ANALYSIS ON q-DEFORMED QUANTUM SPACES

Abstract

A q-deformed version of classical analysis is given to quantum spaces of physical importance, i.e. Manin plane, q-deformed Euclidean space in three or four dimensions, and q-deformed Minkowski space. The subject is presented in a rather complete and selfcontained way. All relevant notions are introduced and explained in detail. The different possibilities to realize the objects of q-deformed analysis are discussed and their elementary properties are studied. In this manner attention is focused on star products, q-deformed tensor products, qdeformed translations, q-deformed partial derivatives, dual pairings, q-deformed exponentials, and q-deformed integration. The main concern of this work is to show that these objects fit together in a consistent framework, which is suitable to formulate physical theories on quantum spaces. *

Klein-Gordon equation in $q$-deformed Euclidean space

“…The q-deformed quantum spaces considered so far are so-called braided Hopf algebras [37]. From this point of view, the two versions of q-translations are nothing else than the representations of two braided co-products ∆ and ∆ on the corresponding commutative coordinate algebras [25]:…”

“…Next, we translate the Hopf algebra axioms in Eqs. ( 226) and ( 227) into corresponding rules for q-translations and q-inversions [25], i. e.…”

“…We can use q-exponentials to calculate q-translations [34]. If we replace the momentum coordinates in the expressions for q-exponentials with derivatives, it applies [25,31,35] exp q (x|∂ y ) ⊲ g(y) = g(x ⊕ y), exp q (x| ∂y ) ⊲ g(y) = g(x ⊕ y),…”

“…In analogy to the undeformed case, q-exponentials satisfy the addition theorems [25,31,32] exp q (x ⊕ y|ip) = exp q (x| exp q (y|ip) ⊛ ip), exp q (ix|p ⊕ p ′ ) = exp q (x ⊛ exp q (x|ip)|ip ′ ), (220) and exp q (x ⊕ y|ip) = exp q (x|exp q (y|ip) ⊛ ip), exp q (ix|p ⊕ p ′ ) = exp q (x ⊛ exp q (x|ip)|ip ′ ). (221)…”

q-Deformed Superalgebras

“…In our previous work [31][32][33][34][35][36][37] attention was concentrated on space-time structures that arise from q-deformation [38]. More concretely, we are interested in q-deformed quantum spaces that could prove useful in physical applications.…”