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Geometric

Algebra

for

Mathematics

and

Physics

-aUniﬁed

Language

Department

of

Physics

and

Astronomy,

Arizona

State

University,

Tempe,

Arizona

85287-1504

David

Hestenes

Abstract

Physics

and

other

applications

of

mathematics

employ

amiscella-

neous

assortment

of

mathematical

tools

in

ways

that

contribute

to

a

fragmentation

of

knowledge.

We

can

do

better!

Research

on

the

de-

sign

and

use

of

mathematical

systems

provides

aguide

for

designing

a

uniﬁed

mathematical

language

for

the

whole

of

physics

that

facilitates

learning

and

enhances

insight.

The

result

of

developments

over

sev-

eral

decades

is

acomprehensive

language

called

Geometric

Algebra

with

wide

applications

to

physics

and

engineering.

This

lecture

is

an

intr0-

duction

to

Geometric

Algebra

with the

goal

of

incorporating

it

into

the

$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}/\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{s}$

curriculum.

I.

Introduction

The

main

subject

of

my

lecture

is

aconstructive

critique

of

the

mathematical

language

used

in

physics

with

an

introduction

to

auniﬁed

language

that

has

been

developed

over

the

last

forty

years

to

replace

it.

The

generic

name

for

that

language

is

Geometric

Algebra

(GA).

The

material

is

developed

in

suﬃcient

detail

to

be

useful

in

instruction

and

research

and

to

provide

an

entree

to

the

published

literature.

After

explaining

the

utter

simplicity

of

the

GA

grammar

in

Section

III,

I

explicate

the

following

unique

features

of

the

mathematical

language:

(1)

GA

seamlessly

integrates

the

properties

of

vectors

and

complex

numbers

to

enable

acompletely

coordinate-ﬀee

treatment

of

$2\mathrm{D}$

physics.

(2)

GA

articulates

seamlessly

with

standard

vector

algebra

to

enable

easy

contact

with

standard

literature

and

mathematical

methods.

(3)

GA

reduces

“

$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}$

,

$\mathrm{d}\mathrm{i}\mathrm{v}$

,

curl

and

all

that”

to

asingle

vector

derivative

that,

among

other

things,

combines

the

standard

set

of

four

Maxwell

equations

into

asingle

equation

and

provides

new

methods

to

solve

it.

Moreover

the

GA

formulation

of

spinors

facilitates

the

treatment

of

rotations

and

rotational

dynamics

in

both

classical

and

quantum

mechanics

without

co

数理解析研究所講究録 1378 巻2004 年7-32

$\epsilon$

ordinates

or

matrices.

GA

provides

fresh

insights

into

the

geometric

structure

of

quantum

mechanics

with

implications

for

its

physical

$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}.19-30$

All

of

this

generalizes

smoothly

to

a

completely

coordinate-ﬀee

language

for

space-

time

physics

and

general

$\mathrm{r}\mathrm{e}1\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}^{1-3,36,37}$

.

The

development

of

GA

has

been

a

central

theme

of

my

own

research

in

theoretical

physics

and

mathematics.

II.

Mathematics

for

Modeling

Physical

Reality

Mathematics

is

taken

for

granted

in

the

physics

curriculum–a

body

of

im-

mutable truths

to

be

assimilated

and

applied.

The profound

inﬂuence

of

math-

ematics

on

our

conceptions

of

the

physical

world

is

never

analyzed.

The

pos-

sibility

that

mathematical

tools

used

today

were

invented

to

solve

problems

in

the

past

and

might

not

be

well

suited

for

current

problems

is

never

considered.

One

does

not

have

to

go

very

deeply

into

the

history

of

physics

to

discover

the

profound

inﬂuence

of

mathematical

invention.

Two

famous

examples

will

suﬃce

to

make

the

point:

The

invention

of

analytic

geometry

and

calculus

was

essential

to

Newton’s

creation

of

classical

$\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{s}.4$

The

invention

of

tensor

analysis

was

essential

to

Einstein’s

creation

of

the

General

Theory

of

Relativity.

Note

my

use

of

the

terms

“invention”

and

“creation”

where

others

might

have

used

the

term

“discovery.”

This

conforms

to

the

epistemological

stance

of

Modeling

$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}^{4,6\triangleleft}$

and

Einstein

himself,

who

asserted

that

scientiﬁc

theories

”cannot

be

extracted

from

experience,

but

must

be

ﬀaely

$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{d}^{n9}$

.

The

point

wish

to

make

by

citing

these

two

examples

is

that

without

essential

mathematical

concepts

the two

theories

would

have

been

literally

inconceivable.

The

mathematical

modeling

tools

we

employ

at

once

extend

and

limit

our

ability

to

conceive

the

world.

Limitations

of

mathematics

are

evident

in

the

fact

that

the

analytic

geometry

that

provides

the

foundation

for

classical mechanics

is

insuﬃcient

for

General

Relativity.

This

should

alert

one

to

the

possibility

of

other

conceptual

limits

in

the

applications

of

mathematics.

Since

Newton’s

day

avariety

of

diﬀerent

symbolic

systems

have

been invented

to

address

problems

in

diﬀerent

contexts.

Figure

1lists

nine

such systems

in

use

today.

Few

scientists

are

proﬁcient

with

all

of

them,

but

each

system

has

advantages

over

the

others

in

some

application

domain. For

example,

for

applications

to

rotations,

quaternions

are

demonstrably

more

eﬃcient

than the

vectorial

and

matrix

methods

taught

in

standard

linear

algebra

courses.

The

diﬀerence

hardly

matters

in

the

world

of

academic

exercises,

but

in

the

aerospace

industry,

for

instance,

where

rotations

are

bread

and

butter,

engineers

opt

for

quaternions.

Each

of

the

mathematical

systems

in

Fig.

1

incorporates

some

aspect

of

ge

ometry.

Taken

together, they

constitute

ahighly

redundant

system

of

multiple

representations

for

geometric

concepts

that

are

essential

in

physics

and

other

applications

of

mathematics.

As

mathematical

language,

this

Babel

of

mathe-

matical

tongues

has

the

following

defects:

1.

Limited

access.

Scientiﬁc

ideas,

methods

and

results

are

distributed

broadly

across

these

diverse

mathematical

systems.

Since

most

scientists

are

Fig.

1.

Multiple

mathematical

systems

contribute

to

the frag-

mentation

of

knowledge,

though

they

have

acommon

geomet-

$\mathrm{r}\mathrm{i}\mathrm{c}$

nexus.

proﬁcient

with

only

afew

of

the

systems,

their

access

to

knowledge

formulated

in

other

systems

is

limited

or

denied.

Of

course,

this

language

barrier

is

even

greater

for

students.

2.

Wasteful

redundancy.

In

many

cases,

the

same

information

is

repre

sented

in

several

diﬀerent

systems,

but

one

of

them

is

invariably

better

suited

than

the others

for

agiven

application.

For

example,

Goldstein’s

textbook

on

$\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{s}^{1}$

gives

three

diﬀerent

ways

to

represent

rotations:

coordinate

matri-

ces,

vectors

and

Pauli

spin

matrices.

The

costs

in

time

and

eﬀort

for

translation

between

these

representations

are

considerable.

3.

Deﬁcient

integration.

The

collection

of

systems

in

Fig.

1is

not

an

integrated

mathematical

structure.

This

is

especially

awkward

in

problems

that

call

for

the

special

features

of

two

or

more

systems.

For,

example,

vector

algebra

and

matrices

are

often

awkwardly

combined

in

rigid

body

mechanics,

while

Pauli

matrices

are

used

to

express

equivalent

relations

in

quantum

mechanics.

4.

Hidden

structure.

Relations

among

scientiﬁc

concepts

represented

in

diﬀerent

symbolic systems

are

diﬃcult

to

recognize

and

exploit.

5.

Reduced

information

density.

The

density

of

information

about

na-

ture

is

reduced

by

distributing

it

over

several

diﬀerent

symbolic

systems.

Evidently

elimination

of

these

defects

will

make

physics

(and

other

scientiﬁc

disciplines)

easier

to

learn

and

apply.

Aclue

as

to

how

that

might

be

done

lies

in

recognizing

that

the

various

symbolic

systems

derive

geometric

interpretations

ﬀom

a

common

coherent

core

of

geometric

concepts. This

suggests

that

one

can

create

auniﬁed

mathematical

language

for

physics

(and

thus

for

alarge

portion

of

mathematics

and

its

applications)

by

designing

it

to

incorporate

an

optimal

representation

of

geometric

concepts.

In

fact,

Hermann

Grassmann

recognized

this

possibility

and

took

it

along

way

more

than

150

years

$\mathrm{a}\mathrm{g}\mathrm{o}.13$ However,

his

program

to

unify

mathematics

was

forgotten

and

his

mathematical

idea

10

were

dispersed,

though

many

of

them

reappeared

in

the

several

systems

of

Fig.

1.

A

century

later

the

program

was

reborn,

with

the

harvest

of

acentury

of

mathematics

and

physics

to

enrich

it.

This

has

been

the

central

focus

of

my

own

scientiﬁc

research.

Creating

auniﬁed

geometric

language

for

physics

and

mathematics

is

aprob-

$\mathrm{l}\mathrm{e}\mathrm{m}$

in

the

design

of

mathematical

systems.

Here

are

some

general

criteria

that

I

have

applied

to

the

design

of

Geometric

Algebra

as

asolution

to

that

problem:

1.

Optimal

algebraic

encoding

of

the

basic

geometric concepts

:

magni-

tude,

direction,

sense

(or

orientation)

and

dimension.

2.

Coordinate-free

methods

to

formulate

and

solve

basic

equations.

3.

Optimal

uniformity

of

method

across

various

domains

(like

as

classical,

quantum

and

relativistic

theories

in

physics)

to

make

common

structures

as

explicit

as

possible.

4.

Smooth

articulation

with

widely

used

alternative

systems

(Fig.

1)

to

ﬃcilitate

access

and

transfer

of

information.

5.

Optimal

computational

eﬃciency.

The

uniﬁed

system

must

be

at

least

as

eﬃcient

as

any

alternative

system

in

every

application.

Obviously,

these

design

criteria

ensure

built-in

beneﬁts

of

the

uniﬁed

lan-

guage.

In

implementing

the

criteria

Ideliberately

sought

out

the

best

available

mathematical

ideas and

conventions.

I

found

that

it

was

frequently

necessary

to

modify

the mathematics

to

simplify

and

clarify

the

physics.

In

the

development

of

any

scientiﬁc

theory,

amajor

task

for

the0-

rists

is

to

construct

amathematical

language

that

optimizes

expres-

sion

of

the

key

ideas

and

consequences

of

the

theory.

Although

existing

mathematics

should

be consulted

in

this

endeavor,

it

should

not

be

incorporated

without

critically

evaluating

its

suitability.

Imight

add

that

the

process

also

works

in

reverse.

Modiﬁcation

of

mathematics

for

the

purposes

of

other

sciences

serves

as

astimulus

for further

development

of

mathematics.

There

are

many

examples

of

this

eﬀect

in

the

history

of

physics.

Perhaps

the

most

convincing evidence

for

validity

of

anew

scientiﬁc

theory

is

successful

prediction

of

asurprising

new

phenomenon.

Similarly,

the

most

impressive

beneﬁts

of

Geometric

Algebra

arise

from

surprising

new

insights

into

the

structure

of

physics

and

other

$\mathrm{s}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{s}.38-43$

The following

Sections

survey

the

elements

of

Geometric

Algebra

and

its

application.

Many

details

and

derivations

are

omitted,

as

they

are

available

elsewhere.

The

emphasis

is

on

highlighting

the

unique

advantages

of

Geometric

Algebra

as

auniﬁed

mathematical

language.

III.

Understanding

Vectors

Arecent

study

on

the

use

of

vectors

by introductory

physics

students

summa-

rized

the

conclusions

in

two

words:

“vector

$avoidanoe.”’ 11$

,

15

Imaintain

that

the

origin

of

thin

serious

problem

lies

not

so

much

in

pedagogy

as

in

the

mathemat-

$\mathrm{i}\mathrm{c}\mathrm{s}$

.

The

fundamental

geometric

concept

of

avector

as

adirected

magnitude

is

not

adequately

represented

in

standard

mathematics.

The

basic

deﬁnitions

of

vector

addition

and

scalar

multiplication

are

essential

to

the

vector

concept

but

not

suﬃcient.

To

complete

the

vector

concept

we

need

multiplication

rules

that

enable

us

to

compare

directions

and

magnitudes

of

diﬀerent

vectors.

A.

The

Geometric

Product

I

take

the

standard

concept

of

a

real

vector

space

for

granted

and

deﬁne

the

geometric

product

ab

for

vectors

$\mathrm{a}$

,

$\mathrm{b}$

,

$\mathrm{c}$

by

the following

rules:

(ab)

$\mathrm{c}=\mathrm{a}(\mathrm{b}\mathrm{c})$

,

associative

(1)

$\mathrm{a}(\mathrm{b}+\mathrm{c})=\mathrm{a}\mathrm{b}+\mathrm{a}\mathrm{c}$

,

left

distributive

(2)

$(\mathrm{b}+\mathrm{c})\mathrm{a}=\mathrm{b}\mathrm{a}+\mathrm{c}\mathrm{a}$

,

right

distributive

(3)

$\mathrm{a}^{2}=|$ $\mathrm{a}$ $|^{2}$

contraction

(4)

where

$|$ $\mathrm{a}$ $|$

is

apositive

scalar

called

the

magnitude

of

$\mathrm{a}$

,

and

$|$

a

$|=0$

implies

that

$\mathrm{a}=0.$

AU

of

these

rules

should

be

familiar

from

ordinary

scalar

algebra.

The

main

diﬀerence

is

absence

of

a

commutative

rule.

Consequently,

left

and

right

dis-

tributive

rules

must

be

postulated

separately.

The

contraction

rule

(4)

is

pecu-

liar

to

geometric

algebra

and

distinguishes

it

from

all

other

associative

algebras.

But

even

this

is

familiar

from

ordinary

scalar

algebra

as

the

relation

of

asigned

number

to

its

magnitude.

$\mathrm{R}\mathrm{o}\mathrm{m}$

the

geometric

product

ab

we

can

deﬁne

two

new

products,

a

symmetric

inner

product

a

$\cdot \mathrm{b}=\frac{1}{2}(\mathrm{a}\mathrm{b}+\mathrm{b}\mathrm{a})=\mathrm{b}\cdot \mathrm{a}$

,

(5)

and

an

antisymmetric

outer

product

a

A

$\mathrm{b}=\frac{1}{2}(\mathrm{a}\mathrm{b}-\mathrm{b}\mathrm{a})=-\mathrm{b}$ $\Lambda \mathrm{a}$

.

(6)

Therefore,

the

geometric

product

has

the

canonical

decomposition

ab

$=\mathrm{a}\cdot \mathrm{b}$

%a

$\Lambda \mathrm{b}$

.

(7)

$\mathrm{R}\mathrm{o}\mathrm{m}$

the

contraction

rule

(4)

it

is

easy

to

prove

that

a

$\cdot$

$\mathrm{b}$

is

scalar-valued,

so

it

can

be

identiﬁed

with

the

standard

Euclidean

inner

product.

The

geometric

signiﬁcance

of

the

outer

product

aA

$\mathrm{b}$

should

also

be

familiar

ﬀfom

the

standard

vector

cross

product

a

$\mathrm{x}\mathrm{b}$

.

The

quantity

a

A

$\mathrm{b}$

is

called

a

bivector,

and

it

cm

be

interpreted

geometrically

as

an

oriented

plane

segment,

as

shown

in

Fig.

2.

It

diﬀers

from

a

$\mathrm{x}\mathrm{b}$

in

being

intrinsic

to

the

plane

containing

a

and

$\mathrm{b}$

,

independent

of

the

dimension

of

any vector

space

in

which

the

plane

lies.

$\mathrm{R}\mathrm{o}\mathrm{m}$

the

geometric interpretations

of

the

inner

and

outer

products,

we

can

infer

an

interpretation

of

the

geometric

product

from

extreme

cases.

For

or-

thogonal

vectors,

we

have

ﬀom

(5)

a

$\cdot \mathrm{b}=0$ $\Leftrightarrow$

ab

$=-\mathrm{b}\mathrm{a}$

.

(8)

12

$\mathrm{F}_{\dot{1}}\mathrm{g}$

.

2.

Bivectors

a

$\Lambda \mathrm{b}$

and

$\mathrm{b}\wedge \mathrm{a}$

represent

plane

segments

of

opposite

orientation

as

speciﬁed

by

a“parallelogram

rule”

for

drawing

the

segments.

On

the

other

hand,

collinear

vectors

determine

aparallelogram

with

vanishing

area

(Fig.

2),

so

from

(6)

we

have

$\mathrm{a}\wedge \mathrm{b}=0$ $\Leftrightarrow$

ab

$=$

ba.

(9)

Thus,

the

geometric

product

ab

provides

a

measure

of

the

relative

direction

of

the

vectors.

Commutativity

means

that

the

vectors

are

colinear.

Anticom-

mutativity

means

that

they

are

orthogonal. Multiplication

can

be

reduced

to

these

extreme

cases

by

introducing

an

orthonormal

basis.

B.

Basis

and

Bivectors

For

$\mathrm{m}$

orthonormal

set

of

vectors

$\{\sigma_{1},\sigma_{2}, \ldots\}$

,

the

multiplicative

properties

can

be

summarized

by

putting

(5)

in

the

form

$\sigma:3$ $\sigma_{j}=\frac{1}{2}(\sigma_{i}\sigma_{j}+\sigma_{j}\sigma:)=\delta_{\dot{l}j}$

(10)

where

$\delta_{\dot{l}\mathrm{j}}$

is

the

usual

Kroenecker

delta.

This

relation

applies

to

aEuclidean

vector

of

any

dimension,

though

for

the

moment

we

focus

on

the

$2\mathrm{D}$

case.

A

unit

bivector

$\mathrm{i}$

for

the

plane

containing

vectors

$\sigma_{1}$

and

$\sigma_{2}$

is

determined

by

the

product

$\mathrm{i}=$

$r_{1^{t}}r_{2}$ $=\sigma_{1}$

A

$\sigma_{2}=-ty_{2^{\mathrm{c}}}y_{1}$

(11)

The

suggestive symbol

$\mathrm{i}$

has

been

chosen

because

by

squaring

(11)

we

ﬁnd

that

$\mathrm{i}^{2}=-1$

(12)

Thus,

$\mathrm{i}$

is

a

truly

geometric

$\sqrt{-1}$

.

We shall

see

that

there

are

others.

From

(11)

we

also

ﬁnd

that

$\sigma_{2}=\sigma_{1}\mathrm{i}=-\mathrm{i}$ $\sigma_{1}$

and

$\sigma_{1}=\mathrm{i}_{\mathrm{f}}2$

.

(13)

In

words,

multiplication

by

$\mathrm{i}$

rotates

the

vectors

through

aright

angle.

It

follows

that

$\mathrm{i}$

rotates

every

vector

in

the

plane

in

the

same

way.

More

generally,

it

follows

that

every

unit

bivector

$\mathrm{i}$

satisﬁes

(12)

and

determines

aunique

plane

in

Euclidean

space.

Each

$\mathrm{i}$

has

two

complementary

geometric

interpretations:

It

oepmsents

a

unique

oriented

area

for

the

plane,

and,

as

an

operator,

it

represents

an

oriented

right

angle

rotation

in

the

plane

13

C.

Vectors

and

Complex

Numbers

Assigning

ageometric

interpretation

to

the

geometric product

is

more

subtle

than

interpreting

inner

and

outer

products

–so

subtle,

in

fact,

that

the

appro-

priate

assignment

has

been

generally

overlooked

until

$\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}1\mathrm{y}^{1}$

.

The

product

of

any

pair

of

unit

vectors

$\mathrm{a}$

,

$\mathrm{b}$

generates

a

new

kind

of

entity

$U$

called

a

rotor,

as

expressed by

the

equation

$U=$ ab.

(14)

The

relative

direction

of

the

two

vectors

is

completely

characterized

by

the

directed

arc

that

relates

them

(Fig. 3),

so

we

can

interpret

$U$

as

representing

that

arc.

The

name

“rotor”

is

justiﬁed

by

the

fact

that

$U$

rotates

aand

$\mathrm{b}$

into

each

other,

as

shown

by

multiplying

(14)

by

vectors

to

get

$\mathrm{b}=\mathrm{a}U$

and

$\mathrm{a}=U\mathrm{b}$

.

(15)

Further

insight

is

obtained

by

noting

that

a

$\cdot$ $\mathrm{b}=\cos\theta$

and

aA

$\mathrm{b}=\mathrm{i}\sin\theta$

,

(16)

where

$\theta$

is

the

angle

from

ato

$\mathrm{b}$

.

Accordingly,

with

the

angle

dependence

made

explicit,

the

decomposition

(7)

enables

us

to

write

(14)

in

the

form

$U_{\theta}=\cos\theta+\mathrm{i}\sin\theta=\mathrm{e}^{\mathrm{i}\theta}$

(17)

It

follows

that

multiplication by

$U_{\theta}$

,

as

in

(15),

will

rotate

any

vector

in

the

$\mathrm{i}$

-plane

through

the

angle

$\theta$

.

This

tells

us

that

we

should

interpret

$U_{\theta}$

as

$\mathrm{a}$

directed

arc

ﬁxed

length

that

can

be

rotated

at

will

on

the

unit

circl\^e

just

as

we

interpret

avector

aas

adirected

line segment

that

can

be

translated

at

will

without

changing

its

length

or

direction

(Fig.

4).

Fig.

3.

Apair

of

unit

vectors

$\mathrm{a}$

,

$\mathrm{b}$

determine

adirected

$a\mathrm{r}\mathrm{r}$

on

the

unit

circle

that

represents

their

product

$U=\mathrm{a}\mathrm{b}$

.

The

length

of

the

arc

is

(radian

measure

of)

the

angle

$\theta$

between

the

vectors.

With

rotors,

the

composition

of $2\mathrm{D}$

rotations

is

expressed

by

the

rotor

product

$U_{\theta}U_{\varphi}=U_{\theta+\varphi}$

(18)

and

depicted

geometrically

in

Fig.

5as

addition

of

directed

arcs

14

$\overline{||}$

a

$|\mathrm{I}$

$2’...’\underline{\mathrm{a}}’..,$

’

$11|||1||$

$|\mathrm{I}|..\cdot..a’$

\prime\prime

”

.,.,

$d’$

$\prime\prime\prime’.\cdot|\prime\prime\prime||1$

”

$\frac{|.\prime’|\mathrm{I}...1||\prime\cdot\prime’|}{\mathrm{a}}...\prime^{-\prime}\prime’\prime’.$

\prime\prime

Fig.

4.

All

directed

arcs

with

equivalent

angles

are

represented

by

asingle

rotor

$U_{\theta}$

,

just

as

line

segments

with

the

same

length

and

direction

are

represented

by

asingle

vector

$\mathrm{a}$

.

–

Fig,

5.

The

composition

of

$2\mathrm{D}$

rotations

is

represented

alge-

braically

by

the

product

of

rotors

and

depicted

geometrically

by

addition

of

directed

arcs.

The

generalization

of

all

this

should

be

obvious.

We

cm

always

interpret

the

product

ab

algebraically

as

acomplex

number

$z$ $=:AU=\lambda e^{\mathrm{i}\theta}=$

ab,

(19)

with

modulus

$|$ $z$ $|=\lambda=|$ $\mathrm{a}$ $||\mathrm{b}|$

.

And

we

can

interpret

$z$

geometrically

as

a

directed

arc

on

acircle

of

radius

$|z|$

(Fig.

6).

It

might

be

surprising

that

this

$\mathrm{g}\infty\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}$

interpretation

never

appears

in

standard

books

on

complex

variables.

Be

that

as

it

may,

the

value

of

the

interpretation

is

greatly

enhanced

by

its

use

in

geometric

algebra.

Fig.

6.

A

complex

number

$z=\lambda U$

with

modulus

Aand

angle

$\theta$

can

be

interpreted

as

a

$\mathrm{d}\dot{\mathrm{u}}$

ected

arc

on

a

circle

of

radius

$\lambda$

.

Its

conjugate

$z^{\mathfrak{j}}=\lambda U^{\uparrow}$

represents

an

arc

with

opposite

orientation

15

The

connection

to

vectors

via

(19)

removes

alot

of

the

mystery

from

complex

numbers

and

facilitates

their

application

to

physics.

For

example,

comparison

of

(19)

to

(7)

shows

at

once

that

the

real

and

imaginary

parts

of

acomplex

number

are

equivalent

to

inner

and

outer

products

of

vectors.

The

complex

conjugate

of

(19)

is

$z’=\lambda U^{\dagger}=$

:A

$e^{-\mathrm{i}\theta}=$

ba,

(20)

which

shows

that

it

is

equivalent

to

reversing

order

in

the

geometric

product.

This

can

be

used

to

compute

the

modulus

of

$z$

in

the

usual

way:

$|z|^{2}=zz^{\mathrm{t}}=)$

h2

$=$

baab

$=\mathrm{a}^{2}\mathrm{b}^{2}=|$ a$|^{2}|$ $\mathrm{b}$ $|^{2}$

(21)

Anyone

who

has

worked

with

complex

numbers

in

applications

knows

that

it

is

usually

best

to

avoid

decomposing

them

into

real

and

imaginary

parts.

Likewise,

in

GA

applications

it

is

usually

best

practice

to

work

directly

with

the

geometric

product

instead

of

separating

it

into

inner

and

outer

products.

GA

gives

complex

numbers

new

powers

to

operate

directly

on

vectors.

For

example,

from

(19)

and

(20)

we

get

$\mathrm{b}=\mathrm{a}^{-}1$

(22)

where

the

multiplicative

inverse

of

vector

ais

given

by

$\mathrm{a}^{-1}=\frac{1}{\mathrm{a}}=\frac{\mathrm{a}}{\mathrm{a}^{2}}=\frac{\mathrm{a}}{|\mathrm{a}|^{2}}$

.

(23)

Thus,

$z$

rotates

and

rescales

ato

get

$\mathrm{b}$

.

This

makes

it

possible

to

construct

and

manipulate

vectorial

transformations

and

functions

without

introducing

a

basis

or

matrices.

This

is

agood

point

to

pause

and

note

some

instructive

implications

of

what

we

have

established

so

far.

Complex

numbers,

especially

equations

(17)

$\mathrm{m}\mathrm{d}$

$(18)$

,

are

ideal

for

dealing

with

plane

trigonometry

and

$2\mathrm{D}$

rotations.

However,

students

in

introductory

science

and

engineering

courses

are

often

denied

access

to

this

powerful

tool,

evidently

because

it

has

areputation

for

being

conceptually

diﬃcult,

and

class

time

would

be

lost

by

introducing

it.

GA

removes

these

barriers

to

use

of

complex

numbers

by

linking

them

to

vectors

and

giving

them

aclear

geometric

meaning.

GA

also

makes

it

possible

to

formulate

and solve

$2\mathrm{D}$

physics

problems

in

terms

of

vectors

without

introducing

coordinates.

Conventional

vector

algebra

cannot

do

this,

in

part

because

the

vector

cross

product

is

deﬁned

only

in

$3\mathrm{D}$

.

That

is

the

main

reason

why

coordinate

methods

dominate

introductory

physics,

computer

science,

engineering,

etc.

The

available

math

tools

are

too

weak

to

do

otherwise.

GA

changes

all

that!

Most

of

the

mechanics

problems

in

introductory

physics

are

$2\mathrm{D}$

problems.

Coordinate

free

GA

solutions

for

the

standard

problems

are

worked

out

in

my

mechanics

$\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{k}.12$

Although

the

treatment

there

is

for

amore

advanced

course

18

it

can

easily

be

adapted

to

the

introductory

level.

The

essential

GA

concepts

for

that

level

have

already

been

presented

in

this

section.

Will

comprehensive

use

of

GA

signiﬁcantly

enhance

student

learning?

We

have

noted

theoretical

reasons

for

believing

that

it

will.

However,

mathematical

reform

at

the

introductory

level

makes

little

sense

unless

it

is

extended

to

the

whole

curriculum.

Taking

physics

as

example,

the

following

sections

provide

strong

justiﬁcation

for

doing

just

that.

We

shall

see

how

simpliﬁcations

at

the

introductory

level

get

ampliﬁed

to

greater

simpliﬁcations

and

surprising

insights

at

the

advanced

level.

IV.

Classical

Physics

with

Geometric

Algebra

This

Section

surveys

the

fundamentals

of

GA

as

amathematical

framework

for

classical

physics

and

demonstrates

some

of

its

unique

advantages.

Detailed

applications

can

be

found

in

the

references.

A.

Geometric

Algebra

for

Physical

Space

The

arena

for

classical

physics

is

a

$3\mathrm{D}$

Euclidean

vector

space

$P^{3}$

,

which

serves

as

amodel

for

“Physical

Space.”

By

multiplication

and

addition

the

vectors

generate

ageometric

algebra

$\mathcal{G}\mathrm{a}=\mathcal{G}(\mathcal{P}^{3})$

.

In

particular,

abasis

for

the

whole

algebra

can

be

generated

from

astanda

$rd$

ﬀame

$\{\sigma_{1},\sigma_{2},\sigma_{3}\}$

,

arighthanded

set

of $\mathrm{o}\mathrm{r}\mathrm{t}\dot{\mathrm{h}}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}$

vectors.

With

multiplication

speciﬁed

by

(10),

the

standard

ﬀame

generates

aunique

trivector

(3-vector)or

pseudoscalar

$i=\sigma_{1}\sigma_{2}\sigma_{3}$

,

(24)

$\mathrm{m}\mathrm{d}$

a

bivector

(2-vect0rbasis

$\sigma_{1}\sigma_{2}=i\sigma_{3}$

,

$\sigma_{2}\sigma_{3}=i\sigma_{1}$

,

$\sigma_{3}\sigma_{1}=i\sigma_{2}$

.

(25)

Geometric

interpretations

for

the

pseudoscalar

and bivector

basis

elements

axe

depicted

in

Figs.

7and

8.

The

pseudoscalar

$i$

has

special

properties

that

facilitate

applications

as

well

as

articulation with standard

vector

algebra.

It

follows

from

(24)

that

$i^{2}=-1$

,

(26)

and

it

follows

from

(25)

that

every

bivector

$\mathrm{B}$

in

$\mathcal{G}_{3}$

is

the

dual

of

avector

$\mathrm{b}$

as

expressed

by

$\mathrm{B}=i\mathrm{b}=\mathrm{b}i$

.

(27)

Thus,

the

geometric

duality

operation

is

simply expressed

as

multiplication

by

the

pseudoscalar

$i$

.

This

enables

us

to

write

the

outer

product

deﬁnd

by

(6)

in

the

form

a

A

b

$=i$

a

x

b.

(24)

17

Fig.

7.

Unit

pseudoscalar

$i$

represents

an

oriented

unit

volume.

The

volume

is

said

to

be

righthanded,

because

$i$

can

be

gener-

ated

from

arighthanded

vector

basis

by

the

ordered

product

$\sigma_{1}\sigma_{2}\sigma_{3}=i.$

Fig.

8.

Unit

bivectors

representing

abasis

of

directed

areas

in

planes

with

orthogonal

intersections

18

Thus,

the

conventional

vector

cross

product

a

$\mathrm{x}\mathrm{b}$

is

implicitly

deﬁned

as

the

dual

of

the

outer

product.

Consequently,

the

fundamental

decomposition

of

the

geometric

product

(7)

can

be

put

in

the

form

ab

$=\mathrm{a}\cdot$ $\mathrm{b}+i$

a

$\mathrm{x}\mathrm{b}$

.

(29)

This

is

the

deﬁnitive

relation

among

vector

products

that

we

need for

smooth

ar-

ticulation

between

geometric

algebra

and

standard

vector

algebra,

as

is

demon-

strated

with

many

examples

in

my

mechanics

$\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{k}.12$

The

elements

in

any

geometric

algebra

are

called

multivectors.

The

special

properties

of

1enable

us

to

write

any

multivector

$M$

in

$\mathcal{G}\mathrm{s}$

in

the

$ex$

anded

form

$M=\alpha+\mathrm{a}+i\mathrm{b}+i\beta,$

(30)

where

$\alpha$

and

Aare

scalars

and

aand

$\mathrm{b}$

are

vectors.

The

main

value

of

this

form

is

that

it

reduces

multiplication

of

multivectors

in

$\mathcal{G}_{3}$

to

multiplication

of

vectors

given

by

(29).

The

expansion

(30)

has the

formal

algebraic

structure

of

a“complex

scalar”

$\alpha+$ $i(3$

added

to

a“complex

vector”

$\mathrm{a}+i\mathrm{b},$

but

any

physical

interpretation

attributed

to

this

structure

hinges

on

the

geometric

meaning

of

$i$

.

Avery

im-

portant

example

is

the

expression

of

the

electromagnetic

ﬁeld

$F$

in

terms

of

an

electric

vector

ﬁeld

$\mathrm{E}$

and

amagnetic

vector

ﬁeld

$B$

:

$F=\mathrm{E}+i\mathrm{B}$

.

(31)

Geometrically,

this

is

adecomposition

of

$F$

into

vector

and bivector

parts.

In

standard

vector

algebra

$\mathrm{E}$

is

said

to

be

apolar

vector

while

$\mathrm{B}$

is

an

axial

vector,

the

two

kinds

of

vector

being distinguished

by

adiﬀerence

in

sign

under

space

inversion.

GA

reveals

that

an

axial

vector

is

just

abivector

represented

by

its

dual,

so

the

magnetic

ﬁeld

in

(31)

is

fully

represented

by

the

complete

bivector

$i\mathrm{B}$

,

rather

than

$\mathrm{B}$

alone.

Thus

GA

makes

the

awkward

distinction

between

polar

and

axial

vectors

unnecessary.

The

vectors

$\mathrm{E}$

and

$\mathrm{B}$

in

(31)

have

the

same

behavior

under

space

inversion,

but

an

additional

sign

change

comes

ﬀom

space

inversion

of

the

pseudoscalar.

To

facilitate

algebraic

manipulations,

it

is

convenient

to

introduce

aspecial

symbol

for

the

operation

(called

reversion)

of

reversing

the

order

of

multiplica-

tion.

The

reverse

of

the

geometric

product

is

deﬁned

by

$(\mathrm{a}\mathrm{b})^{\mathrm{t}}=$

ba.

(32)

We

noted

in

(20)

that

this

is

equivalent

to

complex

conjugation

in

$2\mathrm{D}$

.

Prom

(24)

we

ﬁnd

that

the

reverse

of

the

pseudoscalar

is

$i^{\mathrm{t}}=-i$

.

(33)

Hence

the

reverse

of

an

arbitrary

multivector

in

the

expanded

form

(30)

is

$M^{\uparrow}=\alpha+\mathrm{a}-i\mathrm{b}-iff,$

(34)

16

The

convenience

of

this

operation

is

illustrated

by

applying

it

to

the

electr0-

magnetic

ﬁeld

$F$

in

(31)

and

using

(29)

to

get

$\frac{1}{2}FF^{\mathrm{t}}=\frac{1}{2}(\mathrm{E}+i\mathrm{B})(\mathrm{E}-i\mathrm{B})$ $= \frac{1}{2}(\mathrm{E}^{2}+\mathrm{B}^{2})+\mathrm{E}\mathrm{x}\mathrm{B}$

,

(35)

which

is

recognized

as

an

expression

for

the

energy

and

momentum

density

of

the

ﬁeld.

You

have

probably

noticed

that

the

expanded

multivector

form

(30)

violates

one

of

the

basic

math

strictures

that

is

drilled

into

our

students,

namely,

that

“it

is

meaningless

to

add

scalars

to

vectors,”

not

to

mention

bivectors

and

pseudoscalars.

On

the

contrary,

GA

tells

us

that

such addition

is

not

only

geometrically

meaningful,

it

is

essential

to

simplify

and

unify

the

mathematical

language

of

physics

and

other

applications,

as

can

be

seen

in

many

examples

that

follow.

Shall

we

say

that

this

stricture

against

addition

of

scalars

to

vectors

is

a

misconception

or

even

aconceptual

$\mathrm{v}\mathrm{i}\mathrm{r}\mathrm{u}\mathrm{s}$

?

At

least

it

is

adesign

ﬂaw

in

standard

vector

algebra

that

has

been

almost

universally

overlooked.

As

we

have

just

seen,

elimination

of

the

ﬂaw

enables

us

to

combine

electric

and

magnetic

ﬁelds

into

asingle

electromagnetic

ﬁeld.

And

we

shall

see

below

how

it

enables

us

to

construct

spinors

$fmm$

vectors

(contrary

to

the

received

wisdom

that

spinors

are

more

basic

than

vectors)!

B.

Reﬂections

and

Rotations

Rotations

play

an

essential

role

in

the

conceptual

foundations

of

physics

as

well

as

in

many

applications,

so our

mathematics should

be

designed

to

handle

them

as

eﬃciently

as

possible.

We

have

noted

that

conventional

treatments

employ

an

awkward

mixture

of

vector,

matrix

and

spinor

or

quaternion

methods.

My

purpose

here

is

to

show

how

GA

provides

auniﬁed,

coordinate-ﬀee

treatment

of

rotations

and

reﬂections

that

leaves

nothing

to

be

desired.

The

main

result

is

that

any

orthogonal

transformation

$\underline{U}$

can

be

expressed

in

the

canonical

$fom^{12}$

$\underline{U}\mathrm{x}$ $=\mathit{3}$ $U\mathrm{x}U^{\mathrm{t}}$

,

(36)

where

$U$

is

aunimodular

multivector

called

aversor,

and

the

sign

is

the

parity

of $U$

,

positive

for

arotation

or

negative

for

areﬂection.

The

condition

$U^{\mathrm{t}}U=1.$

(37)

deﬁnes

unimodularity.

The

underbar

notation

serves

to

distinguish

the

linear

operator

$\underline{U}$

from

the

versor

$U$

that

generates

it.

The

great

advantage

of

(36)

is

that

it

reduces the

study

of

linear

operators

to

algebraic

properties

of

their

versors.

This

is

best

understood

from speciﬁc

examples.

The

simplest

example

is

reﬂection

in

aplane

with

unit

normal

$\mathrm{a}$

(Fig.

9),

$\mathrm{x}’=-$

axa

$=-\mathrm{a}(\mathrm{x}_{[perp]}+ \mathrm{x}||)$

a

$=\mathrm{x}_{[perp]}-\mathrm{x}||$

.

(31)

20

To

show

how

this

function

works,

the

vector

$\mathrm{x}$

has

been

decomposed

on

the

right

into

a

parallel

component

$\mathrm{x}||=$

$(\mathrm{x}\cdot \mathrm{a})\mathrm{a}$

that

commutes with

aand

an

orthogonal

component $\mathrm{x}_{[perp]}=(\mathrm{x}\wedge \mathrm{a})\mathrm{a}$

that

anticommutes

with

$\mathrm{a}$

.

As

can

be

seen

below

it

is

seldom

necessary

or

even

advisable

to

make

this

decomposition

in

applications.

The

essential

point

is

that

the

normal

vector

deﬁning

the

direc-

$t\dot{\iota}on$

of

a

plane

also

represents

a

reﬂection

in

the

plane

when

interpreted

as

a

versor.

A

simpler

representation

for

reﬂections

is

inconceivable,

so

it

must

be

the

optimal

representation

for

reﬂections

in

every

application,

as

shown

in

some

important

applications

below.

Incidentally,

the

term

versor

was

coined

in

the

$19^{th}$

century

for

an

operator

that

can

re-verse

adirection.

Likewise,

the

term

is

used

here

to

indicate

ageometric

operational

interpretation

for

amultivector.

Fig,

9.

Reﬂection

in

aplane.

The

reﬂection

(38)

is

not

only

the

simplest

example

of

an

orthogonal

tran&

rot

ation

but

$\mathrm{a}\mathbb{I}$

orthogonal

transformations

can

be

generated

by

reﬂections

of

this

kind.

The

main

result

is

expressed

by

the

following

theorem:

The

product

of

two

reﬂections

is

a

rotation

through

twice

the

angle

between

the

normals

of

the

reﬂecting

planes.

This

important

theorem seldom

appears

in

standard

text-

books,

primarily,

Ipresume,

because

its

expression

in

conventional

formalism

is

so

awkward

as

to

render

it

impractical.

However,

it

is

an

easy

consequence

of

asecond

reﬂection

applied

to

(38).

Thus,

for

aplane

with

unit

normal

$\mathrm{b}$

,

we

have

$\mathrm{x}’=-\mathrm{b}\mathrm{x}’\mathrm{b}=$ baxab $=U\mathrm{x}U^{\mathrm{t}}$

,

(38)

where

anew

symbol

has

been

introduced

for

the

versor

product

$U=$

ba.

The

theorem

is

obvious

$\mathrm{b}\mathrm{o}\mathrm{m}$

the

geometric

construction

in

Fig.

10.

For

an

algebraic

proof

that

the

result

does

not

depend

on

the

reﬂecting

planes,

we

use

(17)

to

write

$U=\mathrm{b}\mathrm{a}=\omega s$ $\xi\theta+\mathrm{i}sin_{2}1\theta=e^{11\theta}2$

,

(40)

where,

anticipating

the

result

from

Fig.

9,

we

denote

the

angle

between

aand

$\mathrm{b}$

by

$\frac{1}{2}\theta$

and

the

unit

bivector

for

the

aA

$\mathrm{b}$

-plane

by

$\mathrm{i}$

.

Next,

we

decompose

$\mathrm{x}$

into

acomponent

$\mathrm{x}_{[perp]}$

orthogonal

to

the

$\mathrm{i}$

-plane

and

acomponent

$\mathrm{x}||$

in

the

plane.

Note

that,

respectively,

the

two

components

commute

(anticommute)

with

$\mathrm{i}$

,

so

$\mathrm{x}_{[perp]}U^{\mathrm{t}}=U^{\mathrm{t}}\mathrm{x}_{[perp]}$

,

$\mathrm{x}||U^{\mathrm{t}}=U\mathrm{x}||$

.

(41)

21

Inserting

this

into

(39)

with

$\mathrm{x}=\mathrm{x}||+\mathrm{x}_{[perp]}$

,

we

obtain

$\mathrm{x}’=U\mathrm{x}U^{\mathrm{t}}=\mathrm{x}_{[perp]}+U^{2}\mathrm{x}||$

.

(42)

These

equations

show how

the

tw0-sided

multiplication

by

the

versor

$U$

picks

out

the

component

of

$\mathrm{x}$

to

be

rotated,

so

we

see

that

one-sided

multiplication

works

only

in

$2\mathrm{D}$

.

As

we

learned

ﬀom

our

discussion

of $2\mathrm{D}$

rotations,

the

versor

$U^{2}=e^{\mathrm{i}\theta}$

rotates

$\mathrm{x}_{[perp]}$

through

angle

$\theta$

,

in

agreement

with

the

half-angle

choice

in

(40).

Fig.

10.

Rotation

as

double

reﬂection,

depicted

in

the

plane

containing

unit

normals

$\mathrm{a}$

,

$\mathrm{b}$

of

the

reﬂecting

planes.

The

great

advantage

of

the

canonical

form

(36)

for

an

orthogonal

transforma-

tion

is

that

it

reduces

the

composition

of

orthogonal

transformations

to

versor

multiplication.

Thus,

composition

expressed

by

the

operator

equation

U2

$L^{r_{1}}=\underline{U}_{3}$

(43)

is

reduced

to

the

product

of

corresponding

versors

$U_{2}U_{1}=U_{3}$

.

(44)

The

orthogonal

transformations form

amathematical

group

with

(43)

as

the

group

composition

law.

The

trouble with

(43)

is

that

abstract

operator

algebra

does

not

provide

away

to

compute $\underline{U}_{3}$

from

given

$\underline{U}_{1}$

and

$\underline{U}_{2}$

.

The

usual

solution

to

this

problem

is

to

represent

the

operators

by

matrices

and

compute

by

matrix

multiplication.

Amuch

simpler

solution

is

to

represent

the

operators

by

versors

and

compute

with

the

geometric

product.

We

have

already

seen

how

the

product

of

reﬂections

represented

by

$U_{1}=$

aand

$U_{2}=\mathrm{b}$

produces

a

rotation

$U_{3}=$

ba.

Matrix

algebra

does

not

provide

such

atransparent

result.

As

is

well

known,

the

rotation

group

is

asubgroup

of

the

orthogonal

group.

This

is

expressed

by

the

fact

that

rotations

are

represented

by

unimodular

versors

of

even

parity,

for

which

the

term

rotor

was

introduced

earlier.

The

composition

of

$2\mathrm{D}$

rotations

is

described

by

the

rotor

equation

(18)

and

depicted

in

Fig.

5.

Its

generalization

to

composition

of

$3\mathrm{D}$

rotations

in

diﬀerent

planes

22

Fig.

11.

Addition

of

directed

arcs

in

$3\mathrm{D}$

depicting

the

product

of

rotors.

is

described

algebraically

by

(44)

and

depicted

geometrically

in

Fig.

11.

This

deserves

some

explanation.

In

$3\mathrm{D}$

a

rotor

is

depicted

as

a

directed

arc

conﬁned

to

a

great

circle

on

the

unit

sphere.

The

product

of

rotors

$U_{1}$

and

$U_{2}$

is

depicted

in

Fig.

11

by

connecting

the

corresponding

arcs

at

a

point

$\mathrm{c}$

where

the

two

great

circles

intersect.

This

determines

points

$\mathrm{a}=\mathrm{c}U_{1}$

and

$\mathrm{b}=U_{2}\mathrm{c}$

,

so

the

rotors

can

be

expressed

as

products

with

a

common

factor,

$U_{1}=$

ca,

$U_{2}=$

bc.

(45)

Hence

(43)

gives

us

$U_{3}=U_{2}U_{1}=(\mathrm{b}\mathrm{c})(\mathrm{c}\mathrm{a})=$