Solvable 4D noncommutative QFT: phase transitions and quest for reflection positivity (original) (raw)
Related papers
2016
We provide further analytical and first numerical results on the solvable λφ 4 4-NCQFT model. We prove that for λ < 0 the singular integral equation has a unique solution, whereas for λ > 0 there is considerable freedom. Furthermore we provide integral formulae for partial derivatives of the matrix 2-point function, which are the key to investigate reflection positivity. The numerical implementation of these equations gives evidence for phase transitions. The derivative of the finite wavefunction renormalisation with respect to λ is discontinuous at λ c ≈ −0.39. This leads to singularities in higher correlation functions for λ < λ c. The phase λ > 0 is not yet under control because of the freedom in the singular integral equation. Reflection positivity requires that the two-point function is Stieltjes. Implementing Widder's criteria for Stieltjes functions we exclude reflection positivity outside the phase [λ c , 0]. For the phase λ c < λ ≤ 0 we show that refining the discrete approximation we satisfy Widder to higher and higher order. This is clear evidence, albeit no proof, of reflection positivity in that phase. 1 We prove in the appendix of [14] that G 0b = 1 is an exact solution of (2) for any λ < 0 and Λ 2 → ∞. This solution seems numerically unstable under small perturbations.
Renormalisation of the Grosse-Wulkenhaar model
Proceedings of 3rd Quantum Gravity and Quantum Geometry School — PoS(QGQGS 2011), 2013
We give an introduction into the problems of local quantum fields and argue, that a quantization of space-time might lead to better behaviour. Next we discuss a special Euclidean φ 4 4-quantum field theory over quantized space-time as an example of a renormalizable field theory. Using a Ward identity, it was possible to prove the vanishing of the beta function for the coupling constant to all orders in perturbation theory. We extend this work and obtain from the Schwinger-Dyson equation a non-linear integral equation for the renormalised two-point function alone. The nontrivial renormalised four-point function fulfils a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations might be the starting point of a nonperturbative construction of a Euclidean quantum field theory on a noncommutative space. We expect to learn about renormalisation from this almost solvable model.
On the Fixed Point Equation of a Solvable 4D QFT Model
Vietnam journal of mathematics, 2015
The regularisation of the λ φ 4 4-model on noncommutative Moyal space gives rise to a solvable QFT model in which all correlation functions are expressed in terms of the solution of a fixed point problem. We prove that the non-linear operator for the logarithm of the original problem satisfies the assumptions of the Schauder fixed point theorem, thereby completing the solution of the QFT model. Keywords quantum field theory • solvable model • Schauder fixed point theorem Mathematics Subject Classification (2010) 81T16 • 81T08 • 47H10 • 46B50
arXiv (Cornell University), 2016
We extend our previous work (on D = 2) to give an exact solution of the Φ 3 D 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 Φ 3 D -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 ≥ 2µ 2 and an isolated fuzzy mass shell around |p| 2 = µ 2 . declare the equations as exact and construct exact solutions. Whereas the Φ 3 6 -Kontsevich model with imaginary coupling constant is asymptotically free [12], our real Φ 3 6 -model has positive β-function. But this is not a problem; there is no Landau ghost, and the theory remains well-defined at any scale! In other words, the real Φ 3 6 -Kontsevich model could avoid triviality. It is instructive to compare our exact results with a perturbative BPHZ renormalisation of the model. In D = 6 dimensions the full machinery of Zimmermann's forest formula is required. We provide in sec. 5 the BPHZ-renormalisation of the 1-point function up to two-loop order. One of the contributing graphs has an overlapping divergence with already 6 different forests. Individual graphs show the full number-theoretic richness of quantum field theory: up to two loops we encounter logarithms, polylogarithms Li 2 and ζ(2) = π 2 6 . The amplitudes of the graphs perfectly sum up to the Taylor expansion of the exact result. The original BPHZ scheme with normalisation conditions at a single scale leads in justrenormalisable models to the renormalon problem which prevents Borel resummation of the perturbation series. We also provide in sec. 5 an example of a graph which shows the renormalon problem. But all these problems cancel as our exact correlation functions are analytic(!) in the coupling constant. Exact BPHZ renormalisation is fully consistent (for the model under consideration)! The most significant result of this paper is derived in sec. 6. Matrix models such as the Kontsevich model Φ 3 D arise from QFT-models on noncommutative geometry. The prominent Moyal space gives rise to an external matrix E having linearly spaced eigenvalues with multiplicity reflecting the dimension D. 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 R D . Euclidean symmetry and invariance under permutations are automatic. The most decisive Osterwalder-Schrader axiom , reflection positivity, amounts for the Schwinger 2-point function to the verification that the diagonal matrix model 2-point function is a Stieltjes function. We proved in [14] that for the D = 2-dimensional Kontsevich model this is not the case. To our big surprise and exaltation, we are able to prove: Theorem 1.1. The Schwinger 2-point function resulting from the scaling limit of the Φ 3 D -QFT model on Moyal space with real coupling constant satisfies reflection positivity in D = 4 and D = 6 dimensions. As such it is the Laplace-Fourier transform of the Wightman 2-point function of a true relativistic quantum field theory [24] (θ, δ are the Heaviside and Dirac distributions). Its Källén-Lehmann mass spectrum ̺( M 2 µ 2 ) [25, 26] is explicitly known and has support on a scattering part with M 2 ≥ 2µ 2 and an isolated fuzzy mass shell around M 2 = µ 2 of non-zero width.
2013
We study quartic matrix models with partition function Z[E, J] = dM exp(trace(JM − EM 2 − λ 4 M 4)). The integral is over the space of Hermitean N ×N-matrices, the external matrix E encodes the dynamics, λ > 0 is a scalar coupling constant and the matrix J is used to generate correlation functions. For E not a multiple of the identity matrix, we prove a universal algebraic recursion formula which gives all higher correlation functions in terms of the 2-point function and the distinct eigenvalues of E. The 2-point function itself satisfies a closed non-linear equation which must be solved case by case for given E. These results imply that if the 2-point function of a quartic matrix model is renormalisable by mass and wavefunction renormalisation, then the entire model is renormalisable and has vanishing β-function. As main application we prove that Euclidean φ 4-quantum field theory on four-dimensional Moyal space with harmonic propagation, taken at its self-duality point and in the infinite volume limit, is exactly solvable and non-trivial. This model is a quartic matrix model, where E has for N → ∞ the same spectrum as the Laplace operator in 4 dimensions. Using the theory of singular integral equations of Carleman type we compute (for N → ∞ and after renormalisation of E, λ) the free energy density (1/volume) log(Z[E, J]/Z[E, 0]) exactly in terms of the solution of a non-linear integral equation. Existence of a solution is proved via the Schauder fixed point theorem. The derivation of the non-linear integral equation relies on an assumption which we verified numerically for coupling constants 0 < λ ≤ 1 π .
Progress in solving a noncommutative quantum field theory in four dimensions
2009
We study the noncommutative φ 4 4-quantum field theory at the self-duality point. This model is renormalisable to all orders as shown in earlier work of us and does not have a Landau ghost problem. Using the Ward identity of Disertori, Gurau, Magnen and Rivasseau, we obtain from the Schwinger-Dyson equation a non-linear integral equation for the renormalised two-point function alone. The non-trivial renormalised four-point function fulfils a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations are the starting point for a perturbative solution. In this way, the renormalised correlation functions are directly obtained, without Feynman graph computation and further renormalisation steps.
Solvable limits of a 4D noncommutative QFT
2013
In previous work we have shown that the (\theta->\infty)-limit of \phi^4_4-quantum field theory on noncommutative Moyal space is an exactly solvable matrix model. In this paper we translate these results to position space. We show that the Schwinger functions are symmetric and invariant under the full Euclidean group. The Schwinger functions only depend on matrix correlation functions at coinciding indices per topological sector, and clustering is violated. We prove that Osterwalder-Schrader reflection positivity of the Schwinger two-point function is equivalent to the question whether the diagonal matrix two-point function is a Stieltjes function. Numerical investigations suggest that this can at best be expected for the wrong sign of the coupling constant. The corresponding Wightman functions would describe particles which interact without momentum transfer. The theory differs from a free theory by the presence of non-trivial topological sectors.
The continuum phase diagram of the 2d non-commutative λϕ 4 model
Journal of High Energy Physics, 2014
We present a non-perturbative study of the λφ 4 model on a non-commutative plane. The lattice regularised form can be mapped onto a Hermitian matrix model, which enables Monte Carlo simulations. Numerical data reveal the phase diagram; at large λ it contains a "striped phase", which is absent in the commutative case. We explore the question whether or not this phenomenon persists in a Double Scaling Limit (DSL), which extrapolates simultaneously to the continuum and to infinite volume, at a fixed noncommutativity parameter. To this end, we introduce a dimensional lattice spacing based on the decay of the correlation function. Our results provide evidence for the existence of a striped phase even in the DSL, which implies the spontaneous breaking of translation symmetry. Due to the non-locality of this model, this does not contradict the Mermin-Wagner theorem.
Infinite dimension reflection matrices in the sine-Gordon model with a boundary
Journal of High Energy Physics, 2012
Using the sine-Gordon model as the prime example an alternative approach to integrable boundary conditions for a theory restricted to a half-line is proposed. The main idea is to explore the consequences of taking into account the topological charge residing on the boundary and the fact it changes as solitons in the bulk reflect from the boundary. In this context, reflection matrices are intrinsically infinite dimensional, more general than the two-parameter Ghoshal-Zamolodchikov reflection matrix, and related in an intimate manner with defects.