Lie Groups: History, Frontiers and Applications. Quantum statistical mechanics and Lie group harmonic analysis, Part A

265 Pages·1980·6.229 MB·English

by  Norman Hurt, Robert Hermann

LIE GROUPS: HISTORY. FRONTIERS AND
APPLICATIONS
EDITORS: M. ACKERMAN, C. BYRNES, H. HARTMAN. R. HERMANN,
C. MARTIN. W. STEIGER. AND N. WALLACH
1. Sophus Lie's 1880 Transformation Group Paper, Translation by M.
Ackerman, Comments by R. Hermann
2. Ricci and Levi-Civita's Tensor Analysis Paper. Translation and Com
ments by R. Hermann
3. Sophus Lie's 1884 Differential Invariants Paper. Translation by M.
Ackerman. Comments by R. Hermann
4. Smooth Compactification of Locally Symmetric Varieties, by A. Ash,
D. Mumford. M Rapoport and Y. Tai
5. Symplectic Geometry and Fourier Analysis. by N. Wallach
6. The 1976 Ames Research Center (NASA) Conference on: The Geometric
Theory of Non-Linear Waves.
7. The 1976 Ames Research Center (NASA) Conference on Geometric
Control Theory.
8. Hilbert's Invariant Theory Papers, Translation by M. Ackerman, Com
ments by R. Hermann
9. Development of Mathematics in the 19th Century, by Felix Klein,
Translated by M. Ackerman, Appendix "Kleinian Mathematics from
an Advanced Standpoint,"· by R. Hermann
10. Quantum Statistical Mechanics and Lie Group Harmonic Analysis. Part
A. by N. Hurt and R. Hermann
LIE GROUPS: HISTORY FRONTIERS
1
AND
APPLICATIO~S
VOLUME X
QUANTUM STATISTICAL MECHANICS AND
LIE GROUP HARMONIC ANALYSIS
PART A
NORMAN HURT
AND
ROBERT HE~MANN
MATH SCI PRESS
Th1& One
1111111111~1111111111~IIIIUIIIIllII
BJHQ-P8H-JJJ1
Copyright e by Norman Hurt and
Robert Hermann
All rights reserved
ISBN: 0-915692-30-9
Liltrary ., C ••• r ... Calaleclac I. PuItUraU• • Data
!iurt, Norman.
Quantum statistical mechanics and lie Group harmonic
analysis.
(~ie ~roups ; v. 10-
Includes bibliographical references.
1. Statistical mechanics. 2. Groups, Theory of.
1. Quantum statistics. 4. Lie groups. 5. Harmonic
analysis. I. Robert, joint author.
~ermann,
II. T1 tle.
QC174.8.HB7 510.1'~.1 80-13949
ISBN0-91S692-30-9 (pt. A)
MATH SCI PRESS
53 JORDAN ROAD
BROOKLINE, HA 02146
Printed in the United States of America
QUANTUM STATISTICAL MECHANICS AND
LIE GROUP HARMONIC ANALYSIS
PART A
PREFP.CE
Ever since the classlc treatlses by Weyl and Wigner, lt ha!> been
clear that there is a close link between group representation theory and
quantum mechanics. This connection remalned dormant for many years; physiclsts
preferred to "invent" their own version, while mathematicians were busy working
out the "pure" theory of group representations. Recently, this relation has
come back into the foreground.
In elementary particle physics, groups have been impressively useful for
phenomenology; such prominent topics as "unitary synunetry", "quarks", and
"broken symmetry" have a group theoretic meaning. Recently, in the ideas of
"gauge fields" the infinite dimensionaZ ~1rcl4'S orlginally defined and studled
by Lie and Cartan have entered.
As quantum mechanics has spread outward into other disciplines (e.g.,
chemistry, electronics, solid state, ... ) the group theoretic baggage has
traveled along and has been found to be very useful. Often these disciplines
require mainly qualltative and structural properties of quantum mechanics-
group theoretic principles are supreme here. Further, the geometric properties
of many such systems (e.g., arystals) are reflected ln the groups that enter.
Thus, material on certain groups (rotation groups, crystallographic groups,
etc.) first developed in the context of atomic physics has been much more
widely applicable.
Statistical meahanias has been one of the most thriving sub-disciplines
of quantum mechanics, has stimulated very deep mathematical work and has led
to great physical advances and insight. By its very nature, it can be split
off from the main body of quantum mechanics and studied separately with lts own
distinctive methods. For this reason, it has attracted the most intensive
mathematical scrutiny, concentrating on such questions as the existence of
certain limits, the possible existence of phase transitions, etc. However,
the practical use of statistical mechanics is dependent on exact or appropri
ate computation of certain mathematical Objects associated with a quantum
system. (In fact, all items of interest usually depend on computation of one
such object, the partition function.) Our aim here is to pursue the study
of these Objects with group-theoretia methodB. In fact, we shall show that
there are close relations between the physical questions and topics of
intense recent interest in the mathematical literature, e.g., the theory of
zeta functions, theta function series, discrete subgroups of Lie groups,
Fourier expansions of functions on Lie groups, etc.
The material which interests us involves ten disciplines:
Statistical mechanics
Lie group representation theory
The theory of discrete subqroups of Lie groups
Analytic number theory
Assymptotics of differential and integral equations
Quantum chemistry
Solid state physics
Analytical mechanics
Functional analysis
Differential geometry
Obviously, in such a cross-disciplinary work we cannot hope to do justice to
everything, and can only skim the surface of much of the material. We will
adopt the approach used ln a "review article" in the sciences (it is not an
art form which is extensively used by mathematicians), with considerably more
detail about the mathematics than is customary in such efforts. We will
emphasize what Paul L~VY called so well, aon~ete functional analY8i8.
Thus, abstraction will mainly be used to carry over formulas and intuition
from finlte to inflnite dimensions, to have "clean" coordinate and basis-free
formulas, utilize geometric intuition, etc. Much of the analysis we do will
be formal, l.e., we freely allow interchange of limits (when it seems reason
able), we are not precise about degrees of differentiability, interchange of
llmits, speciflcatlon of topologles, etc. This is then basically the attitude
taken by a physicist or chemist when using mathematics, with the addition that
we intend to use considerably more sophisticated material on the geometric and
group-theoretic side. One check on this "formal" work is that it leads to
formulas that make sen8e. An experienced mathematician can usually make it
precise. Thus, this monograph might have been titled: Formulas from Statisti
cal Mechanics and Their Relation to Group Theory.
The work had its origin in a set of notes by the first author called
Physics and Homogeneous spaces. It is basically his work. The second author
has made a hobby of writing books explaining the geometric and group-theoretic
side of phYS1CS; his efforts here are in the same direction.
TABLE OF CONTENTS
PREFACE iii
Chapter 1: A GEOMETRIC CODIFICATION OF GIBBSIAN THERMOOYNAMICS
AND STATISTICAL MECHANICS
] Introductjon
2. Thermodynamic Systems 2
3 The Fi rst Law :2
4 The Second Taw
5. The Combined Flrst and Second Lalol. Glbbslan Systems
6. Classlcal Equilibrium Mechanics According to Gibbs
7 The Gibbs Measyres as SoIlltlOos of an Extrt'!!DlltD Problem
8. An Analogy with System Theory
9. Quantum Statistical Mpchanics
10 Classical StatIstical Mcchaolcs for problems of
Analytical Mp.chanics 1..1
11. Quantum Statistical Mechanlcs of AnalytIcal Mechanics
Systems
Bibliography
Chapter 2: THE WEYL-WIGNER-MOYAL THEORY OF QUANTUM STATISTICAL
MECHANICS
) IntrOO"ctlon 11
2. The Weyl Prescri~tion and the Moyal Product l..9.
3. The Weyl Operators as Integral Operators 2..l
4. Traces of the Weyl Operators
S. The Average Value of Quantum Observables 1n pure States
Bibhography
Chapter 3: THE FEYNMAN PATH INTEGRAL AND THE PARTITION FUNCTION
Introdnctloo 29
2 The Trotter Formula and Greeo's ElloctlOO 30
3. Measures and Integrals 31
4. Integrals Deflned by Approxlmatlons so
5. Feynrnan Path Integrals 51
6. The Feynman Path Integral for Newtonlan Pdrtlcl~s un the
Line and Green's FunctIon for the Heat Equation 52
7. Th~ Feynman Theory
8. The Wiener-l(ac Strategy 55
Bibliography 58
Chapter 4: THE HIGH-TEMPERATURE LIMIT AND THE CLASS leAL PARTITION
FllNCl'lON
) Introduction 59
2. The Trotter Formula and the Classlcal Partl.tlOn Formula 60
3 Green's Functlon for HarmonIC Osclliator 62
4 The potential as a PerturbatloD of the Free Hamlltonlan 66
Bibliography 68
v
Vl Contents
ChaIJter S: THE STATISTICAL MECHANICS OF THE QUANTUM HARMONIC
oSCII.IATOR IN TERMS OF LIE GROlTP THEORY
1 Introdllction 69
2. The POlsson Bracket ~lgebra 70
3. Representatlon of ~2 in Terms of Dlfferential
Operators 70
4. SL(2,R) and /¥L 71
ChdfJter 6: THE PARTITION f'UNCTION FOR PHYSICAL SYSTEMS WITH
PISCRETE SPECTRA
J Introd\lctIon 77
2. A Hllbert Space Settlng for Dlscrete Spectra Statistlcal
MechanICS Problems 77
3. Wlgnerlan Symmetry Groups 79
4 The pI aO'lY Rotator 80
:'. Th(t HarmoDlc Oscillator Partltlon Functlon 85
Cha~ter 7: ELLIPTIC DIFFERENTIAL OPERATORS ON COMPACT MANIFOLDS
} Introd\lction 91
2. LInear Dlfferentlal Operators and thelr Symbols 91
:3. The Deflnl tlon of "Elliptlc Dlfferentlal Operators" 94
4. The Symbol and the Symbolic Method. The Parametrix 96
"" ElLlI'tiL Opcrdtor& lind Fredholm Theory 'J'J
6. The Heat Equatlon 103
7. The Larlace-Beltraml Operator on a Rlemannian Manifold 104
8. The Heat Equatlon for Compact Rlemannian SYlmletrlc Spaces 106
9. The Partition Function for the Two-Sphere 110
10. Mulholland's ASymptotlC Expansion for the SpherIcal
Partitlon Functlon and the Resolvent of the Laplace-
Beltraml ~)erator 112
11. Weak SolUtions and "ElliptlC Regularl ty" 114
Blbllography 116
Chapt"'r B: A SURVEY OF CERTAIN RELATIONS BETWEEN STATISTICAL
MECHANICS AND SPECTRAl. TIiEoRY
IntrodllctIon 119
2 GIbb's "Canonica}" Distriblltion 119
3. The Llmitlng Princlpla of Quantum Statlstical Mechanics 122
4. Th~ Planar Rlgld Rotator 124
5. The Free PartIcle and the Harmonlc Osclllator 127
6. Numb~r Theory in Physics 134
7. Some Facts about Compact SymmetrIC Spaces 136
8. StatIstical MechanICS on Symmetric Spaces 139
9. Zeta FunctIons on Compact Lte Groups 147
10. Eillptic Differential Operator AsymPtotics 149
11. Return to the High Temperature Llmit 153
12. Return to the One-Dlmenslonal Free Partlcle of the
Epstein Zeta Funct10n 155
13. Partlcle Movlng In One DimenSlon in a Potentlal V(x) and
the Llmiting Princlple 157
Bibliography 163
Contents
V11
Chapter 9: THE T,lE-THEORETIC FO!!NJ)ATI(]IIS OF THE TRANSFER-MATRIX
METHOD FOR EVALUATING THE PARTITION FUNCTION OF THE
PLANAR ISING MODEL
1. Introduction 167
2. An Ansatz for the Transfer MatriX )68
3. The Two-Dimensional Ising Model 16':}
4. The Ising-Onsager Model as a Problem 1n L1e Group Theory 172
5. Clifford Structures and the SYmmetriC Pa1rs of
Orthogonal L1e Algebras they Generate 174
6. The Algebra1c Structure and General1zat1on of the IS1ng
Model
B1bllOgraphy IB4
Chapter 10: INDUCED REPRESENTATIONS AND HOMOGENEOUS VECTOR BUNDLES
1. Introduction 185
2. Pr1nc1pal Bundles W1 th L1e Graurs as Structure GraUl) and
Their Associated FIber Bund1 18;i
PH
3. Cross-Sect1on of the Associated FIber Spaces as "Covariants"
of the Structure Group 188
4. Vector Bundles 190
5. Tensorial F1elds and the PrInCIpal Tangent Frame Bundle
to a Mani fo] d 190
6. Sp1nor F1elds 1~3
7. Homogeneous Vector Bundles on Homogeneous Spaces. Induced
RepresentatIons 194
~ Frobenius ReCIprocity 1')5
9... The Bore) -Wei 1 Theorem as ,1 Van ant of Frohcnj liS
Reciproc1ty 198
l.O... A General1zed "Borel-Well" Program 202
Chapter 11: THE FOURIER AND lAPLACE TRANSFORM
lInt roduct ion
2. The General1zed Fourler-Lapldce Transform and Inverse
Transform 20'1
] The Classical Fourler Series 206
4. The Fourier Transform on a Compact L1e Group 210
5. The Classical FourIer Transform as the Prototype for a
Fourier Transform for Non-ComDact Lie Groups 216
6. Remarks on the Laplace Transform 218
7. Another InterpretatIon of th~ Laplace Transform 221
8. A Proqram for "Four1er-Laplace" AnalYS1S 1n a L1e Group 222
Biblloqraphy 222
Chapter 12: THE FOURIER TRANSFORM AND INDUCED REPRESENTATIONS
1. Introduct1on UJ
2. Induced Representat10ns 2.2..3.
3. Herm1t1an PaIrIng on Vector Bundles US
4. Group Invariance of Herm1t1an Vector Bundle Pa1rlngs Ufl
5. Unitary Induced RepresentatIons .u..a.
6. Representations Induced from Characters 2.2.:1
7. SL( 2, R) 2..lQ
8. The Foyrier Transform 236
9. The Inverse FourIer Transform 239