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We present some interesting connections between $PT$ symmetry and conformal symmetry.
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Since charge conjugation leaves the coordinates untouched, one could equally have said that in the coordinate sector C P T CPT is compatible with Lorentz invariance. Further discussion of T T transformations in relativistic quantum theory may be found in C. M. Bender and P. D. Mannheim, Phys. Rev. D 84
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Not only are the 𝐄 \mathbf{E} and 𝐁 \mathbf{B} fields reducible under real Lorentz transformations, they are reducible under complex Lorentz transformations as well, since if exp ( i w μ ν M μ ν ) \exp(iw_{\mu\nu}M^{\mu\nu}) does not mix the D ( 1 , 0 ) D(1,0) and D ( 0 , 1 ) D(0,1) components with each other when w μ ν w_{\mu\nu} is real, it does not do so if w μ ν w_{\mu\nu} is complex. This is to be contrasted with representations that contain both left- and right-handed components such as D ( 1 / 2 , 1 / 2 ) D(1/2,1/2) , since here all four components do mix under real Lorentz transformations, and thus continue to do so under complex ones, with the P T PT transformation that takes x μ x_{\mu} to − x μ -x_{\mu} corresponding to a sequence of three complex Lorentz transformations x ′ = x cosh ξ + t sinh ξ x^{\prime}=x\cosh\xi+t\sinh\xi , y ′ = y cosh ξ + t sinh ξ y^{\prime}=y\cosh\xi+t\sinh\xi , z ′ = z cosh ξ + t sinh ξ z^{\prime}=z\cosh\xi+t\sinh\xi , each with a boost angle ξ = i π \xi=i\pi . Thus under a sequence of P T PT transformations and Lorentz boosts with complex boost angle one can transform 𝐄 ( t , 𝐱 ) ± i 𝐁 ( t , 𝐱 ) \mathbf{E}(t,\mathbf{x})\pm i\mathbf{B}(t,\mathbf{x}) first into − [ 𝐄 ( − t , − 𝐱 ) ∓ i 𝐁 ( − t , − 𝐱 ) ] -[\mathbf{E}(-t,-\mathbf{x})\mp i\mathbf{B}(-t,-\mathbf{x})] and then into − [ 𝐄 ( t , 𝐱 ) ∓ i 𝐁 ( t , 𝐱 ) ] -[\mathbf{E}(t,\mathbf{x})\mp i\mathbf{B}(t,\mathbf{x})]
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