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Destruction of Vacuum and Seeing Through Milk

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Destruction of Vacuum and Seeing Through Milk Conceptual problems in QED Part I: Polarization density of the vacuum (photon = classical field) Part II: Bare, physical particles, renormalization (photon = independent particle ) Rainer Grobe Intense Laser Physics Theory Illinois State University Coworkers Prof. Q. Charles Su Dr. Sven Ahrens QingZheng Lv Andrew Steinacher Jarrett Betke Will Bauer Questions Can we visualize the dynamics of QED interactions with space-time resolution? Is this picture correct? Relationship between virtual and real particles? - Dynamics of virtual particles? Peskin Schroeder Fig. 7.8 + Greiner Fig. 1.3a Quantum mechanics: (1) know: f(x,t=0) and h(t) (2) solve: i ∂t f(x,t) = h(t) f(x,t) for the initial state ONLY (3) compute: observables f(t) O f(t) Quantum field theory: (1) know: F(t=0) and H(t) (2) solve: i ∂t fE(x,t) = H(t) fE(x,t) for EACH state fE(x) of ENTIRE Hilbert space (3) compute: observables F(t=0) O( all fE(x,t) ) F(t=0) Charge r(z, t) = F(t=0) – [Y†(z,t), Y(z,t)]/2 F(t=0) density r(z,t) charge operator † 1. example: single electron F(t=0) = bP vac 2 2 2 r(z, t) = – fP(+;z,t) + (SE(+) fE(+;z,t) – SE(–) fE(–;z,t) ) /2 electron’s wave function vacuum’s polarization density (trivial) (not understood) t=0: h f = E f 0 E E fE(-) fE(+) –mc2 mc2 t>0: i ∂t fE(t) = h fE(t) Quick overview (1) Computational approach Steady state vacuum polarization r(z) Space-time evolution of r(z,t) Steady state and time averaged dynamics (2) Analytical approaches Phenomenological model Decoupled Hamiltonians Perturbation theory (3) Applications Coupling r to Maxwell equation Relevance for pair-creation process Relationship to traditional work 2. Example: r(z) for the dressed vacuum state VAC † rpol(z) = VAC – [Y (z), Y(z)]/2 VAC Dirac equation 2 [c s1 pz + mc s3 + Vext (z)] FE(z) = E FE(z) 2 2 rpol(z) = (SE(+) FE(+;z) – SE(–) FE(–;z) ) /2 Width w of external potential Vext(z) determines r(z) V (z) = V exp[-(z/w)2] ext 0 qext (z) - - + + + + - - z w -3 r(z) lCompton= 1/c = 7 10 r(z) w < l -4 l > w r(z) C 10 C -3 10 w=lC/2 w=lC/8 r(z) ~ w exp[-c|z|] r(z) ~ V(z) 3. Example: Dynamics of the polarization density † rpol(z, t) = bare vac – [Y (z,t), Y(z,t)]/2 bare vac 2 2 rpol(z, t) = (SE(+) fE(+;z,t) – SE(–) fE(–;z,t) ) /2 fE(-) fE(+) –mc2 mc2 time-dependent Dirac equation: 2 i ħ ∂t fE(z,t) = [c s3 pz + mc s3 + V(z)] fE(z,t) Temporal evolution of r(x,t) Vext(x) rpol(x,t) movies -1 T rsteady(x) = T ∫ dt r(x,t) Quick overview (1) Computational approach Steady state vacuum polarization r(z) Space-time evolution of r(z,t) Steady state and time averaged dynamics (2) Analytical approaches Phenomenological model Decoupled Hamiltonians Perturbation theory (3) Applications Coupling r to Maxwell equation Relevance for pair-creation process Relationship to traditional work Maxwell Dirac qext(z) Vext(z) rpol(z) Phenomenological model for rpol(z,t) 2 2 3 2 (∂ct –∂z ) rpol(z,t) = 8p c qext(z) c = a /(2p lC ) (guess) exact solution: rpol(z,t) = c [ 2Vext(z) – Vext(z-ct) – Vext(z+ct) ] jpol(z,t) = c c[ Vext(z+ct) – Vext(z–ct) ] if width of Vext > lC => predictions for rpol(z, t) are highly accurate rpol(z,t) and jpol(z,t) for an external point charge ✚ qext(z) = q d(z) r(z,t) = c [2V(z) –V(z-ct) – V(z+ct)] Vext(z) = – 2p q |z| r (z,t) pol exact 75 m odel t3 j(z,t) = c c[V(z+ct) – V(z–ct)] 50 j (z,t) pol t t 3 exact 2 10000 m odel 25 t2 5000 t1 t1 0 ✚ 0 ✚ t -0.5 -0.25 0 0.25 0.5 -5000 1 z [a.u.] t2 -10000 charge conservation ✓ t3 lim r(z=0,t) ∞ ✓ -0.5 -0.25 0 0.25 0.5 L, t∞ z [a.u.] j(z) grows everywhere ✓ Decoupled Hamiltonian model 2 4 2 2 1/2 H(+) = [m c +c p ] + Vext(z) => bound states 2 4 2 2 1/2 H(–) = [m c +c p ] – Vext(z) => scattering states 2 2 rpol(z, t) = (SE(+) fE(+;z,t) – SE(–) fE(–;z,t) ) /2 predictions for rpol(z, t) are highly accurate => transitions between positive and negative Dirac states irrelevant Traditional perturbation theory (1) YE = fE + SE’ fEVextfE / (E-E’) fE + ... (1) YE = fE – SE’ fEVextfE / (E-E’) fE + ... (1) 2 (1) 2 rpol(z, t) = (SE YE (z) – SE YE (z) ) /2 predictions for rpol(z, t) are highly accurate => perturbative approach applicable also to 2 and 3D ? Intermediate summary 2 2 3 2 (∂ct – ∂z ) rpol(z,t) = 8p c qext(z) with c = a /(2p lC ) 4 independent approaches: (steady, dynamics, phenom, decoupled hamiltonian) predict: massless virtual positive particles accumulate around positive charges => Is the energy conserved? => What if real particles are created in addition? => Consistent with traditional QED methods? Quick overview (1) Computational approach Steady state vacuum polarization r(z) Space-time evolution of r(z,t) Steady state and time averaged dynamics (2) Analytical approaches Phenomenological model Decoupled Hamiltonians Perturbation theory (3) Applications Coupling r to Maxwell equation Relevance for pair-creation process Relationship to traditional work Coupled Dirac-Maxwell equation 2 2 rpol(z, t) = (SE(+) fE(+;z,t) – SE(–) fE(–;z,t) ) /2 Dirac equation: 2 i ∂t fE(z,t) = [c s1 [pz – A(z,t)/c] + mc s3 + V(z,t)] fE(z,t) Maxwell equation: 2 2 [∂ct – ∂z ] V(z,t) = 4p r(z,t) 2 2 [∂ct – ∂z ] A(z,t) = 4p j(z,t)/c Energy conservation Etot = Emat(t) + Eint(t) + Efield(t) † 2 Emat (t) = ∫ dz Y (z,t) {c s1 p + s3 mc } Y(z,t) † Eint (t) = q ∫ dz Y (z,t) {V(z,t) – s1 A(z,t)} Y(z,t) -1 2 2 Efield (t) = (8p) ∫ dz { [∂ct A(z,t)] – [∂zV(z,t)] } Temporal gauge: † Eint (t) = – q ∫ dz Y (z,t) s1 A(z,t) Y(z,t) -1 2 Efield (t) = (8p) ∫ dz E (z,t) Energy is conserved despite qpol ∞ qext(x) rpol(x,t) + – Efield Efield + Emat + Eint Emat+ Eint Pair creation regime: rpol = rvac + re-e+ r (z,t) /V 0.001 pol 0 virtual electrons 0.0005 V = 4c2 3c2 2c2 0 – – – – – – 0 ✚- ✚ - -0.0005 +++++++ exact real -0.001 m odel positrons -0.4 -0.2 0 0.2 0.4 z [a.u.] real particles reduce the total charge density induced and real particles obey opposite “force laws” More quantitatively: mass density of real particles Y(z,t) Y(e–) + C Y(e+) m(e-; z, t) vac Y†(e-)Y(e-) vac + † + + effective charge density m(e ; z, t) vac Y (e )Y(e ) vac reff (z) 40 -------- reff(z) + – r (z,t) – r (z,t) m(e ; z) – m(e ; z) pol model 20 r+ (z) ––r (z) r(e ;z,t)tot – r(e ;z,t)vac 0 -20 -40 -0.4 -0.2 0 0.2 0.4 z [a.u.] Traditional pert. QED approach: qext(r) = e d(z) Peskin-Schroeder Eq. 7.93 -2 4 –3 4 2 -1/2 4 2 –1/2 P2 = 4a [k – 4c k (4c -k ) arctan[k(4c -k ) ]] necessary: regularization (P-V or dim.) and charge renormalization + -1 2 -2 -1 + + Eint(z) = (2p) ∫ dk exp(ikz) e k [1 – P2] + ++ + + + + ≅ 2 2 3 ++ + – 2p e z – a e 4p/3 z + ... + + + + + + + + + + + Coulomb vac. pol. correction + + + z Traditional pert. QED approach: 2 2 3 Eint(z) ≅ – 2p e z – a e 4p/3 z + ... Coulomb vac. pol. correction Possible connection to our approach: – ∂2 V (z) = 4p q (z) z ext ext => Vext(z) = – 2p e z ✓ 2 – ∂ z rpol(z) = 8p c qext(z) => rpol(z) = – 4p e c z 2 2 3 + ∂ z Vpol(z) = 4p rpol(z) => Vpol(z) = – 8p /3 e c z = – a e 4p/3 z3 ✓ regularization and charge renormalization NOT necessary (Too) many open questions.... qext(z) = d(z) => infinite plane: lim t∞ r(z,t)∞ 1D ≠ 3D with spatial symmetry (relativity) implications for 2D and 3D: orpol = 8p c qext(r) ?? more contact with traditional methods experimental implications ..... Q.Z. Lv, J. Betke, W. Bauer, Q. Su and R. Grobe, Phys. Rev. Lett. (in preparation) A. Steinacher, J. Betke, S. Ahrens, Q. Su and R. Grobe, Phys. Rev. A 89, 062016 (2014). A. Steinacher, R. Wagner, Q. Su and R. Grobe, Phys. Rev. A 89, 032119 (2014). Conceptual problems in QED Part I: Polarization density of the vacuum (photon = classical field) Part II: Bare, physical particles, renormalization (photon = independent particle ) 5 drawbacks of the S matrix (1) T is built in (2) ds/dw is rate based (3) usually only perturbative (4) no spatial information (5) black box approach What happens inside the interaction zone? The challenge: study QED interactions with space-time resolution ⊙ construct Hamiltonian H ⊙ evolve Y(t=0) to Y(t) by solving i /t Y(t) = H Y(t) ⊙ convert Y(t) into observables ? The problems: ☹ Hilbert space is gigantic ☹ Hamiltonian is “wrong” and requires serious repair ☹ correct physical operators are unknown Electron – positron – photon interactions bp dp ap H0 = S dp Ep bp bp + Sdp Ep dp dp + Sdp wp ap ap Hint = Sdp dk ….
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