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

Statistical physics and light-front quantization J¨ org Raufeisen (Heidelberg U.) • JR and S.J. Brodsky, Phys. Rev. D 70, 085017 (2004) and hep-th/0409157

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 = 2 x − ): 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 P + and � them are � J 3 , � K 3 , � P ⊥ . • The Light Front Hamiltonian � P − , of course, contains interaction terms. J¨ org Raufeisen, Joint Meeting Heidelberg-Li` ege-Paris-Rostock in Spa, Belgium, December 2004

The QCD Phase Diagram (according to MIT) T • So far, all of light-front quantization (figure: M. Alford) takes place at zero temperature and zero density. RHIC QGP However, there is much more: = color superconducting • RHIC (and LHC) probe strongly interacting matter under conditions similar to those in crystal? the early universe. CFL • The region with large baryon chemical potential and gas liq low T cannot be explored in the lab, but the core of neutron stars might be color superconducting. µ compact star nuclear • 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: – � K 3 independent of interaction → frame independent distribution functions – Fewer problems with fermions in numerical approaches (DLCQ) J¨ org Raufeisen, Joint Meeting Heidelberg-Li` ege-Paris-Rostock in Spa, Belgium, December 2004

The Statistical Operator � w odinger equation, ı∂ − | Ψ � = � P − | Ψ � , is replaced by the • In statistical physics, the light-front Schr¨ Light-Front Liouville Theorem : � � i∂ − � � P − , � w = w . For pure states � w = | h �� h | . • In equilibrium, [ � P − , � w ] = 0 → e generators that commute with � P − . w is a function of those Poincar´ – � – The entropy operator ln � w is a linear function of these generators. • Thus � � � P µ − ω � − β ( u µ � µ i � w = exp � J 3 − Q i ) i – u µ is the four velocity of the system and ω its angular velocity. – µ i and � Q i are chemical potentials and conserved charges, respectively. P µ − ω � • In thermodynamics, one usually works in the rest frame, so that u µ � J 3 = � P 0 . P µ − ω � • There is no frame with u µ � J 3 = � P − . Thus, the equal time energy is special among the ten Poincar´ e generators, even on the light front. J¨ org Raufeisen, Joint Meeting Heidelberg-Li` ege-Paris-Rostock in Spa, Belgium, December 2004

Statistical Ensembles • The microcanonical ensemble, � � � MC ∝ δ ( P + − i ) δ ( P − − w 0 p + p − i ) δ (2) ( P ⊥ − p i, ⊥ ) � i i i is appropriate for small systems, such as single hadrons. • The (grand) canonical ensemble (let � P ⊥ = � 0 ⊥ ), � � �� � � � � u + M 2 2 P + + u − h p + φ n/h ( X ) φ ∗ n ′ /h ( X ′ ) | nX �� n ′ X ′ | w = exp − β i − µQ � 2 i h n,n ′ X,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 φ n/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¨ org Raufeisen, Joint Meeting Heidelberg-Li` ege-Paris-Rostock in Spa, Belgium, December 2004

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, � ∂F � S = − . ∂T V • Equilibrium conditions: The five independent parameters T, u κ , µ are Lagrange multipliers that keep the mean value of a quantity constant, while entropy is maximized. – The values of T, u κ , 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 , µ (1) = µ (2) . = u (2) κ 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 ( u µ k µ ) = [exp( βu κ k κ − βµ ) ± 1] − 1 − [exp( βu κ k κ + βµ ) ± 1] − 1 . J¨ org Raufeisen, Joint Meeting Heidelberg-Li` ege-Paris-Rostock in Spa, Belgium, December 2004

Light-Front Quantization of the Fermi Field • The 2-component fermion field operators in the Schr¨ odinger picture are expanded as � � d † ( k, λ ) χ − λ e + ık · r � � d 3 k b ( k, λ ) χ λ e − ık · r + � � � k + Θ( k + ) √ Ψ( r ) = , (2 π ) 3 2 λ � � − λ e − ık · r � � d 3 k λ e + ık · r + � � k + Θ( k + ) � b † ( k, λ ) χ † d ( k, λ ) χ † Ψ † ( r ) √ = , (2 π ) 3 2 λ r ⊥ ) , k = ( k + ,� with σ 3 χ λ = λχ λ , r = ( r − ,� k ⊥ ) . • The creation and annihilation operators obey the anticommutation relations � � � � � b ( k, λ ) , � d ( k, λ ) , � � b † ( k ′ , λ ′ ) d † ( k ′ , λ ′ ) = (2 π ) 3 2 k + δ (3) ( k − k ′ ) δ λ,λ ′ , = so that the anticommutator of the dynamical spinor components at equal light-cone time r + reads α, β ∈ { 1 , 2 } � � Ψ α ( r ) , � � Ψ † = δ α,β δ (3) ( r − r ′ ) . β ( r ′ ) • The entire theory can be formulated in terms of 2-component spinors, but is non-local along the light-cone. J¨ org Raufeisen, Joint Meeting Heidelberg-Li` ege-Paris-Rostock in Spa, Belgium, December 2004

In Medium Green’s Functions of a Fermion • The (time-ordered) Green’s function is defined in terms of Heisenberg field operators as � T + � ψ α ( r 1 ) � ψ † ıG α,β ( r 1 , r 2 ) = β ( r 2 ) � � � ψ α ( r 1 ) � ψ † β ( r 2 ) � Θ( r + 1 − r + 2 ) − � � ψ † β ( r 2 ) � ψ α ( r 1 ) � 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 � � ψ α ( r 1 ) , � � ıG R ψ † � Θ( r + 1 − r + α,β ( r 1 , r 2 ) = � β ( r 2 ) 2 ) � � ψ α ( r 1 ) , � � ψ † ıG A � Θ( r + 2 − r + α,β ( r 1 , r 2 ) = −� β ( r 2 ) 1 ) . • The free Green’s functions in momentum space read k + G (0) R,A � ( k ) = δ α,β k 2 − m 2 ± ı 0sgn( uk ) , α,β � uk � � � k + k + δ ( k 2 − m 2 ) G (0) � α,β ( k ) = δ α,β P k 2 − m 2 − ı sgn( uk ) π tanh . 2 T • There is only one derivative in the numerator, leading to only one pair of fermion doublers on the lattice. J¨ org Raufeisen, Joint Meeting Heidelberg-Li` ege-Paris-Rostock in Spa, Belgium, December 2004

Separation of Quark and Antiquark Distributions • Knowledge of the Green’s function ıG α,β ( r 1 , r 2 ) = � � ψ α ( r 1 ) � 2 ) − � � β ( r 2 ) � ψ † β ( r 2 ) � Θ( r + 1 − r + ψ † ψ α ( r 1 ) � Θ( r + 2 − r + 1 ) enables one to calculate all quantities pertaining to a single particle. • In the limit r + → 0 ± , G α,β ( r 1 , r 2 ) yields the one-particle density matrices for quarks and antiquarks, from which the expectation value of any single-particle operator can be calculated, � � � f (1) d 3 r 1 d 3 r 2 � δ (3) ( r 1 − r 2 ) q α,β ( r 1 , r 2 ) + q α,β ( r 1 , r 2 ) β,α � � F α,β ( r + ) � = � � � , d 3 r q α,α ( r, r ) + q α,α ( r, r ) � with � d 3 r � α ( r ) � f β,γ � F α,β ( r + ) = ψ † ψ γ ( r ) . • The density matrix for quarks is given by ( R = ( r 1 + r 2 ) / 2 , r = r 1 − r 2 ) : � � r ⊥ ) = 1 β ( R + r ψ α ( R − r dr − e + ıp + r − / 2 � � ψ † 2) � � q α,β ( p + , R,� 2) � r + 1 = r + 4 π 2 • Similarly for antiquarks, � r ⊥ ) = ı dr − e + ıp + r − / 2 G α,β ( r + 1 → 0 + , r 1 , r + q α,β ( p + , R,� 2 = 0 , r 2 ) . 4 π J¨ org Raufeisen, Joint Meeting Heidelberg-Li` ege-Paris-Rostock in Spa, Belgium, December 2004