[PPT] - Statistical physics and light-front quantization J org Raufeisen PowerPoint Presentation

SLIDE 1

Statistical physics and light-front quantization

J¨

rg Raufeisen (Heidelberg U.)
JR and S.J. Brodsky, Phys. Rev. D70, 085017 (2004) and hep-th/0409157

SLIDE 2

Introduction: Dirac’s Forms of Hamiltonian Dynamics

Front form: Define initial conditions on

a light-like hypersurface. The evolution in light-cone time x+ is then generated by the Poincare generator P −, i∂−|Ψ = P −|Ψ

Scalar product (x+ = t + z = 2x−):

x·p = 1 2(x+p−+x−p+)− x⊥ p⊥.

Mass shell condition: P +P − −

P 2

⊥ = M 2

In the Front Form, 7 out of 10 Poincare generators are independent of the interaction, among

them are J3, K3, P + and P⊥.

The Light Front Hamiltonian

P −, of course, contains interaction terms.

J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 3

The QCD Phase Diagram (according to MIT)

CFL

liq

QGP T µ

crystal? nuclear

gas

superconducting = color

compact star RHIC

So far, all of light-front quantization

(figure: M. Alford) takes place at zero temperature and zero density. However, there is much more:

RHIC (and LHC) probe strongly interacting matter

under conditions similar to those in the early universe.

The region with large baryon chemical potential and

low T cannot be explored in the lab, but the core of neutron stars might be color superconducting.

In addition, systems such as the wavefunction of a large nucleus may also have statistical
features. (Iancu et al. hep-ph/0410018)
Advantage of the Front Form:

– K3 independent of interaction → frame independent distribution functions – Fewer problems with fermions in numerical approaches (DLCQ)

J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 4

The Statistical Operator w

In statistical physics, the light-front Schr¨
dinger equation, ı∂−|Ψ =

P −|Ψ, is replaced by the Light-Front Liouville Theorem : i∂− w =

P −,

w

.

For pure states w = |hh|.

In equilibrium, [

P −, w] = 0 → – w is a function of those Poincar´ e generators that commute with P −. – The entropy operator ln w is a linear function of these generators.

Thus
w = exp
−β(uµ

P µ − ω J3 −

i

µi Qi)

– uµ is the four velocity of the system and ω its angular velocity.

– µi and Qi are chemical potentials and conserved charges, respectively.

In thermodynamics, one usually works in the rest frame, so that uµ

P µ − ω J3 = P 0.

There is no frame with uµ

P µ − ω J3 = P −. Thus, the equal time energy is special among the ten Poincar´ e generators, even on the light front.

J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 5

Statistical Ensembles

The microcanonical ensemble,
w0

MC ∝ δ(P + −

i

p+

i )δ(P − −

i

p−

i )δ(2)(P⊥ −

i

pi,⊥) is appropriate for small systems, such as single hadrons.

The (grand) canonical ensemble (let

P⊥ = 0⊥),

w =
h
n,n′
X,X′

exp

−β
u+ M 2

h

2P + + u− 2

i

p+

i − µQ

φn/h(X)φ∗

n′/h(X′)|nXn′X′|

is appropriate for large systems, such as nuclei and neutron stars. Here: h=Hamiltonian eigenstate, n=Fock-state number, X=all other variables.

Given a set of LC wavefunctions φn/h(X) (obtained from DLCQ), one can calculate any

expectation value.

Momenta and charges are extensive quantities with a conjugate intensive quantity.

This is not the case for M 2

h and the LC momentum fractions x. J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 6

Thermodynamics

All known thermodynamic relations hold in Light-Front Quantization, e.g.

F = −T ln Tr w.

Defined this way, the Free Energy F is a Lorentz scalar.
So is the entropy,

S = − ∂F ∂T

V

.

Equilibrium conditions: The five independent parameters T, uκ, µ are Lagrange multipliers that

keep the mean value of a quantity constant, while entropy is maximized. – The values of T, uκ, µ are independent of the system size. – Two systems (1) and (2) are in equilibrium with each other, if T (1) = T (2), u(1)

κ

= u(2)

κ , µ(1) = µ(2).

No macroscopic motion is possible in equilibrium. (At least in the absence of vortices.) – The entropy of an ideal gas is maximized for Bose-Einstein and Fermi-Dirac distributions, n(uµkµ) = [exp(βuκkκ − βµ) ± 1]−1 − [exp(βuκkκ + βµ) ± 1]−1 .

J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 7

Light-Front Quantization of the Fermi Field

The 2-component fermion field operators in the Schr¨
dinger picture are expanded as
Ψ(r)

=

λ
d3k

(2π)32 √ k+ Θ(k+)

b(k, λ)χλe−ık·r +

d†(k, λ)χ−λe+ık·r ,

Ψ†(r)

=

λ
d3k

(2π)32 √ k+ Θ(k+)

b†(k, λ)χ†

λe+ık·r +

d(k, λ)χ†

−λe−ık·r

, with σ3χλ = λχλ, r = (r−, r⊥), k = (k+, k⊥).

The creation and annihilation operators obey the anticommutation relations
b(k, λ),

b†(k′, λ′)

=
d(k, λ),

d†(k′, λ′)

= (2π)32k+δ(3)(k − k′)δλ,λ′,

so that the anticommutator of the dynamical spinor components at equal light-cone time r+ reads α, β ∈ {1, 2}

Ψα(r),

Ψ†

β(r′)

= δα,βδ(3)(r − r′).
The entire theory can be formulated in terms of 2-component spinors, but is non-local along

the light-cone.

J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 8

In Medium Green’s Functions of a Fermion

The (time-ordered) Green’s function is defined in terms of Heisenberg field operators as

ıGα,β(r1, r2) = T+ ψα(r1) ψ†

β(r2)

= ψα(r1) ψ†

β(r2)Θ(r+ 1 − r+ 2 ) −

ψ†

β(r2)

ψα(r1)Θ(r+

2 − r+ 1 ).

The average . . . has to be taken with the appropriate ensemble.

In addition, retarded and advanced Green’s functions are defined as the anticommutators

ıGR

α,β(r1, r2)

=

ψα(r1),

ψ†

β(r2)

Θ(r+

1 − r+ 2 )

ıGA

α,β(r1, r2)

= −

ψα(r1),

ψ†

β(r2)

Θ(r+

2 − r+ 1 ).

The free Green’s functions in momentum space read
G(0)R,A

α,β

(k) = δα,β k+ k2 − m2 ± ı0sgn(uk),

G(0)

α,β(k)

= δα,β

P

k+ k2 − m2 − ısgn(uk)πtanh uk 2T

k+δ(k2 − m2)
.
There is only one derivative in the numerator, leading to only one pair of fermion doublers on

the lattice.

J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 9

Separation of Quark and Antiquark Distributions

Knowledge of the Green’s function

ıGα,β(r1, r2) = ψα(r1) ψ†

β(r2)Θ(r+ 1 − r+ 2 ) −

ψ†

β(r2)

ψα(r1)Θ(r+

2 − r+ 1 )

enables one to calculate all quantities pertaining to a single particle.

In the limit r+ → 0±, Gα,β(r1, r2) yields the one-particle density matrices for quarks and

antiquarks, from which the expectation value of any single-particle operator can be calculated, Fα,β(r+) =

d3r1d3r2

f(1)

β,α

qα,β(r1, r2) + qα,β(r1, r2)
δ(3)(r1 − r2)
d3r
qα,α(r, r) + qα,α(r, r)
,

with Fα,β(r+) =

d3r

ψ†

α(r)

fβ,γ ψγ(r).

The density matrix for quarks is given by (R = (r1 + r2)/2, r = r1 − r2) :

qα,β(p+, R, r⊥) = 1 4π

dr−e+ıp+r−/2

ψ†

β(R + r

2) ψα(R − r 2)

r+

1 =r+ 2

Similarly for antiquarks,

qα,β(p+, R, r⊥) = ı 4π

dr−e+ıp+r−/2Gα,β(r+

1 → 0+, r1, r+ 2 = 0, r2). J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 10

The Light-Front Density Matrix and GPDs

qα,β(p+, R,

r⊥) contains all information about unpolarized (q), longitudinally polarized (∆Lq) and transverse spin (∆T q) distributions, qα,β(p+, R, r⊥) = q(p+, R, r⊥)δα,β + ∆Lq(p+, R, r⊥)σ3

α,β

+∆Tq(1)(p+, R, r⊥)σ1

α,β + ∆T q(2)(p+, R,

r⊥)σ2

α,β.

There are actually 8 independent density matrices in QCD, 2 of them T-odd.
The probability distribution for quarks is obtained by setting

r⊥ = 0⊥.

qα,β(p+, R,

r⊥ = 0⊥) contains all information about the impact parameter dependence of PDFs, with b = R⊥. These can be identified with GPDs at ζ = 0. (M. Burkardt)

The quark density matrix contains terms that are off-diagonal in Fock-space (b†d† and bd.)

These contribute to DVCS in different kinematical domains,

k, λ k′, λ′ γ∗(q) γ(q′) P P ′ ζ ≤ X ≤ 1 0 ≤ X ≤ ζ k, λ k′, λ′ γ∗(q) γ(q′) P P ′

J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 11

Covariant Introduction of a Chemical Potential

In equal-time statistics at finite density, Heisenberg operators are defined using the Hamiltonian
P0 =

P 0 − µ Q.

In light-front statistics, the effective Hamiltonian is
H = uκ

P κ − µ Q. This operator propagates the system along its worldline.

In equilibrium, uκ is a constant independent of time and position. The translation generators,

that propagate the along trajectories with constant Q are then

Pκ =

P κ − µ Quκ. The chemical potential modifies the translation generators like a gauge field.

We therefore define Heisenberg operators as,
ψα(r)

= eı b

P−r+/2

Ψα(r)e−ı b

P−r+/2,

ψ†

α(r)

= eı b

P−r+/2

Ψ†

α(r)e−ı b P−r+/2. J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 12

Calculation of Thermodynamic Quantities

Light-front QCD at finite temperature and density combines apparently different fields of

physics.

The Green’s function is not only related to PDFs in the limit r+ → ±0, one can also calculate

thermodynamic quantities from G: – Obtain the net charge Q = q − q as a function of µ, T and V from G. – Integrate ∂Ω ∂µ

V,T

= −Q to obtain the grand-canonical potential.

Given a complete DLCQ solution of a theory, one knows the entire set of light-cone
wavefunctions. This would enable one to also calculate all finite temperature properties of that

theory.

Advantage of DLCQ over Lattice: No problems with dynamical fermions or finite µ.
That makes DLCQ the only known first principle approach applicable to high density field

theory.

J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004

SLIDE 13

Summary

Light-Front quantization has been generalized to finite temperature and density.
Even on the Light-Front, the statistical operator is the exponential of the equal time energy in

the rest frame of the system,

w = exp
−β(uκ

P κ − ω J3 − µ Q)

.

The special meaning of the equal-time energy follows directly from the Poincar´ e algebra.

At T = 0, the System is not in the state with lowest P − (important for SSB.)
The formulation of the theory in terms of the Green’s functions is a new way of looking at

light-front quantization at T = 0 and at T = 0.

We interpret GPDs as light-front density matrices. This is a natural quantum-mechanical

generalization of the parton model. Especially for the wavefunction of a large nucleus, a statistical approach makes sense.

We introduce the chemical potential in a Lorentz-invariant way.
Thermodynamic quantities are accessible in DLCQ, which can be applied even at µ ≫ T.

However, DLCQ yields much more information than needed.

J¨

rg Raufeisen, Joint Meeting Heidelberg-Li`

ege-Paris-Rostock in Spa, Belgium, December 2004