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Bogolubov N.N., Logunov A.A., Todorov I.T. Introduction to Axiomatic Quantum Field Theory
Bogolubov N.N., Logunov A.A., Todorov I.T.  Introduction to Axiomatic Quantum Field Theory









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: Introduction to Axiomatic Quantum Field Theory

: Bogolubov N.N., Logunov A.A., Todorov I.T.

:

At the end of 1960 we made plans to write a monograph about the general principles of quantum field theory and their experimental implications. We intended primarily to give an account of the progress of the theory of dispersion relations since the appearance of the book of Bogolubov, Medvedev and Polivanov ([BMP]. As an introduction we wanted to include a review of the various approaches to axiomatic field theory. This introduction had to cover not only the formulation of Bogolubov, Medvedev and Polivanov, based on the apparatus of functional derivatives of the 5-matrix and the condition of microcausality, but also the field formulation associated with the names of Wightman, Haag, Lehmann, Symanzik, Zimmermann, and others. In the course of the work the tasks (and with them the size) of the introduction grew larger and larger, until eventually it developed into this book.


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$D^{(1)}(x)$ (even invariant function)      334335 511 528
$d_{\chi}$      (see Homogeneous distributions)
$P(\Phi)$ models      620 625626
$SL(2)\otimes SL(2)$      137 478481 533
$SL(2, \mathbb{R})$      156
$Sp(2, \mathbb{R}), Sp(4, \mathbb{R})$      568569
$S^{*}_{V^{+}}$      91 95
$S_{\infty}$      426427
$\alpha *$ (dual of Hermitian elements of $\alpha$)      598
$\alpha$      (see Quasilocal observables Algebra of
$\alpha^{+}(O)$      (see Algebra of
$\alpha^{H} *$      (see Functionals positive)
$\mathcal{D}$      2225 2728 32 48 69 105
$\mathcal{D}^{*}$ (spaces of distributions)      27 48 55 69 85 105
$\mathfrak{L}_{2}$ (space of square-integrable functions)      16 30 3840
$\omega$      (see Domain common)
$\Omega_{0}$(space of finite test-function sequences)$\Omega_{0}^{*}$      287296 310
$\Omega_{1}$ (nuclear space generated by fields)      256
$\Phi^{4}$ interaction, in Yukawa theory      455 04
$\Phi^{4}$ model      468469 582 612620 625- 626
$\textit{1}$ (space of sequences of fast decrease)      32 4344
$\textit{1}$ (space of sequences of fast decrease), f-fold sequences $(\textit{S}_{f})$      119121 197199 303
$\textit{c}$ (spaces of differentiable functions)      1718 2022 25
Abelian (commutative) group      209
Abelian algebra      223 597 607
Absolutely continuous function      37
Acharya, R.      575 632 633
Additive operator      139 239
Adjoint (Hermitian conjugate)      3637 569 586
Adjoint (Hermitian conjugate) in algebra      (see Involution)
Adjoint (Hermitian conjugate) of field operator      253 260 262 529 531532 538
Adjoint representation      224
Adler, S. L.      402 627
Admissible observable      115 116
Advanced distribution      (see Retarded and advanced distributions)
Advanced product $(A_{\chi})$      392399
Affinors      174
Akhiezer, N. I.      37 39 105 627
Akilov, G. P.      15 19 20 21 24 35 37 47 105 630
Aks, S.      0 575 655
Algebra      95 256 584
Algebra of bounded operators $[\beta(H)]$      586 589
Algebra of fields $(\alpha, \mathfrak{F})$      255257 297300 599-601 602 609 610 614 616 624 625
Algebra of open set $[\alpha(O), \mathcal{R}(O) ]$      257258 350 581 596597 599-610
Algebra of quasilocal observables$(\alpha)$      596599
Algebra of unbounded operators      624
Algebra, normed      301 584585
Algebra, nuclear      287289 301
Algebraic approach      4 525 573 577 581611 624625
Algebraic approach, field theory and      582 583 590 597 607611
Algebraic approach, scattering theory in      610611
Almost local field      351 357 364366 374378 384-390
Analytic completion      429
Analytic continuation      332 445446
Analytic extension      509
Analytic vector      608
Analyticity causality and      432433 476
Analyticity in manifold      493495
Analyticity in several variables      467
Analyticity of Fourier transform of retarded function      9198 267268 432434 473 476
Analyticity of Green functions      359360 429449 454460 473 476
Analyticity of Lie group coordinates      214215
Analyticity of Wightman functions      267268 473 476 487 504510
Analyticity on open real set      605606
Analyticity, Lorentz invariance and      434 473497 574
Analyticity, primitive domain of      429 438441 447449
Angular momentum of two-particle state      200204 (see also Spin Poincare Lie
Angular momentum,      147 156 175 205 596 Poincare Lie
Angular momentum, complex      227 (see also Spin Poincare Lie
Angular momentum, orbital      189 516 Poincare Lie
Anomalous commutation relations      474475
Anticommutation relations canonical      566 572 577
Anticommutation relations of $\gamma$-matrices      171
Anticommutation relations of field with creation-annihilation operators      403 410
Anticommutation relations of nonlocal scalar field      511512
Anticommutation relations, discrete transformations and      322
Antiparticles      205208 322 515 517
Antisymmetric unit tensor $(\epsilon_{jkl}, \epsilon_{\mu\nu\lambda\sigma})$      133134 146147
Antiunitary operator      110 139 140 164165 167 241 298 322 515
Antoine, J. P.      240 633
Araki, H.      281 351 466 523 545 573 575 576 604 610 624 625 633 634
Arbuzov, B. A.      499 634
Artin, A.      241 627
Associative algebra      584
Asymptotic completeness      358 378380
Asymptotic completeness,      406 523
Asymptotic conditions      3 357358 362363 365367 381384 389-390 465
Asymptotic fields      358 359 366368 381382 422 430432 465 549
Asymptotic fields, relative weak locality and      5 20
Asymptotic fields, TCP transformation of      514515
Asymptotic states      358 365368 385386 402407 515-517
Asymptotically abelian algebra      625
Automorphism(s)      136 139140 210 596
Automorphism(s), inner, outer      136
Automorphism(s), space-time, in $\Phi^{4}$ model      616618 625
Axial vector      146 154 344 500501
Axiomatic approach      15 362363 460 612 BMP LSZ Algebraic
Axioms, enumerated      117118 142 151 152 248252 256 379
B* -algebra      585 (see also C*-algebra)
Banach algebra (Banach ring)      584585
Banach space      15
Bardakci, K.      466 575 634
Bargmann Hall Wightman theorem      473477 481486 513 518 533 548 574
Bargmann, V.      574
Bargmanns proof, of Wigners theorem      234239 242
Bargmanns theorem      141142
Barton, G.      464 627
Baryon number      122 153 206 352 379 406 600
Baryons      151 406 Proton).
Baumann, K.      351 465 635 669
Belinfante, F. J.      241 635
Bender, C. M.      469 635
Berezansky, Yu. M.      105 608 625 627 635
Berezin, F. A.      577 627 635
Bessel functions      70 81 335 370
Bialynicki Birula, I.      540 636
Bibliographical conventions      6 632633
Bicompactness      213
Bilenky, S. M.      575 636
Bilinear form      (see Functional bilinear Scalar
Bilinear invariant forms of Dirac equation      178 322323 499501
Bilinear invariant forms of Majorana representations      567568
Bispinor      (see Spinor four-component)
Bjorken, J. D.      2 8 460 468 627
Blokhintsev, D. I.      351 508 636
BMP approach      3 400422 430432 450453 465468 525
Bochner Schwartz theorem      269 271
Bochner, S.      495 627
Bogolubov transformation      (see Canonical transformation)
Bogolubov, N. N.      xvii xviii 3 8 72 102 106 146 171 186 241 242 322 333 336 352 399 401 405 409 414 422 429 445 455 465 466 467 468 556 561 574 577 625 628 636 637
Bohm, A.      240 637
Bohr, N.      350 637
Bollini, C.      468 638
Boost      (see Lorentz transformation pure)
Borchers classes      474 517525 575 583 604 609
Borchers classes and      519520 (see also Irreducibility of
Borchers Zimmermann criterion      603 607608
Borchers, H. J.      257 259 292 297 300 350 351 352 517 575 606 607 625 638
Born term      463
Bose operator      274
Bosons, Bose Einstein statistics      194 252 402403 526 564 572
Bound states      465
Boundedness of linear functional      1819
Boundedness of operator      3435 116
Boyce, J. F.      242 672
Bremermann, H. J.      467 638
Brenig, W.      363 464 639
Broken symmetry      475 562563 577
Bros, J.      106 415 467 574 639
Bruhat, F.      252 639
Burgoyne, N.      576 639
C*-algebra      4 581611 614 616 624625
C-convex envelope      508
Cannon, J. T.      617 625 639
Canonical basis, in $d_{\chi}$      340343 564 570
Canonical commutation relations for infinite-component field      566 (see also Anticommutation relations Commutator)
Canonical commutation relations for position and momentum      118 195 Commutator)
Canonical commutation relations for scalar field      305 311 465 557561 Commutator)
Canonical commutation relations in $\Phi^{4}$ model      617 (see also Anticommutation relations Commutator)
Canonical commutation relations in discrete basis      558560 (see also Anticommutation relations Commutator)
Canonical commutation relations inequivalent representations      548 556561 577 Commutator)
Canonical commutation relations of creation-annihilation operators      304305 (see also Anticommutation relations Commutator)
Canonical commutation relations, singular      561 (see also Anticommutation relations Commutator)
Canonical commutation relations, strange representations      475 560561 Commutator)
Canonical commutation relations, Weyl form      559 569 Commutator)
Canonical coordinates      220 227232
Canonical formalism      (see Lagrangian formalism Hamiltonian Canonical
Canonical transformation, linear      559560
Cantor, G.      50 294
Cartan, E.      225
Cartan, E. J.      241 628
Cartesian product      30
Casimir operators      225 (see also Poincare' group invariants Lorentz Casimir
Cauchy inequality      588
Cauchy principal value      5
Cauchy sequence      (see Fundamental sequence)
Causal distribution      (Distributions causal Green causal)
Causal envelope      597 603604
Causal Green function      (see Green functions causal)
Causal shadow      597
Causality      2 247 414 465
Causality in algebraic approach      597
Causality, analyticity and      432433 476
Causality, primitive      258259 597 Microcausality)
Cayley parametrization      479480
Center of algebra      589
Center-of-mass frame      (see Lorentz transformation into
Characteristic subgroup      210
Charge conjugation      131 179180 183 192 206 320323 325326 352 529 568
Charge conjugation in homogeneous distribution formalism      344345
Charge conjugation, space reflection and      325326
Charge renormalization      613
Charge, electric      122 153 206207 250 311318 596 599 600
Charged field      (see Complex field)
Chen, T. W.      460 468 639
Chernikov, N. A.      576 639
Chou Kuang Chao      242 327 640 664
Circumflex, denoting omission      408
Classical theory, algebra of      597
Clifford algebras      241 499 Anticommutation
Closed graph theorem      35
Closed operator      35
Closed subspace      216217
Closure, of operator      37
Cluster decomposition property      151 272282 293 351 531 576
Coherent subspace (Superselection sector)      123127 139 142 250251 312 602 605 607 609
Coherent subspace (Superselection sector), discrete transformations and      326327 (see also Superselection rule)
Commutant      299300 589 601605
Commutation relations      (see Commutator Anticommutation Canonical Anomalous Spin-and-statistics
Commutator in group      221
Commutator in Lie algebra      221222
Commutator in parafield theory      540546
Commutator Kaellen Lehmann representation      330331
Commutator of current with field      417418
Commutator of field with creation-annihilation operators      403
Commutator of free scalar field [D(x)]      302303 328 333335 382 418 554556
Commutator of functional differentiations      409
Commutator of generalized free field      328331
Commutator of Heisenberg and asymptotic fields      465
Commutator of S-operator with creation-annihilation operators      409412
Commutator of vector field      561 (see also Canonical commutation relations Anticommutation
Compact energy, vector of      605606 617
Compact group      143 213 219 225226 600
Compact support, functions of      (see \mathcal{D})
Compactness      213
Compactness, Haag Swieca      573
Compactness, relative      24
Compactness, weak      618
Compatible norms      2122
Compatible topologies      27
Complement, of invariant subspace      217
Complete set, of operators      152153 451452
Complete space      15
Completeness, of field theory      (see Cyclicity)
Completion, in norm      25 33
Complex (charged) field      250 283284 311314 610
Complex conjugation, antiunitary operators and      164
Composite models      379
Cone      74 151 440 488 494
Cone of positive functionals $(\alpha^{*}_{+})$      592 594
Cone, double      606
Conjugate space      1921 2629 105 598
Conjugate, of spinor (Dirac conjugate)      178 183 186
Conjugation      (see Involution)
Connectedness      213
Connectedness, simply      135 213
Constructive field theory      (see Models)
Continuity      212
Continuity of functional      1819 25
Continuity of operator in nuclear space      119120
Continuity of representation      216219
Continuity, absolute      37
Continuity, weak, of field      249
Continuity, weak, of representation      216
Continuous group      (see Topological group)
Continuous spin      150
Convergence      212
Convergence in $\mathcal{S}$ and $\mathcal{S}^{*}$      23 29
Convergence in $\Omega_{0}$      287288
Convergence in $\Omega_{1}$      21 25 6
Convergence in C*-algebra      585586
Convergence in conjugate of countably normed space      2829
Convergence in countably normed space      22
Convergence in norm (uniform)      17 585587
Convergence in normed space      15 17
Convergence in nuclear space      32
Convergence, strong      15 28 32 586 587 615
Convergence, weak      28 32 216 381 586 598 616
Convex set      282
Convolutes $(\Theta_{C^{'})$      7374 375
Convolution      7276 95 104 456459
Convolution multiplication and      73 93 99100
Coordinates, generalized      118
Coulomb law      281
Countably normed space      (see Vector space countably
Coupling constant      450451 455 469 614
Covariance in algebraic approach      596 609
Covariance of analytic functions      473497 574
Covariance of fields      249251 (see also Relativistic invariance SL(2) representations)
Covering group      135
Covering, by open sets      49 213
CP transformation, CP violation      498499 501 Parity)
Creation and annihilation operators      118121 304308 312 315318 320321 336 404 577 613
Creation and annihilation operators for parafield      539544
Creation and annihilation operators in x-space      309 619
Creation and annihilation operators, asymptotic      389390 403
Creation and annihilation operators, commutation relations with fields and S-operator      403 409410
Creation and annihilation operators, majorana representations and      568570 578
Creation and annihilation operators, normal commutation relations, derived      528
Creation and annihilation operators, notation      389
Current(s)      401402 413419 421 441445 577
Current(s) as derivative of S-operator      3 359 401 413 415 430432
Current(s) in Borchers class of field      521522
Current(s) in Yang Feldman equations      358 382 415 417
Current(s), algebras of      402
Current(s), electric      322323 382 413
Current(s), Kaellen Lehmann representation      432
Current(s), reconstruction theorem for      466
Current(s), satisfy axioms      432 514
Current(s), TCP transformation of      514
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