Dresden 2011 – scientific programme
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MM: Fachverband Metall- und Materialphysik
MM 12: Postersitzung I
MM 12.53: Poster
Monday, March 14, 2011, 17:30–19:00, P5
The Unitarily Covariant Formulation of Hedin’s Equations — •Ronald Starke and Georg Kresse — University of Vienna, Faculty of Physics, Comp Mat Phys
The Feynman graph formulary as used e.g. by Hedin’s equations depends on the usage of the spacetime domain. E.g., the well-known GW-approximation is usually formulated as Σ(1,2)=i G(1,2)W(2,1) where G(1,2)W(2,1) denotes a point-wise product in the space time domain, i.e. 1=x1=(x1,t1). For practical calculations, however, it might be useful to work with quantities given as time (frequency) dependent matrices w.r.t. an orbital basis. The resulting expressions for Σ etc. do not carry over directly from the space-time domain. E.g., Σ ji≠i G jiW ij already because W is given in an arbitrary orbital basis by a 4-point quantity W klij. It is, therefore, desirable to reformulate Hedin’s equations such that they hold in an any orbital basis. In analogy to General Relativity, we call this formulation unitarily covariant. For the implementation of such an unitarily covariant formulation, it is necessary to clarify the resulting transformation properties induced by a change of basis in the one-particle Hilbert space H. In particular, it turns out that one has to distinguish between ’upper’ and ’lower’ indices according to the transformation behavior of the respective quantities. The upshot of the fully covariant formulation will be – apart from a certain theoretical insight – the facilitation of explicit calculation especially when it comes to the derivation of useful approximations: the covariant formalism allows for the direct switch to a basis where certain quantities take particularly simple (e.g. diagonal) forms.