Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
Guy F. de Téramond
Universidad de Costa Rica
Escuela de Fı́sica
UCR
Feb 10, 2016
Image credit: Wikipedia
Escuela de Fı́sica UCR, Feb 10, 2016
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
Contents
1 Preamble: Choice of light-front coordinates
3
2 Dirac forms of relativistic dynamics
5
3 Light-front dynamics
7
4 Light-front quantization of QCD
10
5 Semiclassical approximation to QCD in the light front
14
6 The problem of confinement in QCD
18
Escuela de Fı́sica UCR, Feb 10, 2016
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
1
Preamble: Choice of light-front coordinates
• Light-front quantization based on the choice of light-front coordinates to specify initial conditions
tangential to the light cone
x+ ≡ ct + z
light-front time
x+ = 0
x− ≡ ct − z
longitudinal spacial coordinate
x, y
transverse spatial coordinates
• Equal-time quantization based on initial surface
horizontal to the light cone “present”: x, y, z, t = 0
• Causality maintained in LF: boundary conditions
requires information within the causal horizon
• The front form is a natural basis for cosmology:
universe observed along the front of a light wave
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
• LF space coordinates
x+ = x0 + x3
light-front time
x− = x0 − x3
1
2
x⊥ = x , x
longitudinal space variable
transverse space variable
• LF momentum coordinates
p+ = p0 + p3
longitudinal momentum
p− = p0 − p3
1
2
p⊥ = p , p
light-front Hamiltonian
transverse momentum
• Compute p · x to identify Hamiltonian as conjugate to LF time x+
µ
+ −
− +
1
pµ x = 2 p x + p x − p⊥ · x⊥
• Mass-shell condition pµ pµ = p− p+ − p2⊥ = m2 leads to dispersion relation for LF Hamiltonian p−
p2⊥ + m2
p =
p+
−
• All particles are moving in the LF (except for zero modes p+ = 0)
p
p = p2 + m 2 + p 3 > 0
+
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
2
Dirac forms of relativistic dynamics
[P. A. M. Dirac, Forms of relativistic dynamics, Rev. Mod. Phys. 21, 392 (1949)]
• The Poincaré group is the full (dynamical) symmetry group of any form of relativistic dynamics
[P µ , P ν ] = 0
[M µν , P ρ ] = i (g µρ P ν − g νρ P µ )
[M µν , M ρσ ] = i (g µρ M νσ − g µσ M νρ + g νσ M µρ − g νρ M µσ )
• Poincaré generators separated into kinematical and dynamical subgroups
• Kinematical generators act along the initial hypersurface where initial conditions are imposed and
contain no interactions (leave invariant initial surface)
• Dynamical generators are responsible for the evolution of the system and depend on the interactions
(map initial surface into another surface)
• Each front has its Hamiltonian and evolve with a different time, but results computed in any front should
be identical (different parameterizations of space-time: cannot be related by a Lorentz transformation)
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
(a)
ct
(b)
ct
z
z
y
(c)
ct
y
z
y
8-2013
8811A4
• Instant form: initial surface defined by x0 = 0
P 0 , K i = M 0i dynamical,
P, J i = 21 ijk M jk kinematical
• Point form: initial surface is the hyperboloid x2 = κ2 > 0, x0 > 0
P µ dynamical,
M µν kinematical
• Front form: initial surface tangent to the light cone x+ = x0 + x3 = 0
( P±
= P 0 ± P 3)
P − , M −1 = K 1 − J 2 , M −2 = K 2 + J 1 dynamical
P + , P⊥ , M 12 = J 3 , M +1 = K 1 + J 2 , M +2 = K 2 − J 1 , M +− = K 3 kinematical
Escuela de Fı́sica UCR, Feb 10, 2016
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
3
Light-front dynamics
• Hadron with 4-momentum P = (P + , P − , P⊥ ), P ± = P 0 ± P 3 , as bound state of i constituents
• Mass-shell condition P 2 = M 2 leads to dispersion relation for LF Hamiltonian P −
P
•
n-particle bound state with p2i
=
−
m2i ,
p−
i
P2⊥ + M 2
=
,
+
P
P+ > 0
−
p+
,
p
i
i , p⊥i , for each constituent
pi =
p2⊥i + m2i
=
,
+
pi
i
p+
i >0
• Longitudinal momentum P + is kinematical: sum of single particle constituents p+
i of bound state
X
+
P =
p+
p+
i >0
i ,
i
• LF Hamiltonian P − is dynamical: bound-state is arbitrarly off the LF energy shell
P
−
−
n
X
p−
i <0
i
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
• LF evolution given by relativistic Schrödinger-like equation
(x+ = x0 + x3 LF time)
∂
i + |ψ(P )i = P − |ψ(P )i
∂x
where
P2⊥ + M 2
P |ψ(P )i =
|ψ(P )i
+
P
−
• Construct LF invariant Hamiltonian
P 2 = Pµ P µ = P − P + − P2⊥
P 2 |ψ(P )i = M 2 |ψ(P )i
• Since P + and P⊥ are kinematical linear
QM evolution in LF Hamiltonian P − is maintained
Image credit: S. Beane
• State |ψ(P )i is expanded in multi-particle
P − dynamical, F r kinematical
Fock states |ni of the free LF Hamiltonian
X
|ψi =
ψn |ni, |ni = { |uudi, |uudgi, |uudqqi, · · · }, Pn = |ψn |2
n
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
Vacuum in the front-form of dynamics
•
Spectral condition: P +
=
P
+
i pi ,
p+
i
P0
>0
=
√
P2 + M 2
• LF vacuum is the state with P + = 0 and contains
no particles: all other states have P + > 0
• Vacuum of interacting theory is also the trivial vacuum
• Frame independent definition of the vacuum within
the causal horizon: causality maintained
|P|
P 2 |0i = 0
•
LF vacuum is defined at fixed LF time x+
over all x− = x0 − x3 and x⊥
=
x0
+
x3
P−
=
P2⊥ +M 2
P+
• Simple vacuum allows definition of LF wavefunctions:
quantum-mechanical probabilistic interpretation
• Cannot discard zero modes with p+
a = 0, which also lead
to P + = 0 and can mix with trivial vacuum
P+
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
4
Light-front quantization of QCD
• Start with SU (3)C QCD Lagrangian
LQCD = ψ (iγ µ Dµ − m) ψ − 41 Gaµν Ga µν
• Express the hadron four-momentum generator P = (P + , P − , P⊥ ) in terms of dynamical fields
0 ±
ψ+ = Λ+ ψ and A⊥ Λ± = γ γ quantized in null plane x+ = x0 + x3 = 0
!
Z
2
2
m + (i∇⊥ )
2 a
aµ
P − = 12 dx− d2 x⊥ ψ + γ +
ψ
−
A
(i∇
)
Aµ
+
⊥
i∂ +
Z
+ gs dx− d2 x⊥ ψ + γ µ T a ψ+ Aaµ +
Z
gs2
dx− d2 x⊥ cabc cade Abµ Acν AdµAeν
+
4
Z
gs2
1
+
dx− d2 x⊥ ψ + γ + T a ψ+
ψ + γ + T a ψ+
2
2
(i∂ + )
Z
+ gs2
γ
+
dx− d2 x⊥ ψ + γ µ T a Aaµ + T b Abν γ ν ψ+
2
i∂
where the integrals are over initial surface x+
Escuela de Fı́sica UCR, Feb 10, 2016
= 0, where commutation relation for fields are defined
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
• The LF Hamiltonian P − generates LF time translations
ψ+ (x), P
−
∂
= i + ψ+ (x),
∂x
∂
−
A⊥ , P = i + A⊥ (x),
∂x
which evolve the initial conditions for fields to all space-time.
• kinematical generators have no interactions
Z
P+ =
dx− d2 x⊥ ψ + γ + i∂ + ψ+
Z
P⊥ = 21 dx− d2 x⊥ ψ + γ + i∇⊥ ψ+
• Note: In the LF constraint equations determine non-dynamical fields –no dynamical evolution
(Ex. ψ− ) in terms of dynamical fields (Ex. ψ+ )
Escuela de Fı́sica UCR, Feb 10, 2016
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
• Dirac field ψ+ expanded in terms of ladder operators b(q)|0i = d(q)|0i = 0
i
XZ
dq + d2 q⊥ h
−
−iq·x
†
iq·x
p
ψ+ (x , x⊥ )α =
bλ (q)uα (q, λ)e
+ dλ (q) vα (q, λ)e
3
+
(2π)
+
2q
q >0
λ
with
n
o n
o
† 0
† 0
3
+
0+
(2)
0
b(q), b (q ) = d(q), d (q ) = (2π) δ(q − q ) δ
q⊥ − q⊥
P−
P+
P⊥
X Z dq + d2 q⊥ q2 + m2 †
⊥
=
bλ (q)bλ (q) + interactions
(2π)3
q+
λ
X Z dq + d2 q⊥
+ †
=
q
bλ (q)bλ (q)
3
(2π)
λ
X Z dq + d2 q⊥
†
=
q
b
⊥
λ (q)bλ (q)
(2π)3
λ
where q −
=
q2⊥ +m2
q+
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
• Relative partonic coordinates: ki+ = xi P + , p⊥i = xi P⊥i + k⊥i
n
X
xi = 1,
i=1
n
X
k⊥i = 0
i=1
• Fock component ψn (xi , k⊥i , λzi ) depends only on relative coordinates: longitudinal momentum
+
fraction xi = ki /P + , transverse momentum k⊥i and spin λzi
• LFWF ψn is frame independent
P
• Momentum conservation: P + = ni=1 ki+ , ki+ > 0, but LFWF represents a bound-state which is
Pn −
−
off the LF energy shell, P −
i ki < 0
• Express off-shell dependence of bound-state in terms of invariant variable
(invariant mass of constituents)
Mn2
2
= (k1 + k2 + · · · + kn ) =
n
X
i=1
kiµ
2
X k2 + m2
i
⊥i
=
xi
i
• M 2 − Mn2 is the measure of the off-energy shell: key variable which controls the bound state
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
5
Semiclassical approximation to QCD in the light front
[GdT and S. J. Brodsky, PRL 102, 081601 (2009)]
• Reduction of strongly coupled LF multiparticle dynamics to a 1-dim QFT with no particle creation and
absorption: Effective semiclassical approximation
• Central problem is derivation of effective interaction which acts only on the valence sector:
express higher Fock states as functionals of the lower ones
• Advantage: Fock space not truncated and symmetries of the Lagrangian preserved
[H. C. Pauli, EPJ, C7, 289 (1999)
• Compute M 2 from hadronic matrix element
hψ(P 0 )|Pµ P µ |ψ(P )i = M 2 hψ(P 0 )|ψ(P )i
• Find
XZ X
2
2
dxi d k⊥i
M =
n
q
k2⊥q
+
xq
m2q
!
|ψn (xi , k⊥i )|2 + interactions
with phase space normalization
XZ 2
dxi d k⊥i |ψn (xi , k⊥i )|2 = 1
n
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
• In terms of n−1 independent transverse impact coordinates b⊥j , j = 1, 2, . . . , n−1,
!
Z
2
2
n−1
XY
X −∇b⊥q + mq
2
2
∗
M =
ψn (xi , b⊥i ) + interactions
dxj d b⊥j ψn (xi , b⊥i )
x
q
n
q
j=1
with normalization
X n−1
YZ
dxj d2 b⊥j |ψn (xj , b⊥j )|2 = 1.
n j=1
• Semiclassical approximation
2
ψn (k1 , k2 , . . . , kn ) → φn (k1 + k2 + · · · + kn ) , mq → 0
|
{z
}
Mn2 =
P
• For a two-parton system in the mq → 0
2
Mqq
=
k2 +m2
i
⊥i
i
xi
k2⊥
x(1−x)
• Conjugate invariant variable in transverse impact space is
ζ 2 = x(1 − x)b2⊥
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
• To first approximation LF dynamics depend only on the invariant variable ζ , and dynamical properties
are encoded in the hadronic mode φ(ζ)
ψ(x, ζ, ϕ) = e
iLϕ
φ(ζ)
√
X(x)
,
2πζ
where we factor out the longitudinal X(x) and orbital kinematical dependence from LFWF ψ
• Ultra relativistic limit mq → 0 longitudinal modes X(x) decouple (L = Lz )
Z
Z
2
2
p
d
1
d
L
φ(ζ)
√ + dζ φ∗ (ζ) U (ζ) φ(ζ),
+ 2
M2 = dζ φ∗ (ζ) ζ − 2 −
dζ
ζ dζ
ζ
ζ
where effective potential U includes all interaction terms upon integration of the higher Fock states
• LF eigenvalue equation Pµ P µ |φi = M 2 |φi is a LF wave equation for φ
1 − 4L2
d2
− 2−
dζ
4ζ 2
{z
}
|
kinetic energy of partons
+
U (ζ)
| {z }
φ(ζ) = M 2 φ(ζ)
conf inement
• Relativistic and frame-independent LF Schrödinger equation: U is instantaneous in LF time
• Critical value L = 0 corresponds to lowest possible stable solution, the ground state of P 2
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
Comparison of effective potentials in the instant and front forms
• Compare invariant mass in the instant-form in the hadron center-of-mass system P = 0,
2
= 4 m2q + 4p2
Mqq
with the invariant mass in the front-form in the constituent rest frame, kq
2
Mqq
+ kq = 0
k2⊥ + m2q
=
x(1 − x)
obtain
q
q
U = V + 2 p2 + m2q V + 2 V p2 + m2q
2
where p2⊥
=
k2⊥
4x(1−x) ,
m (x−1/2)
p3 = √q
, and V is the effective potential in the instant-form
x(1−x)
• At large distances a linear instant-form potential V implies a harmonic front-form potential U and thus
linear Regge trajectories
[A. P. Trawiński, S. D. Glazek, S. J. Brodsky, GdT, H. G. Dosch, arXiv: 1403.5651]
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
6
The problem of confinement in QCD
• Increase of QCD coupling at low energies implies that an infinite number of quark and gluons are
dynamically coupled
• To compute U one must systematically express higher Fock components as functionals of the lower
ones: Its actual derivation is an open problem
• The QCD Lagrangian in the limit of massless quarks has no scale: still confinement and a mass gap
should emerge from the quantum theory built upon the classical QCD conformal theory
• Vast complexity of QCD confinement problem implies that it could be undecidable: It is not possible to
know, starting from the fundamental QCD Lagrangian, whether the system is gapped or not
[Cubitt, Perez-Garcia and Wolf, Nature 528, 207 (2015)]
• Recent progress: Embedding LF bound-state equations in AdS space (gauge/gravity correspondence)
within a superconformal algebraic structure which determines the confinement dynamics and connects
mesons to baryons
• TOPIC FOR THE NEXT TALK !
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Relativistic Hamiltonian Dynamics in Dirac’s Light-Front Form
Gracias !
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