Quantum Electrodynamics
Contents
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This web site is a summary of QED. Abstract is at the foot of this page.

0. General Formulation

   [0] Action Principle
       (1) Action
       (2) Symmetry

   [1] Classical Lagrangian Formulation
       (1) Discrete System
           (i ) Euler-Lagrange Equation
           (ii) Noether's Theorem for Discrete System
           (iii) Treatment of Complex Variables
                a. Euler-Lagrange Equation
                b. Conserved Charge
       (2) Continuous System
           (i ) Euler-Lagrange Equation
           (ii) Local Field Theory
               a. I()=EEE
               b. I()/ƒ(x)=0́EEE
               c. Noether's Theorem for Local Field Theory
               d. Local Complex Field
                   1. Euler-Lagrange Equation
                   2. Noether's Current
       (3) Example of Classical Lagrangian Formulation
           (i ) Newton's Particles and Euclidean Group
                a. L(x1,x2;y1,y2)=EEE
                b. I(q1,q2)/qij(t)=0́EEE
                c. Symmetry
                    1. O(3) and SO(3)
                        a. Definition
                        b. Parametrization and Generators
                        c. O(N) and SO(N)
                    2. Euclidean Group
                        a. Definition
                        b. Parametrization and Generators
                        c. Linear Representation
                           1. Scalar Representation
                           2. General Case
                    3. Symmetry of Action
                        a. Euclidean Invariance
                        b. Time-Displacement Invariance
                        c. Time-Reversal Invariance
                    4. Application of Noether's Theorem
           (ii) Real Vector Field and Poincare Group
               a. L(x;y)=EEE
               b. I()/ƒ(x)=0́EEE
               c. Symmetry
                   1. Lorentz Group
                       a. Definition and Fundamental Properties
                       b. Parametrization and Generators
                   2. Poincare Group
                       a. Definition
                       b. Parametrization and Generators
                       c. Linear Representation
                          1. Scalar Representation
                          2. Vector Representation
                   3. Symmetry of Action
                   4. Application of Noether's Theorem
           (iii) Complex Scalar Fields and SU(N)
                a. L(x;y)=EEE
                b. I()/ƒ(x)=0́EEE
                c. Symmetry
                    1. SU(N)
                        a. Definition
                        b. Parametrization and Generators
                        c. Example
                           1. SU(1)
                           2. SU(2)
                    2. Symmetry of Action
                        a. Poincare Invariance
                        b. SU(n) Invariance
                    3. Application of Noether's Theorem
                        a. n=1
                        b. n=2

   [2] Classical Hamiltonian Formulation
       (1) Discrete System
           (i ) Velocity
           (ii) Hamiltonian
           (iii) Canonical Equations
           (iv) Action Principle
           (v) Poisson Bracket
           (vi) Canonical Transformations
           (vii) Treatment of Complex Variables (n=2)
                a. Velocity
                b. Hamiltonian
                c. Canonical Equations
                d. Poisson Bracket
                e. Canonical Transformations
       (2) Continuous System
           (i ) Velocity
           (ii) Hamiltonian
           (iii) Canonical Equations
           (iv) Poisson's Bracket
           (v) Local Field Theory
           (vi) Local Complex Field
                a. Velocity
                b. Hamiltonian Density
                c. Canonical Equations
                d. General Complex Field
                e. Canonical Transformations
       (3) Examples of Classical Hamiltonian Formulation
           (i ) Newton's Particles and Euclidean Group
                a. Velocity
                b. Hamiltonian
                c. Canonical Equations
                d. (1)(vi) is to be checked
           (ii) Complex Scalar Fields and SU(N)
                a. Velocity
                b. Hamiltonian Density
                c. Canonical Equations
                d. (2)(vi)d is to be checked

   [3] Canonical Quantization
       (1) Discrete System
           (i ) Quantization Conditions
                a. Equal-Time Commutation-Relations
                b. Heisenberg's Equation of Motion
           (ii) Reconstruction of Canonical Equation
           (iii) Canonical Transformation
           (iv) Treatment of Complex Variables
                a. Quantization Conditions
                    1. Equal-Time Commutation-Relations
                    2. Heisenberg's Equation of Motion
                b. Reconstruction of Canonical Equation
                c. Canonical Transformation
       (2) Continuous System

   [4] Representation and Interpretation
       (1) Discrete System
           (i ) Representation of Canonical Variables
                a. q-Representation
                b. N-Representation
                c. In each representation, qj(t) and pj(t) are given by EEE
           (ii) Probability Interpretation
           (iii) S-Matrix
                a. Definition of Operator S(t2,t1)
                b. Fundamental Properties of S(t2,t1)
                    1. Unitarity
                    2. Dyson's Formula
                    3. Connection between 'in' and 'out'
                c. Probability in Terms of S-Matrix
           (iv) Symmetry
       (2) Continuous System
           (i ) Fock's Representations
           (ii) Probability and S-Matrix


1. Lagrangian Formulation of Classical Electrodynamics

   [1] Gauge Invariant Formulation
       (1) Dirac Matrices
           (i ) Definition
           (ii) Fundamental Properties
           (iii) Representation
       (2) Action
       (3) Euler-Lagrange Equation
       (4) Symmetry
           (i ) Spinor Representation of Poincare Group
               a. Linear Representation of P+
               b. Representation of Is and It
               c. Definition of Full Representation P1/2
           (ii) Physical Vector Representation of Poincare Group
           (iii) Symmetry of Action
               a. Poincare Invariance
               b. Gauge Invariance
           (iv) Application of Noether's Theorem
               a. Poincare Invariance
               b. Gauge Invariance
       (5) Free Fields
           (i ) Free Dirac Field
               a. (x)=EEE
               b. Construction of u()(p) and v()(p)
                   1. Find the Lorentz Transformation
                   2. Define s(p)
                   3. Then u()(p) and v()(p) can be chosen
                   4. Various Formulas
                   5. Representation of u()(p) and v()(p)
               c. b(p)=EEE, d(p)=EEE
           (ii) Free Maxwell Field

   [2] Coulomb Gauge Formulation
       (1) New Variables
       (2) Action
           (i ) I(,)=EEE
           (ii) L(,a;,b)=EEE
           (iii) LD(,)=EEE, LM(A,B)=EEE
       (3) Euler-Lagrange Equation
       (4) Symmetry of Action
           (i ) Translation Invariance
           (ii) Rotation Invariance
           (iii) Spatial Reflection Invariance
           (iv) Time Reversal Invariance
           (v) Gauge Invariance
       (5) Application of Noether's Theorem
           (i ) Translation Invariance
           (ii) Rotation Invariance
           (iii) Gauge Invariance


2. Hamiltonian Formulation of Classical Electrodynamics

   [1] Gauge Invariant Formulation Fails

   [2] Coulomb Gauge Formulation
       (1) Velocity
       (2) Hamiltonian
       (3) Canonical Equations
       (4) Poisson's Bracket
           (i ) {F,G}PB(,a;b)=EEE
           (ii) (d/dt)F((t,*),(t,*);(t,*))=EEE
           (iii) Fundamental Functionals
       (5) Canonical Transformation


3. Canonical Quantization of Electrodynamics

   [1] Quantization Conditions
       (1) Commutation Relations
       (2) Heisenberg's Equation

   [2] Reconstruction of Canonical Equations

   [3] Canonical Transformations
       (1) Transformation Generated by Conserved Quantity
           (i ) Translation
           (ii) Rotation
           (iii) Gauge Transformation
           (iv) Algebra of Generators
       (2) Discrete Transformation
           (i ) Parity
           (ii) Time Reversal
           (iii) Charge Conjugation


4. S-Matrix of Quantum Electrodynamics

   [1] What Operator to Measure?
       (1) Number Density in Momentum-Space
           (i ) Fourier Components
           (ii) Commutation Relations
           (iii) Number Density
           (iv) Transformation of Number Density
                a. Rotation
                b. Parity
                c. Time Reversal
                d. Charge Conjugation
       (2) Fock-Representation of QED
           (i ) EEE
           (ii) Basis of Fock-Space
               a. p(1)(1)e(1),EEE,p(N)(N)e(N)=EEE
               b. e(T1,p;m')=EEE
               c. (,')=EEE

   [2] Feynman's Graph
       (1) Splitting Hamiltonian
       (2) In-Fields
       (3) Wick's Theorem
       (4) Propagators
       (5) Parts of Feynman's Graph
           (i ) External Legs
           (ii) Internal Lines
           (iii) Verteces
           (iv) Trivial Graphs
       (6) S-Matrix in Terms of Feynman's Graph
       (7) Remaining Problems

   [3] LSZ-Formulation
       (1) What Operator to Measure?
       (2) Interpretation
       (3) Reduction Formula
       (4) Gell-Mann-Low's Relation
       (5) Wick's Theorem
       (6) Reduction of External Leg
       (7) Feynman's Graph
           (i ) Parts
           (ii) S-Matrix in Terms of Feynman's Graph

   [4] Renormalization
       (1) Bird's-Eye View
       (2) Algorithm of Virtual Calculation
           (i ) Define the Connected Green's Function
           (ii) Calculate Full Propagators and Proper Vertex Part
           (iii) Verify the Existence of M, Z1, Z2, and Z3
           (iv) Define S~'F, D~'F, and ~
           (v)@Introduce a new variable ee'f
           (vi) Define and Calculate ZR1, ZR2, ZR3, SRF, DRF, and R
           (vii) Represent the Green's Function in Terms of S'F, D'F, , and e
           (viii) Verify the Following Equation: EEE
           (ix) Compare the Kallen-Lehman Representations with (iii)
           (x) Substitute (viii) into [3](7)(ii)
       (3) Practical Calculation
           (i ) KR(p,p',q;e',m')=EEE
           (ii) R(p,p';e',m')σ=EEE
           (iii) SRF(p';e',m')-1u()(p;m')=EEE
           (iv) DRF(q;e',m')ʃ=EEE
           (v) (qq-gʃq2)(q2;e',m')=EEE
           (vi) u'()(p;m')R(p,p;e',m')u()(p;m')=EEE
       (4) Remaining Problems


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yAbstractz I completed this note in 1995 to summarize many pieces into one scheme. In this note, you can see whole of the theory the most briefly. I wondered what renormalization is when I was learning quantum electrodynamics. Then I wondered if it is under the general principle of quantum theory. I showed in this note as Bird's-Eye View that renormalization is not under the principle but is an external prescription. Especially the aspect of 'What Operator to Measure?' holds the key for the problem. If you wonder what role the Kallen-Lehman representation plays, you see the answer at Q-83. I am also proud of the explanation of Feynman's graph in this note. It serves Feynman's graph completely as representations of mathematical formulas. You must surely like sockets. This note does not include path integral method but is based on canonical quantization. This note is not a seiten (canon), so the site-logo is not gold but silver.

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