The other canonical example is the Aharanov-Bohm effect, and we can use this to illustrate the interpretational problem associated with gauge theories, sometimes characterized as a dilemma: failure of determinism or action-at-a-distance (see Healey, 2001).
(For an entry into philosophical problems associated with gauge transformations, see the entry on symmetry and symmetry breaking, especially Section 2.5; and Brading and Castellani (2003).) A hole argument type failure of determinism can often be achieved in field theories, depending, of course, on the specific properties of the field theory.
We refer to [12] the Hardy inequalities associated with nonisotropic gauge induced by the fundamental solution.
The duality mapping associated to a gauge function is defined by (21).
The duality mapping J φ : E → E ∗ associated with a gauge function φ is defined by J φ ( x ) = { f ∗ ∈ E ∗ : 〈 x, f ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f ∗ ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E, where 〈 ⋅, ⋅ 〉 denotes the generalized duality pairing.
The duality mapping J φ : X ⟶ 2 X ∗ associated with a gauge function φ is defined by J φ ( x ) = { f ∗ ∈ X ∗ : 〈 x, f ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f ∗ ∥ = φ ( ∥ x ∥ ), ∀ x ∈ X }, where 〈 ⋅, ⋅ 〉 denotes the generalized duality paring.
A combined theory associated with the gauge group SU 3) ⊗ SU 2) ⊗ U 1) is considered as 'the standard model' of elementary particle physics which was achieved by Glashow, Weinberg and Salam in 1962.
It allows to evaluate NLO QCD and EW cross sections for Drell Yan processes (inclusive), associated Higgs and gauge boson production and single-top quark production in s- and t-channels.
Let E ∗ be the dual space of E. The duality mapping J φ : E → 2 E ∗ associated to a gauge function φ is defined by J φ ( x ) = { f ∗ ∈ E ∗ : 〈 x, f ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f ∗ ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E. In particular, the duality mapping with the gauge function φ ( t ) = t, denoted by J, is referred to as the normalized duality mapping.
Let E* be the dual space of E. The duality mapping J φ : E → 2 E * associated with a gauge function φ is defined by J φ ( x ) = { f * ∈ E * : 〈 x, f * 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f * ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E. In particular, the duality mapping with the gauge function φ(t) = t, denoted by J, is referred to as the normalized duality mapping.
