The geometry of Heisenberg groups : with applications in signal theory, optics, quantization, and...

aptcr 1 'Fhe Sikew Field of Quaternions
I. 1. m lion Af [hte Field of Quat,n ion> i nnd Ett,ni -t Forrmdai
12. F mbednings of C into the QuiterniCns and Naurat niary Groups 7
f:n-7lintear Structures. Syrnplectaic Structures tani O(rintation.
Pauli Arenemens
I. LIter itom orphimns of 12
.5. Th (Orienited Foutation \ngle of the hintner Aut 111r1>ism wih
R spe-C to tht N ;tural \finkowski Netrit 1r
I6i. Link hbet-ten Space-iiie Gouietry aid ELclhian ( immtrv ion to h
(2atr -ion-s 20
(haptp r 2. E nts of the A(atwmr:n of Von--f Bust thops aIml Spin
Represe ntiictins 25
1. 1(e--P ameter 'trCc2ps of Ci (> ) and S () 5
2.2. Pairallels f U titudell and Meridianis on SY3 27
"2. )rOnIe--Parameer Sublroups of S10(21 and h[opf GuiCi le:s 2
"L. 1 Spin Rep -ese itaton
2.5. ThI Inf i zitsml Spii -Rleprseortation
Chaptter 3 tlriil Variab h,s of Sing(dl tir y Frie e itor Fielt , in A i E a icide;
3.1. Th(h (' oiinplex I,inc Blunille Fa
"3. 2. Sinph icti: and Hcr mitiCi Striit cures> il ?
13. I cra1 nat I clitis
3?. C (LVa[( FCCIe IF 07 o11nS e I tJV S, iu s 1
\3n.Vwaw FOi ik I NOW? by T"4I
AT wi ( a imi e rii c Pt Imip;d i idle ii
31. thdamonl atul Periokic IANt Wt n ; 'Amys 69
( hpter L InspUTonWns ( C-t 0t fiernt C1s 齛f iornotlops (.'lsss of
Sn iulait> Frt.e \ci-,or FIdi.tF1-s i0 3 Spac e 71
1. laniinaVplakhiniC (Chnsses i(I Cliractr-istic PrFiltipal funnihlcu )i- Vector
I2 f' St i-il nc ulii of I onl rphisn ' lIass,m' TT
i.L. NOtll [ , UPI Fiat (Arnli-e Rharn (lana,
I S. H,dt-;- h[,r? 穒v D+,c< n+[ >osi ri on t
1TjwA Inw11 1. 11ch - ,r (r 1 AluS w "
J.ord1 Alebra and SL(, 2 I
5.1. Nmatlurl Symplectic Strunto o" a Pll ie n 1-S]a> f)7
.2. '1 hi Noi on of a Heislnherg A]gebr 12
1.3. leisekeilg (rou> 11d its lie Al Aera 115
5.. 1. as a Semi-direct Produc 11
-3.k. ?Heisenberg Algebra Siruci,ur o1 sopff) 121
VG. Ih Spin Group and the SkWoN FiUd of Quatninhont am1 Gov ) 1wd
by ()nly On)t[ HGis-nle Croulp 121
5.. Scala; Prodlct an31d ilinkowNsi Meirics oiln t HOw -i,ea A]r\,a 126
5.X. Symplectic p. Spial Line,a Group} and Loientz !2roup 128
Chapter 6. The IielinIr (Groilp and N?tural (7'-Alg4iras a (A \ fi!l
in 3-SPce 13!
6.1. ThI Heisenberg Gr,oup Bundle of Vncto! Field 132
62. IMoftine Dim-nsional Heisenberg Alg Iobras and ,fini1 D1i)HsioNnal
teisN,r g (GlCroups of ct1or Field 135
6.3. Ilya >s eitbrmiied by H5lomm1)irphilms 1 40
6.J. G(ou1p Algfbras of Infinite [) nsional ieisenbcrlg (Group1s 131
6.). The (? -Group Al-gra and the Twisted i nvoluwohNh. thp \\NVY
Itbra and Mhe PWhisson AlgW%3a 1).
Chapter 7. 1lh SclhrlM dinger Repres-elntaltion and thMi Nh-lu]]e3Qtil
Repq r>ysentw1thlOl 1361
7i. Dw)initin, of 1he Schr6dinger Represej-ntation and] Phase Spae- 1c 131i
7.2. Clar;icleristic Ingr01edienis of the Slchrding0t r Bip1res'enta tion I t66
7.3 Thle nfinitesinmal Schr6din ger Rcpresentation 3aid Phase Space 175
7.. Anroje(:tive Rlepresenta ions of' the Splneuti]iA (roup (onsirulted i a
t1hw Schri-dinger Riprtesentation 176
7.5. A ;ealization of the Mleta)plectic G1-oup and] the Metaplecif
R epresena tion t 1 0
Chqter s. T1'i Heisenberg Group: 2A Basi CG eom3-itric Backgeround of Signal
Analvsis and Gcomeiric Optics 191
8.1. The Notion of a Signal 192
-.2. Tinme-Frequency Analysis and lhe Ui nci(rtainty Principle 193
8.3. Furnher Tools of Tirne-]Irequency Analysis 196
8.4. Reconstruc1ion Forimulae 201
8.N. The Geometry 1Underlying Time-Frequency Analysis 202
8.6. The Radar Ambiguliy tunlction 201
8.7. IThe Stone-von Neumiann Theoremn in 1-iine-Frequency Analysis 205
8.8. Geometric Optics 206
S.9. Holography 210
Chapter 9. Quantization of Quadratic PolynoVniais 21
9.1. Elementary Observations on Information and its Transmission 215
9.2. Pieservation of Information 217
9.3. The Po-isson Algebra, of all Hlomogeineous QuadratVi 1"iyno01ian in
Two Variab-- L 221
9.. .The Quanliza;ion of Inhomogeneous Quadratic Polynomials 230