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Topic 3: Spacetime Geometryand Clifford Algebras

Dr. Bill PezzagliaCSUEB Physics

Updated Nov 28, 2010

1Lecture Series: The Spin of the Matter, Physics 4250, Fall 2010

… for geometry, you know, it the gate of science,and the gate is so low and small that one can only enter it as a little child.

William Kingdon Clifford (1845-1879)

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Index: Rough Draft

A. Dimensional Democracy

B. Grassmann Algebra

C. Clifford Algebra

D. References

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A. Dimensional Democracy

1. Why use Gibbs’ Vectors?

2. Pseudovectors

3. Problems with Gibbs’ Vectors

4. Thinking outside the box

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1. Gibbs Vectors5

1881 Inventor of the vector system we now use.

Oliver Heaviside used these vectors to reformulate Maxwell’s equations

“... a sort of hermaphrodite monster, compounded of the notations of Hamilton and Grassmann” -Tait

Why use Gibbs vectors? 6

• Notational Economy (3 equations in one)• Coordinate free (physics should not depend upon coordinate system)• Encodes isotropy of space

A mathematical language has utility when the metaprinciples of physics (e.g. isotropy) are built into its syntax

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2. Pseudovectors & Pseudoscalars 7

Gibbs’ vector algebra represents a plane by the vector perpendicular to it, which means a “vector” can mean either a line or a plane (ambiguous)

3. Problems with Gibbs Vectors 8

The “problems” with Gibbs vectors are:

• Higher Dimensions: Cross Product won’t generalize to higher dimensions (n.b. 4D for relativity)

• Incompleteness: there are only vectors and scalars. You can’t directly represent a plane.

• Ambiguity: Instead you use the vector perpendicular to a plane. Hence if I give you a vector, are we talking about the directed line, or the plane perpendicular to it?

• Parity Problem: the cross product is not preserved under mirror reflections (the cross product is not really a true vector, rather a “pseudovector”).

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4. Thinking “out of the box” 9

A particular language might build in unquestioned prejudices.

Gibbs’ algebra (and conventional tensors) have “Dimensional Segregation”,

You cannot add different ranked geometries

B. Grassmann Algebra10

Hermann Grassmann(1809-1877)

Inventor of “Linear Algebra”

1844 publishes massive work (which nobody understands)[1 year after quaternions!]

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1. Each Dimension is represented11

2. The Exterior Product12

Note: cannot add a scalar to a vector (unlike quaternions)Note the wedge product is similar to Hamilton’s cross product, except the result is a PLANE, an idea that will extend to any dimension (where as Gibbs or Hamilton cross product does not).

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3. The Dual and Inner Product13

4. Products of Planes and Lines14

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5. Grassmann Calculus 15

Aka “exterior calculus”

(a) Gradient Operator (“nabla”)

Poincare Lemmas

In 3D these are equivalent to:• Div Curl=0• Curl Grad=0

ze

ye

xe

∂∂

+∂∂

+∂∂

=∇ 321 ˆˆˆr

0)(

0)(

=•∇•∇

=∧∇∧∇

anything

anythingrr

rr

(b) Differential Forms 16

Vector Differential

Area “bivector” Differential:

Volume “trivector” Differential:

dzedyedxerd 321 ˆˆˆ ++=r

dzdxeedydzeedxdyeerddA 1332212 ˆˆˆˆˆˆ ∧+∧+∧==r

dxdydzeeerddV 3213 ˆˆˆ ∧∧==r

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(c) Generalized Stoke’s Law 17

These are all basically the same idea, the integral over some “N”dimensional region of the gradient of a thing is equal to the evaluation of the thing on the “N-1”dimensional boundary.

EAdErdV V

rrr•=•∇∫ ∫

∂

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ErdEAdA A

rrrr•=×∇•∫ ∫

∂

∫ −=∇•b

aafbffrd )()(r

∫∫∂

=VV

FdFDifferential Forms:

C. Clifford Algebra18

William Kingdon Clifford(1845-1879)

•Translated Riemann’s work

•Anticipated general relativity

•Died shortly after inventing algebra which combined Hamilton’s and Grassmann’sideas into one form

•Forgotten until recently.

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William Kingdom Clifford (1876) 19

Clifford Algebra has “Dimensional Democracy”, allowing you to add lines to planes

1. That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.

2. That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave

3. That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal

4. That in the physical world, nothing else takes place but this variation, subject (possibly) to the law of continuity.

Unify Phenomena Dimensionally 20

Using Clifford Algebra, get 2 equations in 1

P is the momentum, S is the spin, F is the electromagnetic field

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1. Defining a Clifford Algebra 21

For “N” dimensionshave “N” basis vectors

•They anticommute•Square to +1 12211

2121

+==−=σσσσσσσσ

Njj ,,1}{ L=σ

jijijiij σσσσσσδ +== },{2Or: single rule

2. 3D Clifford Algebra is Pauli Algebra 22

Geometric interpretation of “i” is volumeBivectors are a “quaternion” group

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3. 4D Clifford algebra is Dirac Algebra 23

There are two different “metric signatures” that work for special relativity: (---+) or (+++-)

4. Properties of Clifford Algebra 24

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4. Properties of Clifford Algebra 25

Can do things that Grassmann can’t, like multiplication of two planes gives a plane

Most important, Clifford algebra can do rotations like Hamilton’s quaternions (more later)

5. Geometric Calculus 26

Like quaternions, you have sums of scalars and vectors (and bivectors and trivectors …)

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5. Geometric Calculus 27

You can get 4 Maxwell’s equations in ONE!

( ) ( ) BiEFJFt

rrrr+=−=∇+∂ ρ

( )⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=•∇=+×∇

+−=+×∇−=•∇

=

trivectorBibivectorBEi

vectorJEBscalarE

c

cc

001

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r&

r

r&

r

rρ

6. Example of Utility of Clifford Algebra 28

Left side is using Gibbs vectors, right side using Clifford Algebra

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Continued 29

Solving 4 equations in one can save many steps!

References• The problem on previous two slides was the derivation of the Characteristic

Hypersurfaces for Maxwell’s equations. – The standard (Gibbs) treatment was adapted from Adler, Bazin and

Schiffer, Intro to General Relativity (McGraw-Hill 1965), pp. 108-112– The Clifford Algebra derivation is from W. Pezzaglia, in Lawrynowicz,

Deformations of Mathematical Structures II (1994), pp. 129-134, or hep-th/9211062

• EM in one equation, see Bernard Jancewicz, Multivectors and Clifford Algebra in Electrodynamics (World Scientific1988) p. 78

• William Baylis, Electrodynamics, a Modern Geometric Approach (Birkhauser1999)

• The best quick introduction to Clifford Algebra is David Hestenes:– Space-time Algebra (Gordon & Breach 1966)– New Foundations for Classical Mechanics (Kluwer 1986)– Or, my Ph.D. thesis (I’ll post this on web site)

• The best explanations of Grassmann algebra are usually in the books on Clifford Algebras. However, a standard is Flanders, Differential Forms, Academic Press (1963).

• Another summary of Grassmann algebra would be:D. Fearnley-Sander, American Mathematical Monthly, 86, 809 (1979)

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