Oscillator Representation in Quantum Physics

280 Pages·1995·3.593 MB·English

by  M. Dineykhan, G. V. Efimov, G. Ganbold, S. N. Nedelko (auth.)

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M. Dineykhan G.~Efimov G.Ganbold
S.N.Nedelko
Oscillator ReRresentation
in Quantum Physics
Springer
Authors
M.Dineykhan
G.V.Efimov
G.Ganbold
S.N.Nedelko
BogoliubovLaboratoryofTheoreticalPhysics
JointInstituteforNuclearResearch(JINR)
141980Dubna,Russia
ISBN3-540-59085-4Springer-Verlag Berlin Heidelberg NewYork
CIPdataappliedfor.
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthe
materialisconcerned,specificallytherightsoftranslation,reprinting,re-useofillustra
tions, recitation, broadcasting, reproduction on microfIlms or in any other way, and
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areliableforprosecutionundertheGermanCopyrightLaw.
©Springer-VerlagBerlinHeidelberg1995
PrintedinGermany
Typesetting:Camera-readybytheauthors
SPIN:10127309 55/3142-543210-Printedonacid-freepaper
Contents
1. Introduction.............................................. 1
Part I. The Phase Structure of Quantwn Field Systems
2. Formulation ofthe Method ............................... 9
2.1 Breaking Symmetry in Quantum Field Theory.. ... .... .... 9
2.2 Canonical Quantization and the S-matrix 15
2.3 The Renormalization Group 18
2.3.1 Renormalization Schemes ,.. 18
2.3.2 Renormalization Group Equations. ................. 19
2.4 Unitary Nonequivalent Representations ofthe
Canonical Relations. .................................... 21
2.4.1 An Infinite Number ofDegrees ofFreedom 22
2.4.2 CanonicalTransformations 25
2.4.3 The Van Hove Model 28
2.4.4 Renormalization Group Transformations 28
2.5 The Oscillator Representation Method 30
3. The Phase Structure ofthe (<p2)2 Field Theory in R1+1 .. 33
3.1 The One-Component <p4 Model ,.............. .. .... 34
3.1.1 The Initial Representation for G ~1 ................ 34
3.1.2 The Canonical Transformation. .... .... .... ........ 34
3.1.3 The Phase Structure. ............................. 36
3.2 The O(N)-Invariant Model (<p2)2 in Rl+1 •• .••• ..•• .••. ..•• 39
3.2.1 Initial Representation. ............................ 40
3.2.2 The Canonical Transformation. .................... 40
3.2.3 The Phase Structure. ............................. 42
4. The Phase Structure of the Three-Dimensional <p4 Theory 45
4.1 The Hamiltonian <p4 in R2+1 ............................. 45
4.2 The Canonical Transformations 47
4.3 The Symmetric Model 50
4.3.1 The BS-phase. ... ............................ .... 51
4.3.2 The S-phases 52
VI
4.4 The Model with SSB in the Initial Representation 53
4.4.1 The BS-phase " .... .... .. .. ........ . 54
4.4.2 The S-phase ............ 54
4.5 The O(N)-Invariant Model in R2+l ... .................... 55
4.5.1 The Hamiltonian ofthe Model. .. .............. .... 55
4.5.2 The Canonical Transformation ,..... 56
4.5.3 The Phase Structure. ............................. 60
4.6 Stability Under Change ofthe R-scheme in the
Initial Representation ................................... 65
4.6.1 Subtraction at Zero External Momentum and
Arbitrary "Mass" in Propagators. .................. 67
4.6.2 Dimensional Regularization and the Minimal
Subtraction Scheme. .............................. 68
4.6.3 DimensionalRegularizationandZero-MomentumSub-
traction ......................................... 69
5. The Four-Dimensional 11'4 Theory 71
5.1 The Hamiltonian (<p4)4 and Canonical Transformations.. .... 71
5.2 The Symmetric Phases.. ..... ... .... .... .... ........ .... 73
5.3 DynamicalSymmetry Breaking 77
5.4 The Asymptotically Free Model.. .... .................... 78
5.5 ACorrelationBetween the PhaseStructureandUV Divergences 79
6. The 11'4 Theory at Finite Temperatures ................... 81
6.1 The Two-DimensionalModel. .... .... ................ .... 81
6.1.1 The Hamiltonian at Zero Temperature. ......... .... 81
6.1.2 The Thermo Field Dynamics. ...................... 83
6.1.3 The Symmetric Model 86
6.1.4 The Model with SSB in the Initial Representation. ... 87
6.2 The Three-Dimensional Model <p4 at Finite Temperatures ... 91
6.2.1 The Thermal Vacuum Representation. .............. 91
6.2.2 The Symmetric Model 93
6.2.3 The Case ofSSB in the Initial Representation. ....... 96
6.3 The Systems in Rl+1 and R2+1 ..........•.........•••..• 98
7. The Two-Dimensional Yukawa Theory 99
7.1 The Hamiltonian and Renormalization 100
7.2 The Canonical Transformation 101
7.3 The Phase Structure 104
7.3.1 The Pure YukawaInteraction 104
7.3.2 The Yukawa Model with Boson Self-Interaction 109
7.4 Comparison with Other Approaches 115
7.4.1 The Renormalized Formulation 116
7.4.2 The Regularized Formulation 118
References 121
VII
Part II. The Gaussian Equivalent Representation of Functional
Integrals in Quantum Physics
8. Path Integrals in Quantum Physics 127
8.1 Gaussian Path Integrals 127
8.2 A Short Historical Review on the Path Integration Method .. 131
8.3 The Feynman Path Integral Formalismin
Quantum Mechanics 134
8.4 PIs as Solutions ofDifferential Equations 138
8.5 Path Integrals in Quantum Field Theory 142
9. The Gaussian Equivalent Representation of
Functional Integrals 145
9.1 A General Description ofthe Method 145
9.2 The GER Method for Quantum Statistics 150
10. The Polaron Problem 157
10.1 Introduction 157
10.2 The Polaron Path Integral Formulated in d-dimensions 159
10.3 Application ofthe GER Method to the d-dimensional'Polaron 161
10.4 Bounds for the Polaron Ground-State Energy in d Dimensions162
10.5 Particular Case ofFeynman's Estimation 165
10.6 Corrections to the Leading Term ofthe Energy 166
10.7 Scaling Relations 168
10.8 Numerical Results 169
10.8.1 The Weak Coupling Limit 170
10.8.2 The Strong Coupling Regime 174
10.8.3 The Intermediate Coupling Range 177
11. The Character ofthe Phase Transition in Two- and Three-
Dimensional cp4 Theory 179
11.1 A Statement ofthe Problem 179
11.2 The Renormalized Lagrangianofthe <p~3 - Model 180
11.3 The Effective Potential in <p~3 - Theory 181
11.4 The Leading-Order Termofthe EP as the "Cactus-type" Po-
tential in <p~ .....•.........•................••......... 183
11.5 The Non-Gaussian Correction in <p~ ........•..........•••. 184
11.6 The Strong Coupling Regime in <p~ ....•...••...•.....•••• 187
12. Wave Propagation in Randomly Distributed Media 189
12.1 The Green Function ofthe Wave Equation 189
12.2 Calculation ofthe PI by the GER Method 192
12.3 The Green Function for Large Distances 195
VIII
13. Bound States in QFT 197
13.1 The Mass ofBound States " 197
13.2 The Nonrelativistic Limit 201
References 203
Part III. Oscillator Representation in Quantum Mechanics
14. The Oscillator in Quantum Mechanics 209
15. The Oscillator Representation in Rd ............•......... 215
15.1 Hamiltoniansin the Oscillator Representation 216
15.2 The Second Correction 218
15.3 The General Case " 220
16. The Oscillator Representation in the Space R3 .....•...•. 223
16.1 Large Distances 224
16.1.1 Small Distances 225
16.2 Formulation ofthe Problem 226
16.2.1 The Ground State Energy in the Zeroth and
Second Approximations 228
16.2.2 Radial Excitations 229
16.2.3 The Upper Estimate 230
16.2.4 The Parameter D and the Oscillator Basis 231
17. Anharmonic Potentials 235
17.1 Anharmonic Potentials in R1 •••..•••.........•••••.•.•.. 236
17.2 Anharmonic Potentials in R3 240
17.2.1 Power-Law Potentials 241
17.2.2 The Logarithmic Potential. 245
18. Coulomb-Type Potentials '" 247
18.1 The Coulomb Potential 247
18.2 The Screened Coulomb Potential 248
18.2.1 The Critical Screening Length 250
19. The Relativized Schrodinger Equation 251
19.1 Examples 254
20. Three-Body Coulomb Systems 257
20.1 The Three-Body Hamiltonianfor the Ground State 258
20.2 The Hamiltonian in the Oscillator Representation 263
20.2.1 The Accuracy ofthe Zeroth Approximation 265
20.2.2 The Stability ofThree Unit-Charge Systems 267
IX
20.3 Mesic Molecules ofLight Nuclei in the
Oscillator Representation 270
20.3.1 The Binding Energy ofthe Mesic Molecules of
Light Nuclei 271
20.3.2 The Stability ofMesic Molecules ofLight Nuclei 273
References 277
1. Introduction
The investigation ofmost problemsofquantum physics leads to the solution
ofthe Schrodinger equation with an appropriate interaction Hamiltonian or
potential. However, the exact solutions are known for rather a restricted set
of potentials, so that the standard eternal problem that faces us is to find
the best effective approximation to the exact solution of the Schrodinger
equation under consideration. In the most general form, this problem can
be formulated as follows. Let a total HamiltonianH describing a relativistic
(quantum field theory) or a nonrelativistic (quantum mechanics) system be
given. Our problem is to solve the Schrodinger equation
Hlftn= Enlftn,
i.e., to find the energy spectrum {En} and the proper wave functions {lftn}
=
including the'ground state or vacuum lfto 10). The main idea of any ap
proximation technique is to find a decomposition
in such a way that Ha describes our physical system in the "closest to H"
manner, and the Schrodinger equation
=
HolJt.(O) E(O)lJt.(O)
n n n
can be solved exactly. The interaction Hamiltonian HI is supposed to give
smallcorrections to the zero approximationwhich can be calculated.
In this book, we shall consider the problem of a strong coupling regime
in quantum field theory, calculations ofpath or functional integrals over the
Gaussian measure and spectral problemsin quantum mechanics. Let us con
sider these problems briefly.
The real problems ofquantum field theory which will be studied in this
book can be exemplified by a one-component scalar field. The total Hamil
tonian is
Ifthecouplingconstant9issmallenoughonecanchooseinthedecomposition
H=Ho+HI