tag:blogger.com,1999:blog-3832136017893749497.post5625836889079823560..comments2023-09-29T08:49:30.765-04:00Comments on Elliptic Composability: Florin Moldoveanuhttp://www.blogger.com/profile/01087655914212705768noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-3832136017893749497.post-7693254601784519222015-02-11T13:06:14.147-05:002015-02-11T13:06:14.147-05:00The deformation Δa(q) = a(q)⊗q^c + q^{−c}⊗a(q) is ...The deformation Δa(q) = a(q)⊗q^c + q^{−c}⊗a(q) is related to Bogoliubov operators. We may write <br /><br />q^c = Ie^{2πiφ} = I(cos(2πφ) + sin(2πφ)),<br /><br />for φ a parameter related to acceleration. Of course in addition<br /><br />(q^c)^{-1} = Ie^{-2πiφ} = I(cos(2πφ) - sin(2πφ)).<br /><br />This gives <br /><br />Δa(q) = a(q)⊗I(cos(2πφ) + sin(2πφ)) + (cos(2πφ) - sin(2πφ))I⊗a(q)<br /><br />= (a(q)⊗I + I⊗a(q))cos(2πφ) + (a(q)⊗I - I⊗a(q))sin(2πφ)<br /><br />We may of course form the Hermitian conjugate<br /><br />Δa^†(q) = a^†(q)⊗q^{-c} + q⊗a^†(q)<br /><br />= (a^†(q)⊗I + I⊗a^†(q))cos(2πφ) + (a^†(q)⊗I - I⊗a^†(q))sinh(2πφ).<br /><br />The commutator [Δa(q), Δa^†(q)] is<br /><br />[Δa(q), Δa^†(q)] = ([a(q), a^†(q)] ⊗I + I⊗[a(q), a^†(q)])(cos^2(2πφ) + sin^2(2πφ)),<br /><br />which gives a unit product.<br /><br />The hyperbolic trigonometric functions obtained from φ --- > iφ are squeeze parameters and we have a C* form of the Bogoliubov operators<br /><br />b = au + a^†v, b^† = a^†u + av<br /><br />for u = cosh(2πφ) and v = sinh(2πφ). The commutator of b and b^† is <br /><br />[b, b^†] = u^2 – v^2 = 1.<br /><br />The q-deformed algebra has the same u^2 – v^2 = 1. This then means we have the operators employed in computing Hawking and Unruh radiation.<br /><br />LCLawrence Crowellhttps://www.blogger.com/profile/12090839464038445335noreply@blogger.com