Contents

I Preliminaries 51 Physical theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1 Consequences of Newtonian dynamical and measurement theories 82 The physical arena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Symmetry and groups . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Topological spaces and manifolds . . . . . . . . . . . . . . 202.3 The Euclidean group . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 The construction of Euclidean 3-space from the Euclidean group 30

3 Measurement in Euclidean 3-space . . . . . . . . . . . . . . . . . . . . 323.1 Newtonian measurement theory . . . . . . . . . . . . . . . . . . . 323.2 Curves, lengths and extrema . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 The Functional Derivative . . . . . . . . . . . . . . . . . . . . . . 363.3.1 An intuitive approach . . . . . . . . . . . . . . . . . . . . 363.3.2 Formal de!nition of functional differentiation . . . . . . . 42

3.4 Functional integration . . . . . . . . . . . . . . . . . . . . . . . . 534 The objects of measurement . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1 Examples of tensors . . . . . . . . . . . . . . . . . . . . . . . . . 554.1.1 Scalars and non-scalars . . . . . . . . . . . . . . . . . . . 554.1.2 Vector transformations . . . . . . . . . . . . . . . . . . . . 564.1.3 The Levi-Civita tensor . . . . . . . . . . . . . . . . . . . . 574.1.4 Some second rank tensors . . . . . . . . . . . . . . . . . . 59

4.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.1 Vectors as algebraic objects . . . . . . . . . . . . . . . . . 654.2.2 Vectors in space . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 The metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3.1 The inner product of vectors . . . . . . . . . . . . . . . . 744.3.2 Duality and linear maps on vectors . . . . . . . . . . . . . 754.3.3 Orthonormal frames . . . . . . . . . . . . . . . . . . . . . 81

4.4 Group representations . . . . . . . . . . . . . . . . . . . . . . . . 854.5 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

II Motion: Lagrangian mechanics 925 Covariance of the Euler-Lagrangian equation . . . . . . . . . . . . . . 946 Symmetries and the Euler-Lagrange equation . . . . . . . . . . . . . . 97

6.1 Noether#s theorem for the generalized Euler-Lagrange equation . 976.2 Conserved quantities in restricted Euler-Lagrange systems . . . . 101

6.2.1 Cyclic coordinates and conserved momentum . . . . . . . 1016.2.2 Rotational symmetry and conservation of angular momen-

tum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.2.3 Conservation of energy . . . . . . . . . . . . . . . . . . . . 1086.2.4 Scale Invariance . . . . . . . . . . . . . . . . . . . . . . . 109

6.3 Conserved quantities in generalized Euler-Lagrange systems . . . 1126.3.1 Conserved momenta . . . . . . . . . . . . . . . . . . . . . 1126.3.2 Angular momentum . . . . . . . . . . . . . . . . . . . . . 1146.3.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.3.4 Scale invariance . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177 The physical Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.1 Galilean symmetry and the invariance of Newton#s Law . . . . . 1207.2 Galileo, Lagrange and inertia . . . . . . . . . . . . . . . . . . . . 1227.3 Gauging Newton#s law . . . . . . . . . . . . . . . . . . . . . . . . 128

8 Motion in central forces . . . . . . . . . . . . . . . . . . . . . . . . . . 1348.1 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.1.1 Euler#s regularization . . . . . . . . . . . . . . . . . . . . 1388.1.2 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . 140

8.2 General central potentials . . . . . . . . . . . . . . . . . . . . . . 1438.3 Energy, angular momentum and convexity . . . . . . . . . . . . . 1458.4 Bertrand#s theorem: closed orbits . . . . . . . . . . . . . . . . . . 1488.5 Symmetries of motion for the Kepler problem . . . . . . . . . . . 152

8.5.1 Conic sections . . . . . . . . . . . . . . . . . . . . . . . . 1558.6 Newtonian gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16110 Rotating coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

10.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16610.2 The Coriolis theorem . . . . . . . . . . . . . . . . . . . . . . . . . 170

11 Inequivalent Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . 17211.1 General free particle Lagrangians . . . . . . . . . . . . . . . . . . 17211.2 Inequivalent Lagrangians . . . . . . . . . . . . . . . . . . . . . . . 175

2

x p.

III Conformal gauge theory 181

11.2.1 Are inequivalent Lagrangians equivalent? . . . . . . . . . 17811.3 Inequivalent Lagrangians in higher dimensions . . . . . . . . . . . 179

12 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18212.1 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18212.2 Relativistic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 18412.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18912.4 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 19012.5 Relativistic action with a potential . . . . . . . . . . . . . . . . . 191

13 The symmetry of Newtonian mechanics . . . . . . . . . . . . . . . . . 19613.1 The conformal group of Euclidean 3-space . . . . . . . . . . . . . 19713.2 The relativisic conformal group . . . . . . . . . . . . . . . . . . . 20313.3 A linear representation for conformal transformations . . . . . . 204

14 A new arena for mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 20714.1 Dilatation covariant derivative . . . . . . . . . . . . . . . . . . . 20814.2 Consequences of the covariant derivative . . . . . . . . . . . . . . 21114.3 Biconformal geometry . . . . . . . . . . . . . . . . . . . . . . . . 21214.4 Motion in biconformal space . . . . . . . . . . . . . . . . . . . . . 21514.5 Hamiltonian dynamics and phase space . . . . . . . . . . . . . . 216

14.5.1 Multiparticle mechanics . . . . . . . . . . . . . . . . . . . 21814.6 Measurement and Hamilton#s principal function . . . . . . . . . . 22014.7 A second proof of the existence of Hamilton#s principal function . 22414.8 Phase space and the symplectic form . . . . . . . . . . . . . . . . 22714.9 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

14.9.1 Example 1: Coordinate transformations . . . . . . . . . . 23614.9.2 Example 2: Interchange of and . . . . . . . . . . . . . 23814.9.3 Example 3: Momentum transformations . . . . . . . . . . 238

14.10Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . 23915 General solution in Hamiltonian dynamics . . . . . . . . . . . . . . . . 241

15.1 The Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . 24115.2 Quantum Mechanics and the Hamilton-Jacobi equation . . . . . 24215.3 Trivialization of the motion . . . . . . . . . . . . . . . . . . . . . 243

15.3.1 Example 1: Free particle . . . . . . . . . . . . . . . . . . . 24615.3.2 Example 2: Simple harmonic oscillator . . . . . . . . . . . 24715.3.3 Example 3: One dimensional particle motion . . . . . . . 249

3

(3)( )

soso p, q

IV Bonus sections 25116 Classical spin, statistics and pseudomechanics . . . . . . . . . . . . . . 251

16.1 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25116.2 Statistics and pseudomechanics . . . . . . . . . . . . . . . . . . . 25416.3 Spin-statistics theorem . . . . . . . . . . . . . . . . . . . . . . . . 258

17 Gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26117.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26217.2 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

17.2.1 The Lie algebra . . . . . . . . . . . . . . . . . . . . 27017.2.2 The Lie algebras . . . . . . . . . . . . . . . . . . . 27417.2.3 Lie algebras: a general approach . . . . . . . . . . . . . . 276

17.3 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 28017.4 The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . 28517.5 The Hodge dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28817.6 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 28917.7 The Levi-Civita tensor in arbitrary coordinates . . . . . . . . . . 29117.8 Differential calculus . . . . . . . . . . . . . . . . . . . . . . . . . 292

17.8.1 Grad, Div, Curl and Laplacian . . . . . . . . . . . . . . . 29417.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

4

33

Part I

PreliminariesGeometry is physics; physics is geometry. It is human nature to unify our expe-rience, and one of the important specializations of humans to develop languageto describe our world. Thus, we !nd the unifying power of geometric descriptiona powerful tool for spotting patterns in our experience, while at the same time,the patterns we !nd lead us to create new terms, concepts, and most impor-tantly, pictures. Applying geometric abstractions to model things in the world,we discover new physics. Creating new pictures, we invent new mathematics.

Our ability to make predictions based on perceived patterns is what makesphysics such an important subject. The more mathematical tools we have athand, the more patterns we can model, and therefore the more predictions wecan make. We therefore start this adventure with some new tools, Lie groupsand the calculus of variations. We will also cast some old familiar tools in a newform.

[DESCRIBE EACH]With this extension of the familiar calculus, along with some new ways to

look at curves and spaces, we will be able to demonstrate the naturalness andadvantages of the most elegant formulation of Newton#s laws of mechanics: thephase space formulation due to Euler, Lagrange, Hamilton and many others, inwhich the position and the momentum of each particle are treated as indepen-dent variables coupled by a system of !rst-order differential equations.

We will take a gradual approach to this formulation. Our starting pointcenters on the idea of space, an abstraction which started in ancient Greece.Basing our assumptions in our direct experience of the world, we provisionallywork in a -dimensional space, together with time. Because we want to associatephysical properties with objects which move in this space rather than with thespace itself, we demand that the space be homogeneous and isotropic. This leadsus to the construction of (among other possibilities) Euclidean -space. In orderto make measurements in this space, we introduce a metric, which allows us tocharacterize what is meant by uniform motion. We also treat the descriptionof matter from a fundamental approach, carefully de!ning the mathematicalstructures that might be used to model the stuff of the world. Finally, wedevelop techniques to describe the motion of matter.

Each of these developments involves the introduction of new mathematics.

5

!

! "

"

Fv

EB

BE J

=

1= 0

1+ =

4

=

dynamical laws

1. Physical theories

md

dt

c

d

dt

c

d

dt

!

c

H" i"

t".

The description of uniform motion leads to the calculus of variations, the de-scription of matter leads to a discussion of vectors and tensors, and our analysisof motion requires techniques of differential forms, connections on manifolds,and gauge theory.

Once these tools are in place, we derive various well-known and not so well-known techniques of classical (and quantum) mechanics.

Numerous examples and exercises are scattered throughout.[MORE HERE]Enjoy the ride!

Within any theory of matter and motion we may distinguish two conceptuallydifferent features: dynamical laws and measurement theory. We discuss each inturn.

By , we mean the description of various motions of objects,both singly and in combination. The central feature of our description is gen-erally some set of dynamical equations. In classical mechanics, the dynamicalequation is Newton#s second law,

or its relativistic generalization, while in classical electrodynamics two of theMaxwell equations serve the same function:

The remaining two Maxwell equations may be regarded as constraints on theinitial !eld con!guration. In general relativity the Einstein equation gives thetime evolution of the metric. Finally, in quantum mechanics the dynamical lawis the Schrdinger equation

which governs the time evolution of the wave function,

6

! " %

%%%

measurement theory

x

u v u v

j k

x

j x

k x

34

=

=

=

=

( ) = (3 4 5 )

= 3

= 4

= 5

.

,

, ,

x

y

z

l. lx, y, z m, m, m

x

ly

lz

l

Several important features are implicit in these descriptions. Of course thereare different objects $ particles, !elds or probability amplitudes $ that must bespeci!ed. But perhaps the most important feature is the existence of some arenawithin which the motion occurs. In Newtonian mechanics the arena is Euclidean-space, and the motion is assumed to be parameterized by universal time.Relativity modi!ed this to a -dimensional spacetime, which in general relativitybecomes a curved Riemannian manifold. In quantum mechanics the arena isphase space, comprised of both position and momentum variables, and againhaving a universal time. Given this diverse collection of spaces for dynamicallaws, you may well ask if there is any principle that determines a preferred space.As we shall see, the answer is a quali!ed yes. It turns out that symmetry givesus an important guide to choosing the dynamical arena.

A is what establishes the correspondence between cal-culations and measurable numbers. For example, in Newtonian mechanics theprimary dynamical variable for a particle is the position vector, While thedynamical law predicts this vector as a function of time, we never measure avector directly. In order to extract measurable magnitudes we use the Euclideaninner product,

If we want to know the position, we specify a vector basis ( say) forcomparison, then specify the numbers

These numbers are then expressed as dimensionless ratios by choosing a lengthstandard, If is chosen as the meter, then in saying the position of a particleis a we are specifying the dimensionless ratios

A further assumption of Newtonian measurement theory is that particles movealong unique, well-de!ned curves. This is macroscopically sound epistemology,

7

V! 3=

| " " " "d x

"V.

gauge theory

1.1. Consequences of Newtonian dynamical and measurement theories

since we can see a body such as a ball move smoothly through an arc. However,when matter is not continuously monitored the assumption becomes suspect. In-deed, the occasional measurements we are able to make on fundamental particlesdo not allow us to claim a unique path is knowable, and quantum experimentsshow that it is incorrect to assume that unique paths exist.

Thus, quantum mechanics provides a distinct example of a measurementtheory $ we do not assume unique evolution. Perhaps the chief elements ofquantum measurement theory is the Hermitian inner product on Hilbert space:

and its interpretation as the probability of !nding the particle related to inthe volume As noted in the preceeding paragraph, it is incorrect to assume aunique path of motion. The relationship between expectation values of operatorsand measurement probabilities is a further element of quantum measurementtheory.

The importance of the distinction between dynamical laws and measurementtheories will become clear when we introduce the additional element of symmetryin the !nal sections of the book. In particular, we will see that the techniquesof allow us to reconcile differences between the symmetry of adynamical law and the symmetry of the associated measurement theory. Weshall show how different applications of gauge theory lead to the Lagrangianand Hamiltonian formulations of classical mechanics, and eventually, how a smallchange in the measurement theory leads to quantum mechanics.

One of our goals is to develop a systematic approach to !nding dynamical lawsand measurement theories. This will require us to examine some mathemat-ical techniques, including functional analysis, group theory and gauge theory.Nonetheless, some features of our ultimate methods may be employed immedi-ately, with a more sophisticated treatmenr to follow. With this in mind, we nowturn to a development of Newton#s law from certain prior ideas.

Our starting point is geometry. Over two thousand years ago, Aristotle askedwhether the space occupied by an object follows the object or remains where itwas after the object moves away. This is the conceptual beginning of abstractspace, independent of the objects in it. The idea is clearly an abstraction,

8

11

any

convenient

Brief statement about Popper and Berkeley.

and physicists have returned again and again to the inescapable fact that weonly know space through the relationships between objects. Still, the idea of acontinuum in which objects move may be made rigorous by considering the fullset of possible positions of objects. We will reconsider the idea in light of somemore contemporary philosophy.

Before beginning our agruments concerning spacde, we de!ne another ab-straction: the particle. By a particle, we mean an object sufficiently small anduncomplicated that its behavior may be accurately captured by specifying itsposition only. This is our physical model for a mathematical point. Naturally,the smallness of size required depends on the !neness of the description. Formacroscopic purposes a small, smooth marble may serve as a model particle, butfor the description of atoms it becomes questionable whether such a model evenexists. For the present, we assume the existence of effectively point particles,and proceed to examine space. It is possible (and desirable if we take seriouslythe arguements of such philosophers as Popper and Berkeley ), to begin withour immediate experience.

Curiously enough, the most directly accessible geometric feature of the worldis time. Our experience is a near-continuum of events. This is an immediateconsequence of the richness of our experience. In fact, we might de!necontinuous or nearly continuous element of our experience $ a succession ofcolors or a smooth variation of tones $ as a direction for time. The fact thatwe do not rely on any one particular experience for this is probably because wechoose to label time in a way that makes sense of the most possible experienceswe can. This leads us to rely on correlations between many different experiences,optimizing over apparently causal relationships to identify time. Henceforward,we assume that our experience unfolds in an ordered sequence.

A simple experiment can convince us that a 3-dim model is fordescribing that experience. First, I note that my sense of touch allows metrace a line down my arm with my !nger. This establishes the existence ofa single continuum of points which I can distinguish by placing them in 1-1correspondence with successive times. Further, I can lay my hand &at on myarm, experiencing an entire 2-dim region of my skin. Finally, still holding myhand against my arm, I cup it so that the planar surface of my arm and theplanar surface of my hand are not in contact, although they still maintain acontinuous border. This establishes the usefulness of a third dimension.

A second, similar experiment makes use of vision. Re&ecting the 2-dim

9

==

=

=

=

1 2 3

1 2 3

1 2 3

1 1 1 2 2 2 3 3 3

1 2 3 4

Exercise 1.1.

Exercise 1.2.

x .a a a . . .

x .b b b . . .

z .c c c . . .

w,

w .a b c a b c a b c . . .

w,w .d d d d . . .

What idea of spatial relation can we gain from the senses ofsmell and taste?

What is the dimension of the world of touch?

nature of our retina, visual images appear planar $ I can draw lines betweenpairs of objects in such a way that the lines intersect in a single intermediatepoint. This cannot be done in one dimension. Furthermore, I am not presentedwith a single image, but perceive a succession in time. As time progresses, I seesome images pass out of view as they approach others, then reemerge later. Suchan occultation is easily explained by a third dimension. The vanishing objecthas passed on the far side of the second object. In this way we rationalize thedifference between our (at least) 2-dim visual experience and our (at least) 3-dimtactile experience.

As we all know, these three spatial dimensions together with time providea useful model for the physical world. Still, a simple mathematical proof willdemonstrate the arbitrariness of this choice. Suppose we have a predictive physi-cal model in 3-dim that adequately accounts for various phenomena. Then thereexists a completely equivalent predictive model in any dimension. The proof fol-lows from the proof that there exist 1-1 onto maps between dimensions, whichwe present !rst.

For simplicity we focus on a unit cube. For any point in the three dimensionalunit cube, let the decimal expansions for the Cartesian coordinates be

We map this point into a 1-dim continuum, by setting

This mapping is clearly 1-1 and onto. To map to a higher dimension, we takeany given

10

n n

n n

n n

n n n n

i

i

Exercise 1.3.

1 1 +1 2 +1

2 2 +2 2 +2

3 3 +3 2 +3

2 3

=

=

=

=

( )

( )

3 ( )

3

0 999;0 999

000 999

(999 345 801 )

= 999345801

x .d d d . . .

x .d d d . . .

x .d d d . . .

x .d d d . . .

x t

w t

x $, %, &

,,

, , , . . .

w . . . .

and partition the decimal expansion,

...

Now suppose we have a physical model making, say, a prediction of the positionof a particle 3-dim as a function of time,

Applying the mapping gives a 1-dim sequence,

containing all the same information.Thus, any argument for a three dimensional model must be a pragmatic

one. In fact, even though this mapping is never continuous, there might existmodels in other dimensions that display useful properties more effectively thanour usual 3-dim models.

11

Here is an example that shows how descriptions in different di-mensions can reveal different physical information about a body. Consider thedescription of an extended body. In a three dimensional representation, wemight specify a parameter family of positions, together with suit-able ranges for the parameters. Alternative, we may represent this as a singlenumber as follows. Divide a region of -space into 1 meter cubes; divide eachcube into 1000 smaller cubes, each one decimeter on a side, and so on. Num-ber the 1 meter cubes from to number the decimeter cubes within eachmeter cube from to and so on. Then a speci"c location in space may beexpressed as a sequence of numbers between and

which we may concatenate to give

"n

n

3

1 2 1 2 3

2. The physical arena

Exercise 1.4.

Exercise 1.5.

= 999345801 274

=

( )10 ( ( ) ( ) ( ))

w . . . .

R

x a a . . . a .b b b . . .

w tx x t , y t , z t .

homogeneous, isotropic scale invariant

This is clearly a 1-1, onto map. Now, for an extended body, choose a pointin the body. About this point there will be a smallest cube contained entirelywithin the body. The speci"cation of this cube is a decimal expansion,

Additional cubes which together "ll the body may be speci"ed. Disuss theoptimization of this list of numbers, and argue that an examination of a suitablyde"ned list can quickly give information about the total size and shape of thebody.

Devise a scheme for mapping arbitrary points in to a singlereal number. Hint: The essential problem here is that the decimal expansionmay be arbitrarily long in both directions:

Try starting at the decimal point.

Devise a 1-1, onto mapping from the 3-dim position of a particleto a 1-dim representation in such a way that the number is always within

of the component of

!nite

We have presented general arguments that we can reconcile our visual and tac-tile experiences of the world by choosing a 3-dim model, together with time. Westill need to specify what we mean by a space. Returning to Aristotle#s question,we observe that we can maintain the idea of the 'space( where an object was asindependent of the body if we insist that 'space( contain no absolute informa-tion. Thus, the orientation of a body is to be a property of the body, not of thespace. Moreover, it should not matter whether a body is at this or that locationin space. This notion of space lets us specify, for example, the relative nearnessor farness of two bodies without any dependence on the absolute positions of thebodies. These properties are simply expressed by saying that space should haveno preferred position, direction or scale. We therefore demand a 3-dim spacewhich is and .

12

""

group

"

" ""

1

1 1

1

1 2 3 1 2 3

1 2 3 1 2 3 1 2 3

2.1. Symmetry and groups

!

{{ ! } "}"

" !!

! !

" " ! ! " " !" ! " ! " ! " !

==

=

( ) = ( )

1 4

= 1 1

1 11 1 11 1 1

1

1 (1 ( 1)) = 1 = (1 1) ( 1)

1 ( 1 ( 1)) = 1 = (1 ( 1)) ( 1)

G,

e,e G.

g, AA ,

g , A A.g g e. g G, g G

g g e.

g , g G, g G g g g .

g , g , g G, g g g g g g .

S, .

B, B , ,.

etc.

Mathematically, it is possible to construct a space with any desired symmetryusing standard techniques. We begin with a simple case, reserving more involvedexamples for later chapters. To begin, we !rst de!ne a mathematical objectcapable of representing symmetry. We may think of a symmetry as a collectionof transformations that leave some essential properties of a system unchanged.Such a collection, of transformations must have certain properties:

1. We may always de!ne an identity transformation, which leaves the sys-tem unchanged:

2. For every transformation, taking the system from description to an-other equivalent description there must be another transformation, de-noted that reverses this, taking to The combined effect of the twotransformations is therefore We may write:

3. Any two transformations must give a result which is also achievable by atransformation. That is,

4. Applying three transformations in a given order has the same effect if wereplace either consecutive pair by their combined result. Thus, we haveassociativity:

These are the de!ning properties of a mathematical . Precisely, a groupis a set, of objects together with a binary operation satisfying propertiesWe provide some simple examples.

The binary, or Boolean, group, consists of the pairwhere is ordinary multiplication The multiplication table is therefore

Naturally, is the identity, while each element is its own inverse. Closure isevident by looking at the table, while associativity is checked by tabulating alltriple products:

13

$$$

$

n

n

n n

n n n

n n n

n n n

n n n

{ ! } )

) ! *

)

* ! !

** *

) ) ) ! *) ) ) ! *

) ) ) !) ) ) !

!! *

!

De!nition 2.1.

De!nition 2.2. Theorem 2.3.

Remark 1.

=0 1 2 1

=+ +

+ +

+

= ( )

+ + + +

+ +

+ +

+ +

( ) = ( + ) =+ + + +

+ + + +

( ) = ( + ) =+ + + +

+ + + +

+ ++ + 2

( ) = ( + ) = + +

( ) = ( + )

=+ + + + 2+ + 2 + + 2

= + +

The pair is therefore a group.There is another way to write the Boolean group, involving modular addition.

We de!ne

Addition mod always produces a group with elements:

14

B

S n S, , , . . . , n . n n , S.

a, b S

a ba b a b < n

a b n a b n

n n

G S,

na b < n a b S, a b n a b n S.

a n a.

a b < n, b c < n

a b < n, b c n

a b n, b c n

a b c a b ca b c a b c < n

a b c n a b c n

a b c a b ca b c a b c < n

a b c n a b c n

a b < n, b c > n,n < a b c < n.

a b c a b c a b c n

a b c a b c n

a b c n a b c < na b c n a b c n

a b c n

Let be the set of consecutive integers beginning zero,Addition modulo (or mod ), is cyclic addition on

That is, for all

where is the usual addition of real numbers.

The pair is a group.

For proof, we see immediately that multiplication mod is closed,because if then while if thenZero is the additive identity, while the inverse of is Finally, to checkassociativity, we have four cases:

The "rst case is immediate because

In the second case, we note that since and we must haveTherefore,

2one

n n n

n n n

n n n

n n n

n

+

) ) ! ) !) ) ) ! !

) ) ! ) !) ) ) ! !

)

!

{ }-

-

+ + 2

2 + + 3

( ) = ( + ) = + +

( ) = ( + ) = + +

( ) = ( + ) = + + 2

( ) = ( + ) = + + 2

2

0 10 0 11 1 0

0 1 1 1

=

=

n < a b c < n

n a b c < n

a b c a b n c a b c n

a b c a b c n a b c n

a b c a b n c a b c n

a b c a b c n a b c n

G n

.

,

S a, b

a bab

a e.

a ba a bb b

For the "nal case, we have two subcases:

In the "rst subcase,

while in the second subcase,

Therefore, is an -element group.

Now, returning to our discussion of the Boolean group, consider additionmod The multiplication table is

Notice that if we rename the elements, replacing and we repro-duce the multiplication of the Boolean group. When this is the case, we say wehave two representations of the same group. We make this idea more precisebelow. First, we prove that, while there may be different representations, thereis only group with two elements. To prove this, suppose we pick any setwith two elements, and write the general form of the multiplicationtable:

One of these must be the identity; without loss of generality we chooseThen

15

isomorphism

De!nition 2.4.

1 2

1 2 1 2

1 2 3

1 1

2 2

3 3

1 2 3

2

-

!

) .

. )

)

.

- { } ) { }

{{ } .} .

1 1

= ( ) = ( )

( ) ( ) = ( )

= = ( )

= ( )

= ( )

= ( )

=

( ) = 0 ( ) = 1= ( ) = ( 0 1 )

=

b a,b

a ba a bb b a

a , b

G S, H T, '', G H. '

g , g G,

' g ' g ' g g

G H, G H

'g g g . h ' g

g,

h ' g

h ' g

h ' g

h h h

'' a ' b

G , a, b H , ,

G a, b, c , , ,G

Let and be two groups and let be aone-to-one, onto mapping, between and Then is an if itpreserves the group product in the sense that for all in

When there exists an isomporphism between and then and are saidto be to one another.

Finally, since must have an inverse, and its inverse cannot be we must !llin the !nal spot with the identity, thereby making its own inverse:

Comparing to the boolean table, we see that a simple renaming,reproduces the boolean group. The relationship between different representa-tions of a given group is made precise by the idea of an .

isomorphism

(2.1)

isomorphic

The de!nition essentially means that provides a renaming of the elementsof the group. Thus, suppose Thinking of as the new namefor and setting

eq.(2.1) becomes

Applying the group product may be done before or after applying with thesame result. In the Boolean case, for example, setting andshows that and are isomorphic.

Now consider a slightly bigger group. We may !nd all groups with threeelements as follows. Let where the group operation,remains to be de!ned by its multiplication table. In order for to be a group,

16

% & % &"

" "" "

1

1 1

1 1

=

)

=

=

=

( ) = ( )

=

=

=

.

.

.

. .

. . . .. . . .

. .

.

a e.

e b ce e b cb bc c

z, cx

y

x c z

y c z

x c y c

cc

x c c y c c

x c c y c c

x e y e

x y

x y

e b ce e b cb b c ec c e b

one of the elements must be the identity. Without loss of generality, we pickThen the multiplication table becomes

Next, we show that no element of the group may occur more than once in anygiven row or column. To prove this, suppose some element, occurs in thecolumn twice. Then there are two distinct elements (say, for generality, and

such that

From this we may write

But since must be invertible, we may multiply both sides of this equation byand use associativity:

in contradiction with and being distinct elements. The argument for anelement occurring twice in the same row is similar.

Returning to the three-element multiplication table, we see that we have nochoice but to !ll in the remaining squares as

thereby showing that there is exactly one three element group.Many groups are already familiar:

17

{ } !

{ }

!

.{ .} { .}

.

! ! ! !

x

y

xy

x y xy

x y xy

x x xy y

y y xy x

xy xy y x

Example 2.5.

Example 2.6.

Exercise 2.1.

Exercise 2.2.

Exercise 2.3.

Exercise 2.4.

= ++ 0 + = + 0 = + ( ) = 0

+ ( + ) = ( + ) +

= mod = + )

= ++ 0 + = + 0 =

+ ( ) = 0 + ( + ) = ( + ) +

= 3 + 1 + 04 + 001 + 0005 + 00009 +

= =+1

: ( ) ( )

: ( ) ( )

: ( ) ( )

8

G Z, ,a, b, c a b R a a a a a

a b c a b c Gp, p

a b p n a b np .

G R, ,a, b, c a b R a a a

a a a b c a b c GQ,

! . . . . . . . .

ab a b.

G S, G S , ,S S. nn

n

R x, y x, y

R x, y x, y

R x, y x, y

e R R Re e R R RR R e R RR R R e RR R R R e

Of course, the real numbers form a !eld, which is a much nicer object thana group.

In working with groups it is generally convenient to omit the multiplicationsign, writing simply in place of

18

Let the integers under addition. For all integerswe have (closure); (identity);

(inverse); (associativity). Therefore, is a group. Theintegers also form a group under addition mod where is any integer (Recallthat if there exists an integer such that

Let the real numbers under addition. For all realnumbers we have (closure); (identity);

(inverse); (associativity). Therefore, isa group. Notice that the rationals, are not a group under addition becausethey do not close under addition:

A subgroup of a group is a group withthe same product, such that is a subset of Prove that a group withelements has no subgroup with elements. (Hint: write the multiplication tablefor the element subgroup and try adding one row and column.)

Find all groups (up to isomorphism) with four elements.

Show that the three re#ections of the plane

together with the identity transformation, form a group. Write out the multi-plication table.

Find the -element group built from the three dimensional re-#ections and their products.

n

{ }

!

!

! "

"

! !

! ! !!

!

' ( ' (' (

$$' ( ) )

' (' (' (' (

2.2. Lie groups

Example 2.7.

Example 2.8.

Example 2.9.

G R, ,R

R

n Vn n

x, y(

x x ( y (

y x ( y (

xy

( (( (

xy

R( (( (

, ( , ! ,

R ( R ( (( (

( ( ( ( ( ( ( (

( ( ( (

= +

1

( ):

= cos sin

= sin + cos

=cos sinsin cos

=cos sinsin cos

[0 2 )

( ) ( ) =cos sinsin cos

cos sinsin cos

=cos cos sin sin cos sin sin cossin cos + cos sin sin sin + cos cos

=cos ( + ) sin ( + )sin ( + ) cos ( + )

While groups having a !nite number of elements are entertaining, and even !nduse in crystalography, most groups encountered in physics have in!nitely manyelements. To specify these elements requires one or more continuous parameters.We begin with some familiar examples.

19

The real numbers under addition, form a Lie groupbecause each element of provides its own label. Since only one label is required,is a -dimensional Lie group.

The real, -dim vector space under vector addition is an-dim Lie group, since each element of the group may be labeled by real

numbers.

: Rotations in the plane. Consider a rotation of vectorsthrough an angle

which we may write as

The transformation matrices

where is normal matrix multiplication, form a group. To see this, considerthe product of two elements,

aa a b a b

x x a

aa

a

2 2 2

3

3+

!

{{ } }

2( ) (2 )

2 ( )= +

3

( ) = +

= =

manifolds topological space

!.R ( R ! ( ,

Abelian.

R (l x y .

T

V

T T , V , T T T

Exercise 2.5.

Example 2.10.

Example 2.11.

2.2.1. Topological spaces and manifolds

We can give a precise de!nition of a Lie group if we !rst de!ne the useful class ofspaces called . The de!nition relies on the idea of a .This section is intended only to give the reader an intuitive grasp of these terms.Thus, while the de!nitions are rigorous, the proofs appeal to familiar propertiesof lines and planes. For example, a rigorous form of the proof of the equality oftwo topologies for the plane given below requires detailed arguments involvingterms we do not introduce here such as limit points and metrics. A completetreatment takes us too far a!eld, but a few of the many excellent presentationsof these topics are listed in the references.

20

so the set is closed under multiplication as long as we consider the addition ofangles to be addition modulo We immediately see that the inverse to anyrotation is a rotation by and the associativity of (modular)addition guarantees associativity of the product. Notice that rotations in theplane commute. A group in which all products commute is called

Show that the -dim rotation group preserves the Euclid-ean length,

Rotations in -dim. These depend on three parameters, whichmay be de"ned in various ways. We could take an angle of rotation about eachof the three Cartesian coordinate axes. A more systematic approach is to use theEuler angles. The simplest parameterization, however, is to specify a unit vectorgiving the axis of rotation, and an angle representing the amount of the rotationaround that axis. The latter method involves the elegant use of quaternions,pioneered by Klein in the 19th century. We will give a complete treatment ofthis Lie group when we begin our study of Lie algebras.

Translations in 3-dim. Consider the translations of 3-space,given by vector addition:

where is any 3-vector,

The pair is a Lie group.

' (

{ }2

0

0

2 2

0

( )

!

!

!

n

n

P

De!nition 2.12.

open setsclosed

! !

!

{ }

{ | }|!

( )( ) ( ) ( )

[ ]

=1 1

0

= ( ) 0

( ) = ( ) + ( ) 1 + 1

( )

S, S* , S

A, B * , A B * .

A * ,A

* .

', S * .

S

a, b ,a, , , b ,

a, b .

A

An,n

n .R . *

* U a, b a, b R, >

U a, b a, b x, y < x y