Confinement and Higher Fock States in -Front Holography

Guy F. de Teramond´

Universidad de Costa Rica

LIGHT CONE 2010 Applications of Light-Cone Coordinates to Highly Relativistic Systems

Dallas, May 23 - 27, 2011

ct ct ct

z z z

y y y

1-2011 1-2011 1-2011 8811A1 8811A2 8811A3

LC 2011 2011, Dallas, May 23, 2011 Page 1 1 Introduction

2 Light Front Dynamics

Semiclassical Approximation to QCD in the Light Front

3 Light-Front Holographic Mapping of Wave Equations

Higher Spin Modes in AdS Space

Dual QCD Light-Front

Bosonic Modes and Spectrum

Fermionic Modes and Spectrum

4 Light-Front Holographic Mapping of Current Matrix Elements

Mapping Form Factors

Nucleon Elastic Form Factors

Nucleon Transition Form Factors

5 Confinement Interaction and Higher Fock States

Space and Time-Like Pion Form Factor

Pion Transition Form Factor

LC 2011 2011, Dallas, May 23, 2011 Page 2 1 Introduction

Gauge Correspondence and Light-Front QCD

Recent developments inspired by the AdS/CFT correspondence [Maldacena (1998)] between gravity in • AdS space and conformal field theories in physical space-time have led to an analytical semiclassical approximation to light-front QCD, which provides physical insights into its non-perturbative dynamics

LF quantization is the ideal framework to describe hadronic structure in terms of constituents: simple • vacuum structure allows to compute matrix elements diagonal in particle number

Calculation of matrix elements P + q J P requires boosting the hadronic from P to • h | | i | i P + q : boosts are trivial in LF | i Invariant Hamiltonian equation for bound states similar structure of AdS equations of motion: direct • connection of QCD and AdS/CFT methods possible

Isomorphism of SO(4, 2) group of conformal transformations with generators P µ,M µν,Kµ,D, • with the group of isometries of AdS5, a space of maximal symmetry, negative curvature and a four-dim

boundary: (Dim isometry group of AdSd+1 is (d + 1)(d + 2)/2)

LC 2011 2011, Dallas, May 23, 2011 Page 3 AdS5 metric: • 2 2 R µ ν 2 ds = 2 ηµνdx dx dz |{z} z | {z } − LAdS LMinkowski

A distance LAdS shrinks by a warp factor z/R • as observed in Minkowski space (dz = 0): z LMinkowski LAdS ∼ R

Since the AdS metric is invariant under a dilatation of all coordinates xµ λxµ, z λz, the • → → variable z acts like a scaling variable in Minkowski space

µ Short distances xµx 0 map to UV conformal AdS5 boundary z 0 • → → µ 2 Large confinement dimensions xµx 1/Λ map to large IR region of AdS5, z 1/ΛQCD, • ∼ QCD ∼ thus there is a maximum separation of and a maximum value of z

Use the isometries of AdS to map the local interpolating operators at the UV boundary of AdS into the • modes propagating inside AdS

LC 2011 2011, Dallas, May 23, 2011 Page 4 2 Light Front Dynamics

Different possibilities to parametrize space-time [Dirac (1949)] • Parametrizations differ by the hypersurface on which the initial conditions are specified. Each evolve • with different “times” and has its own Hamiltonian, but should give the same physical results

ct Forms of Relativistic Dynamics: dynamical vs. kinematical generators [Dirac (1949)] •

Instant form: hypersurface defined by t = 0, the familiar one z

• y H, K dynamical, L, P kinematical 1-2011 8811A1

ct

Point form: hypersurface is an hyperboloid • µ µν z P dynamical,M kinematical y

1-2011 8811A2 Front form: hypersurface is tangent to the light cone at τ = t + z/c = 0 ct • − x y + z P ,L ,L dynamical,P , P⊥,L , K kinematical z

y

1-2011 8811A3

LC 2011 2011, Dallas, May 23, 2011 Page 5 Semiclassical Approximation to QCD in the Light Front

[GdT and S. J. Brodsky, PRL 102, 081601 (2009)]

Compute 2 from LF Lorentz invariant Hamiltonian equation for the relativistic bound state • M µ − + 2  2 PµP ψ(P ) = P P P ψ(P ) = ψ(P ) | i − ⊥ | i M | i Relevant variable in the limit of zero masses (dual to the invariant mass) • n−1 r x X ζ = xjb 1 x ⊥j − j=1 the x-weighted transverse impact coordinate of the spectator system (x active quark)

For a two-parton system ζ2 = x(1 x)b2 • − ⊥

To first approximation LF dynamics depend only on the invariant variable ζ, and hadronic properties • are encoded in the hadronic mode φ(ζ) from

z φ(ζ) ψ(x, ζ, ϕ) = eiL ϕX(x) √2πζ + − factoring angular ϕ, longitudinal X(x) and transverse mode φ(ζ) (P , P⊥, Jz commute with P )

LC 2011 2011, Dallas, May 23, 2011 Page 6 z Ultra relativistic limit mq 0 longitudinal modes X(x) decouple (L = L ) • → | |  2 2  Z p d 1 d L φ(ζ) Z 2 = dζ φ∗(ζ) ζ + + dζ φ∗(ζ) U(ζ) φ(ζ) M −dζ2 − ζ dζ ζ2 √ζ

where the confining forces from the interaction terms are summed up in the effective potential U(ζ)

µ 2 LF eigenvalue equation PµP φ = φ is a LF wave equation for φ • | i M | i

2 2  d 1 4L  2 2 − 2 + U(ζ) φ(ζ) = φ(ζ) −dζ − 4ζ | {z } M | {z } confinement kinetic energy of partons

Effective light-front Schrodinger¨ equation: relativistic, frame-independent and analytically tractable • Eigenmodes φ(ζ) determine the hadronic mass spectrum and represent the probability amplitude to • find n-massless partons at transverse impact separation ζ within the at equal light-front time

Semiclassical approximation to light-front QCD does not account for particle creation and absorption •

LC 2011 2011, Dallas, May 23, 2011 Page 7 3 Light-Front Holographic Mapping of Wave Equations

Higher Spin Modes in AdS Space

Description of higher spin modes in AdS space (Frondsal, Fradkin, Vasiliev, Metsaev) • Spin-J in AdS represented by totally symmetric rank J tensor field ΦM ···M • 1 J M µ  Action for spin-J field in AdSd+1 in presence of dilaton background ϕ(z) x = (x , z) • Z  0 0 0 1 d ϕ(z) NN M1M MJ M S = d x dz √g e g g 1 g J DN ΦM ···M DN 0 ΦM 0 ···M 0 2 ··· 1 J 1 J 0 0  2 M1M MJ M µ g 1 g J ΦM ···M ΦM 0 ···M 0 + − ··· 1 J 1 J ···

where DM is the covariant derivative which includes parallel transport

Physical hadron has plane-wave and polarization indices along 3+1 physical coordinates • −iP ·x ΦP (x, z)µ ···µ = e Φ(z)µ ···µ , Φzµ ···µ = = Φµ µ ···z = 0 1 J 1 J 2 J ··· 1 2 µ 2 with four-momentum Pµ and invariant hadronic mass PµP =M

LC 2011 2011, Dallas, May 23, 2011 Page 8 Construct effective action in terms of spin-J modes ΦJ with only physical degrees of freedom • [Dosch, Brodsky and GdT (in preparation); Lyubovitskij et al., operator construction (in preparation)]

Introduce fields with tangent indices • J M1 M2 MJ  z  Φˆ A A ···A = e e e ΦM M ···M = ΦA A ···A 1 2 J A1 A2 ··· AJ 1 2 J R 1 2 J

Find effective action • Z  0 0 0 1 d ϕ(z) NN µ1µ µJ µ S = d x dz √g e g η 1 η J ∂N Φˆ µ ···µ ∂N 0 Φˆ µ0 ···µ0 2 ··· 1 J 1 J 0 0  2 µ1µ µJ µ µ η 1 η J Φˆ µ ···µ Φˆ µ0 ···µ0 − ··· 1 J 1 J upon µ-rescaling

Variation of the action gives AdS wave equation for spin-J mode ΦJ = Φµ ···µ • 1 J " d−1−2J  ϕ   2# z e (z) µR 2 ∂z ∂z + ΦJ (z) = ΦJ (z) − eϕ(z) zd−1−2J z M

J with Φˆ J (z) = (z/R) ΦJ (z) and all indices along 3+1

LC 2011 2011, Dallas, May 23, 2011 Page 9 Dual QCD Light-Front Wave Equation z ζ, ΦP (z) ψ(P ) ⇔ ⇔ | i [GdT and S. J. Brodsky, PRL 102, 081601 (2009)]

−3/2+J ϕ(z)/2 Upon substitution z ζ and φJ (ζ) ζ e ΦJ (ζ) in AdS WE • → ∼ " d−1−2J  ϕ   2# z e (z) µR 2 ∂z ∂z + ΦJ (z) = ΦJ (z) − eϕ(z) zd−1−2J z M

find LFWE (d = 4)  2 2  d 1 4L 2 − + U(ζ) φJ (ζ) = M φJ (ζ) −dζ2 − 4ζ2

with 1 1 2J 3 U(ζ) = ϕ00(z) + ϕ0(z)2 + − ϕ0(z) 2 4 2z and (µR)2 = (2 J)2 + L2 − − AdS Breitenlohner-Freedman bound (µR)2 4 equivalent to LF QM stability condition L2 0 • ≥ − ≥ Scaling dimension τ of AdS mode Φˆ J is τ = 2 + L in agreement with twist scaling dimension of a • two parton bound state in QCD and determined by QM stability condition

LC 2011 2011, Dallas, May 23, 2011 Page 10 Bosonic Modes and Meson Spectrum

Soft wall model: linear Regge trajectories [Karch, Katz, Son and Stephanov (2006)]

Choose Dilaton ϕ(z) = +κ2z2 [ϕ(z) = κ2z2 does not provides area law for the Wilson loop] • − Effective potential: U(z) = κ4ζ2 + 2κ2(L + S 1) • − R 2 Normalized eigenfunctions φ φ = dζ φ(z) = 1 • h | i | | s 1+L 2n! 1/2+L −κ2ζ2/2 L 2 2 φn,L(ζ) = κ ζ e L (κ ζ ) (n+L)! n

Eigenvalues 2 = 4κ2 (n + L + S/2) • Mn,L,S 0.8 ΦHΖL HaL ΦHΖL HbL 0.5 0.6

0.4 0.0

0.2 -0.5 Ζ Ζ 0.0 0 2 4 6 8 10 0 2 4 6 8 10

2 2 LFWFs φn,L(ζ) in physical space time for dilaton exp(κ z ): a) orbital modes and b) radial modes

LC 2011 2011, Dallas, May 23, 2011 Page 11 4κ2 for ∆n = 1 4κ2 for ∆L = 1 2κ2 for ∆S = 1

PC J JPC 0-+ 1+- 2-+ 3+- 4-+ 1-- 2++ 3-- 4++ n=3 n=2 n=1 n=0 6 6 n=3 n=2 n=1 n=0

f2(2300) 4 4 π2(1880) π(1800) a4(2040) ρ(1700) f2(1950) f (2050) 2 4 M M2 π (1670) ω(1650) ρ π(1300) 2 3(1690) 2 b(1235) ω (1670) 2 3 ρ(1450) ω(1420) a2(1320) f2(1270) π 0 ρ(770) 0 ω(782) 0 1 2 3 4 0 1234 2-2011 9-2009 8796A5 L 8796A1 L

Regge trajectories for the π (κ = 0.6 GeV) and the I =1 ρ-meson and I =0 ω-meson families (κ = 0.54 GeV)

LC 2011 2011, Dallas, May 23, 2011 Page 12 Fermionic Modes and Baryon Spectrum [Hard wall model: GdT and S. J. Brodsky, PRL 94, 201601 (2005)] [Soft wall model: GdT and S. J. Brodsky, (2005), arXiv:1001.5193]

From Nick Evans

Nucleon LF modes • s 2+L 2n! 3/2+L −κ2ζ2/2 L+1 2 2 ψ+(ζ)n,L = κ ζ e L κ ζ (n + L)! n s 3+L 1 2n! 5/2+L −κ2ζ2/2 L+2 2 2 ψ−(ζ)n,L = κ ζ e Ln κ ζ √n + L + 2 (n + L)!

Normalization Z Z • 2 2 dζ ψ+(ζ) = dζ ψ−(ζ) = 1

Eigenvalues • 2 = 4κ2 (n + L + 1) Mn,L,S=1/2 “Chiral partners” • N(1535) M = √2 MN(940)

LC 2011 2011, Dallas, May 23, 2011 Page 13 4κ2 for ∆n = 1 Same multiplicity of states for and ! 4κ2 for ∆L = 1 2κ2 for ∆S = 1

n=3 n=2 n=1 n=0 (a) (b) 6 n=3 n=2 n=1 n=0 Δ(2420) 2 M N(2200) 4

Δ(1950) Δ(1905) N(1710) N(1680) Δ(1920) 2 N(1720) Δ(1600) Δ(1910) N(1440) Δ(1232)

N(940) 0 0 1 2 3 4 0 1 2 3 4 9-2009 8796A3 L L

Regge trajectories for positive parity N and ∆ baryon families (κ = 0.5 GeV)

LC 2011 2011, Dallas, May 23, 2011 Page 14 4 Light-Front Holographic Mapping of Current Matrix Elements [S. J. Brodsky and GdT, PRL 96, 201601 (2006), PRD 77, 056007 (2008)], PRD 78 025032 (2008)]

EM transition matrix element in QCD: local coupling to pointlike constituents • ψ(P 0) J µ ψ(P ) = (P + P 0)µF (Q2) h | | i 0 µ µ where Q = P P and J = eqqγ q − EM hadronic matrix element in AdS space from coupling of external EM field propagating in AdS with • extended mode Φ(x, z) Z 4 ϕ(z) M ∗ d x dz √g e A (x, z)ΦP 0 (x, z)←→∂ M ΦP (x, z) 4 4 0  0µ 2 (2π) δ P P µ P + P F (Q ) ∼ − How to recover hard pointlike scattering at large Q out of soft collision of extended objects? • [Polchinski and Strassler (2002)]

+ Mapping of J elements at fixed light-front time: ΦP (z) ψ(P ) • ⇔ | i

LC 2011 2011, Dallas, May 23, 2011 Page 15 QCD Drell-Yan-West electromagnetic FF in impact space [Soper (1977)] • n−1 Z n−1 2 X Y 2 X  X  2 F (q ) = dxjd b⊥j eq exp iq⊥ xkb⊥k ψn(xj, b⊥j) · | | n j=1 q k=1

+ 2 1 Consider a two-quark π Fock state ud with eu = and e = • | i 3 d 3 Z 1 Z 2 2 2 iq⊥·b⊥(1−x) + b b Fπ (q ) = dx d ⊥e ψud/π(x, ⊥) 0 + with normalization Fπ (q =0) = 1

Integrating over angle • ! Z 1 Z r 2 2 dx 1 x F + (q ) = 2π ζdζJ0 ζq − ψ (x, ζ) π x(1 x) x ud/π 0 − where ζ2 = x(1 x)b2 − ⊥

LC 2011 2011, Dallas, May 23, 2011 Page 16 Compare with electromagnetic FF in AdS space [Polchinski and Strassler (2002)] • Z 2 3 dz 2 F (Q ) = R V (Q, z)Φ + (z) z3 π

where V (Q, z) = zQK1(zQ)

Use the integral representation • r ! Z 1 1 x V (Q, z) = dx J0 ζQ − 0 x

Compare with electromagnetic FF in LF QCD for arbitrary Q. Expressions can be matched only if • LFWF is factorized φ(ζ) ψ(x, ζ, ϕ) = eiMϕX(x) √2πζ Find •  −3/2 p ζ X(x) = x(1 x), φ(ζ) = Φ(ζ), z ζ − R →

LC 2011 2011, Dallas, May 23, 2011 Page 17 Form factor in soft-wall model expressed as τ 1 product of poles along vector radial trajectory • − (twist τ = N + L) [Brodsky and GdT, Phys.Rev. D77 (2008) 056007] 1 F (Q2) =  Q2  Q2   Q2  1 + M2 1 + M2 1 + M2 ρ ρ0 ··· ρτ−2

where 2 4κ2(n + 1/2) [negative SL (dashed line) positive TL dilaton (continuous)] Mρn → → Correct scaling incorporated in the model • “Free current” V (Q, z) = zQK (zQ) infinite radius (mauve), no pole structure in time-like region • 1 → “Dressed current” non-perturbative sum of an infinite number of terms finite radius (blue) • → 2 FΠIq M 1.0 2 2 4 1 2 4 M GeV Q F IQ MIGeV M HbL I M HaL 1.2 p 3 2 ‡ 2 0.8 M 4 Κ Hn + 1  2L Ρ 1700 1.0 H L

0.6 0.8 2 ΡH1450L

0.6 0.4 1 0.4 M2 ‡ 4 Κ2Hn + 1L 0.2 0.2 Q2 IGeV2M ΡH770L n 0.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0 5 10 15 20 25 30 35 0 1 2 Pion form factor (τ = 2) form factor (τ = 3) VM radial trayectories

LC 2011 2011, Dallas, May 23, 2011 Page 18 Nucleon Form Factors

Light Front Holographic Approach [Brodsky and GdT] • EM hadronic matrix element in AdS space from non-local coupling of external EM field in AdS with • fermionic mode ΨP (x, z) Z 4 ϕ(z) M A d x dz √g e ΨP (x, z) eA Γ AM (x, z)ΨP (x, z)

4 4 0  0 0 µ (2π) δ P P µ ψ(P ), σ J ψ(P ), σ ∼ − h | | i

Effective AdS/QCD model: additional term in the 5-dim action • [Abidin and Carlson, Phys. Rev. D79, 115003 (2009)] Z 4 ϕ(z) M N  A B η d x dz √g e Ψ eA eB Γ , Γ FMN Ψ

Couplings η determined by static quantities

Generalized Parton Distributions in gauge/gravity duals • [Vega, Schmidt, Gutsche and Lyubovitskij, Phys.Rev. D83 (2011) 036001] [Nishio and Watari, arXiv:1105.290]

LC 2011 2011, Dallas, May 23, 2011 Page 19 Compute Dirac proton form factor using SU(6) flavor symmetry • Z dz F p(Q2) = R4 V (Q, z)Ψ2 (z) 1 z4 +

Nucleon AdS • 2+L s κ 2n! 7/2+L L+1 2 2 κ2z2/2 Ψ (z) = z L κ z e− + R2 (n + L)! n

Normalization (F p(0) = 1,V (Q = 0, z) = 1) • 1 Z dz R4 Ψ2 (z) = 1 z4 + Bulk-to-boundary propagator [Grigoryan and Radyushkin (2007)] • Z 1 2 2 2 dx Q κ2z2x/(1 x) V (Q, z) = κ z x 4κ2 e− − (1 x)2 0 − Find • 1 F p(Q2) = 1  Q2  Q2  1 + 2 1 + 2 Mρ Mρ0 with 2 4κ2(n + 1/2) Mρn →

LC 2011 2011, Dallas, May 23, 2011 Page 20 Nucleon Transition Form Factors

n=0,L=0 n=1,L=0 Compute spin non-flip EM transition N(940) N ∗(1440): Ψ Ψ • → + → + Transition form factor • Z p 2 4 dz n=1,L=0 n=0,L=0 F1 ∗ (Q ) = R Ψ (z)V (Q, z)Ψ (z) N→N z4 + + p  Orthonormality of Laguerre functions F1 ∗ (0) = 0,V (Q = 0, z) = 1 • N→N Z 4 dz n0,L n,L R Ψ (z)Ψ (z) = δ 0 z4 + + n,n

Find • Q2 2 p 2 2√2 MP F1 ∗ (Q ) = N→N 3  Q2  Q2  Q2  1 + 2 1 + 2 1 + 2 Mρ M 0 M 00 ρ ρ with 2 4κ2(n + 1/2) Mρn →

LC 2011 2011, Dallas, May 23, 2011 Page 21 Data from I. Aznauryan, et al. CLAS (2009)

LC 2011 2011, Dallas, May 23, 2011 Page 22 5 Confinement Interaction and Higher Fock States [S. J. Brodsky and GdT (in progress)]

Is the AdS/QCD confinement interaction responsible for quark pair creation? • Only interaction in AdS/QCD is the confinement potential • In QFT the resulting LF interaction is a 4-point effective interaction wich leads to qq qq, q qqq, • → → qq qq and q qqq → →

Create Fock states with extra quark-antiquark pairs. • No mixing with qqg Fock states (no dynamical ) • Explain the dominance of quark interchange in large angle elastic scattering • [C. White et al. Phys. Rev D 49, 58 (1994)

Effective confining potential can be considered as an instantaneous four-point interaction in LF time, • similar to the instantaneous exchange in LC gauge A+ = 0. For example Z + a + a − 4 − 2 ψγ T ψ 1 ψγ T ψ Pconfinement κ dx d ~x⊥ + 4 + ' P (∂/∂⊥) P

LC 2011 2011, Dallas, May 23, 2011 Page 23 Space and Time-Like Pion Form Factor [GdT and S. J. Brodsky, arXiv:1010.1204]

Higher Fock components in pion LFWF • π = ψ qq τ=2 + ψ qqqq τ=4 + | i qq/π| i qqqq/π| i ··· corresponding to interpolating operators = ψγ+γ5ψ and = ψγ+γ5ψψψ O O Expansion of LFWF up to twist 4 • κ = 0.54 GeV, Γρ = 130, Γρ0 = 400, Γρ00 = 300 MeV,Pqqqq = 13%

0.6 2 2 2 2 Π MΡ log FΠIQ M Q F IQ M 2

0.5

1 2 0.4 MΡ¢

2 0 M ¢¢ 0.3 Ρ

0.2 -1

0.1 -2 2 2 Q2 GeV2 Q GeV 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5

LC 2011 2011, Dallas, May 23, 2011 Page 24 Meson Transition Form-Factors

[S. J. Brodsky, Fu-Guang Cao and GdT, arXiv:1005.39XX]

Pion TFF from 5-dim Chern-Simons structure [Hill and Zachos (2005), Grigoryan and Radyushkin (2008)] • Z Z 4 LMNP Q d x dz  AL∂M AN ∂P AQ

4 (4) 2 µνρσ (2π) δ (pπ + q k) Fπγ(q ) µ(q)(pπ)νρ(k)qσ ∼ − p 2 −κ2z2/2 Take Az Φπ(z)/z, Φπ(z) = 2Pqq κ z e , Φπ Φπ = Pqq • ∝ h | i p  Find φ(x) = √3fπx(1 x), fπ = Pqq κ/√2π • − 1 Z h 2 2 2 i 2 2 4 φ(x) −PqqQ (1−x)/4π f x Q Fπγ(Q ) = dx 1 e π √3 1 x − 0 − normalized to the asymptotic DA [P = 1 Musatov and Radyushkin (1997)] qq → 2 2 2 Large Q TFF is identical to first principles asymptotic QCD result Q Fπγ(Q ) = 2fπ • → ∞ The CS form is local in AdS space and projects out only the asymptotic form of the pion DA •

LC 2011 2011, Dallas, May 23, 2011 Page 25 Sz =+1,Lz = 1 γ γ − γ∗(q) 1/2 − 1/2 u − γ(k) π0 (a) u¯ Sz = 1,Lz =+1 +1/2 γ − γ

Sz =+1,Lz = 1 γ∗(q) γ γ −

u γ(k) +1/2 +1/2 u¯ 1/2 0 1/2 − (b) − u π Sz = 1,Lz =+1 u¯ γ − γ

0 Twist-two (a) and twist-four contribution (b) to γγ∗ π . →

LC 2011 2011, Dallas, May 23, 2011 Page 26 Feta pion-gamma transition form factor, Q2Fpigamma TFF eta prime

0.30 0.30 0.40 BaBar BaBar BaBar CLEO CLEO 0.35 CLEO 0.25 CELLO 0.25 Free current; Twist 2 Free current; Twist 2 ) Free current; Twist 2 Dressed current; Twist 2 ) 0.30 )

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Dressed current; Twist 2 0.20 0.20 Dressed current; Twist 2+4 Dressed current; Twist 2+4 Dressed current; Twist 2+4 0.25 (GeV )(GeV ) )(GeV 0.15 2 0.15 0.20 2 2 (Q

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LC 2011 2011, Dallas, May 23, 2011 Page 27 ct ct ct

z z z

y y y

1-2011 1-2011 1-2011 8811A1 8811A2 8811A3

“ Working with a front is a process that is unfamiliar to physicists. But still I feel that the mathematical simplification that it introduces is all-important. I consider the method to be promising and have recently been making an extensive study of it. It offers new opportunities, while the familiar instant form seems to be played out ” P.A.M. Dirac (1977)

LC 2011 2011, Dallas, May 23, 2011 Page 28