Topic 3: Spacetime Geometry and Clifford Algebras

Transcription

1 Lectue Seies: The Spin of the Matte, Physics 4250, Fall 200 Topic 3: Spacetime Geomety and Cliffod Algebas D. Bill Pezzaglia CSUEB Physics Updated Nov 28, 200 fo geomety, you know, it the gate of science, and the gate is so low and small that one can only ente it as a little child. William Kingdon Cliffod ( )

2 Index: Rough Daft 3 A. Dimensional Democacy B. Gassmann Algeba C. Cliffod Algeba D. Refeences A. Dimensional Democacy 4. Why use Gibbs Vectos? 2. Pseudovectos 3. Poblems with Gibbs Vectos 4. Thinking outside the box 2

3 . Gibbs Vectos 5 88 Invento of the vecto system we now use. Olive Heaviside used these vectos to efomulate Maxwell s equations... a sot of hemaphodite monste, compounded of the notations of Hamilton and Gassmann -Tait Why use Gibbs vectos? 6 Notational Economy (3 equations in one) Coodinate fee (physics should not depend upon coodinate system) Encodes isotopy of space A mathematical language has utility when the metapinciples of physics (e.g. isotopy) ae built into its syntax 3

4 2. Pseudovectos & Pseudoscalas 7 Gibbs vecto algeba epesents a plane by the vecto pependicula to it, which means a vecto can mean eithe a line o a plane (ambiguous) 3. Poblems with Gibbs Vectos 8 The poblems with Gibbs vectos ae: Highe Dimensions: Coss Poduct won t genealize to highe dimensions (n.b. 4D fo elativity) Incompleteness: thee ae only vectos and scalas. You can t diectly epesent a plane. Ambiguity: Instead you use the vecto pependicula to a plane. Hence if I give you a vecto, ae we talking about the diected line, o the plane pependicula to it? Paity Poblem: the coss poduct is not peseved unde mio eflections (the coss poduct is not eally a tue vecto, athe a pseudovecto ). 4

5 4. Thinking out of the box 9 A paticula language might build in unquestioned pejudices. Gibbs algeba (and conventional tensos) have Dimensional Segegation, You cannot add diffeent anked geometies B. Gassmann Algeba 0 Hemann Gassmann ( ) Invento of Linea Algeba 844 publishes massive wok (which nobody undestands) [ yea afte quatenions!] 5

6 . Each Dimension is epesented 2. The Exteio Poduct 2 Note: cannot add a scala to a vecto (unlike quatenions) Note the wedge poduct is simila to Hamilton s coss poduct, except the esult is a PLANE, an idea that will extend to any dimension (whee as Gibbs o Hamilton coss poduct does not). 6

7 3. The Dual and Inne Poduct 3 4. Poducts of Planes and Lines 4 7

8 5. Gassmann Calculus 5 Aka exteio calculus (a) Gadient Opeato ( nabla ) Poincae Lemmas + eˆ x + e y = eˆ ˆ 2 3 z In 3D these ae equivalent to: Div Cul=0 Cul Gad=0 ( anything) = 0 ( anything) = 0 (b) Diffeential Foms 6 Vecto Diffeential Aea bivecto Diffeential: Volume tivecto Diffeential: = eˆ dx + eˆ dy + e dz d ˆ da = d = eˆ eˆ 2dxdy + eˆ 2 eˆ 3dydz + eˆ 3 eˆ dzdx dv = d 3 = eˆ eˆ 2 e ˆ 3dxdydz 8

9 (c) Genealized Stoke s Law 7 These ae all basically the same idea, the integal ove some N dimensional egion of the gadient of a thing is equal to the evaluation of the thing on the N- dimensional bounday. V A d 3 E = da E = b a V A d A E d E d f = f ( b) f ( a) Diffeential Foms: V df = V F C. Cliffod Algeba 8 William Kingdon Cliffod ( ) Tanslated Riemann s wok Anticipated geneal elativity Died shotly afte inventing algeba which combined Hamilton s and Gassmann s ideas into one fom Fogotten until ecently. 9

10 William Kingdom Cliffod (876) 9. That small potions of space ae in fact of a natue analogous to little hills on a suface which is on the aveage flat; namely, that the odinay laws of geomety ae not valid in them. 2. That this popety of being cuved o distoted is continually being passed on fom one potion of space to anothe afte the manne of a wave 3. That this vaiation of the cuvatue of space is what eally happens in that phenomenon which we call the motion of matte, whethe pondeable o etheeal 4. That in the physical wold, nothing else takes place but this vaiation, subject (possibly) to the law of continuity. Cliffod Algeba has Dimensional Democacy, allowing you to add lines to planes Unify Phenomena Dimensionally 20 Using Cliffod Algeba, get 2 equations in P is the momentum, S is the spin, F is the electomagnetic field 0

11 . Defining a Cliffod Algeba 2 Fo N dimensions have N basis vectos { σ j } j =, L, N They anticommute σσ 2 = σ σ 2 Squae to + σ σ = σ σ = δ = { σ, σ } = σ σ + σ σ ij O: single ule i j i j i j 2. 3D Cliffod Algeba is Pauli Algeba 22 Geometic intepetation of i is volume Bivectos ae a quatenion goup

12 3. 4D Cliffod algeba is Diac Algeba 23 Thee ae two diffeent metic signatues that wok fo special elativity: (---+) o (+++-) 4. Popeties of Cliffod Algeba 24 2

13 4. Popeties of Cliffod Algeba 25 Can do things that Gassmann can t, like multiplication of two planes gives a plane Most impotant, Cliffod algeba can do otations like Hamilton s quatenions (moe late) 5. Geometic Calculus 26 Like quatenions, you have sums of scalas and vectos (and bivectos and tivectos ) 3

14 5. Geometic Calculus 27 You can get 4 Maxwell s equations in ONE! ( + ) F = ( ρ J ) F = E + ib t E = ρ B + E& c = + = i( E + B& ) c = 0 i B = 0 c scala J vecto bivecto tivecto 6. Example of Utility of Cliffod Algeba 28 Left side is using Gibbs vectos, ight side using Cliffod Algeba 4

15 Continued 29 Solving 4 equations in one can save many steps! Refeences 30 The poblem on pevious two slides was the deivation of the Chaacteistic Hypesufaces fo Maxwell s equations. The standad (Gibbs) teatment was adapted fom Adle, Bazin and Schiffe, Into to Geneal Relativity (McGaw-Hill 965), pp The Cliffod Algeba deivation is fom W. Pezzaglia, in Lawynowicz, Defomations of Mathematical Stuctues II (994), pp , o hepth/92062 EM in one equation, see Benad Jancewicz, Multivectos and Cliffod Algeba in Electodynamics (Wold Scientific988) p. 78 William Baylis, Electodynamics, a Moden Geometic Appoach (Bikhause999) The best quick intoduction to Cliffod Algeba is David Hestenes: Space-time Algeba (Godon & Beach 966) New Foundations fo Classical Mechanics (Kluwe 986) O, my Ph.D. thesis (I ll post this on web site) The best explanations of Gassmann algeba ae usually in the books on Cliffod Algebas. Howeve, a standad is Flandes, Diffeential Foms, Academic Pess (963). Anothe summay of Gassmann algeba would be: D. Feanley-Sande, Ameican Mathematical Monthly, 86, 809 (979) 5