Table Of ContentTHE
INTERNATIONAL SERIES
OF
MONOGRAPHS ON PHYSICS
SERIES EDITORS
J. BIRMAN City University of New York
S. F. EDWARDS University of Cambridge
R. FRIEND University of Cambridge
C. H. LLEWELLYN-SMITH University College London
M. REES University of Cambridge
D. SHERRINGTON University of Oxford
G. VENEZIANO CERN, Geneva
International Series of Monographs on Physics
117. G.E.Volovik:The universe in a helium droplet
116. L.Pitaevskii,S.Stringari:Bose{Einstein condensation
115. G.Dissertori,I.G.Knowles,M.Schmelling:Quantum chromodynamics
114. B.DeWitt:The global approach to quantum fleld theory
113. J.Zinn-Justin:Quantum fleld theory and critical phenomena, Fourth edition
112. R.M.Mazo:Brownian motion: (cid:176)uctuations, dynamics, and applications
111. H.Nishimori:Statistical physics of spin glasses and information processing: an
introduction
110. N.B.Kopnin:Theory of nonequilibrium superconductivity
109. A.Aharoni:Introduction to the theory of ferromagnetism, Second edition
108. R.Dobbs:Helium three
107. R.Wigmans:Calorimetry
106. J.Ku˜bler:Theory of itinerant electron magnetism
105. Y.Kuramoto,Y.Kitaoka:Dynamics of heavy electrons
104. D.Bardin,G.Passarino:The standard model in the making
103. G.C.Branco,L.Lavoura,J.P.Silva:CP violation
102. T.C.Choy:Efiective medium theory
101. H.Araki:Mathematical theory of quantum flelds
100. L.M.Pismen:Vortices in nonlinear flelds
99. L.Mestel:Stellar magnetism
98. K.H.Bennemann:Nonlinear optics in metals
97. D.Salzmann:Atomic physics in hot plamas
96. M.Brambilla:Kinetic theory of plasma waves
95. M.Wakatani:Stellarator and heliotron devices
94. S.Chikazumi:Physics of ferromagnetism
91. R.A.Bertlmann:Anomalies in quantum fleld theory
90. P.K.Gosh:Ion traps
89. E.Sim¶anek:Inhomogeneous superconductors
88. S.L.Adler:Quaternionic quantum mechanics and quantum flelds
87. P.S.Joshi:Global aspects in gravitation and cosmology
86. E.R.Pike,S.Sarkar:The quantum theory of radiation
84. V.Z.Kresin,H.Morawitz,S.A.Wolf:Mechanisms of conventional and high Tc
superconductivity
83. P.G.deGennes,J.Prost:The physics of liquid crystals
82. B.H.Bransden,M.R.C.McDowell:Charge exchange and the theory of ion{atom
collision
81. J.Jensen,A.R.Mackintosh:Rare earth magnetism
80. R.Gastmans,T.T.Wu:The ubiquitous photon
79. P.Luchini,H.Motz:Undulators and free-electron lasers
78. P.Weinberger:Electron scattering theory
76. H.Aoki,H.Kamimura:The physics of interacting electrons in disordered systems
75. J.D.Lawson:The physics of charged particle beams
73. M.Doi,S.F.Edwards:The theory of polymer dynamics
71. E.L.Wolf:Principles of electron tunneling spectroscopy
70. H.K.Henisch:Semiconductor contacts
69. S.Chandrasekhar:The mathematical theory of black holes
68. G.R.Satchler:Direct nuclear reactions
51. C.M¿ller:The theory of relativity
46. H.E.Stanley:Introduction to phase transitions and critical phenomena
32. A.Abragam:Principles of nuclear magnetism
27. P.A.M.Dirac:Principles of quantum mechanics
23. R.E.Peierls:Quantum theory of solids
The Universe
in a Helium Droplet
GRIGORY E. VOLOVIK
Low Temperature Laboratory,
Helsinki University of Technology
and
Landau Institute for Theoretical Physics, Moscow
.
CLARENDON PRESS OXFORD
2003
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FOREWORD
It is often said that the problem of the very small cosmological constant is
the greatest mystery in cosmology and in particle physics, and that no one has
any good ideas on how to solve it. The contents of this book make a lie of that
statement. The material in this monograph builds upon a candidate solution to
theproblem,oftendubbed‘emergence’.Itisasolutionsosimpleanddirectthat
it can be stated here in this foreword. Visualize the vacuum of particle physics
asifitwereacoldquantumliquidinequilibrium.Thenitspressuremustvanish,
unless it is a droplet{in which case there will be surface corrections scaling as
an inverse power of the droplet size. But vacuum dark pressure scales with the
vacuum dark energy, and thus is measured by the cosmological constant, which
indeed scales as the inverse square of the ‘size’ of the universe. The problem is
‘solved’.
But there is some bad news with the good. Photons, gravitons, and gluons
must be viewed as collective excitations of the purported liquid, with dispersion
laws which at high energies are not expected to be relativistic. The equivalence
principle and gauge invariance are probably inexact. Many other such ramifl-
cations exist, as described in this book. And experimental constraints on such
deviant behavior are extremely strong. Nevertheless, it is in my opinion not out
ofthequestionthatthedi–cultiescaneventuallybeovercome.Iftheyare,itwill
mean that many sacrosanct beliefs held by almost all contemporary theoretical
particle physicists and cosmologists will at the least be severely challenged.
This book summarizes the pioneering research of its author, Grisha Volovik,
andprovidesasplendidguideintothismostlyunexploredwildernessofemergent
particle physics and cosmology. So far it is not respectable territory, so there is
danger to the young researcher venturing within{working on it may be detri-
mental to a successful career track. But together with the danger will be high
adventure and, if the ideas turn out to be correct, great rewards. I salute here
those who take the chance and embark upon the adventure. At the very least
they will be rewarded by acquiring a deep understanding of much of the lore of
condensed matter physics. And, with some luck, they will also be rewarded by
uncoveringaradicallydifierentinterpretationoftheprofoundproblemsinvolving
the structure of the very large and of the very small.
Stanford Linear Accelerator Center James D. Bjorken
August 2002
PREFACE
Topology is a powerful tool for gaining the most important information on
complicated many-body systems in a very economic way. Topological classiflca-
tionofdefects{vortices,domainwalls,monopoles{allowsustoelucidatewhich
defect is stable and what will be the result of the fusion of two defects, with-
out resort to any equations. In many cases there are no simple equations which
govern such processes, while numerical simulations from the flrst principles {
from the Theory of Everything (provided that such a theory exists) { are highly
time consuming and not conclusive because of lack of generality. A number of
difierent vortices with an intricate structure of the multi-component order pa-
rameter have been experimentally observed in the super(cid:176)uid phases of 3He, but
themathematicswhichisusedtotreatthemisassimpleastheequation1+1=0.
This equation demonstrates that the collision of two singular vortices gives rise
toacontinuousvortex-skyrmion,orthattwosolitonwallsannihilateeachother.
Another example is the Fermi surface: it is stable because it is a topological
defect { a quantized vortex in momentum space. Again, without use of the mi-
croscopic theory, only from topology in the momentum space, one can predict
all possible types of behavior of the many-body system at low energy which do
not depend on details of atomic structure. The system is either fully gapped, or
the Fermi surface is developed, or, what has most remarkable consequences, a
singularpointinthemomentumspaceevolves{theFermipoint.IfaFermipoint
appears, as happens in super(cid:176)uid 3He-A, at low energies the system is governed
by a quantum fleld theory describing left-handed and right-handed fermionic
quasiparticles interacting with efiective gauge and gravity flelds. Practically all
the ingredients of the Standard Model emerge, together with Lorentz invariance
andotherphysicallaws.Thissuggeststhatmaybeourquantumvacuumbelongs
to the same universality class, if so, the origin of the physical laws could be un-
derstoodtogetherwithsomepuzzlessuchasthecosmologicalconstantproblem.
Inthisbookwediscussthegeneralconsequencesfromtopologyonthequan-
tumvacuuminquantumliquidsandtheparallelsinparticlephysicsandcosmol-
ogy.Thisincludestopologicaldefects;emergentrelativisticquantumfleldtheory
andgravity;chiralanomaly;thelow-dimensionalworldofquasiparticleslivingin
thecoreofvortices,domainwallsandother‘branes’;quantumphasetransitions;
emergent non-trivial spacetimes; and many more.
Helsinki University of Technology Grigory E. Volovik
November 2002
CONTENTS
1 Introduction: GUT and anti-GUT 1
I QUANTUM BOSE LIQUID
2 Gravity 11
2.1 Einstein theory of gravity 11
2.1.1 Covariant conservation law 12
2.2 Vacuum energy and cosmological term 12
2.2.1 Vacuum energy 12
2.2.2 Cosmological constant problem 14
2.2.3 Vacuum-induced gravity 15
2.2.4 Efiective gravity in quantum liquids 15
3 Microscopic physics of quantum liquids 17
3.1 Theory of Everything in quantum liquids 17
3.1.1 Microscopic Hamiltonian 17
3.1.2 Particles and quasiparticles 18
3.1.3 Microscopic and efiective symmetries 18
3.1.4 FundamentalconstantsofTheoryofEverything 20
3.2 Weakly interacting Bose gas 21
3.2.1 Model Hamiltonian 21
3.2.2 Pseudorotation { Bogoliubov transformation 22
3.2.3 Low-energy relativistic quasiparticles 23
3.2.4 Vacuum energy of weakly interacting Bose gas 23
3.2.5 Fundamental constants and Planck scales 25
3.2.6 Vacuum pressure and cosmological constant 26
3.3 From Bose gas to Bose liquid 27
3.3.1 Gas-like vs liquid-like vacuum 27
3.3.2 Model liquid state 28
3.3.3 Real liquid state 29
3.3.4 Vanishingofcosmologicalconstantinliquid4He 29
4 Efiective theory of super(cid:176)uidity 32
4.1 Super(cid:176)uid vacuum and quasiparticles 32
4.1.1 Two-(cid:176)uid model as efiective theory of gravity 32
4.1.2 Galilean transformation for particles 32
4.1.3 Super(cid:176)uid-comoving frame and frame dragging 33
4.1.4 Galilean transformation for quasiparticles 34
4.1.5 Momentum vs pseudomomentum 35
4.2 Dynamics of super(cid:176)uid vacuum 36
4.2.1 Efiective action 36
viii
4.2.2 Continuity and London equations 37
4.3 Normal component { ‘matter’ 37
4.3.1 Efiective metric for matter fleld 37
4.3.2 External and inner observers 39
4.3.3 Is the speed of light a fundamental constant? 40
4.3.4 ‘Einstein equations’ 41
5 Two-(cid:176)uid hydrodynamics 42
5.1 Two-(cid:176)uid hydrodynamics from Einstein equations 42
5.2 Energy{momentum tensor for ‘matter’ 42
5.2.1 Metric in incompressible super(cid:176)uid 42
5.2.2 Covariant and contravariant 4-momentum 43
5.2.3 Energy{momentum tensor of ‘matter’ 44
5.2.4 Particle current and quasiparticle momentum 44
5.3 Local thermal equilibrium 45
5.3.1 Distribution function 45
5.3.2 Normal and super(cid:176)uid densities 45
5.3.3 Energy{momentum tensor 46
5.3.4 Temperature 4-vector 47
5.3.5 When is the local equilibrium impossible? 47
5.4 Global thermodynamic equilibrium 48
5.4.1 Tolman temperature 48
5.4.2 Global equilibrium and event horizon 49
6 Advantages and drawbacks of efiective theory 51
6.1 Non-locality in efiective theory 51
6.1.1 Conservation and covariant conservation 51
6.1.2 Covariance vs conservation 52
6.1.3 Paradoxes of efiective theory 52
6.1.4 No Lagrangian in classical hydrodynamics 53
6.1.5 Novikov{Wess{Zumino action for ferromagnets 54
6.2 Efiective vs microscopic theory 56
6.2.1 Does quantum gravity exist? 56
6.2.2 What efiective theory can and cannot do 56
6.3 Super(cid:176)uidity and universality 58
II QUANTUM FERMIONIC LIQUIDS
7 Microscopic physics 65
7.1 Introduction 65
7.2 BCS theory 66
7.2.1 Fermi gas 66
7.2.2 Model Hamiltonian 67
7.2.3 Bogoliubov rotation 68
7.2.4 Point nodes and ‘relativistic’ quasiparticles 69
7.2.5 Nodal lines are not generic 70
ix
7.3 Vacuum energy of weakly interacting Fermi gas 70
7.3.1 Vacuum in equilibrium 70
7.3.2 Axial vacuum 71
7.3.3 Fundamental constants and Planck scales 73
7.3.4 Vanishing of vacuum energy in liquid 3He 74
7.3.5 Vacuum energy in non-equilibrium 74
7.3.6 Vacuum energy and cosmological term 75
7.4 Spin-triplet super(cid:176)uids 76
7.4.1 Order parameter 76
7.4.2 Bogoliubov{Nambu spinor 77
7.4.3 3He-B { fully gapped system 78
7.4.4 Bogoliubov quasiparticle vs Dirac particle 79
7.4.5 Mass generation for Standard Model fermions 80
7.4.6 3He-A { super(cid:176)uid with point nodes 81
7.4.7 Axiplanar state { (cid:176)at directions 82
7.4.8 3He-A { Fermi surface and Fermi points 83
1
7.4.9 Planar phase { marginal Fermi points 84
7.4.10 Polar phase { unstable nodal lines 85
8 Universality classes of fermionic vacua 86
8.1 Fermi surface as topological object 87
8.1.1 Fermi surface is the vortex in momentum space 87
8.1.2 p-space and r-space topology 89
8.1.3 Topological invariant for Fermi surface 90
8.1.4 Landau Fermi liquid 91
8.1.5 Collective modes of Fermi surface 91
8.1.6 Volume of the Fermi surface as invariant 92
8.1.7 Non-Landau Fermi liquids 93
8.2 Systems with Fermi points 94
8.2.1 Chiral particles and Fermi point 94
8.2.2 Fermi point as hedgehog in p-space 95
8.2.3 Topological invariant for Fermi point 96
8.2.4 TopologicalinvariantintermsofGreenfunction 97
8.2.5 A-phase and planar state: the same spectrum
but difierent topology 99
8.2.6 Relativistic massless chiral fermions emerging
near Fermi point 99
8.2.7 Inducedelectromagneticandgravitationalflelds 100
8.2.8 Fermi points and their physics are natural 101
8.2.9 Manifolds of zeros in higher dimensions 103
9 Efiective quantum electrodynamics in 3He-A 105
9.1 Fermions 105
9.1.1 Electric charge and chirality 105
9.1.2 Topological invariant vs chirality 106
x
9.1.3 Efiective metric viewed by quasiparticles 106
9.1.4 Super(cid:176)uid velocity in axial vacuum 107
9.1.5 Spin from isospin, isospin from spin 108
9.1.6 Gauge invariance and general covariance
in fermionic sector 109
9.2 Efiective electromagnetic fleld 109
9.2.1 Why does QED arise in 3He-A? 109
9.2.2 Running coupling constant 111
9.2.3 Zero-charge efiect in 3He-A 111
9.2.4 Light { orbital waves 112
9.2.5 Doesoneneedthesymmetrybreakingtoobtain
massless bosons? 113
9.2.6 Are gauge equivalent states indistinguishable? 114
9.3 Efiective SU(N) gauge flelds 114
9.3.1 Local SU(2) from double degeneracy 114
9.3.2 Role of discrete symmetries 115
9.3.3 W-boson mass, (cid:176)at directions, supersymmetry 116
9.3.4 Difierentmetricsfordifierentfermions.Dynamic
restoration of Lorentz symmetry 117
10 Three levels of phenomenology of super(cid:176)uid 3He 118
10.1 Ginzburg{Landau level 119
10.1.1 Ginzburg{Landau free energy 119
10.1.2 Vacuum states 120
10.2 London level 120
10.2.1 London energy 120
10.2.2 Particle current 121
10.2.3 Parameters of London energy 121
10.3 Low-temperature relativistic regime 122
10.3.1 Energyandmomentumofvacuumand‘matter’ 122
10.3.2 Chemical potential for quasiparticles 123
10.3.3 Double role of counter(cid:176)ow 123
10.3.4 Fermionic charge 124
10.3.5 Normal component at zero temperature 125
10.3.6 Fermi surface from Fermi point 126
10.4 Parameters of efiective theory in London limit 128
10.4.1 ParametersofefiectivetheoryfromBCStheory 128
10.4.2 Fundamental constants 129
10.5 How to improve quantum liquid 130
10.5.1 Limit of inert vacuum 130
10.5.2 Efiective action in inert vacuum 131
10.5.3 Einstein action in 3He-A 132
10.5.4 Is G fundamental? 132
10.5.5 Violation of gauge invariance 133
