Here is my library of handwritten notes, started since I was an undergraduate.
It is the product of countless late-night and weekend hours of independent study.
It is inspired by my undergraduate physics teacher, Dr. Paul Stevenson,
who taught me classical mechanics and quantum field theory.
His vast library of handwritten notes were concise yet meticulous such that anyone
could read them and learn something new in a short time.
Since these notes will ultimately become lecture material for physics courses,
I have also composed exercises which I thought were instructive as a student,
and are scattered throughout these notes.
While these notes are written for myself, I have made it a point to be
readable by others. All commentary appearing in these notes are my own,
and their correctness cannot be guaranteed. Unlinked items refer to notes that are not
suitable for public viewing (i.e. chicken scratch).
Radial Coulomb problem ▪ Summary ▪ Spin-orbit interaction ▪ Spinor spherical harmonics ▪ Fine structure of hydrogen ▪ Zeeman effect in alkali atoms ▪ Normal Zeeman effect ▪ Stark effect/Electric polarizability
Isotropic 3D harmonic oscillator ▪ Infinite spherical potential ▪ Delta-function potentials
Feynman-Hellmann theorem ▪ Generalized virial theorem ▪ Normalization of bound states ▪ Quantum mechanical sum rules
Angular momentum coupling schemes (LS vs. jj) ▪ Atomic units ▪ Level scheme of helium ▪ Nuclear motion ▪ Structure of helium-like atoms ▪ Excited states of helium ▪ Weak field Zeeman effect
Pauli's theory of spin ▪ Wigner-Eckart theorem ▪ Jenkins-Manohar-Trott minimal coupling trick ▪ Action principle for charged particle in EM field ▪ Quantum mechanical motion in EM field ▪ Probability and current density in A≠0 ▪ Charged particle in a homogeneous magnetic field ▪ Landau's algebraic solution ▪ Coherent states ▪ Aharanov-Bohm effect
Klein-Gordon Coulomb problem ▪ Spin from relativity (Dirac equation) ▪ Rotational invariance of Dirac hamiltonian ▪ Properties of Dirac spin matrix ▪ Parity invariance of Dirac equation ▪ Coupling of Dirac's electron to EM field ▪ Quadratic from of Dirac equation ▪ Salpeter hamiltonian ▪ Plane wave solutions to free Dirac equation ▪ Dirac hole theory ▪ Separation of Dirac equation for spherically symmetric systems ▪ Free spherical Dirac waves ▪ Solution to Dirac-Coulomb problem
Time-independent perturbation theory ▪ Time-dependent evolution of states ▪ Adiabatic approximation ▪ Pictures of quantum mechanics ▪ Dirac's interaction picture ▪ Time dependent perturbation theory ▪ Leading order amplitude ▪ Applications ▪ Next-to-leading order amplitude ▪ Explicit time-dependent perturbation ▪ Operator formalism ▪ Transition amplitudes for long times: the S-matrix
Stationary wavepackets ▪ Partial wave projection of wavepackets ▪ Time development of 3D wavepackets ▪ Time development of partial wavepackets ▪ Summary
Lorentz transformations ▪ Generators of Lorentz group ▪ Rotation/boost matrices ▪ Vectors in hyperpolar coordinates ▪ Poincaré group ▪ Levi-Civita tensors ▪ Pauli-Lubanski vector
Casimir operators ▪ Classification of states ▪ Representations of Lorentz group ▪ Deriving the spin matrices ▪ Transformation law of spinors ▪ Rotation/boost matrices for spin-1/2 fields ▪ Spin 4-vector
Lagrangian and hamiltonian ▪ Plane-wave solutions to Weyl wave equation ▪ Weyl u and v spinors ▪ Properties of Weyl u and v spinors ▪ Canonical quantization ▪ Quantum Hamiltonian ▪ Noether's theorem for Weyl spinors ▪ Intrinsic internal symmetry and fermion charge ▪ Canonical energy-momentum angular momentum tensors
Forming the 4-component Dirac spinor ▪ Noether's theorem for Dirac spinors ▪ Dreiner-Haber-Martin stacking/splitting formulas ▪ Plane wave solutions, u and v spinors ▪ Properties of Dirac u and v spinors ▪ Lorentz transformations of Dirac spinors ▪ P and C conjugation of Dirac fields ▪ Non-relativistic limit of Dirac spinor bilinears ▪ Angular momentum in terms of ladder operators
Gamma matrices in various representations ▪ Conjugation identites ▪ Clifford algebra ▪ Algebra with σ^{μν} ▪ Trace identites ▪ Basis of sixteen 4x4 hermitian matrices ▪ Chisholm expansion ▪ Hodge dual identites ▪ Sirlin expansion ▪ Identities for products of spinor chains ▪ Fierz identities
Forming the 4-component Majorana fermion ▪ Plane wave expansion ▪ Parity of Majorana fermions ▪ Mojarana condition on u and v spinors ▪ Majorana fermion bilinears ▪ Propogators ▪ Completeness relations ▪ Example
Free fermion propagator with complex mass ▪ Full fermion propagator ▪ LSZ reduction formula ▪ Projectors for fermion self energy function ▪ Parametrizations for vertex functions ▪ Gordon identity ▪ Projectors for vertex form factors ▪ Electromagnetic moments
Field strengths and currents ▪ Maxwell action and equations of motion ▪ Lorentz and discrete transformations ▪ Helmholtz decomposition ▪ Natural-SI units conversion ▪ Second Noether's theorem for Maxwell theory ▪ Massive/massless polarization 4-vectors
Radiation gauge condition ▪ Polarization 3-vectors ▪ Coulomb gauge propagator ▪ Including interaction with sources ▪ Coulomb gauge analysis of Higgs mechanism
Hamiltonian mechanics ▪ Classification of constraints ▪ Particle in strong B field ▪ Particle on a sphere ▪ Canonical quantization of EM field
Faddeev-Popov method ▪ Gauge fixing ▪ BRST symmetry of action
Abelian case ▪ Deriving the R_{ξ} gauge propagator ▪ Nonabelian case: Real representation ▪ Obtaining generators for real representation ▪ Transformation law of shifted fields ▪ Identities ▪ Nonabelian case: Complex representation ▪ Canonical quantization in Fermi-ξ gauges ▪ BRST symmetry
Pure Yang-Mills ▪ Yang-Mills Higgs: real representation
General linear gauge conditions ▪ Full propagators ▪ Effective action for gauge fields
Conservation laws from functional methods ▪ Scalar electrodynamics ▪ Quantum electrodynamics ▪ Ward identitiy for π^{+}γ→π^{+}γ ▪ Ward identity for τ→eγ ▪ ABJ axial anomalies ▪ Connection to BPST instanton ▪ 't Hooft vertex for SU(2)
Lie and Graded algebras ▪ Graded Lie algebra ▪ Graded matrices ▪ Supernumbers ▪ Matrix representation of Grassmann generators ▪ Superanalytic functions ▪ Calculus of Grassmann variables ▪ Integration over Grassmann variables ▪ Review of symplectic structure and canonical quantization ▪ Classical mechanics of fermionic systems ▪ Coherent states for fermionic oscillators▪ Path integral representation of Dirac-fermionic oscillator
Super-Poincaré algebra ▪ Vanishing of the vacuum energy ▪ Representations of Super-Poincaré algebra ▪ Classification of irreps
Superspace ▪ Identities for Grassmann coordinates ▪ Scalar superfield ▪ Supersymmetry transformations ▪ Transformation law of superfields ▪ Covariant derivatives ▪ Chiral superfield ▪ Properties of chiral superfields ▪ Vector superfield ▪ Gauge transformation of vector superfield
Building a SUSY lagrangian ▪ D-terms from chiral superfields ▪ F-terms from chiral superfields ▪ Superfield strength tensor ▪ F-terms from superfield strength tensor ▪ D-terms from superfield strength tensor ▪ Supersymmetric QED ▪ Generalization to non-abelian gauge symmetries
MSSM Field content ▪ Superpotential SUSY interactions ▪ Gauge interactions ▪ Scalar potential from superpotential ▪ Soft SUSY breaking in MSSM
The Darboux method ▪ Operator formalism of SUSY QM ▪ The "symmetry" of SUSY (on shell transformations) ▪ Superspace formulation of SUSY QM ▪ Superfield ▪ Off shell transformation law for component fields ▪ Properties of superfields ▪ SUSY covariant derivative ▪ Building a supersymmetric action ▪ Example
Definitions and conventions ▪ Partial fraction expansion ▪ Feynman parametric representation of scalar integrals ▪ Feynman parametric representation of tensor integrals ▪ Feynman parametric representation of Passarino-Veltman functions ▪ Feynman parametric representation of scalar integrals with repeated propagators ▪ Feynman parametric representation of tensor integrals with repeated propagators ▪ Relationship to standard Passarino-Veltman functions ▪ Short-circuiting identities ▪ Passarino-Veltman symmetric tensor identities ▪ Covariant decomposition of shifted tensor integrals
UV divergent parts ▪ IR divergent parts of scaleless functions ▪ Derivatives of Passarino-Veltman functions ▪ Behavior of Passarino-Veltman functions at infinity
Logarithm and Veltman function ▪ Analytic continuation of the logarithm ▪ Multivariable Veltman function ▪ Dilogarithm (Spence) function ▪ Numerical evaluation of the dilogarithm ▪ Bennakker-Denner continuation of the dilogarithm
Scalar A_{0} function ▪ Scalar B_{0} function ▪ DiscB function ▪ Taylor and asymptotic series of DiscB ▪ Imaginary part of B_{0} ▪ Kinematic invariances of C_{0} ▪ Scalar C_{0} ▪ Notes on numerical evaluation of C_{0} ▪ Imaginary part of C_{0}▪ Kinematic invariances of D_{0} ▪ Scalar D_{0} ▪ Imaginary part of D_{0}
Two vanishing external momenta ▪ One vanishing external momentum ▪ No vanishing external momenta
IR divergent box 12 ▪ IR divergent box 13 ▪ IR divergent box 16
Tensor A functions ▪ Recursion reduction for B functions ▪ Iterative reduction for B functions ▪ Passarino-Veltman reduction formula for B functions ▪ Analytic formula for B_{1...1} functions ▪ Special cases for B_{1...1} functions ▪ Analytic representation of B_{1...1} for r = −1 ▪ Reduction formula for B functions of negative rank ▪ Appendix: singular integrals I_{r}(a)
Lorentz decomposition of C functions ▪ Passarino-Veltman reduction formula for C functions ▪ Case det(Z) ≠ 0 ▪ Case det(Z) = 0, and X_{0j} ≠ 0 ▪ Case det(Z) = 0, and X_{0j} = 0 ▪ Case all Z_{ij} vanishing, and f_{k} ≠ 0 ▪ Case all Z_{ij} vanishing, and f_{k} = 0 ▪ Explicit result for det(Z) = det(X) = 0 ▪ Reduction formula for C functions of negative rank ▪ Case IR-divergent tensor triangle 1,4,5,6 ▪ Case IR-divergent tensor triangle 2,3
Lorentz decomposition of D functions ▪ Passarino-Veltman reduction formula for D functions ▪ Covariant decomposition of pinched C functions ▪ Case det(Z) ≠ 0 ▪ Case det(Z) = 0, and X_{0j} ≠ 0 ▪ Case det(Z) = 0, and X_{0j} = 0 ▪ Case all Z_{ij} vanishing, and f_{k} ≠ 0 ▪ Case all Z_{ij} vanishing, and f_{k} = 0 ▪ Explicit integration for det(Z) = 0, and X_{0j} = 0 ▪ Reduction formula for D functions of negative rank
Path integral formulation of QM ▪ Exact solution for free particle ▪ Exact solution for harmonic oscillator ▪ Gelfand-Yaglom formula ▪ Table of Gaussian integrals
Quantum time correlation functions ▪ Path integral as generating functional ▪ Perturbation theory for functional integrals ▪ Quantum equations of motion ▪ Conservation laws
Quantum effective action ▪ Mass dimensions of Green functions ▪ Renormalization of Green functions
Constructive definition ▪ Effective potential of harmonic oscillator ▪ Effective potential from effective action ▪ Tadpole evaluation of V_{eff} ▪ Diagramatic evaluation of V_{eff} ▪ Spontaneously broken φ^{4} theory ▪ Effective potential of Yukawa theory ▪ Effective potential in gauge theory ▪ Path integral evaluation of V_{eff} ▪ Extending path integral method to all orders ▪ Effective potential in scalar QED ▪ Summary of one loop formulas ▪ Effective potential approximation to pole mass ▪ Two loop effective potential of φ^{4} theory
Nielsen's identity in BG-R_{ξ} gauge ▪ Verification at one loop▪ Gauge dependence of the effective potential ▪ Gauge dependence of the effective action ▪ One-loop correction to VEV ▪ Gauge dependence of the sphaleron rate ▪ Gauge dependence of T_{c} and φ_{crit} ▪ Gauge independence of sign of curvature ▪ Gauge invariant analasis of the electroweak phase transition
Kink ▪ Scalar vortex ▪ Derrick's theorem ▪ 'tHooft-Polyakov monopole ▪ Fermions in a kink soliton ▪ Non-topological solitons (Q-balls)
WKB in quantum mechanics ▪ Tunneling through a potential barrier ▪ Single-variable tunneling ▪ Tunneling in potentials with classical degeneracy ▪ Multivariable WKB approximation ▪ Multivariable tunneling ▪ Decay of a false vacuum in scalar field theory
SU(2) Yang-Mills review ▪ Periodic potentials ▪ Chern-Simons current ▪ Euclideanization of Maxwell action ▪ Euclideanization of Yang-Mills action ▪ Importance of axial-temporal gauge ▪ Instantons in pure SU(N) Yang-Mills theory ▪ Bogomolny bound ▪ Solving the equations of motion ▪ Evaluation of euclidean action ▪ Visualizing the Yang-Mills instanton solution ▪ Temporal-axial gauge ▪ Electroweak instantons ▪ Nonexistence of electroweak equations of motion
Electroweak sphaleron at T=0 ▪ Sphaleron energy functional ▪ Electroweak sphaleron at T≠0
Systems with classical degeneracy ▪ Systems with classical metastable points
1-loop determinant in covariantly constant BG field ▪ Schwinger's proper-time representation ▪ Heat equation ▪ Heat kernel evaluation of anomalous dimensions ▪ Zeta function regularization ▪ Determinants and phase shifts ▪ Gelfand-Yaglom formalism in 1D ▪ Radial Gelfand-Yaglom formalism ▪ Functional determinant around bounce solution: False vacuum decay ▪ Heat kernel expansion and large L behavior
Free field theory ▪ Canonical ensemble ▪ Density operator ▪ Quantum correlation functions ▪ Imaginary time propagator ▪ Wick's theorem at finite temperature ▪ One loop self energy in φ^{4} theory ▪ One loop self energy in φ^{3} theory ▪ High temperature expansion of J_{B} and J_{F} ▪ Bessel function representation of J_{B} and J_{F} ▪ Path integral formulation
Schwinger-Keldysh formalism ▪ Generating functional ▪ Real time self energy calculation φ^{4} theory and φ^{3} theory ▪ Photon Debye mass
Effective potentials in φ^{4} theory ▪ High temperature expansion in general theory ▪ Landau analysis of electroweak phase transition ▪ Latent heat ▪ Gauge boson thermal masses ▪ Daisy ring resummation ▪ Dimensional reduction to high-T effective theory
Langevin equations ▪ Fokker-Planck equation ▪ Smoluchowski equation ▪ Transition rates at finite temperature ▪ Kramer barrier crossing
Field content and Lagrangian ▪ Higgs sector ▪ Pure Higgs Feynman rules ▪ Gauge-Higgs sector ▪ Gauge-Higgs Feynman rules ▪ Yukawa interactions ▪ Yukawa Feynman rules ▪ Including Dirac neutrino masses ▪ Gauge sector ▪ Gauge self interaction Feynman rules ▪ Gauge fixing: R_{ξ} gauge ▪ Propagators in R_{ξ} gauge ▪ Gauge-ghost, Higgs-ghost Feynman rules ▪ Electroweak neutral current ▪ Neutral current Feynman rules ▪ Charged currents ▪ CKM Matrix ▪ Charged current Feynman rules ▪ CKM phases ▪ Jarlskog invariant ▪ Standard parametrization of CKM matrix
Classification tree of symmetries ▪ Custodial isospin ▪ Chiral anomalies of abelian currents ▪ Baryon and lepton currents
Z width ▪ Z-γ interference ▪ Z pole: left-right asymmetry ▪ Z pole: forward-backward asymmetry ▪ Higgs decay modes (H→ff, H→WW*/ZZ*, H→γγ/gg) ▪ Higgs production at LHC (WH, ZH, WW→H, ZZ→H, gg→H) ▪ Reduction to Fermi theory
Classical and quantum treatments ▪ Transition rates within the dipole approximation ▪ Oscillator and transition strengths ▪ General formulae: electric multipole ▪ Selection rule for E1 transitions ▪ Lifetime of 1s_{1/2} (F=1) state of hydrogen
Leading order amplitude ▪ Energy regimes (Thompson/Rayleigh/fluorescence)
Photoionization cross section ▪ Born and dipole approximation ▪ Born Approximation ▪ With spherical Coulomb waves ▪ Comparision with data
Construction of harmonics of definite L ▪ Construction of harmonics in multipole basis ▪ Figures
Construction ▪ Properties ▪ Rayleigh expansion for vector plane waves ▪ Multipole expansion for vector fields
Radiation Hamiltonian ▪ Evaluation of electromagnetic 2^{J}-pole matrix element ▪ Radiation Hamiltonian in terms of static multipole moments (Siegert's theorem) ▪ Table of static multipole moments ▪ Summary ▪ Emission amplitudes and decay rates ▪ Resonant absorption amplitudes and cross sections ▪ Multipole amplitudes and transition rates ▪ Single particle electric 2^{J}-pole matrix element ▪ Single particle magnetic 2^{J}-pole matrix element ▪ Selection rule for E-J and M-J transitions ▪ Alignment vs polarization
Thompson formula ▪ Moller scattering
Spin from relativity ▪ Coupling to EM field
Levels of approximation summary ▪ Scattering off Coulomb field LO ▪ Scattering off Coulomb field NLO ▪ Scattering off a recoiling (point-like) proton ▪ Rosenbluth's formula for elastic scattering ▪ Interpretation of F_{1}(0) and F_{2}(0) form factors ▪ Interpretation of Sachs for factors G_{E}(q^{2}) and G_{M}(q^{2})
Calculation of bremsstrahlung amplitude ▪ Classical bremsstrahlung
e^{+}e^{−}→μ^{+}μ^{−} ▪ Compton scattering ▪ Tests of QED ▪ The Hamiltonian of QED
Lagrangian ▪ Matrix-valued gauge fields ▪ Trace theorems for SU(N) ▪ Faddeev-Popov quantization ▪ Feynman rules ▪ Ward identity in non-abelian gauge theory
Renormalized lagrangian ▪ Feynman rules of renormalized QCD ▪ One loop quark self energy ▪ One loop gluon vacuum polarization ▪ One loop ghost self energy ▪ One loop ghost-gluon vertex ▪ Deriving and solving the renormalization group equations ▪ Scale-setting problem in QCD
Kinematics and the origin of IR singularities ▪ Setting up ▪ Tree level cross section: e^{+}e^{−}→qq ▪ Wavefunction renormalization ▪ Quark vertex function ▪ Real emission e^{+}e^{−}→qqg ▪ Total inclusive cross section
Conventions ▪ Laboratory frame kinematics ▪ Relationship of invariant variables to lab. frame variables ▪ Interrelationships of invariant variables ▪ One body lepton phase space for DIS ▪ Physical region ▪ Equations for m=0 ▪ Matching to elastic limit (Rosenbluth formula)
LSZ derivation of DIS scattering amplitude ▪ Leptonic tensor ▪ Hadronic tensor ▪ DIS differential cross section ▪ EM and EW cross section in lab. frame ▪ Parton model calculation ▪ Neutrino deep inelastic scattering
Setting up ▪ Evaluation of H-transverse; real emission ▪ Plus distributions ▪ Evaluation of H-transverse; virtual correction ▪ Evaluation of H-longitudinal, real emission
C, P, T of mesons ▪ Meson octet ▪ Kaons: τ-θ puzzle, P-violation ▪ GIM mechanism ▪ Baryons ▪ Parity doubling ▪ Vector meson dominance
Symmetries of QCD ▪ Chiral symmetry breaking ▪ Chiral lagrangian to O(p^{2}) ▪ Quark masses ▪ Feynman rules ▪ Coupling to nucleons ▪ Coupling to electromagnetism ▪ Pion decay ▪ Chiral lagrangian to O(p^{4}) ▪
Pion self energy ▪ Pion EM form factor
Schrödinger equation for potential well ▪ Bound states ▪ Continuum solutions ▪ Phase shifts ▪ Jost function ▪ Eikonal cross section
Representation theory of SO(3) ▪ Orbital angular momentum in polar coordinates ▪ Spherical harmonics ▪ Spherical law of cosines ▪ Legendre functions ▪ Rayleigh expansion ▪ Gegenbauer expansion ▪ Normalization of Bessel functions ▪ Confluent hypergeometric function
Lab frame kinematics ▪ COM frame kinematics ▪ Mandelstam plane ▪ Connecting lab and COM frame
Normalization/completenss conventions ▪ Linear algebra review ▪ Asymptotic theory ▪ Møller and scattering operators ▪ Time evolution operator ▪ Reduction to one-body potential scattering ▪ Operator formulation of scattering theory ▪ Classical cross section ▪ Quantum cross section ▪ Optical theorem ▪ Symmetries and conservation laws ▪ Partial wave series ▪ Unitarity circle and Argand diagrams ▪ Green/resolvent operator ▪ Lippmann-Schwinger equation ▪ Translation operator (off shell T-matrix) ▪ Relation of transition operator to S-matrix
Quantum scattering problem ▪ Time evolution of scattering wave packet ▪ Quantum scattering cross section ▪ Partial wave series ▪ Partial wave amplitude, differential and total cross section ▪ Unitarity relation ▪ Partial wave scattering amplitude ▪ Scattering phase shifts ▪ Threshold behavior
The Born series ▪ Calculating in the Born approximation ▪ Properties of the Born level approximation ▪ Yukawa potential at Born level ▪ Distribution of charge ▪ Effective range approximation ▪ Bethe's expression for the effective range approximation ▪ WKB approximation for phase shifts ▪ Review of elementary optics (diffraction) ▪ Eikonal approximation ▪ Relationship of Born and Eikonal series ▪ Eikonalization of Coulomb amplitude ▪ N/D method ▪ Calogero's variable phase method ▪ Scattering on singular potentials
Partial wave Coulomb scattering ▪ Regular Coulomb Wavefunction ▪ Irregular Coulomb Wavefunction ▪ Full Coulomb scattering wave ▪ Partial wave expansion of scattering solutions ▪ Non-existence of partial wave series for Coulomb amplitude ▪ Taylor's theorem for Coulomb phase shifts ▪ Behaviour of the Born series for Coulomb scattering ▪ Superposition of Coulomb and short range potentials
Radial Schrödinger equation ▪ Regular scattering solutions ▪ Irregular scattering solutions ▪ Jost functions ▪ Analyticity of S_{l} ▪ Threshold behavior and effective range approximation ▪ Bound states ▪ Resonances ▪ Dispersion relations
Hilbert spaces ▪ Scattering stationary states with spin ▪ Scattering amplitudes and cross sections ▪ Optical theorem ▪ Rotational invariance with spin ▪ Parity invariance with spin ▪ Invariant amplitude approach ▪ Polarization and the density matrix ▪ Partial waves (fixed-axis) ▪ Helicty formalism ▪ Parity invariance for helicty amplitudes ▪ Wigner D matrix ▪ Partial wave (helicity amplitudes) ▪ Parity invariance (partial wave helicty amplitudes) ▪ Partiy transformation of one particle states
Hilbert spaces for identical particles ▪ Scattering of two identical particles ▪ Effect on Coulomb scattering
Model system ▪ Channel Hamiltonians ▪ Hilbert spaces of multichannel scattering ▪ Multichannel scattering operator ▪ Cross sections ▪ Threshold behavior for inelastic cross sections ▪ e^{−}-H scattering
Ricatti-Bessel functions ▪ Analytic continuation and circuit relations ▪ Summary ▪ Power series ▪ Theorem for analytic continuation ▪ Analytic properties of regular/irrgular/Jost functions ▪ Analytic properties of S-matrix/Mandelstam symmetry
Complex angular momentum ▪ Regge poles ▪ Lehmann Ellipse ▪ Sommerfeld-Watson Transformation ▪ Relation to resonance widths ▪ Mandelstam representation
Index of refraction ▪ Beer-Lambert's law ▪ Sommerfeld Enhancement
2→2 kinematics: Mandelstam variables ▪ Boundary of physical region ▪ Flux ▪ One body phase space ▪ Two body phase space (COM frame) ▪ 2→2 kinematics in lab. frame ▪ Two body phase space (lab. frame) ▪ Crossing symmetry ▪ Three body phase space ▪ Kinematics for Regge theory ▪ Five particle kinematics ▪ Kinematics for inclusive processes
Kramer-Krönig relation ▪ Optical theorem ▪ Unitarity bubble diagrams for scattering amplitudes ▪ Factorization of scattering amplitudes ▪ Branch cuts from unitarity relations ▪ Single variable dispersion relations ▪ Subtracted dispersion relations ▪ Mandelstam representation in relativistic scattering
Partial wave decomposition ▪ Elastic unitarity ▪ Threshold behavior ▪ Karplus curve ▪ Lehmann Ellipse ▪ Field theoretic studies of partial waves ▪ Froissart-Gribov projection ▪ Froissart bound on total cross section
Sommerfeld-Watson transformation ▪ Regge signature ▪ Singularities of signatured partial wave amplitudes ▪ Regge pole phenomenology ▪ Soft diffraction ▪ Dolen-Horn-Schmid duality and FESR ▪ Analytic properties of Regge trajectory functions ▪ Dual diagrams and overlapping channels ▪ Multi-Regge theory ▪ Mueller's optical theorem
LSZ Asymptotic Theory ▪ Matching onto non-relativistic potential scattering ▪ Analyticity of field theoretic amplitudes ▪ Many body phase space (recurrence relation) ▪ Narrow width approximation ▪ Three body decays ▪ Cutkosky's cutting rules/Largest time equation ▪ Landau's equations/Singularities of Feynman integrals
Resonance duality ▪ Euler Gamma and Beta functions ▪ Spectrum in the 2→2 Veneziano model ▪ Residues of the Veneziano B_{4} function ▪ Regge behavior of Veneziano amplitude ▪ Zeros of the Veneziano amplitude ▪ Symmetric form of the full Veneziano amplitude ▪ Shapiro-Virasoro form
Case of 5 particles ▪ N-particle generalization ▪ Factorization ▪ Bardakçi-Ruegg form ▪ Möbius transformations ▪ Anharmonic ratio ▪ Koba-Nielsen form ▪ Extended integration range
Coherent states ▪ Formulation of operator factorization ▪ Factorization of N-particle amplitude ▪ Residues in operator formalism ▪ Verification of operator formalism
Virasoro operator formalism ▪ Vertex factor ▪ Cyclic symmetry ▪ More on the field operator
Representation theory of SO(2,1) ▪ Realization of SO(2,1) on Fock space ▪ Transformation properties of Q^{μ}(τ) ▪ Conformal generators (Virasoro operators) ▪ Verifying Virasoro algebra ▪ Conformal transformations ▪ Properties of Virasoro operators ▪ On-shell conditions
Twist operator ▪ Internal symmetry: Chan-Paton ▪ Spin
Factorization of 4-point Virasoro amplitude ▪ Regge behavior of Virasoro amplitude