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Harald Grosse

Publications and source records attributed to Harald Grosse.

At least 19 recordsLinked to original sources
arXiv2025

Relationship between a $\Phi^4$ matrix model and harmonic oscillator systems

A Hermitian $\Phi^4$ matrix model with a Kontsevich-type kinetic term is studied. It was recently discovered that the partition function of this matrix model satisfies the Schr\"odinger equation of the $N$-body harmonic oscillator, and that eigenstates of the Virasoro operators can be derived from this partition function. We extend these results and obtain an explicit formula for such eigenstates in terms of the free energy. Furthermore, the Schr\"odinger equation for the $N$-body harmonic oscillator can also be reformulated in terms of connected correlation functions. The $U(1)^N$-symmetry allows us to derive loop equations.

hep-th↗
arXiv2025

A Quantum Energy Inequality for a Non-commutative QFT

We present a quantum energy inequality (QEI) for quantum field theories formulated in non-commutative spacetimes, extending fundamental energy constraints to this generalized geometric framework. By leveraging operator-theoretic methods inspired by the positivity map of Waldmann et al. \cite{waldmannpos}, we construct linear combinations of deformed operators that generalize the commutative spacetime techniques of Fewster et al., \cite{Few98}. These non-commutative analogs enable us the derivation of a lower bound on the deformed averaged energy density, ensuring the stability of the underlying quantum field theory. Our result establishes rigorous constraints on the expectation values of the deformed (non-commutative) energy density, reinforcing the physical consistency of non-commutative models while preserving core principles of quantum field theory.

hep-th↗
arXiv2023

Real symmetric $Φ^4$-matrix model as Calogero-Moser model

We study a real symmetric $Φ^4$-matrix model whose kinetic term is given by $\mathrm{Tr}( E Φ^2)$, where $E$ is a positive diagonal matrix without degenerate eigenvalues. We show that the partition function of this matrix model corresponds to a zero-energy solution of a Schödinger type equation with Calogero-Moser Hamiltonian. A family of differential equations satisfied by the partition function is also obtained from the Virasoro algebra.

hep-th↗
arXiv2023

Integrability of $Φ^4$ Matrix Model as $N$-body Harmonic Oscillator System

We study a Hermitian matrix model with a kinetic term given by $ Tr (H Φ^2 )$, where $H$ is a positive definite Hermitian matrix, similar as in the Kontsevich Matrix model, but with its potential $Φ^3$ replaced by $Φ^4$. We show that its partition function solves an integrable Schrödinger-type equation for a non-interacting $N$-body Harmonic oscillator system.

math-ph↗
arXiv2022

A Laplacian to compute intersection numbers on $\bar{\mathcal{M}}_{g,n}$ and correlation functions in NCQFT

Let $F_g(t)$ be the generating function of intersection numbers on the moduli spaces $\bar{\mathcal{M}}_{g,n}$ of complex curves of genus $g$. As by-product of a complete solution of all non-planar correlation functions of the renormalised $Φ^3$-matrical QFT model, we explicitly construct a Laplacian $Δ_t$ on the space of formal parameters $t_i$ satisfying $\exp(\sum_{g\geq 2} N^{2-2g}F_g(t))=\exp((-Δ_t+F_2(t))/N^2)1$ for any $N>0$. The result is achieved via Dyson-Schwinger equations from noncommutative quantum field theory combined with residue techniques from topological recursion. The genus-$g$ correlation functions of the $Φ^3$-matricial QFT model are obtained by repeated application of another differential operator to $F_g(t)$ and taking for $t_i$ the renormalised moments of a measure constructed from the covariance of the model.

math-ph↗
arXiv2019

Solution of the self-dual $Φ^4$ QFT-model on four-dimensional Moyal space

Previously the exact solution of the planar sector of the self-dual $Φ^4$-model on 4-dimensional Moyal space was established up to the solution of a Fredholm integral equation. This paper solves, for any coupling constant $λ>-\frac{1}π$, the Fredholm equation in terms of a hypergeometric function and thus completes the construction of the planar sector of the model. We prove that the interacting model has spectral dimension $4-2\frac{\arcsin(λπ)}π$ for $|λ|<\frac{1}π$. It is this dimension drop which for $λ>0$ avoids the triviality problem of the matricial $Φ^4_4$-model. We also establish the power series approximation of the Fredholm solution to all orders in $λ$. The appearing functions are hyperlogarithms defined by iterated integrals, here of alternating letters $0$ and $-1$. We identify the renormalisation parameter which gives the same normalisation as the ribbon graph expansion.

math-ph↗
arXiv2019

Solution of all quartic matrix models

We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp(-{N}\,\mathrm{Tr}(E\Phi^2+(\lambda/4)\Phi^4)) d\Phi$ on Hermitian ${N}\times{N}$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar. It was previously established that the large-$N$ limit of the second moment (the planar two-point function) satisfies a non-linear integral equation. By employing tools from complex analysis, in particular the Lagrange-B\"urmann inversion formula, we identify the exact solution of this non-linear problem, both for finite $N$ and for a large-${N}$ limit to unbounded operators $E$ of spectral dimension $\leq 4$. For finite $N$, the two-point function is a rational function evaluated at the preimages of another rational function $R$ constructed from the spectrum of $E$. Subsequent work has constructed from this formula a family $\omega_{g,n}$ of meromorphic differentials which obey blobbed topological recursion. For unbounded operators $E$, the renormalised two-point function is given by an integral formula involving a regularisation of $R$. This allowed a proof, in subsequent work, that the $\lambda\Phi^4_4$-model on noncommutative Moyal space does not have a triviality problem.

math-ph↗
arXiv2017

The $Φ^3_4$ and $Φ^3_6$ matricial QFT models have reflection positive two-point function

We extend our previous work (on $D=2$) to give an exact solution of the $Φ^3_D$ large-$\mathcal{N}$ 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 $|p|^2 \geq 2μ^2$ and an isolated fuzzy mass shell around $|p|^2=μ^2$.

math-ph↗
arXiv2017

Exact solution of matricial $Φ^3_2$ quantum field theory

We apply a recently developed method to exactly solve the $Φ^3$ matrix model with covariance of a two-dimensional theory, also known as regularised Kontsevich model. Its correlation functions collectively describe graphs on a multi-punctured 2-sphere. We show how Ward-Takahashi identities and Schwinger-Dyson equations lead in a special large-$\mathcal{N}$ limit to integral equations that we solve exactly for all correlation functions. Remarkably, these functions are analytic in the $Φ^3$ coupling constant, although bounds on individual graphs justify only Borel summability. The solved model arises from noncommutative field theory in a special limit of strong deformation parameter. The limit defines ordinary 2D Schwinger functions which, however, do not satisfy reflection positivity.

math-ph↗
arXiv2016

Trace Formulas for a Class of non-Fredholm Operators: A Review

We review previous work on spectral flow in connection with certain self-adjoint model operators $\{A(t)\}_{t\in \mathbb{R}}$ on a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$, and the index of the operator $D_{A}^{}= (d/d t) + A$ acting in $L^2(\mathbb{R}; \mathcal{H})$, where $A$ denotes the operator of multiplication $(A f)(t) = A(t)f(t)$. In this article we review what is known when these operators have some essential spectrum and describe some new results in terms of associated spectral shift functions. We are especially interested in extensions to non-Fredholm situations, replacing the Fredholm index by the Witten index, and use a particular $(1+1)$-dimensional model setup to illustrate our approach based on spectral shift functions.

math.AP↗
arXiv2015

Solvable 4D noncommutative QFT: phase transitions and quest for reflection positivity

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 $λ_c \approx -0.39$. 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 $λ_c<λ\leq 0$ 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.

hep-th↗
arXiv2015

On the fixed point equation of a solvable 4D QFT model

The regularisation of the $λϕ^4_4$-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.

math-ph↗
arXiv2015

On a spectral flow formula for the homological index

Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators {A(t)} parametrized by the real line. Then under certain conditions \cite{RS95} that include the assumption that the operators {D(t)= D+A(t)} all have discrete spectrum then the spectral flow along the path { D(t)} can be shown to be equal to the index of d/dt+D(t) when the latter is an unbounded Fredholm operator on L^2(R, H). In \cite{GLMST11} an investigation of the index=spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general. In \cite{CGPST14a} the question of what happens when the Fredholm condition is dropped altogether was investigated. In these circumstances the Fredholm index is replaced by the Witten index. In this paper we take the investigation begun in \cite{CGPST14a} much further. We show how to generalise a formula known from the setting of the L^2 index theorem to the non-Fredholm setting. Our main theorem gives a trace formula relating the homological index of \cite{CaKa:TIH} to an integral formula that is known, for a path of selfadjoint Fredholms with compact resolvent and with unitarily equivalent endpoints, to compute spectral flow. Our formula however, applies to paths of selfadjoint non-Fredholm operators. We interpret this as indicating there is a generalisation of spectral flow to the non-Fredholm setting.

math.FA↗
arXiv2014

Anomalies of Dirac type operators on Euclidean space

We develop by example a type of index theory for non-Fredholm operators. A general framework using cyclic homology for this notion of index was introduced in a separate article [CaKa13] where it may be seen to generalise earlier ideas of Carey-Pincus and Gesztesy-Simon on this problem. Motivated by an example in two dimensions in [BGG+87] we introduce in this paper a class of examples of Dirac type operators on R^{2n} that provide non-trivial examples of our homological approach. Our examples may be seen as extending old ideas about the notion of anomaly introduced by physicists to handle topological terms in quantum action principles with an important difference, namely we are dealing with purely geometric data that can be seen to arise from the continuous spectrum of our Dirac type operators.

math.FA↗
arXiv2014

Construction of the Φ^4_4-quantum field theory on noncommutative Moyal space

We review our recent construction of the $ϕ^4$-model on four-dimensional Moyal space. A milestone is the exact solution of the quartic matrix model $Z[E,J]=\int dΦ\exp(tr(JΦ- EΦ^2 -(λ/4) Φ^4))$ in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of $E$. 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. Its numerical solution reveals a second-order phase transition at $λ_c\approx-0.396$ and a phase transition of infinite order at $λ=0$. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. They are only sensitive to diagonal matrix correlation functions, and clustering is violated. The Schwinger 2-point function is reflection positive iff the diagonal matrix 2-point function is a Stieltjes function. Numerically this seems to be the case for coupling constants $λ\in [λ_c,0]$.

math-ph↗
arXiv2013

Solvable limits of a 4D noncommutative QFT

In previous work we have shown that the (θ->\infty)-limit of ϕ^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.

math-ph↗
arXiv2013

Slavnov-Taylor identities, non-commutative gauge theories and infrared divergences

In this work we clarify some properties of the one-loop IR divergences in non-Abelian gauge field theories on non-commutative 4-dimensional Moyal space. Additionally, we derive the tree-level Slavnov-Taylor identities relating the two, three and four point functions, and verify their consistency with the divergent one-loop level results. We also discuss the special case of two dimensions.

hep-th↗