The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges

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1 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

2 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, , (2016). arxiv: [BCRV17] The universal C*-algebra of the electromagnetic field II. Topological charges and spacelike linear fields. Lett. Math. Phys., 107, , (2017). arxiv: 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

3 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

4 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

5 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

6 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

7 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 dd n = {f D n+1 : df = 0} i.e. closed form in D n+1 Poincaré Lemma : D n D 4 n s.t. = ( 1) n+1 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 : δf = 0} s.t. δd n C n 1 co-closed n-forms in particular C 1 = {f D 1 : div f = 0} divergence free 1-forms δd 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. δh = f Moreover supp(df ), supp( f ), supp(δf ) supp(f ) Local action of d, and δ P + D n D n s.t. (P, f ) f P := f P 1 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

8 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 x l s,γ(x). = 1 Then l s,γ C 1 and supp(l s,γ) (O 0 + γ) L. 0 dt s(x γ(t)) γ(t) If dx s(x) 0 there is no f D 2 with support in L s.t. l s,γ = δf, 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

9 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(δh), h D 2 A conserved current is a linear map j : D 1 g j(g) s.t. δj(s) = j(ds) = 0, s D 0 1 st Maxwell equation (using e.m. field F) (using e.m. vector potential A) df(τ) := F(δτ) = 0 F (δτ) = A(δ 2 τ) = 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(δdg), g D 1 Current conservation δj(s) = F (d 2 s) = 0 δj(s) = A(δd 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

10 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, δh 1 = f 1, δh 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

11 The C*-algebras of the e.m. field: definition Take the *-algebra generated by the formal unitary operators V (a, g). = e iaa(g) with a R, g C 1 and quotient w.r.t. the ideal given by the relations V (a 1, g)v (a 2, g) = V (a 1 + a 2, g), V (a, g) = V ( a, g), V (0, g) = 1 (1) V (a 1, g 1 )V (a 2, g 2 ) = V (1, a 1 g 1 + a 2 g 2 ) if supp(g 1 ) supp(g 2 ) (2) V (a, g), V (a 1, g 1 ), V (a 2, g 2 ) = 1 for any g if supp(g 1 ) supp(g 2 ) (3) (1): algebraic properties of unitary one-parameter groups a V (a, g) (2): spacelike linearity and locality properties of A (3): the symbol, is the group commutator so V (a 1, g 1 ), V (a 2, g 2 ) are central element of topological nature we call topological charges The *-algebra generated by the V (a, g) and relations (1) to (3) has a C*-norm induced by all of its GNS reps; its completion w.r.t. this norm, is the universal C*-algebra of the electromagnetic field denoted by V Strongly regular states: (a 1,..., a n) ω(v (a 1, g 1 ) V (a n, g n)) smooth Then π(v (a, g)) = e iaaπ(g) are unitaries and A π(g) are self-adjoint on a stable common core, including the GNS vector Ω From (2), A π(g) are spacelike linear, i.e. it is true on the common core a 1 A π(g 1 ) + a 2 A π(g 2 ) = A π(a 1 g 1 + a 2 g 2 ), whenever supp(g 1 ) supp(g 2 ) Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 11/27

12 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 12/27

13 Linear symplectic forms and absence of topological charges Given a regular pure states ω on V with GNS (π, H, Ω), not required to be the vacuum, exists A π(g) self-adjoint as above from (3), the commutator [A π(g 1 ), A π(g 2 )] is affiliated with the centre of the weak closure π(v) as multiples of the identity and we may define, for (wick) disjoint-support 1-forms σ π(g 1, g 2 ). = i Ω, [A π(g 1 ), A π(g 2 )] Ω, g 1, g 2 C 1, supp(g 1 ) supp(g 2 ) depending on the state ω, σ π may be a bilinear and skew symmetric, i.e. a symplectic form on C 1 ; in general σ π is only spacelike linear, see (2) if σ π vanishes for any pair of test functions g 1, g 2 C 1 having supports in spacelike separated, linked loop-shaped regions, then the corresponding topological charges vanish In fact it is possible to prove, using co-cohomology: 1. For any loop-shaped region L and any function g C 1, supp(g) L, exists a loop function l s,γ, as above, in the same co-cohomology class 2. Let g 1, g 2 C 1 supported in L 1 and L 2, spacelike separated loop-shaped linked regions with retracts γ 1 linked to γ 2 ; then there are two loop functions l s1,γ 1, l s2,γ 2 co-cohomologous to g 1, g 2, s.t. σ π(g 1, g 2 ) = σ π(l s1,γ 1, l s2,γ 2 ), for any bilinear σ π. Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 13/27

14 Linear symplectic forms and absence of Topological Charges 3. For given loop γ, the co-cohomology classes are fixed by the class values which are given by the integral κ. = dx s(x) R of the scalar functions s D 0 in the definition of l s,γ. Thus for given loops γ 1, γ 2, the expression σ π(l s1,γ 1, l s2,γ 2 ) is proportional to the product κ 1 κ It is possible to deform any two disjoint simple loops γ 1, γ 2 in R 4 to corresponding disjoint simple loops β 1 and β 2 on the time-zero plain R 3 s.t. σ π(l s1,γ 1, l s2,γ 2 ) = σ π(l s 1,β 1, l s 2,β 2 ). 5. For fixed product κ 1 κ 2, the values of σ π depend only on the homology class of β 1 in R 3 \supp(β 2 ). Hence, for a symplectic form σ π linear in both entries, i.e. whose representation gives a linear vector potential A π on C 1, the topological charges vanish: Proposition Let L 1, L 2 be two spacelike separated loop-shaped regions which can continuously be retracted to spacelike linked loops γ 1 and γ 2, respectively. Moreover, let σ π be linear in both entries. Then, for any g 1, g 2 C 1 having support in L 1, respectively L 2, one has σ π(g 1, g 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 14/27

15 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 15/27

16 Non-trivial Topological Charges and spacelike linearity in vacuum Merging the electric and magnetic parts of a free e.m. field in a non-linear manner gives a new field with spacelike linearity and topological charge The Haag-Kastler net coincides with the net of the original e.m. field then this is a reinterpretation of the theory giving non trivial topological charges Regular, quasi-free, vacuum state ω 0 on V with GNS denoted by (π 0, H 0, Ω 0 ) gives both a vanishing electric current j 0 (h), for every h D 1 and a linear free vector potential A 0 in the representation π 0 Remind that j(g) = A(δdg) for g D 1, this result follow by using Reeh-Schlieder theorem and Källén-Lehmann representation of Wightman two-point functions For g C 1 and G D 2 a co-primitive of g we get the constant tensor G µν =. dx G µν (x) depends only on g, invariant under translations and covariant under Lorentz transformations of g A 0 (δ G) also depend only on g C 1 but not on the co-primitive G In fact, if δk = G G for k D 3 and δg = δg = g, it holds A 0 (δ δk) = A 0 (δd k) = j 0 ( k) = 0 in vacuum Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 16/27

17 Let θ ± be the characteristic functions of the positive respectively negative reals and let G 2. = G µν G µν. Suppose for simplicity that g C 1 is supported in a connected region then we obtain a topological potential A T defining A T (g). = θ +(G 2 ) A 0 (δg) + θ (G 2 ) A 0 (δ G). For 1-forms g 1, g 2 C 1 with (connected) spacelike separated, linked supports, i.e. supp(g 1 ) supp(g 2 ), we have the commutator [A T (g 1 ), A T (g 2 )] = ( θ +(G 1 2 )θ+(g 2 2 ) + θ (G 1 2 )θ (G 2 2 ) ) (G1, G 2 ) 1 + ( θ +(G 1 2 )θ (G 2 2 ) θ+(g 2 2 )θ (G 1 2 ) ) (G1, G 2 ) 1 where is the commutator function of the free Maxwell field The term (G 1, G 2 ), studied in the cited result [Roberts77], makes the commutator [A T (g 1 ), A T (g 2 )] non-trivial and we have non-trivial topological charges, by spacelike linearity, in vacuum with or without currents Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 17/27

18 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 18/27

19 Non-trivial Topological Charges with electric current It is in general not possible to couple any given current to a Wightman field via a field equation [Araki, Haag, Schroer 1961] The vector potential, being spacelike linear is not a Wightman field so the inhomogeneous Maxwell equation does have solutions for almost any given current We may obtain a spacelike linear vector potential carrying a topological charge by combining a current j with the previous spacelike linear A T For any given j, look for a spacelike linear potential A J in a regular vacuum representation of the universal algebra V, satisfying the inhomogeneous Maxwell equation A J (δdh) = j(h), h D 1 Problem: A J is already defined on the subspace δdd 1 of C 1 ; how to extend A J to all C 1 and obtain its localization from the one of j Lemma Let g = δdh δdd 1 and call h D 1 the pre-image of g δdd 1. (i) h is uniquely determined by g up to elements of dd 0. (ii) Let O supp(g) be s.t. the complement of Õ =. (O + V +) (O + V ) has 4 trivial first homology, H 1 (R \Õ) = {0}. There exist pre-images h of g having support in any given neighbourhood of Õ. (iii) The map from δdd 1 into the classes D 1 /dd 0 of pre-images is continuous. Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 19/27

20 Denote by g elements in the subspace δdd 1 C 1, i.e. g C 1 and exists h D 1 s.t. g = δdh We extend A J to C 1 by { A J (g) =. j(h ) if g = g 0 otherwise A J (g) is well defined by Lemma (i), since j(ds) = 0, s D 0 Then, using Lemma (iii), it is possible to extend consistently A J to elements not in δdd and prove the existence of a regular vacuum state on V with a spacelike linear vector potential A J in its GNS representation with A J (g) = A J (g ) = j(h ), g C 1 and A J (g) = 0 otherwise But the topological charge of the potential A J associated with linked spacelike separated loop-shaped regions, are trivial. In fact the loop-shaped region L and L. = {L + V +} {L + V } L both have as continuous retract the simple spacelike loop γ; so R 4 \ L is homotopic to R 4 \γ and for their homology groups holds, by using Alexander duality H 1 (R 4 \ L) H 1 (R 4 \γ) H 2 (γ) H 2 (S 1 ) = {0} so by (ii) of Lemma, the component g of g supported in L has a pre-image h supported in any given neighbourhood of L Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 20/27

21 thus if g 1, g 2 C 1 have supports in spacelike separated loop-shaped regions L 1 and L 2 respectively, one obtains [A J (g 1 ), A J (g 2 )] = [A J (g 1 ), A J (g 2 )] = [j(h 1 ), j(h 2 )] = 0 However, combining the previous results for A T with the one of A J we obtain non-trivial topological charges with electric currents (similar to definition of s-products in the Wightman framework of quantum field theories [Borchers 1984]) (π T, H T, Ω T ) be a regular vacuum representation with non-trivial topological charge but trivial electric current and (π J, H J, Ω J ) be a regular vacuum representation with non-trivial electric current but trivial topological charge Construct the representation π TJ on H T H J putting for a R, g C 1 π TJ (V (a, g)). = π T (V (a, g)) π J (V (a, g)). Restricting the algebra π TJ (V) to H TJ = πtj (V) Ω TJ H T H J, where. Ω TJ = ΩT Ω J one obtains a regular vacuum representation (π TJ, H TJ, Ω TJ ) of V with generating function ω TJ (V (a, g)). = Ω T, π T (V (a, g))ω T Ω J, π J (V (a, g))ω J, a R, g C 1 Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 21/27

22 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 22/27

23 Topological charges of multiplets of electromagnetic fields Discuss the appearance of topological charges for multiplets of e.m. fields, that is non-abelian gauge theories, in short distance (scaling) limit: in the asymptotically free case the fields become non-interacting and transform as tensors under the adjoint action of some global gauge group Consider the case of two e.m. fields with corresponding intrinsic vector potentials defined on C 1 C 1. The resulting *-algebra generated by the unitaries V 2 (a, g) with a R and g C 1 C 1 has a C*-norm induced by all of its GNS representations; its completion w.r.t. this norm denoted by V 2 is the universal C*-algebra in the scaling limit V 2 is not isomorphic to the completion of V V since the two types of fields need not commute with each other We show that there exist regular vacuum representations of V 2 with non-trivial topological charge i.e. for g 1, g 2 C 1 C 1 with supp supp(g 1 ) supp(g 2 ) and a 1, a 2 R it holds [A(g 1 ), A(g 2 )] 0 Observe that now the underlying electromagnetic fields are (linear) Wightman fields so we shall proceed using a generalized free field Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 23/27

24 Denote up and down subspaces of C 1 C 1 by C u = C 1 {0} and C d = {0} C 1 with elements g u, g d and primitives G u, G d respectively For g C 1 and classes of co-primitives G D 2 s.t. g = δg we use the scalar product on the one-particle space of the free Maxwell field. G 1, G 2 0 = (2π) 3 dp θ(p 0 ) δ(p 2 ) (p Ĝ1(p))(p Ĝ2(p)), G 1, G 2 D 2 for fixed 1 ζ 1 we give a sesquilinear form on C C 1 C 1 g 1, g 2 ζ. = G1u, G 2u 0 + G 1d, G 2d 0 + ζ G 1u, G 2d 0 ζ G 1d, G 2u 0 hence, ζ defines a positive (semidefinite) scalar product on C C 1 C 1 with a corresponding regular quasi-free vacuum states ω ζ on V 2 with generating function ω ζ (V 2 (a, g)). = e a2 g,g ζ /2, a R, g C 1 C 1 In the GNS representations (π ζ, H ζ, Ω ζ ), we have that A ζ (g) is a (linear) Wightman field on C 1 C 1 with a global internal symmetry group SO(2) Nevertheless, the potential A ζ carries a non-trivial topological charge obtained by the commutator [A ζ (g 1 ), A ζ (g 2 )] = ( g 1, g 2 ζ g 2, g 1 ζ ) 1 = ( (G 1u, G 2u ) + (G 1d, G 2d ) + ζ (G 1u, G 2d ) ζ (G 1d, G 2u )) 1 similarly to the case of A T. Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 24/27

25 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 25/27

26 Conclusions and outlook Representations with topological charges at finite scale in asymptotically free, non-abelian gauge theories Topological charges for massive theories: look not to depend on masses, so are intrinsically related to the non-massive e.m. field Relation between Topological charges and electric charge and currents: e.g. can we measure the electric charge by the topological one?... Universal algebra for models with e.m. field in low dimensional spacetime Superselection of topological charges Connection representations and potential systems [CRV2012, 2015] for the intrinsic potential with topological charges Universal algebra and topological aspects of non-abelian, local-gauge theories How topological charges would manifest themselves experimentally e.g. by interference patterns,... Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 26/27

27 Thank You for Your Attention Fabio Ciolli, Dipartimento di Matematica Università di Roma Tor Vergata The universal C*-algebra of the electromagnetic field: spacelike linearity and topological charges 27/27