1) a'lf. Therefore, a'l' is also an eigenvector of H, with eigenvalue flm (A.1 ). In like manner, a"'l' is an eigenvector of H, with eigenvalue ohm (A.+ 1). Thus, a, the lowering operator, Jowers the eigenvalue of the wave function, and thus the particle's energy, by one unit of fico when it operates on the wave function. Similarly, the raising operator a" raises the energy by one unit. As was mentioned above, the movement of energy in a system is of primary interest in physics. Therefore, these operators, which move energy to and from physical states in discrete units are fundamental to modern physics. The raising and lowering operations can be repeatedly applied to a wave function, but not indefinitely for the lowering operator. For a fixed 'If, H 'I' = Aflco 'I' H a'l' = (A. 1 )fico a'lf, Ha2'1' = (A. 2) fico a2'1f, Han'l' = (A. n) 'hco an'lf. Negative energy makes no sense, so the eigenvalues must be bounded below. To determine the lower bound on the possible energies of the system, the inner product of H'lf with 'I' is computed.
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38 Thus, the eigenvalues of H, and therefore the energies possible for the system to attain, are bounded below by Therefore, there exists avo. the ground state, such that and 1 H a* '1'0 = t'lro (1 +2) a*'l'o, 1 H a*" '1'0 = flro (n + 2) a*"'l'o. From these relations, it is clear that successively operating on the ground state wave function by a* increases the energy of the system unit by unit. The quantization of energy is obvious, as energy can only be added or subtracted in units of1iw. This is a result of the comm[Jtation relation [H ,a*] = flroa*. Defining a set of eigenfunctions 'l'n = an '1'0 would make sense, but, if the ground state is normalized to 1, the higher states can also be normalized to one by defining: a*" yo 'l'n = _r: v nl
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39 Because the wave functions 'l'n have distinct eigenvalues, they are orthogonal [9; p.55], and if they are normalized, the result will be a complete, infinite, orthonormal basis for the function space of the Hamiltonian. To show that the 'l'n are normalized, several facts are needed. so 1 1 n(l)(a*a +2) 'l'n = fl(l) (n +2) 'l'n Note that the fl(l) and the i are on both sides of the equation and can be cancelled. Then, a* a 'l'n = n 'l'n This relation is one of those that is needed. Now, note 'lin an yo .Jni a* 'lin = in 'l'n1 Thus, vn 'l'n = a* 'l'n1 or ...J n+ 1 'l'n+ 1 = a* 'l'n This is another needed relation. A similar one can be found for a. So and a* a 'l'n = n 'l'n a*a 'l'n = 'l'n a*a 'l'n = ...Jn a* 'l'n1 a* avn = a* vn 'l'n1' avn = "" 'l'n1
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Now that there are several useful relations, the proof of normalization is straightforward. Begin with the inner product of 'l'n with itself. = (nvn.'l'n) = (a*avn.'l'n) = ( avn.Cl'lfn) = (Vii 'l'n1 {ii 'l'n1 ) = n(vn1.'1'n1) Thus, ('lfn,'lfn) = ('l'n1.'1'n1). and since ('lfQ,'IfO} = 1, all the eigenvectors are normalized. 9.4 Dirac Notation Dirac established a notation that is widely used. The notation as it applies to the above development of the quantum mechanical harmonic oscillator is defined as In> is the normalized eigenvector of a* a with eigenvalue n. Therefore, an yo In>= 'l'n_ r. v n! The above wor1< can be represented using this notation. 1 H 'l'n =fiw (n +2) 'l'n. becomes 1 H In> =1'lw (n +2) In>. Cl'lfn = {ii 'l'n1 becomes a In>= Tn ln1>. a*vn = ..V n+ 1 'l'n+ 1 becomes a* In>= ..Vn+1 ln+1>. In Dirac notation, the dual space of the space spanned by the wave functions is denoted . Also, = X and = A.. Also, )*= ( ...Jm lm1>)* = ...Jm
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41 = rm = rm Sm1 n if = Sm,n+1 if a*ln> is performed. These are the same, so the order of operation is irrelevant.
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42 10.0 Extension of the Harmonic Oscillator Boson Operators. It is useful to consider [2; p. 50] a system of independent harmonic oscillators Oa.. Each one will have its own energy, so it will have its own Hamiltonian operator Ha. and eigenfunction/eigenvalue pairs. Each one will also have its independent raising and lowering operators aa. and aa.*. That these operators are independent of one another yields the following commutation relations. [aa,.ap] = [aa*.ap*J = 0, [aa.ap*] = aa,p1 The total energy of the system is the sum of all the energies of the oscillators, so the Hamiltonian of the system is the sum of the individual Hamiltonians. H =.:::; T a + Ua 1 H = n(l)a (aa *aa. + 2) .... The eigenfunctions of the entire system are indexed 11 ,2,3, ... > where 1 is the eigenfunction for the first oscillator, 2 is the eigenfunction for the second oscillator, etc. The ground state is denoted 10,0,0, ... > = 10>, and it is the lowest energy state for the entire system. This means that the ground state for the system is the state in which all ttie independent oscillators are at their lowest energy state. There are no cross terms in the eigenfunctions, so there can be no coupling between oscillators. The system eigenfunctions are denoted This system eigenfunction has 9scillator 1 in state 1n1 > with n1 raisings in energy from its ground state. Oscillator 2 has n2 raisings in energy, etc. The expected relation holds when the total system Hamiltonian operates on the system eigenfunction. 1 H 1n1 ,n2,n3, ... nk, ... > = .::::;. 1'i(l)a. (na. + 2) 1n1 ,n2,n3,"k> 01
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43 So the energy of the system is the sum of all the units of energy added to each oscillator to raise it to its current state from the ground state. Each energy unit, or excitation, is called a boson. In electromagnetic field theory, the boson is a photon. In solid state physics, the boson is a phonon, the vibration of a lattice. The operators a and a* are therefore called boson operators ..
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44 11.0 Construction of suC2l Using Boson Operators In classical physics; angular momentum is defined to be the cross product between the radius vector and the linear momentum vector: L = r x p. Therefore, Lj = Xk PI XI Pk. for j,k,l cyclic. Earlier, it was shown that {9t3,x} forms a Ue algebra, su(2), with the antisymmetric tensor as the structure constants. The angular momentum vectors hold the same relationship as any basis vectors for :Jt3: {Lj.Lk} = jkl L1. Therefore, when converting from classical to quantum mechanics [2; p. 51,52], the quantum mechanical operators {Ji} corresponding to the classical functions {Li} for angular momentum must satisfy [Jj. Jk] = ifl jkl Jl. If the position operator Ci is selected to be multiplication on the left by position Xi, and the momentum operator Pi is selected to then XJ a!k) forj,k,lcylic. With these definitions, the structure constants may be verified to be the antisymmetric tensor, and the quantum mechanical operators form another representation of su(2). (:n)2 a a a a = T ( x 2ax3 x 3ax1 x 2ax3 x 1 ax3 
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.45 In like manner, [J2, J3] = in J1, and [J3, J1] = in J2. Therefore, the angular momentum operators form another Lie algebra isomorphic to {9t3,x} and the algebra formed by the Pauli spin matrices. Now choose a system of units such that fi = 1. Then [Jj, Jk] = i jkl J1. Define J+ = J1 + iJ2, J_ = J1 iJ2, Jo = J3. Then the commutators may be evaluated.to see what.Ue algebra these operators generate. The Ue algebra is, in fact, another representation of su(2). = JoJ+J+JO = J3J1 + iJ3J2J1J3iJ2J3 = J3J1 J1J3 + i (J3J2J2J3) = [J3,J1] + i [J3,J2l [Jo.J1 = JoJJ.Jo = J3J1 iJ3J2J1J3 + iJ2J3 = J3J1 J1J3 + i (J2J3J3J2) = [J3.J1] + i fJ2J3]
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. = (J1 iJ2) =J_ [J+,J1 = J_J+ = (J1 + iJ2) (J1 iJ2) (J1 iJ2) (J1 + iJ2) = J12 + iJ1J2 + J2 2 (J12 iJ2J1 + + J22). = i J1J2 + J1J2) = i (2 [J2,J1]) = i 2 < i J3) = 2J3 ... 2Jo 46 These structure constants determine the algebra generated by the operators J+, J_, and Jo, and it is su(2). Now this same algebra will be generated using the boson operators a and a*. Let there be two harmonic oscillator states in a system with a1 .a2, a 1 *, and a2* as boson operators With the familiar relationships as follows. [a1,a2] = [a1*,a2*] = 0 [ai,aj*] = Sij1 Now define the J operators in terms of the boson operators. Put Then the structure constants may be verified to be the same as those seen above. [JQ,J+] =[J+,JO] =[a1*a2, i a1*a1 i a2*a2] ={
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= i (a1" (a2a2* a2"a2) a2 + a1" (a1 a1" a1"a1) a2) 1 =2(a1"a2+a1"a2) [Jo,J1 = [J_, Jol 1 = [a2"a1, 2 (a1"a1 S2"a2)] 1 1 =t2[a2"a1, a1"a1]2[a2"a1, a2"a2]) = i ([a2"a1, a2"a2][a2"a1, a1"a1]) =i (S2"a1S2*a2a2"a2a2"a1 a2"a1a1"a1 + a1"a1a2"a1) 1 = 2 (a2"(a2"a2a2a2")a1 + a2"(a1"a1 a1 a1")a1) =i (a2"a1 +a2"a1) =J_ =2Jo So, since these representations have the same structure constants as above, they also represent su(2). This representation is a useful one in physics. To see this, act on a wave function 1n1n2>. Recall aln> = {f, lnb and a*ln>= ..Jn+1 ln+1>. J+ln1 n2> = a1"a21n1 n2> J+ln1n2> = ..Jn1+1 ...fri21n1+1 ;n21> 47 Therefore, the operator J+ takes one boson, or quanta of energy, from state two and puts it in state one. The operator reverses the process.
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Jln1n2> = a2*a1 1n1n2> J_ln1n2> = {n1 vn2+1 ln11,n2+1> The operator Jo finds the difference in the number of excitations in the two states. If the sign in Jo is changed, and the operator is doubled, the operator P = a1*a1 + a2*a2 is created. This operator counts the number of states! 48 Sometimes it is more convenient to label the states n1 and n2 differently, especially when the interesting aspects of a system include the angular momentum of the particles. This algebraic representation is based on angular momentum in a fundamental way, so the relabeling of states makes sense here. Define m = i (n1 n2) and j =i (n1 + n2). The quantity j is the absolute value of the total angular momentum. The quantity m is the z component of the total angular momentum. In terms of the original boson operators a1* and a2*, ljm>
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49 Then, substituting j and m directly into the equations derived above in n1 and n2 with n1 = j + mandn2=jm, J+ln1n2> = v"'n21n1+1, n21>, J+U m> = i+m+ 1 .J jm lj, m+ 1 >. The final state U. m+1> is seen by noting tt1e following. n1+n2 h' h (n1+1) + (n21) n1+n2 1 2 w 1c goes to 2 2 = J n1n2 h' h t (n1+1)(n21) n1n2+2 1 m = 2 w 1c goes o 2 2 = m + Slmilar1y, Jln1.n1>..,.Jn1(n2+1) 1n11, n2+1> (jm+1) lj, m1>, Joln1 n2> = (n1 n2) 1n1 n2> goes to goes to and goes to JoU m> = m li m>, Pln1n2> = (n1 + n2) ln1n2> Pln1 n2> = 2j U m>. The operators J+ and J_ act as shift operators on the angular momentum. The plus operator increases the z component of the angular momentum by one unit and the minus operator decreases it. Thus, these are the "raising" and "lowering" operators for the z component of angular momentum. The Jo operator pulls out the value of the z component of the angular momentum, and the P operator pulls out the total angular momentum. This representation has n1+n2+1 = 2j+1 dimensions (including the ground state). The states are li m> such that m = j. 1 ... +j, since m is the z component of j. In fact, this representation describes each possible state a system may have if it has a total angular momentum of j. The number of possible states, or configurations, the system may have is 2j+ 1, so this representation is called a 2)+ 1 multiplet.
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50 In the most simple case, for j = n 1;n2 the neutron and the proton are represented as two states of a doublet, the nucleon. This case has some special names associated with it as it represents no spatial or temporal structure, but instead represents internal symmetries. The value j is called the isospin, and it is relabelled I. The value m is l3 or lz, and is the Z component of the isospin.
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51 12.0 Conclusion The above sections show some of the utility of Lie theory in the study of both classical and modern physics. The Lie algebras specifically studied in this paper, su(2) and the Heisenberg algebra, are only two. examples of the many Lie structures, both algebras and groups, that are useful in physics. In fact, the study of nuclear physics uses the Lie group SU(4), the st_rong interactions are described by SU(3), and the electroweak interaction is modelled by SU(2) x U(1 ). Work done now in advanced topics such as cosmic strings and grand unification theory uses Lie theory for modeling. The applicability of Lie groups and algebras to such a vast array of physical problems leads to questions about the basic structure of the physical world. Because the Lie products are noncommutative, much of the successful modeling of the physical world has noncommutative, rather than commutative products. The many areas of study in applications of Lie theory will provide meaning and intuition to this noncommutivity, as well as uncover m()re truths about the physical world.
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References [1] N. Jacobson, Basic algebra I, 2nd ed., W. H. Freeman and Company, New York, 1985. This source was used throughout this section. [2] D. H. Sattinger, 0. L. Weaver, Lie groups and algebras with applications to physics. geometry. and mechanics, SpringerVerlag New York, Inc., New 1986 [3] P. M. Cohn, Lie groups, Cambridge at the University Cambridge, UK, 1968 [4] A. N. Kolmogorov, S. V. Fomin, Introductory real analysis, Dover Publications, Inc., New York, 1970 [5] N. Jacobson, Lie algebras, Dover Publications, Inc., New York, 1962 52 [6] M. L. Curtis, Matrix groups, 2nd ed., SpringerVerlag New York Inc., New York, 1984 [7] F. T. Avignone, Ill, Lecture notes from Modem Physics class PHYS 503 Fall, 1983, at University of South Carolina, Columbia, SC was a primary source for this section. This material is in most mechanics books. [8] F. W. Constant, Theoretical physics. AddisonWesley Publishing Company, Inc., Reading, Massachusetts, 1954. Chapter 10, Advanced dynamics, is an excellent source for this material. [9] K. Gottfried, Quantum mechanics Vol1, W. A. Benjamin, Inc., New York, 1966

