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Noncommutative quantum field theory
Author(s) -
Grosse H.,
Wulkenhaar R.
Publication year - 2014
Publication title -
fortschritte der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.201400020
Subject(s) - noncommutative geometry , mathematics , mathematical analysis , matrix (chemical analysis) , quartic function , matrix function , eigenvalues and eigenvectors , fredholm determinant , diagonal matrix , pure mathematics , mathematical physics , diagonal , symmetric matrix , quantum mechanics , physics , geometry , materials science , composite material
We summarize our recent construction of the ϕ 4 ‐model on four‐dimensional Moyal space. This is achieved by solving the quartic matrix model for a general external matrix in terms of the solution of a non‐linear equation for the 2‐point function and the eigenvalues of that matrix. The β‐function vanishes identically. For the Moyal model, the theory of Carleman type singular integral equations reduces the construction to a fixed point problem. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. The Schwinger 2‐point function is reflection positive iff the diagonal matrix 2‐point function is a Stieltjes function.