The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges Fabio Ciolli Dipartimento di Matematica Università di Roma “Tor Vergata” Algebraic Quantum Field Theory: Where Operator Algebra meets Microlocal Analysis Palazzone, Cortona, June 5th, 2018 Fabio Ciolli, Dipartimento di Matematica Università di Roma “Tor Vergata” The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 1/27
Joint project with D. Buchholz, G. Ruzzi and E. Vasselli The C*-algebra of the e.m. field, defined by the e.m. potential on the 4-dim Minkowski spacetime [Buchholz ESI lectures 2012] Also based on a former key result by J. Roberts [Roberts 77]: the commutator of the e.m. field F with its Hodge dual field ⋆ F supported respectively on two surfaces whose boundaries are causally disjoint but linked together, is not vanishing Remind: ⋆ interchange the electric and magnetic parts of F [BCRV16] The universal C*-algebra of the electromagnetic field. Lett. Math. Phys. , 106 , 269–285, (2016). arXiv:1506.06603 [BCRV17] The universal C*-algebra of the electromagnetic field II. Topological charges and spacelike linear fields. Lett. Math. Phys. , 107 , 201–222, (2017). arXiv:1610.03302 A related project The net of causal loops and connection representations for abelian gauge theories on a 4-dim globally hyperbolic spacetime using a net of local C*-algebras with loops supported observables joint work with G. Ruzzi and E. Vasselli [CRV12], [CRV15] Fabio Ciolli, Dipartimento di Matematica Università di Roma “Tor Vergata” The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 2/27
An AQFT roadmap for QED by structural algebraic properties Define the Universal C*-algebras of e.m. (observable) field V that will appear in any theory incorporating electromagnetism (vacuum or non-trivial current) e.g. QED V defines a net on a poset of regions of R 4 , with covariance and causality but may not contain the dynamic of the theory Fix an interesting (pure, vacuum, . . . ) state ω . The GNS representation ( π ω , H ω , Ω) on the algebras V \ ker π ω gives the dynamical information of the net and distinguishes different theories We obtain any e.m. theory that satisfy the Haag-Kastler axioms: the physical content of a theory is encoded in its observable net Linearity on test functions Models in Haag-Kastler axioms do not require an a priori unrestrained condition of linearity of the field over the set of test functions This is just a matter of convenience, e.g. symplectic spaces, Weyl algebras, Wightman framework Other examples of non linearity also appear in other contests of AQFT: definition of CFT and perturbative AQFT Fabio Ciolli, Dipartimento di Matematica Università di Roma “Tor Vergata” The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 3/27
Topological charges and spacelike linearity for the e.m. field Following [Roberts 77], Topological charges result from commutators of the intrinsic (gauge invariant) vector potential A in spacelike separated, topologically non-trivial regions These commutators are in the center of the algebra V : existence in [BCRV16], non trivial examples in [BCRV17] The vector potential A is well defined in all regular, pure states of V but topological charges vanish if A is unrestrained linear on the test functions: hence no topological charges in the Wightman framework Nevertheless, we exhibit regular vacuum states with non-trivial topological charges: the vector potential is homogeneous and spacelike linear on test functions Such states also exist in the presence of non-trivial electric currents We also exhibit topological charges for theories with several e.m. potentials depending linearly on the test functions, e.g. scaling (short distance) limit of non-abelian gauge fields with asymptotic freedom Fabio Ciolli, Dipartimento di Matematica Università di Roma “Tor Vergata” The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 4/27
Outline The C*-algebras of the e.m. field Linear symplectic forms and absence of topological charges Non-trivial topological charges and spacelike linearity in vacuum Non-trivial topological charges with spacelike linearity and electric current Topological charges of multiplets of electromagnetic fields Conclusions and outlook Fabio Ciolli, Dipartimento di Matematica Università di Roma “Tor Vergata” The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 5/27
Outline The C*-algebras of the e.m. field Linear symplectic forms and absence of topological charges Non-trivial topological charges and spacelike linearity in vacuum Non-trivial topological charges with spacelike linearity and electric current Topological charges of multiplets of electromagnetic fields Conclusions and outlook Fabio Ciolli, Dipartimento di Matematica Università di Roma “Tor Vergata” The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 6/27
Notations Minkowski spacetime R 4 , signature (+ , − , − , − ) and causal disjointness ⊥ D n := D n ( R 4 ) real smooth compactly supported n -forms on R 4 in particular D 2 are valued in the antisymmetric tensors of rank two d : D n → D n + 1 s.t. dd = 0 differential d D n = { f ∈ D n + 1 : df = 0 } i.e. closed form in D n + 1 Poincaré Lemma ⋆ ⋆ = ( − 1 ) n + 1 ⋆ : D n → D 4 − n s.t. Hodge dual operator δ := − ⋆ d ⋆ s.t. δ : D n → D n − 1 s.t. δδ = 0 co-differential C n := C n ( R 4 ) = { f ∈ D n : δ f = 0 } s.t. δ D n ⊂ C n − 1 co-closed n -forms in particular C 1 = { f ∈ D 1 : div f = 0 } divergence free 1-forms δ D n = C n − 1 dual version of the Poincaré Lemma a primitive of f ∈ D n is g ∈ D n − 1 s.t. dg = f a co-primitive of f ∈ D n is h ∈ D n + 1 s.t. δ h = f Moreover supp ( df ) , supp ( ⋆ f ) , supp ( δ f ) ⊆ supp ( f ) Local action of d , ⋆ and δ P ↑ ( P , f ) �→ f P := f ◦ P − 1 + × D n → D n s.t. action of Poincaré group leaves the space C 1 of divergence-free 1-forms globally invariant Fabio Ciolli, Dipartimento di Matematica Università di Roma “Tor Vergata” The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 7/27
Topological non-trivial regions and Loop functions Loop-shaped region L : open, bounded and contains some spacelike (pointwise, hence simple) loop [ 0 , 1 ] ∋ t �→ γ ( t ) i.e. γ is a deformation retract of L and therefore is homotopy equivalent (i.e. homotopic) to L Linked loop-shaped regions: Hopf link and Whitehead link Loop function: for any L with γ we may choose O 0 a small neighbourhood of the origin such that ( O 0 + γ ) ⊂ L and s ∈ D 0 a real scalar function with supp ( s ) ⊆ O 0 and define � 1 x �→ l s ,γ ( x ) . = dt s ( x − γ ( t )) ˙ γ ( t ) 0 Then l s ,γ ∈ C 1 and supp ( l s ,γ ) ⊆ ( O 0 + γ ) ⊂ L . � If dx s ( x ) � = 0 there is no f ∈ D 2 with support in L s.t. l s ,γ = δ f , i.e. l s ,γ is co-closed but not co-exact in this region Fabio Ciolli, Dipartimento di Matematica Università di Roma “Tor Vergata” The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 8/27
The e.m. field F and the intrinsic gauge-invariant e.m. potential A The e.m. quantum field is a linear map F : D 2 ∋ h �→ F ( h ) The e.m. intrinsic vector potential is a map A : C 1 ∋ f �→ A ( f ) s.t. F ( h ) . = A ( δ h ) , h ∈ D 2 A conserved current is a linear map j : D 1 ∋ g �→ j ( g ) s.t. δ j ( s ) = j ( ds ) = 0 , s ∈ D 0 1 st Maxwell equation (using e.m. field F ) (using e.m. vector potential A ) F ( δτ ) = A ( δ 2 τ ) = 0 , dF ( τ ) := F ( δτ ) = 0 τ ∈ D 3 by the Local Poincaré Lemma and independence from co-primitives 2 nd Maxwell equation j ( g ) = F ( dg ) j ( g ) = A ( δ dg ) , g ∈ D 1 Current conservation δ j ( s ) = F ( d 2 s ) = 0 δ j ( s ) = A ( δ d 2 s ) = 0 , s ∈ D 0 Fabio Ciolli, Dipartimento di Matematica Università di Roma “Tor Vergata” The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 9/27
Locality for F vs locality for A For the e.m. field F Given h 1 and h 2 in D 2 with spacelike separated supports supp ( h 1 ) ⊥ supp ( h 2 ) ⇒ [ F ( h 1 ) , F ( h 2 )] = 0 , For the intrinsic e.m. vector potential A Given f 1 and f 2 in C 1 such that supp ( f 1 ) × supp ( f 2 ) i.e. separated by double cones (or by opposite characteristic wedges) the independence from the co-primitive allows to choose two spacelike separated co-primitives: supp ( f 1 ) × supp ( f 2 ) ⇒ ∃ h 1 , h 2 ∈ D 2 , δ h 1 = f 1 , δ h 2 = f 2 s.t. h 1 ⊥ h 2 obtaining a stronger form of Locality for A supp ( f 1 ) × supp ( f 2 ) ⇒ [ A ( f 1 ) , A ( f 2 )] = [ F ( h 1 ) , A ( h 2 )] = 0 For f 1 and f 2 in C 1 with supp ( f 1 ) ⊥ supp ( f 2 ) but not supp ( f 1 ) × supp ( f 2 ) ⇒ [ A ( f 1 ) , A ( f 2 )] � = 0 Fabio Ciolli, Dipartimento di Matematica Università di Roma “Tor Vergata” The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 10/27
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