Recent progress in reducible QED Jan Naudts Universiteit Antwerpen, Belgium Torun, June 2017 ◮ ... a rigorous quantum field theory ◮ ... new mathematical results, lengthy calculations ◮ ... 3 claims, partially supported by the mathematics CLAIM: Emergence of the Coulomb forces CLAIM: Convergent S-matrix expansion CLAIM: Existence of the wave function of the vacuum 1/ 17
Outline I Reducible QED II Emergence of the Coulomb forces III Bound states IV The S-matrix V The wave function of the vacuum state 2/ 17
I Reducible QED ◮ Formalism ◮ History ◮ World view (limited to QED) WARNING: This is not the standard formalism of QED Abandon the axiom that representations of the canonical (anti-)commutation relations are irreducible. The reducible representation is an integral of irreducible ones. Objection: Reducible representations are partly classical. 3/ 17
Formalism ◮ Reduction is done by integration over the wave vector k in R 3 . ◮ For a given wave vector k the system is purely quantum mechanical: ◮ the wave vector k labels the irreducible representation. ◮ Operators X k act in the k -th Fock space. ◮ Wave functions ψ k belong to Fock space. They are properly normalized: || ψ k || = 1 for each wave vector separately! ◮ Field operators depend on both x and k . ◮ ˆ A α ( x ) expressed in terms of creation and annihilation operators ◮ 2 degrees of freedom (transverse photons only) ◮ Dirac currents ˆ j µ ( x ) defined by the free Dirac equation 4/ 17
For instance 1 � � e − ik ph a H + e ik ph ˆ 2 λε ( H ) µ x µ ˆ µ x µ ˆ a † A α ( x ) = α ( k ph ) H + 1 � � e − ik ph a V + e ik ph 2 λε ( V ) µ x µ ˆ µ x µ ˆ a † α ( k ph ) . V k ph is the wave vector of the photon field. λ is a free parameter introduced for dimensional reasons What strikes is that the integration over the wave vector is missing. It is postponed to the evaluation of quantum expectations. 5/ 17
History Work by Marek Czachor (Gdansk) and collaborators ◮ M. Czachor, Non-canonical quantum optics, J. Phys. A 33 , 8081 (2000). Attempts to make this work into a rigorous theory fail. ◮ Unpublished Alternative formulation ◮ J. Naudts, On the Emergence of the Coulomb Forces in Quantum Electrodynamics, Advances in High Energy Physics 2017, 7232798 (2017). ◮ + several papers on the archive. � Czachor et al use a resolution of the identity I = R 3 d E k . This is replaced here by the requirement that || ψ k || = 1 for any k . In addition, only transversely polarized photons are taken into account. 6/ 17
World view (limited to QED) Euclidean space is filled with quantum harmonic oscillators, one pair at each site to cover horizontal and vertical polarization of light waves. Mathematically, this corresponds with a product of a classical 3-d space with a two-d quantum space. The corresponding algebra of observables is an integral over R 3 of copies of the algebra of observables of a pair of quantum harmonic oscillators. The electron is described by a set of 16 states including vacuum, electron/positron, spin up/down. These states can be conceived as local excitations of the vacuum state | 0 , 0 � × |∅� . 7/ 17
II Emergence Erik Verlinde (Amsterdam): gravity is an emergent force ◮ E. P . Verlinde, On the Origin of Gravity and the Laws of Newton, JHEP 1104, 029 (2011), arXiv:1001.0785. ◮ E. P . Verlinde, Emergent Gravity and the Dark Universe, arXiv:1611.02269. Entropic argument: area law supplemented with a volume law CLAIM: ◮ Longitudinal and scalar photons are not needed in QED. ◮ Coulomb attraction/repulsion results from interaction between dressed charges. Two results: ◮ General argument; ◮ Theorem. 8/ 17
General argument • Use the temporal (Hamiltonian, Weyl) gauge : A 0 ( x ) ≡ 0 ◮ M. Creutz, Quantum Electrodynamics in the temporal gauge, Ann. Phys. 117 , 471–483 (1979). ◮ Obvious choice if only transversely polarized photons exist. ◮ Drawback: No manifest Lorentz covariance • There exists a unitary transformation ˆ V which removes the charge operator from Gauss’ equation ∇ · ˆ E ( x ) = − µ 0 c ˆ ∇ · ˆ j 0 ( x ) ⇒ E ( x ) = 0 . Use this transformation here in the opposite direction. • ˆ A α ( x ) in absence of interaction, ˆ A ′ α ( x ) in presence of interactions. 9/ 17
The emergent picture Electric field operators ˆ E ′′ α ( x ) belonging to the emergent picture are defined by α ( x ) + µ 0 c ∂ � 1 ˆ α ( x ) = ˆ U ( − x 0 )ˆ ˆ j 0 ( y , 0 )ˆ E ′′ E ′ U ( x 0 ) . d y 4 π ∂ x α | x − y | ◮ They satisfy Gauss’ law ∇ · ˆ ∇ · ˆ E ′′ ( x ) = − µ 0 c ˆ E ′ ( x ) = 0. j 0 ′ ( x ) . resp. ˆ j 0 ′′ ( x ) = ˆ ◮ Maxwell’s equations are satisfied with j 0 ′ ( x ) and α ( x ) = − 1 ∂ � � ˆ ˆ α ( x ) − ˆ j ′′ E ′′ E ′ α ( x ) . µ 0 c ∂ x 0 CONCLUSION: There exist two equivalent descriptions of the same time-dependent charge distribution ˆ j ′ 0 ( x ) , one with Coulomb forces, one without. 10/ 17
III Bound states The total energy of the electron field decreases by being dressed with an appropriate transversely polarized photon field. H ph + ˆ H el + ˆ Let H = ˆ H I . ρ el ( k ) = ρ el ( 0 ) exp ( −| k | 2 / 2 σ 2 ) THEOREM Let be given with ρ el ( 0 ) > 0 but small. There exist wave functions ψ for which � ˆ ph � + � ˆ H H I � < 0 and � ph � ψ | ˆ H el ψ � k ph , k = ρ el ( k ) � ω ( k ) . d k The proof shows that the binding is optimal for long-wavelength photons. 11/ 17
Sketch of the proof Take a wave function of the form 1 � � ψ k ph , k = ρ ( k ph ) ρ el ( k ) τ m , 0 ( k ph , k ) | m , 0 � × |{ 1 }� m = 0 � + 1 − ρ ( k ph ) ρ el ( k ) | 0 , 0 � × |∅� with τ m , n ( k ph , k ) either 1 or 0, with ρ el ( − k ) = ρ el ( k ) , with τ ( 0 , 0 ) = 1 − τ ( 1 , 0 ) , � ph ρ ( k l 3 with d k ph ) = 1. It describes an electron field with initially with spin up, entangled with a single photon with horizontal polarization. 12/ 17
Choose τ 1 , 0 ( k ph , k ) = Θ( k · ε ( H ) ( k ph )) (Heaviside’s function). Then � ˆ H I � is of the form � ph ρ ( k � ˆ − l 3 I � = ph ) w ( k ph ) H d k with � l 3 d k U ( H ) ( k ph ) ph , − k ) τ 1 , 0 ( k ph , k ) w ( k = and U ( H ) ( k ph , − k ) ≥ 0 if and only if k · ε ( H ) ( k ph ) ≥ 0. Hence, the interaction is attractive in a half-space of wave vectors k . The remainder of the proof is straightforward. 13/ 17
Analogy with the polaron problem In Solid State Physics, a conduction electron in a polarizable crystal can form a bound state with a cloud of quantized lattice vibrations (phonons). This bound state is called a polaron . ◮ The charge of the electron is assumed to be fully screened. Here, the assumption is that Coulomb forces do not exist. ◮ The Fröhlich Hamiltonian is similar to the Hamiltonian of QED. A huge body of knowledge exists about the Fröhlich polaron. ◮ J. T. Devreese and A. S. Alexandrov, Fröhlich polaron and bipolaron: recent developments, Rep. Prog. Phys. 72 , 066501 (2009); arXiv:0904.3682. 14/ 17
Discussion The interaction between transversely polarized photons and the electron field peaks in the long wavelength limit. Conjecture: The interaction induces repulsion between regions with charges of equal sign and attraction between regions of opposite sign. Further work is needed to validate this statement. The analogy with the polaron gives partial support. A time-dependent analysis is still missing. 15/ 17
IV The S-matrix Perturbation expansion in the usual manner (Neumann series, Dyson trick, Wick theorem) ψ f = e iy 0 ˆ Suitable choice of phase shift ˆ h U ( x 0 , y 0 ) e − ix 0 ˆ h ψ i h removes all divergences — verified up to second order in q ◮ Calculate vacuum polarization, vacuum fluctuations, self-energy → Faraday rotation, scalar phase shift, Larmor precession 16/ 17
V The wave function of the vacuum state ψ i = ψ f yields ˆ H I ψ = ˆ Definition: h ψ . Solution is ψ vac . ψ vac can be calculated using a perturbation expansion � � � � ε ( H ) α | 1 , 0 � + ε ( V ) vac a | 0 , 0 �|∅� + α | 0 , 1 � |{ s , t }� + · · · ψ = b α ( s , t ) s , t = ↑ , ↓ α ◮ coefficients depend on momenta ◮ second order result reliable at low energies ◮ contribution to dark energy and dark mass only in momentum space ◮ · · · 17/ 17
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