Basis Light-Front Quantization Approach
to
Quantum Field Theory
Xingbo Zhao
With Anton Ilderton,
Heli Honkanen, Pieter Maris,
James Vary, Stan Brodsky
Department of Physics and
Astronomy
Iowa State University
Ames, USA
INT Workshop INT-13-52W, Seattle, Feb. 11-22, 2013
Outline
• Basis Light-Front Quantization (BLFQ) [Vary et al ’10, Honkanen et al ‘11]
– For spectrum and wavefunction of the eigenstates of quantum
field theory (ultimate goal: hadrons & nuclear structure)
• Time-dependent Basis Light-Front Quantization (tBLFQ)
– For time evolution of quantum field configurations (ultimate
goal: hadrons & nuclear dynamics)
2
Basis Light-front Quantization
• BLFQ: approach for quantum field theory
– Nonperturbative
– First-principles
– Numerical
• BLFQ solves for
– Eigenspectrum of quantum field theory
– Wavefunction for eigenstates
• BLFQ is based on Light-front dynamics for
– Boost invariance in wavefunction
– Simple vacuum structure
–…
3
Light-front Dynamics
[Dirac 1949]
• Time is redefined t = x 0 ® x + º x 0 + x3
Light front
2 HAMILTONIAN DYNAMICS 2 HAMILTONIAN
DYNAMICS
0
3 19
• Hamiltonian
is thus H = P 0 ®
P ºP -P
• Quantization and initial condition are specified on
the equal light-front time plane
• Subsequent evolution is given by
¶
1 +
i + y (x ) = P y (x + )
4
¶x
2
The two four-volume element s are relat ed by t he Jacobian J (x) = ||∂x/ ∂x||, particu
Figure
1: Dirac’s
three forms of Hamilt onian dynamics.
Figure 1: Dirac’s t hree forms of Hamilt
onian
dynamics.
The two four-volume elements are relat ed by t he Jacobian J (x) = ||∂x/ ∂x||, part icularly
General Procedure for BLFQ
1. Derive Hamiltonian from Lagrangian by
Legendre transform
2. Construct BLFQ basis a
3. Form Hamiltonian matrix in BLFQ basis a ' P - a
4. Diagonalize Hamiltonian Pie. solve P - b = Pb- b
5. Evaluate observables O º b Ô b
5
General Procedure for BLFQ
1. Derive Hamiltonian from Lagrangian by
Legendre transform
2. Construct BLFQ basis a
3. Form Hamiltonian matrix in BLFQ basis a ' P - a
4. Diagonalize Hamiltonian Pie. solve P - b = Pb- b
5. Evaluate observables O º b Ô b
6
Example: Obtain LF QED Hamiltonian
• QED Lagrangian
• Derived Light-front Hamiltonian
kinetic energy terms
vertex
interaction
instantaneous
photon
interaction
instantaneous
fermion
interaction
7
General Procedure for BLFQ
1. Derive Hamiltonian from Lagrangian by
Legendre transform
2. Construct BLFQ basis a
3. Form Hamiltonian matrix in BLFQ basis a ' P - a
4. Diagonalize Hamiltonian Pie. solve P - b = Pb- b
5. Evaluate observables O º b Ô b
8
Basis Construction
1. Fock-space expansion
e.g.
2. For each Fock particle:
– Transverse ( x ^ = x1,2) 2D-HO basis, labeled by n,m quantum
number (as well as HO basis parameter b = MW )
– Longitudinal ( x - = x 0 - x 3 ) plane-wave basis
e.g. eg = e Ä g
e
e
e
e
g
g
g
g
e
=
{n
,m
,
k
,
l
}
g
=
{n
,
m
,
k
,
l
}
with
and
9
Set of Transverse 2D HO Modes for n=4
m=0
m=1
m=3
m=2
m=4
J.P. Vary, H. Honkanen, J. Li, P. Maris, S.J. Brodsky, A. Harindranath,
G.F. de Teramond, P. Sternberg, E.G. Ng and C. Yang, PRC 81, 035205 (2010).
10
ArXiv:0905:1411
Basis Reduction
• “Pruning” based on symmetry of Hamiltonian:
Net fermion
number:
Longitudinal component of
total angular momentum:
Longitudinal momentum:
ån
f
i
=Nf
i
å (m + s ) = J
i
i
åk
i
=K
i
• Truncation:
Fock sector truncation
åk
i
=K
i
å[ 2n + | m | +1] £ N
i
i
i
max
i
z
General Procedure for BLFQ
1. Derive Hamiltonian from Lagrangian by
Legendre transform
2. Construct BLFQ basis a
3. Form Hamiltonian matrix in BLFQ basis a ' P - a
4. Diagonalize Hamiltonian Pie. solve P - b = Pb- b
5. Evaluate observables O º b Ô b
12
QED Hamiltonian in BLFQ Basis
Basis parameters: Nmax=2, K=1.5, b=me, Nf=1,
a ' PQED
a
e
e + eg sectors
eg
(MeV)
e
eg
0.3482
(Kinetic energy)
-0.0119
-0.0119
0.9139
(Kinetic energy)
13
General Procedure for BLFQ
1. Derive Hamiltonian from Lagrangian by
Legendre transform
2. Construct BLFQ basis a
3. Form Hamiltonian matrix in BLFQ basis a ' P - a
4. Diagonalize Hamiltonian Pie. solve P - b = Pb- b
5. Evaluate observables O º b Ô b
14
QED Eigenstates in BLFQ Basis: Solution
P
• Eigenspectrum of QEDin the small basis (Nf=1):
Pb-(MeV) M b
(MeV)
e
eg
phys
scat
eb
eg b
0.348
0.511
-0.9998
-0.021
0.914
1.054
-0.021
-0.9998
• Both bound state and scattering state spectra are
obtained
15
invariant mass (MeV)
invariant mass (MeV)
QED Eigenspectrum in Larger Basis
Nmax=K-1/2
Nmax=K-1/2
• Single electron(bound state) + eγ scattering states (continuum)
• Larger basis covers wider QED spectrum on both ultraviolet and
infrared ends (of the scattering states)
16
General Procedure for BLFQ
1. Derive Hamiltonian from Lagrangian by
Legendre transform
2. Construct BLFQ basis a
3. Form Hamiltonian matrix in BLFQ basis a ' P - a
4. Diagonalize Hamiltonian Pie. solve P - b = Pb- b
5. Evaluate observables O º b Ô b
17
Evaluate Electron g-2 with BLFQ Approach
• Electron anomalous magnetic moment
g-2
ae º
2
• Leading contribution to ae is from QED
[Schwinger 1948]
a æ
1 ö
ae =
ça =
÷
2p è 137 ø
• ae is electron Pauli form factor at zero-moment transfer limit:
ae = F2 (q 2 ® 0)
2
• In BLFQ , ae = ephys F̂2 (q ® 0) ephys
18
Numerical Results for Electron g-2
Major update to: H. Honkanen, P. Maris, J.P. Vary, S.J. Brodsky,
Phys. Rev. Lett. 106, 061603 (2011)
=
1
8p 2
= 0.11254...
ae / e2
•
•
•
As Nmax ∞, results approach Schwinger result
Less than 1% deviation from Schwinger’s result (by linear extrapl.)
Convergence over wide range of ω’s (by a factor of 25!)
19
Electron Parton Distribution Function
• Sum rules ò dxH (x,0,0) = F1 (0) and ò dxE(x, 0, 0) = F2 (0) satisfied
• For H(x,0,0) the peak around x=1 expected from pert. theory
• For E(x,0,0) as basis size increases, results approach pert. theory
20
BLFQ compared to Lattice Gauge Theory
• Complimentary to Lattice Gauge Approach with the
following advantages:
– (Boost invariant) LF-wavefunction is directly accessible
– Convenience in calculating observables such as parton
distribution function
– Time evolution of quantum field configurations can be
straightforwardly calculated
21
Time-dependent Basis Light-front Quantization
• For time-dependence of processes, esp:
– system with strong interaction
– system under strong background field
– system under time-dependent background field
• Numerically evolve quantum field
configurations according to LF Schrodinger eq.
¶
1 +
i + y (x ) = P y (x + )
¶x
2
22
Example: Nonlinear Compton Scattering
• Nonlinear Compton scattering: simplest laser-particle scattering
–
e + ng (laser) ® e'+ g '
• Space-time structure
x-
• Two effects: acceleration and radiation
23
Example: Nonlinear Compton Scattering
• Nonlinear Compton scattering: simplest laser-particle scattering
–
e + ng (laser) ® e'+ g '
• Space-time structure
x-
Laser beam trajectory
• Two effects: acceleration and radiation
24
General Procedure for tBLFQ
1.
2.
3.
4.
Derive the Hamiltonian from the Lagrangian
Switch to the interaction picture
Prepare the initial (‘in’) state
Evolve the initial state through the time-evolution
operator until the background field subsides
5. Project the scattering final state onto ‘out’ states
(constructed
of PQ E D , | β i . out of QED eigenstates) and obtain Smatrix element
−
✓
S = I hout |T+ exp
x +f
i
−
2
Z
◆
VI | i n i I .
(14)
x−
0
Similar t o ‘in’ st at es, ‘out ’ st at es are also superposit ion
of physical part icles and can t hus be const ruct ed out of
A µ (x − )
25
General Procedure for tBLFQ
1.
2.
3.
4.
Derive the Hamiltonian from the Lagrangian
Switch to the interaction picture
Prepare the initial (‘in’) state
Evolve the initial state through the time-evolution
operator until the background field subsides
5. Project the scattering final state onto ‘out’ states
(constructed
of PQ E D , | β i . out of QED eigenstates) and obtain Smatrix element
−
✓
S = I hout |T+ exp
x +f
i
−
2
Z
◆
VI | i n i I .
(14)
x−
0
Similar t o ‘in’ st at es, ‘out ’ st at es are also superposit ion
of physical part icles and can t hus be const ruct ed out of
A µ (x − )
26
Example: Obtain LF QED Hamiltonian
• QED Lagrangian
• Derived Light-front Hamiltonian
kinetic energy terms
vertex
interaction
instantaneous
photon
interaction
instantaneous
fermion
interaction
27
QED in background EM field
• Replace
P
=
P
•
QED +
and obtain
= PQED
+V
28
A Simple Laser Field Profile
• Key properties:
―
―
―
―
is treated classically
is in lightcone gauge,
is uniform in x1,2 and light=front time x+
depends on x: electric field in
longitudinal (x-) direction
― a0 describes the field strength
― l- describes the laser field’s spatial frequency in x29
QED in background EM field
• Replace
P
=
P
•
QED +
and obtain
= PQED
+V
30
General Procedure for tBLFQ
1.
2.
3.
4.
Derive the Hamiltonian from the Lagrangian
Switch to the interaction picture
Prepare the initial (‘in’) state
Evolve the initial state through the time-evolution
operator until the background field subsides
5. Project the scattering final state onto ‘out’ states
of PQ E D , | β i .
(constructed
out of QED eigenstates) and obtain Smatrix element
x
✓
◆
−
S = I hout |T+ exp
i
−
2
Z
+
f
VI | i n i I .
(14)
x−
0
Similar t o ‘in’ st at es, ‘out ’ st at es are also superposit ion
of physical part icles and can t hus be const ruct ed out of
−
A µ (x − )
31
Switch to the Interaction Picture
• Time-dependent term treated as “interaction” part
P - = PQED
+
= PQED
+ V(x + )
¶
1 ¶
1
+
+
+
• i + y (x ) = P y (x ) ® i + y (x ) = VI y (x + )
I
¶x
2
¶x
2
I
• Different splitting from traditional perturb. theory
•
b ' VI (x + ) b is needed, where QED eigenstates b
are found by BLFQ
32
General Procedure for tBLFQ
1.
2.
3.
4.
Derive the Hamiltonian from the Lagrangian
Switch to the interaction picture
Prepare the initial (‘in’) state: a single physical electron
Evolve the initial state through the time-evolution
operator until the background field subsides
5. Project the scattering final state onto ‘out’ states
(constructed
out of QED eigenstates) and obtain Sof PQ E D , | β i .
matrix element
−
✓
S = I hout |T+ exp
x +f
i
−
2
Z
◆
VI | i n i I .
(14)
x−
0
Similar t o ‘in’ st at es, ‘out ’ st at es are also superposit ion
of physical part icles and can t hus be const ruct ed out of
A µ (x − )
33
General Procedure for tBLFQ
1.
2.
3.
4.
Derive the Hamiltonian from the Lagrangian
Switch to the interaction picture
Prepare the initial (‘in’) state: a single physical electron
Evolve the initial state through the time-evolution
operator until the background field subsides
5. Project the scattering final state onto ‘out’ states
(constructed
out of QED eigenstates) and obtain Sof PQ E D , | β i .
matrix element
−
✓
S = I hout |T+ exp
x +f
i
−
2
Z
◆
VI | i n i I .
(14)
x−
0
Similar t o ‘in’ st at es, ‘out ’ st at es are also superposit ion
of physical part icles and can t hus be const ruct ed out of
A µ (x − )
34
General Procedure for tBLFQ
1.
2.
3.
4.
Derive the Hamiltonian from the Lagrangian
Switch to the interaction picture
Prepare the initial (‘in’) state: a single physical electron
Evolve the initial state through the time-evolution
operator until the background field subsides
5. Project the scattering final state onto ‘out’ states
(constructed
of PQ E D , | β i . out of QED eigenstates) and obtain Smatrix element
−
✓
S = I hout |T+ exp
x +f
i
−
2
Z
◆
VI | i n i I .
(14)
x−
0
Similar t o ‘in’ st at es, ‘out ’ st at es are also superposit ion
of physical part icles and can t hus be const ruct ed out of
A µ (x − )
35
Results: Nonlinear Compton Scattering
• Laser profile:
with a0 = 10, l- = 2MeV
• Basis space: Nmax=8, K: 1.5+3.5+5.5 three segments
• Initial state (x+=0): ground state electron in K=1.5 segment
time (x+) evolution
invariant mass (MeV)
invariant mass (MeV)
36
Results: Nonlinear Compton Scattering
Ground states in K=1.5,3.5,5.5 segments Excited states in K=1.5,3.5,5.5 segments
time (x+) evolution
invariant mass (MeV)
invariant mass (MeV)
• Electron is
accelerating
and
radiating
• Entire process is nonperturbative (initial state changes significantly)
37
Invariant Mass of the System
• Invariant mass increase reflects energy injection by laser field
• Experimentally accessible by measuring pme and pgm
38
Conclusion and Outlook
• Basis Light-Front Quantization (BLFQ) approach
– Directly solve for wavefunction of eigenstates of quantum field systems
– Application to QED generates realistic electron wavefunction, which
reproduces the Schwinger result
• Time-dependent Basis Light-Front Quantization (tBLFQ) approach
– Real time formalism in BLFQ allows for extension to time-dependent regime
– Wavefunction (“snapshot”) of system accessible at any intermediate time
– Application to nonlinear Compton scattering simulates the electron
acceleration and radiation coherently in the nonperturbative regime
• Renormalization; construct (multi-)particle “in” and “out” states
• Apply to QCD and study hadron structure (e.g. parton distribution
function) and production in scattering process
• Mesons and baryons projects: underway…
39
Thank you!
40
Backup Slides
41
Under what conditions do we require a quark-based
description on nuclear structure?
“Quark Percolation in Cold and Hot Nuclei”
= 8* -+#) >? +! #)
; " <! '* !
-( $. + &'! %
( ) *+
/ 0120130145016
7 8( $9 : -8! %
+$!
Spin content of the proton
Nuclear form factors
DIS on nuclei – Bjorken x > 1
Nuclear Equation of State
Probes with Q > 1 GeV/c
') %
+$, +&'( %
+ -+) . %
"
7 8( $9 ; " <! '* !
! " #$%&'! %
( ) *+
H.J. Pirner and J.P. Vary,
Phys. Rev. C. 84, 015201(2011);
arXiv: nucl-th/1008.4962
42
J.P. Vary, Proc. VII Int’l Seminar on High Energy Physics Problems,
"Quark Cluster Model of Nuclei and Lepton Scattering Results,"
Multiquark Interactions and Quantum Chromodynamics, V.V. Burov, Ed.,
Dubna #D-1, 2-84-599 (1984) 186 [staircase function for x > 1]
43
See also: Proceedings of HUGS at CEBAF1992, & many conf. proceedings
Comparison between Quark-Cluster Model and JLAB data
Data:
K.S. Egiyan, et al., Phys. Rev. Lett. 96, 082501 (2006)
Theory: H.J. Pirner and J.P. Vary, Phys. Rev. Lett. 46, 1376 (1981)
and Phys. Rev. C 84, 015201 (2011); nucl-th/1008.4962;
M. Sato, S.A. Coon, H.J. Pirner and J.P. Vary, Phys. Rev. C 33, 1062 (1986)
44
Steps to implement BLFQ
• Enumerate Fock-space basis subject to symmetry constraints
• Evaluate/renormalize/store H in that basis
• Diagonalize (Lanczos)
• Iterate previous two steps for sector-dep. renormalization
• Evaluate observables using eigenvectors (LF amplitudes)
• Repeat previous 4 steps for new regulator(s)
• Extrapolate to infinite matrix limit – remove all regulators
• Compare with experiment or predict new experimental results
Above achieved for QED test case – electron in a trap
H. Honkanen, P. Maris, J.P. Vary, S.J. Brodsky,
Phys. Rev. Lett. 106, 061603 (2011)
Improvements: trap independence, (m,e) renormalization, . . .
45
X. Zhao, H. Honkanen, P. Maris, J.P. Vary, S.J. Brodsky, in prep’n
46
Discretized Light Cone Quantization (c1985)
Basis Light Front Quantization*
f ( x ) = å [ fa ( x ) aa+ + f a* ( x ) aa ]
a
where { aa } satisfy usual (anti-) commutation rules.
Furthermore, f a ( x ) are arbitrary except for conditions :
Orthonormal:
Complete:
ò fa ( x) fa ( x ) d x = daa
å fa ( x) fa ( x') = d ( x - x')
*
'
3
*
'
3
a
=> Wide range of choices for f a ( x )and our initial choice is
fa ( x) = Ne
ik + x -
Yn,m (r,j ) = Ne
ik + x -
f n,m (r) c m (j )
*J.P. Vary, H. Honkanen, J. Li, P. Maris, S.J. Brodsky, A. Harindranath, G.F. de Teramond,
P. Sternberg, E.G. Ng and C. Yang, PRC 81, 035205 (2010). ArXiv:0905:1411
Challenges and Solution
• Challenges
– Covariant perturbation theory calculates S-matrix between in- and
out-states with infinite evolution time in-between
– Nontrivial transform between results in BLFQ basis and momentum
basis (often used in perturbative calculation):
1.
2.
3.
Integration over HO wave function needed
Different normalization for basis states,
Kronecker delta (BLFQ basis) vs. Dirac delta (momentuem basis)
Nmax truncation exclusive for BLFQ basis
• Solution -- Lightfront (LF) perturbation theory in BLFQ basis
– Able to calculate transition amplitude per unit time
– Allows for comparison with nonpert. calculation on the level of
transition matrix element of the laser field  ' V L |  , where | 
QED
P
and |  ' are eigenstates of  (adopt the interaction picture)
48
tBLFQ Basis Construction
8
{ K , M j , Nf }
{ K , M j , Nf }
1
3
2
Diagonalise
−
PQED
{ K , M j , Nf }
{ K , M j , Nf }
BLFQ basis state | ↵ i =
= |βi
Diagonalise
tBLFQ basis state
−
PQED
{ K , M j , Nf }
{ K , M j , Nf }
1
| e i ⇠ | k e, me, n e, λ e i
2
| eγ i ⇠ | k e, me, n e, λ e i ⌦| k γ , mγ , n γ , λ γ i
3
k e = K et c
k e+ k γ = K et c
| ephys i
k= K et c
49
Comparison with Perturbation Theory
invariant mass (MeV)
50
Light-front vs Equal-time Quantization
[Dirac 1949]
equal
time dynamics
vs DYNAMICS
light front dynamics19
2 HAMILTONIAN
DYNAMICS
2 HAMILTONIAN
¶
i j (t) = H j (t)
¶t
H b = Eb b
1
¶
1 +
i + j (x ) = P j (x + )
¶x
2
Figure
1: Dirac’s
three forms of Hamilt onian dynamics.
Figure 1: Dirac’s t hree forms
of Hamilt
onian dynamics.
P - b = Pb- b
element sJ are
ed by
t hepart
Jacobian
The two four-volume elementsThe
aretwo
relatfour-volume
ed by t he Jacobian
(x) relat
= ||∂x/
∂x||,
icularlyJ (x) = ||∂x/ ∂x||, particularly
4
d4x t =
J (x)
x.Jacobian
We shallonly
keepimplicit
t rack ly.
of t The
he Jacobian
only implicit ly. The t hree-volum
d4x = J (x) d4x. We shall keep
rack
of tdhe
t hree-volume
element dω0 is t reat ed correspondingly.
element dω0 is t reat ed correspondingly.
t hebe
above
consideratofions
be independent
of t his reparamet rizat ion. Th
All t he above considerationsAll
must
independent
t hismust
reparamet
rizat ion. The
expressions
like t he Lagrangian
be xexpressed
in t erms of eit53
her x or x
fundament al expressions like fundamental
t he Lagrangian
can be expressed
in t erms of can
eit her
or x.
There
is however
one
subt le point
. By
mat t er
convenience one defines the hyperspher
There is however one subt le point
. By
mat t er of
convenience
one
defines
theofhypersphere
Basis Functions for Single Particle States
[Vary et al ’10, Honkanen et al ‘11]
• Optimal basis is chosen to speed up numerical calculation
+
^
+
^
• e; p , p = e; p Ä e; p
• Plane wave basis for longitudinal direction: e; p+ ~ exp(ip+ x- )
• Harmonic oscillator basis for transverse direction: e; p^ ~ e; n, m
– n, m : eigenstates of 2D-harmonic oscillator (HO) of frequency ω
•
e; pe+, pe^ ® e; pe+, ne, me
eg ; pe+, pe^, pg+, pg^ ® eg ; pe+, ne, me, pg+, ng , mg
• In each Fock sector we truncate states with total HO quantum
number beyond Nmax =åi (2ni + mi +1)
– Larger Nmax -> larger basis -> more realistic results
-> numerically more expensive
Outline for BLFQ approach
• Set up Hilbert space by Fock space expansion:
|ephysical ñ = a|eñ+b|eg ñ +c|egg ñ+d|eg ee ñ +¼
• Calculate the Hamiltonian matrix in the Fock Space:
iH j
H = T +V
– T: Kinetic energy term for each particle in each Fock sector
– V: Interaction term coupling different states (and different sectors)
E.g.,
, See Young Li’s talk for more details
• Diagonalize H and obtain eigenvalues and eigenstates
• Extract observables from the eigenstates: ephysical
O º ephysical Ô ephysical
Ô
is the quantum operator for O
Elementary vertices in LF gauge
QED & QCD
QCD
56
57
Time-dependent Basis Light-front Quantization
• A numerical non-perturbative approach for time-dependent
problems in quantum field theory
• Solves the generalized wave-equation for time-evolution of
quantum field configurations j (x + )
Generalized wave-eq.
¶
i + j (x + ) = P+ j (x + )
¶x
BLFQ
tBLFQ
P Fi = P
0
+
0,i
+
Fi
Provide basis
¶
i + j (x + ) I = VI j (x + )
¶x
I
• Works in the interaction picture: P+ (x + ) = P+0 +V(x + )
• Typical applications: strong field laser physics, heavy-ion physics…
59
Solving Nonlinear Compton Scattering in tBLFQ
1. Write down the Hamiltonian P+:
P+ (x + ) = P+QED +V LAS (x + )
2. Solve P+QED Fi = P+i Fi for the tBLFQ basis Fi
3. Prepare initial state j (0)
- physical electron: the ground state of P+QEDwith nf=1
4. Calculate matrix elements for VLAS
Fj V
LAS
+
(x ) Fi
I
=e
i( P+j -P+i )x +
F j V LAS (x + ) Fi
5 . Solve for the generalized wave-equation numerically
¶
i + Fi j (x + ) I = å Fi V LAS F j
¶x
j
I
F j j (x + )
x
æ
ö
Y(x ) I = U(x ,0) Y(0) I = T exp ç -i ò VIL (x + ')dx + '÷ Y(0)
è 0
ø
+
+
I
+
® (1- iVIL (x + )Dx + ) (1- iVIL (x2+ )Dx + )(1- iVIL (x1+ )Dx + ) Y(0)
I
I
60
Nonpert. Vs Pert. Laser Matrix Element
Nonperturbative evaluation
1. Diagonalize PQED for |  ' |  in BLFQ basis
2. Compute V L in BLFQ basis
3. Sandwich V L with |  ' |  and obtain
' VL | 
nonpert
' V L | 
Perturbative evaluation
1. Diagonalize Pkinetic for |  ' |  in BLFQ basis
2. Convert|  ' |  to momentum basis
3. Evaluate |  ' |  from |  ' |  using LF
perturbation theory
0
0
0
0
0
0

1
Q
|  '  1 
V
 | 0 '
kinetic

'

P

i
0
0




1
Q
|   1 
V
 | 0
kinetic


P

i
0
0




These calculations are actually done in
momentum basis (no basis truncation).
4. Sandwich V L with |  '
' VL |
pert
 0 ' V Q

|
and keep terms of leading order in V L , V Q :
1
 ' P
kinetic
0
 i 0
V L |  0   0 ' V L*
1
 P
kinetic
0
 i 0
V Q | 0

61
Evolution of Invariant Mass of the System

1
, a0  0.5me
13700
Nmax=24
Nmax=16
Nmax=8

1
, a0  0.5me
13700

1
, a0  0.5me
13700
• Invariant mass increases with time as laser field “pumps”
energy in
• As Nmax increases better agreements are achieved between
calculations based on laser matrix elements from LF. pert. and
nonpert. methods, intermediate truncation effects are
removed gradually in the nonperturbative case
• Quasi-linear dependence on x+ is expected in the perturbative
regime
62
Evolution of Excited States for Nmax=24
x+=10MeV-1

1
, a0  0.5me
13700
x+=80MeV-1
x+=40MeV-1

1
, a0  0.5me
13700

1
, a0  0.5me
13700
• Evolution of all excited states in the basis are tracked
• Excited states are being populated as time increases
– Decent agreement between nonpert. and lf. pert. laser matrix elements
• Peak structure emerges for transitions conserving energy
–
–
–
–
Only transitions conserving (light-front) energy keep increasing with time
Transitions not conserving (light-front) energy oscillate with time
Peak location agrees with covariant perturbation theory
63
Peak width consistent with energy-time uncertainty principle
Application to Scattering Process
of PQ− E D , | β i .
x0
• S-matrix
✓
S = I hout |T+ exp
x +f
i
−
2
Z
◆
VI | i n i I .
(14)
x−
0
−
1.
Construct
“in”
state
out
of
QED
eigenstates
A µ (xand
) use
Similar t o ‘in’ st at es, ‘out ’ st at es are also superposit ion
y t(0)
= in
as initial
of physical
part iclesstate
and can
husI be
const
I ruct ed out of
t he eigenst at es of PQ E D , | β i .
y (0) until the background field subsides at x +
2.
Evolve
f
[Combine red and blue t ext int o somet hing short and
sexy: comparison
between
we do andfinal
what state
is nor- y (x +f )
and obtain
thewhat
scattering
mally done, in pert . t heory.]
+
y
(x
) onto t“out”
3. usProject
states
out of
Let
compare t hisf approach
o t he usual
scat t(constructed
ering
−
approach in pert urbat ion t heory. T here, t he init ial st at e
e
QED eigenstates) and obtain S-matrix element
would be an asympt ot ic st at e, describing free part icles at
−
x + = − 1 , under t he usual assumpt ion t hat t he coupling
swit ches o↵ at large t imes. T his ‘in’ st at e, | i i would be
evolved through all t ime, from t he infinit e past t o t he infinit e fut ure. T he evolved st at e would t hen be project ed
FIG. 1. An illust rat ion of
elect ron ent ers a backgrou
emit s a phot on. A ft er em
accelerat ed unt il it 64leaves t