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Second International Summer School of the GDR PH-QCD “Correlations between partons in nucleons”. N. Cédric Lorcé. Multidimensional pictures of the nucleon (1/3). IFPA Liège. June 30-July 4, 2014, LPT, Paris-Sud University, Orsay, France. Outline. Introduction Tour in phase space - PowerPoint PPT Presentation
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Cédric LorcéIFPA Liège
Multidimensional pictures of the nucleon (1/3)
June 30-July 4, 2014, LPT, Paris-Sud University, Orsay, France
Second International Summer School of the GDR PH-QCD
“Correlations between partons in nucleons”
N
Outline
• Introduction• Tour in phase space• Galileo vs Lorentz• Photon point of view
Lecture 1
Introduction
10-14m 10-15m 10-18m10-10m
d
u
Atom Nucleus Nucleons
Quarks
Atomic physics
Nuclear physics
Hadronic
physics
Particle physics
Proton
Neutron
Up
Down
Structure of matter
1/33
Introduction
Elementary particles
Matter Interaction
Graviton ?
2/33
Introduction
Relevant degrees of freedom depend on typical energy scale
Degrees of freedom
« Resolution »
3/33
Introduction
Nucleon pictures
« Naive » Realistic
3 non-relativistic heavy quarks
Indefinite # of relativistic light quarks
and gluons
MN = Σ mconst (~2%) + Eint (~98% !)
Quantum Mechanics + Special Relativity = Quantum Field Theory
Brout-Englert-Higgs mechanism
QCD
4/33
MN ~ Σ mconst
Introduction
At the energy frontier
Goal : understanding the nucleon internal structure
Why ?
LHC
Quark & gluon distributions ?
Proton Proton
5/33
Introduction
At the intensity frontier
Why ?
Nucleon shape & size ?Nucleon spin structure ?
Nucleon
Electron
…
Goal : understanding the nucleon internal structure
6/33
E.g.
Introduction
How ?
Deep Inelastic Scattering Elastic Scattering
Deeply Virtual Compton ScatteringSemi-Inclusive DIS
Proton « tomography
»
Phase-space distribution
Goal : understanding the nucleon internal structure
7/33
[Gibbs (1901)]
State of the system
Phase space
Classical Mechanics
Position
Mom
en
tum
Particles follow well-defined trajectories
8/33
[Gibbs (1902)]
Phase-space density
Phase space
Statistical Mechanics
Position
Mom
en
tum
Position-space density
Momentum-space density
Phase-space average
9/33
[Wigner (1932)][Moyal (1949)]
Wigner distribution
Phase space
Quantum Mechanics
Position
Mom
en
tum
Position-space density
Momentum-space density
Phase-space average
10/33
[Wigner (1932)][Moyal (1949)]
Wigner distribution
Phase space
Quantum Mechanics
Position
Mom
en
tum
Hermitian (symmetric) derivative
Fourier conjugate variables
11/33
Phase space
Harmonic oscillator
Heisenberg’s uncertainty
relations
Wigner distributions have applications in:
• Nuclear physics • Quantum chemistry• Quantum molecular dynamics• Quantum information• Quantum optics• Classical optics• Signal analysis• Image processing• Quark-gluon plasma• …
Quasi-probabilistic
12/33
Phase space
In quantum optics, Wigner distributions are « measured » using homodyne tomography
Idea : measuring projections of Wigner distributions from different directions
Binocular vision in
phase space !
2D 2D
3D
Find the hidden 3D picture
13/33
[Lvovski et al. (2001)][Bimbard et al.
(2014)]
Phase-space/Wigner distribution
Phase space
Quantum Field Theory
Covariant Wigner operator
[Carruthers, Zachariasen (1976)]
[Ochs, Heinz (1997)]
Equal-time Wigner operator
Time ordering ?
Scalar fields
14/33
Phase space
~ internal motion center-of-mass motion
Nucleons are composite systems
Interesting Not so interesting
15/33
[Ji (2003)][Belitsky, Ji, Yuan (2004)]
Phase space
in position space
Localized state in momentum space
Phase-space compromise
Intrinsic phase-space/Wigner distribution
Identified with intrinsic variables
Breit frame
CoM momentum
CoM position
16/33
[Ji (2003)][Belitsky, Ji, Yuan (2004)]
Phase space
in position space
Localized state in momentum space
Phase-space compromise
Intrinsic phase-space/Wigner distribution
Identified with intrinsic variables
Breit frame
CoM momentum
CoM position
Time translation
Same energy !
17/33
Galilean symmetry
All is fine as long as space-time symmetry is Galilean
Position operator can be defined
18/33
Lorentz symmetry
But in relativity, space-time symmetry is Lorentzian
Position operator is ill-defined !
Lorentz contraction
Creation/annihilation of pairs
No separation of CoM and internal coordinates
Further issues :
Spoils (quasi-) probabilistic
interpretation
19/33
Instant-form dynamics Light-front form dynamics
Forms of dynamics
Space-time foliation
[Dirac (1949)]
« Space » = 3D
hypersurface
« Time » = hypersurface
label
Light-front components
Time
Space
Energy
Momentum
20/33
t4, z4
t3, z3
t2, z2
Instant form vs light-front form
t1, z1
Ordinary point of view
t5, z5
t6, z6
t7, z7
21/33
Instant form vs light-front form
Photon point of view
x+
x+
x+
x+x+
x+
x+
22/33
Instant form vs light-front form
Initial frame
23/33
Instant form vs light-front form
Boosted frame
24/33
Instant form vs light-front form
(Quasi) infinite-momentum frame
25/33
Transverse boost
Longitudinal momentum plays the role of mass in the transverse plane
Light-front operators
[Kogut, Soper (1970)]
Transverse space-time symmetry is Galilean
Transverse position operator can be defined !
26/33
Quasi-probabilistic interpretation
What about the further issues with Special Relativity ?
Transverse boosts are Galilean No transverse Lorentz contraction !
No sensitivity to longitudinal Lorentz contraction !
is conserved and positive
Drell-Yan frame
Particle number is conserved in Drell-Yan frame
Conclusion : quasi-probabilistic interpretation is possible !
27/33
[Ji (2003)][Belitsky, Ji, Yuan (2004)]
Non-relativistic phase space
in position space
Localized state in momentum space
Identified with intrinsic variables
CoM momentum
CoM position
Time translation
28/33
Equal-time Wigner operator
Intrinsic phase-space/Wigner distribution
Relativistic phase space
in position space
Localized state in momentum space« CoM »
momentum
« CoM » position
Equal light-front time Wigner operator
Intrinsic relativistic phase-space/Wigner distribution
Space-time translation
[C.L., Pasquini (2011)]
[Soper (1977)][Burkardt (2000)][Burkardt (2003)]
[Ji (2003)][Belitsky, Ji, Yuan (2004)]
Boost invariant !
Normalization
29/33
NB : is invariant under light-front boosts
Relativistic phase space
Our intuition is instant form and not light-front form
It can be thought as instant form phase-space/Wigner distribution in IMF !
2+3D
Longitudinal momentum
Transverse momentum
Transverse position
In IMF, the nucleon looks like a pancake
[C.L., Pasquini (2011)]
30/33
Link with parton correlators
General parton correlator
Phase-space/Wigner distributions are Fourier transforms
31/33
Space-time translation
Link with parton correlators
Adding spin to the picture
Leading twist
Dirac matrix ~ quark polarization
32/33
Link with parton correlators
Adding spin and color to the picture
Wilson line
Gauge transformation
Very very very complicated object not directly measurable
Let’s look for simpler measurable correlators
33/33
Summary
• Understanding nucleon internal structure is essential• Concept of phase space can be generalized to QM and QFT• Relativistic effects force us to abandon 1D in phase space
Lecture 1
2+3D
Transverse momentum
Transverse position
