The Global Approach to Quantum Field Theory 2-Volume Set

by DeWitt, Bryce

• ISBN13: 9780198510932
• ISBN10: 0198510934
• Format: Hardcover
• Copyright: 2003-05-01
• Publisher: Clarendon Press

Summary

This new volume takes a complete look at how classical field theory, quantum mechanics and quantum field theory are interrelated. It takes a global approach and discusses the importance of quantization by relating it to different theories such as tree amplitude and conservation laws. There are special chapters devoted to Euclideanization and renormalization, space and time inversion and the closed-time-path formalism.

Table of Contents

I CLASSICAL DYNAMICAL THEORY
Fundamentals	3	(10)
Classical Dynamical Systems	3	(1)
Space of Histories, Functional Differentiation	4	(2)
Examples of Simple Functionals. Notational Abbreviations	6	(3)
Condensed and Supercondensed Notations	9	(2)
Changing Topology	11	(1)
Comments on Chapter 1	12	(1)
Reference	12	(1)
Dynamics and Invariance Transformations	13	(17)
Action Functional and Dynamical Equations	13	(1)
Invariance Transformations	14	(3)
Commutator of Invariance Transformations	17	(1)
Gauge Algebras. Gauge Groups. Orbits	18	(4)
Functional Differential Equation for the Structure Functions	22	(2)
Further Simplification of Notation	24	(1)
Physical Observables	25	(1)
Gauge Groups and Manifest Covariance	26	(2)
Comments on Chapter 2	28	(1)
References	29	(1)
Small Distrubances and Green's Functions	30	(16)
The Equations of Small Disturbances. Jacobi Fields	30	(1)
The Structure of i, S,j	31	(1)
Other Identities Satisfied by the Coefficient Matrices	32	(1)
Supplementary Conditions	33	(2)
Retarded and Advanced Green's Functions of F	35	(1)
Equality of Left and Right Green's Functions	36	(1)
Reciprocity Relations	37	(1)
Explicit Structure of F	38	(2)
Relation between the Green's Functions of J and F	40	(2)
The Green's Functions Off Shell. Coherent Green's Functions	42	(2)
Landau Green's Functions	44	(1)
Transformation Laws for the Green's Functions	44	(1)
Comments on Chapter 3	45	(1)
The Peierls Bracket	46	(14)
Disturbances in Physical Observables	46	(1)
The Reciprocity Relation for Physical Observables	47	(1)
Invariance Properties of DAB	47	(2)
The Peierls Bracket and the Supercommutator Function	49	(1)
The Bracket Identities	50	(1)
Standard Canonical Systems. Equivalence of Peierls and Poisson Brackets	51	(3)
The Wronskian Operator	54	(2)
The Cauchy Problem for Jacobi Fields	56	(2)
Comments on Chapter 4	58	(1)
Reference	59	(1)
Finite Disturbances. Tree Theorems. Asymptotic Fields	60	(18)
Differential and Integro-differential Equations for Finite Disturbances	60	(3)
Iterative Solutions. Tree Functions	63	(1)
Background Field. Transformation Laws	64	(2)
Changes in P and k	66	(1)
Gauge Transformations	67	(1)
Tree Amplitudes and Tree Theorems	67	(2)
Tree Theorems When the Configuration Space is not a Vector Space	69	(1)
Change of Chart	70	(3)
Asymptotic ``In'' and ``Out'' Fields	73	(4)
Comments on Chapter 5	77	(1)
Reference	77	(1)
Conservation Laws	78	(39)
Alternative Action Functional	78	(1)
Internal Sources	78	(1)
Killing Fields, Currents, and Charges	79	(2)
The Charge Group	81	(1)
Flow Invariance of the Charges	82	(1)
Fundamental Lemma	83	(2)
A Special Transformation	85	(2)
The Peierls Bracket (qA, qB). The View from the Remote Past or Future	87	(3)
The View from Spatial Infinity	90	(3)
Compact Universes and Linearization Constraints	93	(1)
Pseudocovariants	94	(1)
Externally Impressed Fields	95	(1)
Charges in the Presence of External Fields	96	(2)
Peierls Bracket for the Charges	98	(1)
Differences form the Previous Theory	98	(3)
External Gravitational Field. Diffeomorphism Gup	101	(2)
Stress-energy Density	103	(4)
Killing Vector Fields. The Poincare Group	107	(1)
Energy-momentum Vector and Angular Momentum Tensor	108	(1)
Local Conformal Group	109	(1)
Conformal Killing Vector Fields. The Global Conformal Group	110	(1)
Conformal Charges and their Peierls Brackets	111	(1)
Comments on Chapter 6	112	(1)
References	113	(4)
II THE HEURISTIC ROAD TO QUANTIZATION. THE QUANTUM FORMALISM AND ITS INTERPRETATION
Classical Theory of Measurement	117	(14)
System. Apparatus. Coupling	117	(1)
Apparatus Inertia	118	(2)
Disturbance in the Apparatus	120	(1)
Design of the Coupling. Accuracy of the Measurement	121	(1)
Compensation Term	122	(1)
Stern-Gerlach Experiment	123	(2)
Electric Field Measurement	125	(5)
Comments on Chapter 7	130	(1)
Reference	130	(1)
Quantum Theory of Measurement	131	(14)
Measurement of Two Observables	131	(1)
Mutual Uncertainties	132	(1)
A Universal Uncertainty Principle	133	(1)
The Heuristic Quantization Rule	134	(2)
A Preferred Basis	136	(2)
Relative States. Good Measurements	138	(2)
Many Worlds	140	(1)
Unobservability of the Splits	141	(3)
Comments on Chapter 8	144	(1)
References	144	(1)
Interpretation of the Quantum Formalism I	145	(17)
Reality	145	(1)
Signalling by Permutations	146	(1)
Equal Likelihood	147	(2)
Unequal Probabilities	149	(1)
Irrational Probabilities	150	(2)
Expectation Value. Single System vs. an Ensemble	152	(1)
Relative Frequency Function. The Ensemble Interpretation of Quantum-mechanical Probabilities	153	(3)
Density Operator. Decoherence	156	(3)
Imperfect Measurements	159	(2)
Comments on Chapter 9	161	(1)
Reference	161	(1)
The Schwinger Variational Principle and the Feynman Functional Integral	162	(16)
Transition Amplitudes	162	(1)
The Schwinger Variational Principle	162	(3)
External Sources and Chronological Products	165	(3)
The Operator Dynamical Equations and the Measure Functional	168	(3)
Functional Fourier Analysis. The Feynman Functional Integral	171	(2)
Comments on the Feynman Integral	173	(2)
The Schwinger Variational Principle Revisited	175	(1)
Expressions Involving S,i	176	(1)
Comments on Chapter 10	177	(1)
References	177	(1)
The Quantum Mechanics of Standard Canonical Systems	178	(17)
Problems with the Naive Quantization Rule	178	(1)
The Position Mapping	178	(2)
Vector Operators	180	(3)
The Momentum Operator	183	(1)
Restriction to a Local Chart	184	(2)
Overlapping Charts. Transformation of Coordinates	186	(2)
The Position Representation	188	(1)
The Projection m-form	189	(1)
The Momentum Operator in the Position Representation	190	(2)
The Schrodinger Equation	192	(2)
Remarks on Global Consistency	194	(1)
Comments on Chapter 11	194	(1)
Reference	194	(1)
Interpretation of the Quantum Formalism II	195	(18)
Tracing Out	195	(1)
The Model System	196	(2)
Density Operator	198	(1)
Localization. Sharp Decoherence	199	(2)
Discussion	201	(1)
Coarse Graining. Decoherence Function	202	(1)
Interpretation of the Diagonal Elements	203	(2)
Emergence of Classicality	205	(2)
Many Worlds Again. Prbability as a Phenomenological Concept	207	(2)
Comments on Chapter 12	209	(1)
References	209	(4)
III EVALUATION AND APPROXIMATION OF FEYNMAN FUNCTIONAL INTEGRALS
The Functional Integral for Standard Canonical Systems	213	(21)
The Path Integral	213	(1)
The Point-to-point Amplitude. The Action Function	213	(3)
Formal Computation of det G[x] in the Lagrangian Formalism	216	(1)
Formal Computation of det G[x, p] in the Hamiltonian Formalism	217	(2)
Ambiguity in the Path Integral	219	(1)
Homotopy	220	(1)
The Universal Covering Space. Covering Translations	221	(2)
The Relation of Homotopy to Homology and Cohomology	223	(2)
The Path Integral in C	225	(2)
Fundamental Domains	227	(1)
Partial Amplitudes in C. The Total Amplitude	228	(1)
Combination Law	229	(1)
The Hamiltonian Form of the Path Integral and the Schrodinger Equation	230	(2)
Homotopy in Quantum Field Theory	232	(1)
Comments on Chapter 13	233	(1)
References	233	(1)
Approximation and Evaluation of the Path Integral	234	(17)
The Van Vleck-Morette Determinant	234	(2)
Jacobi Fields and Green's Functions for the Trajectory xc	236	(3)
Determinantal Relations	239	(2)
The Loop Expansion	241	(2)
The WKB Approximation	243	(3)
Other Boundary Conditions	246	(1)
WKB Approximation	247	(3)
Comments on Chapter 14	250	(1)
References	250	(1)
The Nonrelativistic Particle in a Curved Space	251	(24)
From the Hamiltonian Path Integral to the Lagrangian Path Integral	251	(1)
The Nonrelativistic Particle in a Curved Space	252	(1)
Covariant Variation	253	(1)
Covariant Differentiation with Respect to t	254	(1)
The Dynamical Equations	255	(1)
Covariant Functional Differentiation	255	(2)
The Measure for the Lagrangian Path Integral	257	(2)
Computation of HQ	259	(3)
Normalization of the Path Integral	262	(1)
A Two-loop Calculation	263	(2)
Analysis of the Jacobi Field Operator	265	(2)
The Morse Index	267	(1)
Morse Index Theorem	268	(3)
Application to Path Integration	271	(1)
The Generalized Morse Index	272	(2)
Comments on Chapter 15	274	(1)
References	274	(1)
The Heat Kernel	275	(16)
History	275	(1)
Geodesic Normal Coordinates	275	(2)
Coincidence Limits	277	(2)
Caustics	279	(2)
Riemannian Connection. The World Function	281	(2)
Auxiliary Geometrical Quantities. Modified Covariant Derivative	283	(1)
Further Coincidence Limits	284	(1)
Heat-kernel Expansion. Recursion Relations	285	(1)
Comments on Chapter 16	286	(1)
References	287	(4)
IV LINEAR SYSTEMS
Linear Boson Fields in Stationary Backgrounds	291	(29)
The Scalar Field	291	(1)
Stationary Backgrounds	292	(1)
The Field Equations	293	(2)
The Energy	295	(1)
Energy Bounds	296	(1)
Mode Functions	297	(1)
Alternative Representation	298	(2)
Multiple Roots	300	(1)
Wronskian Relations	301	(1)
Zero Roots	302	(2)
Wronskian Relations Again	304	(3)
General Solution. Supercommutator Function. Energy	307	(1)
Matrix Identities	307	(2)
Mode Functions for the Scalar Field	309	(1)
The Massless Scalar Field in a Compact Universe	310	(2)
The Vector Field	312	(3)
Rescaling of Time and Energy. Canonical Form for the Energy	315	(3)
Conformally Invariant Scalar and Vector Fields	318	(1)
Comments on Chapter 17	319	(1)
Quantization of Linear Boson Fields	320	(21)
Green's Functions	320	(1)
Quantization	321	(1)
Super Hilbert and Fock Spaces	322	(5)
Nonstationary Backgrounds and Inequivalent Vacua	327	(1)
Nonuniqueness of ∂/∂x	328	(2)
Vacua Defined by Symmetry Properties	330	(1)
The Feynman Propagator	330	(2)
Feynman Propagators for the Scalar and Vector Fields	332	(3)
When the Energy is Unbounded from Below	335	(2)
When K-1 M is Fully Diagonalizable and has Pure Imaginary Eigenvalues	337	(2)
Comments on Chapter 18	339	(1)
References	340	(1)
Linear Fermion Fields. Stationary Backgrounds	341	(26)
Local Lorentz Frames	341	(1)
The Dirac Matrices	342	(2)
Spin Structures	344	(1)
Space Inversions and Pin Structures	345	(2)
Spinor Fields	347	(2)
The Spin Connection	349	(1)
Generalized Spin or Pin Structures	350	(1)
Real Representations	351	(1)
Lagrange Function. Conformal Invariance. Majorana Representation	351	(2)
Stress-energy Density	353	(2)
Leibniz Rule. Matrix Identities. Canonical Form for the Energy	355	(2)
Energy-momentum Conservation and the Poincare Group	357	(2)
The Dirac Operator	359	(1)
Stationary Backgrounds. A Special Field of Local Lorentz Frames	360	(2)
A Model Stem	362	(1)
Mode Functions	362	(3)
When ---iB is not Positive Definite	365	(1)
Comments on Chapter 19	365	(1)
References	366	(1)
Quantization of Linear Fermion Fields	367	(10)
Green's Functions	367	(1)
Quantization	367	(1)
Fock Space	368	(2)
Hole Theory	370	(2)
The Feynman Propagator	372	(4)
Comments on Chapter 20	376	(1)
Linear Fields in Nonstationary Backgrounds	377	(19)
``In'' and ``Out'' Regions	377	(2)
Bogoliubov Relations	379	(2)
``In'' and ``Out'' Fock Spaces and the S-matrix	381	(1)
Particle Production and Annihilation Amplitudes	382	(3)
One-particle Scattering Amplitudes	385	(1)
Nonsingularity of α	386	(1)
Unitarity. The Vaccum Persistence Amplitude	387	(2)
Unitarily Inequivalent Fock Spaces	389	(2)
Green's Function Representation of eiW. The Feynman Propagator	391	(2)
Functional Integral Representation of eiW	393	(1)
Comments on Chapter 21	394	(1)
References	395	(1)
Linear (or Linearized) Fields Possessing Invariant Flows	396	(13)
The Electromagnetic Field in a Stationary Curved Background	396	(1)
Other Linear or Linearized Fields on Stationary Backgrounds	397	(2)
Mode Functions for F and F	399	(2)
Mode-function Decompositions of G and k	401	(1)
Quantization	402	(1)
Fields on Nonstationary Backgrounds	402	(2)
Green's Function Representation of eiW	404	(2)
Comments on Chapter 22	406	(3)
V NONLINEAR FIELDS
The Effective Action, the S-matrix, and Slavnov-Taylor Identities	409	(47)
Anticipation of Counter Terms	409	(1)
Correlation Functions	410	(1)
The Effective Action	411	(2)
The Relation of Γ to W. The Legendre Transform	413	(1)
The Correlation Functions as Tree Functions	413	(1)
Structure of the Effective Action	414	(2)
The Loop Expansion	416	(3)
Real-valuedness of the Counter Terms. Perturbative Renormalizability	419	(2)
Asymptotic Fields	421	(1)
Asymptotic States	422	(2)
Wave-packet States	424	(2)
The S-matrix and the Scattering Operator	426	(1)
The Lehmann-Symanzik-Zimmermann Theorem	427	(3)
Mode Functions for the Effective Action	430	(2)
The Effective Action as the Generator of Quantum Dynamics	432	(1)
Coherent States as Relative Vacua	433	(3)
Evaluation of (+, rel vac-, rel vac)	436	(2)
Expression of the S-matrix in Terms of Quantum Tree Functions	438	(1)
The Cluster Decomposition Principle	439	(1)
Use of a (Super)Classical Background	440	(2)
Construction of the Relative S-matrix	442	(2)
Structure of the Relative S-matrix	444	(2)
When the Configuration Space is Not a Supervector Space	446	(3)
Relation to the Classical Tree Theorems	449	(1)
Slavnov-Taylor Identities	450	(2)
Current Algebra	452	(1)
Relation to the Effective Action. Finiteness of the Current Operator	453	(1)
Comments on Chapter 23	454	(1)
References	455	(1)
Gauge Theories I. General Formalism	456	(36)
Structure of the Space of Field Histories	456	(1)
Fibre-Adapted Coordinated Patches	457	(2)
A Metric for Φ/G	459	(1)
Vilkovisky's Connection	460	(2)
Properties of Vilkovisky's Connection	462	(1)
The Functional Integral for ``In-Out'' Amplitudes	463	(2)
Properties of the Jacobian J[Y]	465	(2)
A Special Choice for Ω and a New Measure Functional	467	(1)
Explicit Form for the New Measure	468	(2)
Consistency of Eq. (24.80)	470	(1)
Ghosts	471	(2)
Relation to the Functional G. The Batalin---Vilkovisky Equation	473	(1)
Counter Terms, the Measure Functional, and the Quantum BV Equation	474	(2)
The Slavnov Operator and BRST Transformations	476	(1)
Cohomology of the Slavnov Operator	477	(1)
Loop Decomposition of the Measure	478	(2)
The Role of the Measure Functional	480	(2)
The Effective Action	482	(2)
The Zinn-Justin Equation. Proof of Eq. (24.125)	484	(1)
The Yang-Mills Field in Four Dimensions	485	(3)
Rigidity of the Gauge Group	488	(1)
Renomalization Constants	489	(1)
Comments on Chapter 24	490	(1)
References	491	(1)
Gauge Theories II. Background Field Methods. Scattering Theory	492	(21)
Invariants. The Quantum Slavnov Operator	492	(1)
Integrating Out the Ghosts	493	(1)
Reduced Effective Action	494	(1)
Introduction of a Background Field	495	(1)
Gauge Fixing	496	(1)
Loop Expansion of the Reduced Effective Action	496	(2)
Structure of the Reduced Effective Action	498	(2)
The Full Quantum Shell	500	(1)
Alternative Loop Expansions	500	(1)
The S-matrix	501	(2)
Mode Functions for the Effective Action	503	(2)
S-matrix Relative to an Arbitrary Background	505	(2)
Use of the (Super)Classical Background	507	(2)
A Special Phenomenon	509	(1)
Use of Γ[y] in Constructing the S-matrix	510	(1)
Comments on Chapter 25	511	(1)
Reference	512	(1)
Case-I Gauge Theory without Ghosts. Description of Cases II and III	513
Geodesic Normal Fields	513	(1)
A New Effective Action	514	(2)
The Illusory Ghost	516	(2)
The Loop Series	518	(1)
Rules for Differentiating the Measure	519	(1)
Invariance of the Loop Graphs	520	(2)
Renormalization	522	(1)
Difficulties in Applying the LSZ Theorem	523	(1)
Case-II Systems	524	(1)
Uncertain Cohomology of the Slavnov Operator	525	(1)
Decomposition of the Effective Action	525	(1)
Scattering Theory	525	(1)
Case-III Systems	526	(2)
Comments on Chapter 26	528	(1)
Reference	528
Index	1
Volume 2
VI Tools for Quantum Field Theory. Applications
The Heat Kernel	531	(1)
Formal Representation of the Feynman Propagator in the Nonstationary Case	531	(1)
Schwinger's Representation and the Heat Kernel	532	(1)
Some Technical Details. Coincidence Limits	533	(1)
Geodetic Parallel Displacement	534	(2)
The Cartan-Killing Metric	536	(1)
Application to the Lorentz Group	537	(3)
Traces	540	(1)
The Heat-kernel Expansion	541	(1)
Expansion for the Feynman Propagator	542	(2)
Difficulties in the Nonflat Case	544	(2)
Heat-kernel Representation of eiW	546	(1)
Dimensional Regularization	547	(2)
δW for the Spinor Field	549	(2)
Heat Kernel for the Spinor Field	551	(1)
Dimensional Regularization in the Spinor Case	552	(2)
Zeta Function Regularization	554	(2)
Zeta Function Representation of W	556	(2)
Electromagnetic Background	558	(1)
Comments on Chapter 27	558	(1)
References	559	(1)
Vacuum Currents. Anomalies	560	(1)
Formal Invariances of the Vacuum Persistence Amplitude	560	(1)
The Trace Anomaly	561	(3)
Alternative Derivation of the Trace Anomaly	564	(1)
Conformal Invariance of the Finite Local Terms	565	(1)
Trace Anomaly for the Massless Spinor Field	566	(2)
Chiral Coupling	568	(2)
Traces	570	(2)
Gauge Invariance of the Classical Theory	572	(1)
Vaccum Persistence Amplitude and Heat Kernel	572	(2)
Asymptotic Expansion	574	(1)
The Zeta Function and its Gauge Transformation Law	575	(2)
The Chiral Anomaly	577	(1)
Robustness of the Chiral Anomaly	578	(2)
Computation of the Chiral Anomaly	580	(2)
The Wess-Zumino Consistency Condition	582	(2)
Comments on Chapter 28	584	(1)
References	585	(1)
More Vacuum Phenomena	586	(1)
Particle Production by Weak Backgrounds	586	(1)
Explicit Expressions for Tμν (u~, u)o and jαμ(u~, u)o	587	(2)
Average Number of Particles Producted	589	(3)
Energy Production	592	(1)
No Particle Production by Freely Propagating Linear Background Waves	592	(1)
Expectation Values of the Current Operators	593	(1)
Physical Renormalization of the Current Operators	594	(1)
The Case of the Scalar Field. Lack of Uniqueness when n=4	595	(3)
The Spinor Field. Massless Fields	598	(1)
Conformally Invariant Fields in Conformally Flat Spacetimes	599	(1)
Conformally Invariant Scalar Field	600	(3)
De Sitter and Anti-de Sitter Spacetimes	603	(1)
Einstein Static Spacetime	604	(1)
Alternative Vacua. Rindler Mode Functions	605	(3)
Stress Energy in the Rindler Vacuum	608	(1)
Connection Between Rindler and Minkowski Quanta. Bogoliubov Coefficients	609	(2)
Inverse Bogoliubov Coefficients	611	(1)
Formal Pair Production and Annihilation Amplitudes	612	(1)
Unitary Inequivalence of Minkowski and Rindler Folk Spaces	613	(1)
Thermality of the Minkowski Vacuum Relative to Rindler Fock Space	614	(3)
Properties of the Rindler Gas	617	(1)
Thermal Wightman Function	618	(2)
Detectors	620	(2)
Accelerating Detectors. The Unruh Effect	622	(2)
Comments on Chapter 29	624	(1)
References	624	(1)
Black Hole Vacua. Hawking Radiation	625	(1)
Scalar Field in a Black-hole Metric	625	(2)
Change of Coordinates. Mode Functions	627	(2)
Normalization	629	(1)
Kruskal Coordinates	630	(2)
Extended Mode Functions	632	(2)
Mode Functions in Region II	634	(1)
Mode Functions with ω2 < μ2	635	(1)
Analytic Continuation in ω	635	(2)
Quasi-bound States	637	(1)
Zeroes of α (l, ω)	637	(2)
The Boulware Vacuum	639	(1)
The Hartle-Hawking Vacuum	640	(1)
Properties of the Hartle-Hawking Vacuum	641	(2)
Computation of (Trenμν) for Large r	643	(3)
Reactions of a Monopole Detector to the Hartle-Hawking and Boulware Vacua	646	(1)
Black Holes Formed by Collapse	646	(1)
``In'' Mode Functions	646	(3)
``Out'' Mode Functions	649	(2)
Bogoliubov Coefficients	651	(3)
Pair Production	654	(1)
Wightman Function	655	(1)
Hawking Radiation	656	(1)
Temperature and Entropy of a Black Hole	657	(1)
Black-hole Luminosity	657	(1)
Decay and Ultimate Fate of Black Holes	658	(2)
Comments on Chapter 30	660	(1)
References	660	(2)
The Closed-time-path or ``In-in'' Formalism	662	(1)
Functional Integral for Expectation Values	662	(2)
Properties of the System Described by the Action SY	664	(1)
The Jacobi Field Operator. Mode Functions	665	(2)
Wronskian Relations and Supercommutator Function	667	(1)
Other Green's Functions	668	(2)
Return to the Fields Y+ and Y-	670	(2)
External Sources. Causality	672	(1)
The Measure	673	(1)
The ``In-In'' Effective Action	674	(2)
``In-in'' Formalism for Gauge Theories	676	(1)
Connection 1-form and Horizontal Projection Operator	677	(1)
Vilkovisky's Connection	678	(1)
Geodesic Normal Fields	679	(1)
The Effective Action	680	(1)
Alternative Propagator	681	(1)
The Measure	682	(1)
Discussion	683	(1)
Comments on Chapter 31	684	(1)
References	685	(4)
Special Topics
Euclideanization and Renormalization	689	(1)
Momentum Space	689	(1)
Vertex Functions	690	(1)
Graphical Rules in Momentum Space	691	(1)
Contribution of the Measure in One-loop Graphs	692	(4)
Wick Rotation	696	(1)
Euclideanization	697	(1)
Boundary Conditions	698	(1)
Failure of Euclideanization in the Case of Quantum Gravity	699	(1)
Graphs and Subgraphs	700	(1)
The Pole Operation	701	(1)
Counter Terms	702	(2)
Minimal Subtraction and Finiteness	704	(2)
Comments on Chapter 32	706	(1)
References	706	(1)
Canonical Transformations. Space Inversion and Time Reversal	707	(1)
Infinitesimal Canonical Transformations	707	(1)
Finite Canonical Transformations	708	(1)
Space Inversion	709	(2)
Examples of Systems Invariant Under Space Inversion	711	(1)
Quantum Theory	712	(1)
Pseudoscalar Field	713	(1)
Massive Vector Field in Four Dimensions	714	(1)
Massive Antisymmetric Field in Four Dimensions	714	(1)
Massive Spinor Field in Four Dimensions	714	(3)
Time Reversal	717	(1)
Transformation of the Jacobi Field Operator	718	(2)
Quantum Theory	720	(1)
Chiral Couplings	721	(1)
Time Reversal of Asymptotic Fields	721	(2)
Massive Antisymmetric Field in Four Dimensions	723	(1)
Massive Spinor Field in Four Dimensions	723	(2)
Massless Fields	725	(1)
CP Invariance for Massless Fields	726	(1)
The Relation Between Helicity, Chirality, and Charge	727	(2)
CP Invariance for Massive Fields	729	(1)
The PT Theorem for Even-dimensional Spacetimes	729	(1)
Comments on Chapter 33	729	(2)
Quantum Electrodynamics	731	(1)
Action and Gauge Group	731	(1)
Jacobi Field Operator	732	(1)
One-loop Effective Action	733	(1)
Electrodynamic β-function	734	(1)
Constant Fields	735	(2)
Computation of the Trace	737	(1)
Renormalization. Scattering of Light by Light	738	(2)
Pair Production	740	(2)
Effective Field Equations	742	(3)
Vacuum Polarization	745	(1)
Mass Operator	745	(2)
Anomalous Magnetic Moment	747	(5)
Atomic Level Shifts	752	(2)
Comments on Chapter 34	754	(1)
References	754	(1)
The Yang-Mills and Gravitational Fields	755	(1)
Action and Gauge Group for the Yang-Mills Field	755	(1)
Classical Field Equation and Jacobi Field Operator	756	(1)
The Operators &J and F and the Invariant Metric	757	(1)
Traces of the a2 Coefficients and the One-loop Effective Action	758	(1)
Yang-Mills β-function and Asymptotic Freedom	759	(1)
Stress-energy Density of the Yang-Mills Field	760	(1)
Yang-Mills Trace Anomaly	761	(2)
Constant Fields	763	(3)
Evidence for Confinement	766	(1)
Yang-Mills Charge	767	(2)
Careful Definition of G and Φ	769	(3)
Axial Gauge	772	(1)
Homotopy of Gauge Transformations. Cartan-Maurer Invariant	773	(1)
Evaluation of I ([D])	774	(4)
Chern-Simons Secondary Characteristic	778	(2)
Chern Number	780	(1)
The Total Amplitude. Instantons	781	(2)
Action and Gauge Group for the Gravitational Field	783	(1)
Classical Field Equation and Jacobi Field Operator	784	(1)
The Operators J and F and the Invariant Metric	785	(2)
Traces of the a2 Coefficients and the One-loop Effective Action	787	(1)
One-loop Finiteness	788	(1)
Gravitational Stress-energy Density	789	(1)
Energy, Momentum, and Angular Momentum in the Gravitational Field	790	(2)
Topological Questions	792	(1)
Coupled Yang-Mills and Gravitational Fields	792	(2)
Comments on Chapter 35	794	(1)
References	795	(4)
Examples. Simple Exercises in the use of the Global Formalism
The Nonrelativistic Particle in Flat Space	799	(1)
Action Functional and Green's Functions	799	(1)
Momentum and Energy	800	(1)
The Case Ji = O	801	(1)
The Transition Amplitude in the General Case	802	(3)
The Momentum-to-momentum Transition Amplitude	805	(1)
Variation of the Sources	806	(1)
A Simple Fermi System	807	(1)
Action Function and Green's Functions	807	(1)
Eigenvectors of y	808	(1)
The Energy	808	(1)
A Pure Basis	809	(1)
An Alternative Representation	810	(1)
The Functional Integral Representation of (y, ty1, 1)	811	(3)
Evaluation of the Functional Integral	814	(3)
Other Transition Amplitudes	817	(2)
The Average (Super)Classical Trajectory	819	(1)
Propagator of the Fermi System	820	(2)
A Fermi doublet	822	(1)
Action Function and Green's Functions	822	(1)
A Symmetrical Basis	823	(2)
The Energy	825	(1)
Coherent States	825	(1)
The Coherent-state Representation	826	(2)
Transition Amplitudes	828	(3)
The Stationary Trajectory	831	(2)
Variation of the Sources	833	(2)
Fermi Multiplet	835	(1)
Action Functional and Green's Functions	835	(1)
The Case of Even N	836	(1)
The Case of Odd N	837	(1)
Hamiltonian and Path Integrals	838	(1)
The Fermi Oscillator	839	(1)
Action Functional and Green's Functions	839	(1)
Mode Functions and Hamiltonian	840	(1)
Basis Supervectors	841	(1)
Coherent States	842	(1)
The Functional Integral Representation of (a*, ta1, t1)	843	(2)
Direct Evaluation of the Functional Integral	845	(1)
The Importance of Endpoint Contributions	846	(1)
The Stationary Trajectory as a Matrix Element	847	(1)
The Feynman Propagator	847	(2)
The Bose Oscillator	849	(1)
Action Functional and Green's Functions	849	(1)
Eigenvectors	850	(1)
Mode Functions and Hamiltonian	850	(2)
Energy Eigenvectors	852	(1)
Coherent States	852	(2)
Hamilton-Jacobi Theory	854	(2)
The Point-to-point Amplitude	856	(1)
The Functional Integral Representation (a*, ta1, t1)	857	(1)
The Stationary Path Between Coherent States	858	(1)
The Feynman Propagator	859	(1)
Energy Eigenfunctions	860	(2)
A Fourth-order System	862	(1)
Action Functional and Green's Functions	862	(1)
Wronskian Operator and Mode Functions	863	(1)
Hamiltonian	864	(1)
Wightman Function and Feynman Propagator	865	(1)
Pauli-Villars Regularization	866	(1)
Fourth-order Quantum Gravity	867	(1)
A Model for Ghosts	868	(1)
Action Functional and Green's Functions	868	(1)
Wronskian Operator and Mode Functions	869	(1)
Decomposition of the Dynamical Variables	870	(1)
Energy and Hamiltonian	871	(1)
Use of Ghosts in Gauge Theory	871	(2)
Free Scalar Field in Flat Spacetime	873	(1)
Mode Functions	873	(1)
Green's Functions	874	(2)
Massive Vector Field in Four-dimensional Flat Spacetime	876	(1)
Green's Functions	876	(1)
Mode Functions	877	(3)
Massive Antisymmetric Tensor Field	880	(1)
Action and Field Equation	880	(1)
Minkowski Spacetime	880	(1)
Mode Functions in Four Dimensions	881	(2)
Massive Symmetric Tensor Field in Flat Spacetime	883	(1)
Action and Field Equation	883	(1)
Jacobi Field Operator and Feynman Propagator	884	(1)
Mode Functions in Four Dimensions	885	(1)
Spin	886	(2)
Massive Spinor Field in Flat Spacetime	888	(1)
Action, Field Equation, and Feynman Propagator	888	(1)
Spinor Boosts	888	(1)
Polarization Spinors in Four Dimensions	889	(3)
Mode Functions	892	(2)
Spin	894	(1)
Massive spin-3/2 Field in Flat Spacetime	895	(1)
Action and Field Equation	895	(1)
Jacobi Field Operator and Feynman Propagator	896	(1)
Auxiliary Matrices when n = 4	897	(2)
Mode Functions and Polarization Vector-spinors	899	(1)
Wronskian Operator and Mode-function Relations	900	(1)
Spin	901	(1)
Electromagnetic Field in Flat Spacetime	902	(1)
The Special Character of Masslessness	902	(1)
Invariant-flow Vectors and Supplementary Conditions	902	(1)
Polarization Vectors	903	(1)
The Operator F and its Mode Functions	904	(1)
Spin. Helicity	905	(1)
Massless Symmetric Tensor Field in Flat Spacetime	906	(1)
Invariant-flow Vectors and Supplementary Condition	906	(1)
The Operator F and its Mode Functions	907	(1)
Physical Mode Functions	908	(1)
Spin. Helicity	909	(1)
Massless Spinor Field in Four-dimensional Flat Spacetime	910	(1)
Polarization Spinors	910	(1)
Properties of the Polarization Spinors	911	(2)
Mode Functions	913	(1)
Spin. Helicity	914	(1)
Massless Spin 3/4 Field in Four-dimensional Flat Spacetime	915	(1)
Invariant Flow Vectors and Supplementary Condition	915	(1)
The Operators F	916	(1)
Polarization Vector-spinors	917	(1)
Mode Functions	917	(2)
Spin. Helicity	919	(1)
Renormalization Group and Spontaneous Symmetry Breaking in the λY4 Model	920	(1)
The λY4 Model	920	(1)
Jacobi Field Operator, Heat Kernel, and Effective Action	921	(2)
Renormalization Group	923	(2)
Self-consistency Conditions	925	(2)
Energy Zero Point and Physical Mass	927	(1)
Sponteaneous Symmetry Breaking	928	(2)
The Relativistic Particle in Minkowski spacetime	930	(1)
Action Functional and Dynamical Equations	930	(1)
Gauge Invariance	931	(1)
Green's Functions	931	(1)
The Measure	932	(2)
The Ghost Determinant	934	(1)
The Functional Integral	935	(1)
Consistency Check	936	(2)
Evaluation of the Path Integral	938	(1)
Interpretation of the Path Integral	939	(1)
Change of Variables	940	(2)
Further Reduction of the Functional Integral	942	(1)
Elimination of λ	943	(2)
A Simple Soluble Nonlinear Model	945	(1)
Action Functional and Gauge Invariance	945	(1)
Dynamical Equations and Conserved Quantities	946	(1)
Energy	946	(1)
Jacobi Field Operator and Ghost Operator	947	(3)
Quantization	950	(1)
Dynamics	951	(1)
Quantum Mechanics on a Circle	952	(1)
Lagrangian and Periodicity Conditions	952	(1)
Homotopy	953	(1)
Energy Eigenfunctions	954	(2)
Quantum Mechancis on a Klein Bottle	956	(1)
Periodicity Conditions and Lagrangian	956	(1)
Energy Eigenfunctions	956	(1)
Homotopy	957	(2)
Ghosts for Ghosts	959	(1)
Linearly Dependent Flow Vectors	959	(1)
Nonsingular Operators	960	(1)
Mode Functions	961	(3)
Green's Function Relations	964	(2)
Nonstationary Background	966	(3)
Massless Antisymmetric Tensor Field	969	(1)
Flow Vectors	969	(1)
Matris and Operators	970	(1)
Four-dimensional Minkowski Spacetime	971	(1)
Heat Kernels and Vacuum Persistence Amplitude	972	(4)
Appendix A: Superanalysis	976	(1)
Supernumbers	976	(1)
Superanalytic Functions	977	(1)
Functions of Real Variables	978	(1)
Integration	979	(2)
Supervector Spaces	981	(2)
Bases	983	(1)
Change of Basis	984	(1)
Shifting Indices. The Supertranspose	985	(2)
Dual Supervector Spaces	987	(1)
Dual Bases	988	(1)
The Supertrace and Superdeterminant	989	(1)
Integration over Rcm x Ran	990	(1)
Transformation of Variables	991	(3)
Gaussian Integrals	994	(2)
Supermanifolds	996	(2)
Structures on Supermanifolds	998	(1)
Super Lie Groups	999	(1)
Super Hilbert Spaces	1000	(1)
Linear Operators. Superalgebras	1001	(1)
Physical Observables	1002	(1)
Basic Inequalities for Ordinary Hilbert Space. Averages and Variances	1003	(1)
Supercommutator and Antisupercommutator	1004	(2)
Appendix B	1006	(1)
The Structure Functions to All Orders	1006	(4)
Appendix C	1010	(1)
The Case in Which is not Supersymmetric	1010	(4)
Appendix D: Properties of Vilkovisky's Connection	1014	(1)
Horizontal Geodesics	1014	(1)
Derivation of Eq. (24.40)	1014	(1)
Vertical Geodesics	1015	(1)
Derivation of Eq. (24.41)	1016	(2)
Appendix E: Analytic Continuation in Dimension	1018	(1)
Compilation of Familiar Equations	1018	(1)
Axioms for Analytic Continuation in (Euclidean) Dimensions	1019	(1)
Illustrative Examples	1020	(5)
Integrals over Euclidean Spaces of Even Negative Dimension	1025	(1)
External Parameters	1026	(2)
Integrals of Tensors	1028	(2)
Change of Integration Order	1030	(1)
Integration Over Minkowski Spacetime	1031	(4)
Analytic Continuation in the Presence of Dirac Matrices	1035	(1)
Analytic Continuation of Traces	1036	(2)
Appendix F: Conformally Flat Spacetimes	1038	(1)
Conformal Transformations	1038	(1)
Conformal Flatness in Two Dimensions	1039	(1)
Conformal Flatness in Three Dimensions	1040	(1)
Conformal Flatness in Dimensions Greater Than Three	1040	(2)
Spacetimes of Isotropic Curvature	1042
Index	1