Abstract
We extend our previous work (on D=2) to give an exact solution of the ΦD3 large-N matrix model (or renormalised Kontsevich model) in D=4 and D=6 dimensions. Induction proofs and the difficult combinatorics are unchanged compared with D=2, but the renormalisation – performed according to Zimmermann – is much more involved. As main result we prove that the Schwinger 2-point function resulting from the ΦD3-QFT model on Moyal space satisfies, for real coupling constant, reflection positivity in D=4 and D=6 dimensions. The Källén–Lehmann mass spectrum of the associated Wightman 2-point function describes a scattering part |p|2≥2μ2 and an isolated broadened mass shell around |p|2=μ2.
Highlights
The Kontsevich model [1,2] is of paramount importance because it elegantly proves Witten’s conjecture [3] about the equivalence of two approaches to quantum gravity in two dimensions:
In [21] two of us (H.G.+R.W.) have shown that translating the type of scaling limit considered for the matrix model correlation functions back to the position space formulation of the Moyal algebra leads to Schwinger functions of an ordinary quantum field theory on RD
We proved in [14] that for the D = 2-dimensional Kontsevich model this is not the case
Summary
The Kontsevich model [1,2] is of paramount importance because it elegantly proves Witten’s conjecture [3] about the equivalence of two approaches to quantum gravity in two dimensions:. The initial step for the Kontsevich model is, in a special limit of large matrices coupled with an infinitely strong deformation parameter, an integral equation solved by Makeenko and Semenoff [17] From this point of origin we explicitly solved all correlation functions. Our point of view is to define quantum field theory by quantum equations of motion, i.e. Schwinger–Dyson equations These equations can formally be derived from the partition function, but we forget the partition function, declare the equations as exact and construct exact solutions. Whereas the 36-Kontsevich model with imaginary coupling constant is asymptotically free [12], our real 36-model has positive β-function This is not a problem; there is no Landau ghost, and the theory remains well-defined at any scale! The most significant result of this paper is derived in sec
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